See also notes on:

  • SMT
  • prolog
  • datalog

TPTP

TPTP Thousands of Problems for Theorem Provers is an incredible resource Quick Guide

A couple different input formats. As a human, tff is probably the best. Typed first order formula. cnf is painful, fof is untyped and you’re likely to screw it up imo. tff offers built in ints, which is cool.

TPTP in latex Axiom List Highlights:

  • PUZ - puzzles
  • GEO - geometry
  • GRP - group theory
  • CAT - category theory
  • SET - set theory
  • NUM - number stuff robinson arithmetic
  • TOP
  • DAT - datatypes

Automated reasoning book

  • puzzles
  • circuit design (other sythesis problems?)
  • program verification

Using for higher order or typed systems

I took a big dump of the axioms because they are hard to peruse. You can find them in my notes. They are very interesting.

Methodology

I find I am often making inconsistent theories because I am trying to be too cute and avoid writing burdensome and inefficient side conditions. A neat thing though is that

Sanity checks:

  1. Make sure your axioms can’t derive a contradiction. Run vampire or eprover and it should hopefully hang or produce a model
  2. Make sure there are at least two elements that haven’t collapsed. I find I tend to make theories that collapse
  3. If you have a simpler seeming axiom than the one that feels to most trustworthy, write them both and ask for equivalence to be proven. If it works, then you can be a little more sure that your simpler axiom is correct.

Puzzles

% https://www.tptp.org/TPTP/QuickGuide/Problems.html
%------------------------------------------------------------------------ 
%----All (hu)men are created equal. John is a human. John got an F grade. 
%----There is someone (a human) who got an A grade. An A grade is not 
%----equal to an F grade. Grades are not human. Therefore there is a 
%----human other than John. 
fof(all_created_equal,axiom,( 
    ! [H1,H2] : 
      ( ( human(H1) 
         & human(H2) ) 
     => created_equal(H1,H2) ) )). 

fof(john,axiom,( 
    human(john) )). 

fof(john_failed,axiom,( 
    grade(john) = f )). 

fof(someone_got_an_a,axiom,( 
    ? [H] : 
      ( human(H) 
      & grade(H) = a ) )). 

fof(distinct_grades,axiom,( 
    a != f )). 

fof(grades_not_human,axiom,( 
    ! [G] : ~ human(grade(G)) )). 

fof(someone_not_john,conjecture,( 
    ? [H] : 
      ( human(H) 
      & H != john ) )). 
%-------------------------------------------------------------------- 
% a silly axiomatization of add and theorem.
% using typed formula

tff(num_type,type,
    num: $tType ).
tff(add_type,type,
    add: ( num * num ) > num ).
tff(zero_type,type,
    zero: num ).
tff(add_zero_axiom,axiom,
    ! [X: num] : add(X, zero) = X ).
tff(add_comm_axiom,axiom,
    ! [X: num, Y : num] : add(X, Y) = add(Y,X) ).
tff(mytheorem, conjecture, ? [X : num] : add(zero, X) = X).

% https://www.tptp.org/cgi-bin/SeeTPTP?Category=Problems&Domain=DAT&File=DAT001=1.p
tff(list_type,type,
    list: $tType ).

tff(nil_type,type,
    nil: list ).

tff(mycons_type,type,
    mycons: ( $int * list ) > list ).

tff(sorted_type,type,
    sorted: list > $o ).

tff(empty_is_sorted,axiom,
    sorted(nil) ).

tff(single_is_sorted,axiom,
    ! [X: $int] : sorted(mycons(X,nil)) ).

tff(recursive_sort,axiom,
    ! [X: $int,Y: $int,R: list] :
      ( ( $less(X,Y)
        & sorted(mycons(Y,R)) )
     => sorted(mycons(X,mycons(Y,R))) ) ).

tff(check_list,conjecture,
    sorted(mycons(1,mycons(2,mycons(4,mycons(7,mycons(100,nil)))))) ).

Array axioms. Also axioms for int collections, int map, heaps. They all boil to pretty similar stuff though

% https://www.tptp.org/cgi-bin/SeeTPTP?Category=Axioms&File=DAT001=0.ax
tff(array_type,type,
    array: $tType ).

tff(read_type,type,
    read: ( array * $int ) > $int ).

tff(write_type,type,
    write: ( array * $int * $int ) > array ).

tff(ax1,axiom,
    ! [U: array,V: $int,W: $int] : ( read(write(U,V,W),V) = W ) ).

tff(ax2,axiom,
    ! [X: array,Y: $int,Z: $int,X1: $int] :
      ( ( Y = Z )
      | ( read(write(X,Y,X1),Z) = read(X,Z) ) ) ).
% https://www.tptp.org/cgi-bin/SeeTPTP?Category=Axioms&File=DAT001=0.ax
tff(array_type,type,
    array: $tType ).

tff(read_type,type,
    read: ( array * $int ) > $int ).

tff(write_type,type,
    write: ( array * $int * $int ) > array ).

tff(ax1,axiom,
    ! [U: array,V: $int,W: $int] : ( read(write(U,V,W),V) = W ) ).

tff(ax2,axiom,
    ! [X: array,Y: $int,Z: $int,X1: $int] :
      ( ( Y = Z )
      | ( read(write(X,Y,X1),Z) = read(X,Z) ) ) ).
fof(ground, axiom,
    edge(1,2) & edge(2,3)
).
%fof(path1, axiom,
%    ![X,Y]: (edge(X,Y) => path(X,Y))
%).
%fof(path2, axiom,
%    ![X,Y,Z]:  ((edge(X,Y) & path(Y,Z)) => path(X,Z))
%).
fof(clark, axiom,
    ![X,Z]: (((?[Y]: (edge(X,Y) & path(Y,Z))) | edge(X,Z)) <=> path(X,Z))
).

fof(myquery, conjecture,
    ?[X,Y]: path(X,Y)
   %path(1,2)
).

Math

Groups Lattices kleene algebra, kat

Categories Set theory https://en.wikipedia.org/wiki/List_of_set_identities_and_relation Geometry

antiderivatives


thf(manitype, type, mani : $tType). % manifold

thf(consttype, type, const : $real > mani > $real ).

thf(xtype, type, x : mani > $real ). % coordinate function
thf(ytype, type, y : mani > $real ). % coordinate function
thf(rtype, type, r : mani > $real ). % coordinate function

thf(plustype, type, plus : (mani > $real) > (mani > $real) > mani > $real ). % coordinate function

thf(plusaxioms,axiom, (plus @ X @ Y)  = (plus @ Y @ X)).


thf(consttype, type, const : $real > mani > $real ).

%thf(dtype, type, d : (mani > $real) >  ).
thf(sintype, type, sin : (mani > $real) > mani > $real).

%thf(powtype,type, > )

thf ^ [X] : X is a lambda [a,b,c] is a built in tuple. $i is a universe of individuals $o is bools $true $false $real $int

echo '
%re
thf(cox_type, type, cos : $real >  $real).
thf(const_type, type, const : $real > $real > $real).
thf(id_type, type, id : $real  > $real).
thf(exp_type, type, exp : $real >  $real).
thf(deriv_type, type, d : ($real > $real) > $real > $real).

%thf(dx, axiom, ^[X : $real] : (d @ id) = ^[X]:  (d @ ([]))).
thf(dx, axiom, (d @ (^[X : $real] : X)) = (^[X : $real]: 1)).


%thf(antideriv, conjecture, ?[F : $real > $real] (![X : $real] : ( (exp @ (const @ X)) = d @ (F @ X) )).
' > /tmp/calc.p
#eprover-ho --auto /tmp/calc.p --conjectures-are-questions
vampire /tmp/calc.p

ATP as a Logical Framework

I am used to in the prolog and datalog context to consider the clauses not as classical logical clauses, but instead a description of inference rules. This is the attitude in Miller & Nadathur and The Logic of Logic Programming

It seems to me that there is nothing preventing this interpretation for ATP as well. There are a numbr of axiom sets in the TPTP library that have thsi flavor

~ a  
is the same as
   a
-------
This is how we express a goal.


~a | b is

   a
------
   b


~a | ~c | b

is

 a   c
------
   b

Putting together clauses (resolution) is interpreted as putting together inference rules to make proof fragments.

% finite sets either by AC axioms or some primitive
cnf(and_comm, axiom, and(A,B) = and(B,A)).
cnf(and_assoc, axiom, and(A,and(B,C)) = and(and(A,B),C)).


cnf(not_intro ,axiom,  ~ turnstile(and(C, not(A)), B) | turnstile(C , and(A,B)).

Multiple consequence logic. Smiley https://en.wikipedia.org/wiki/Multiple-conclusion_logic

Symbolic Execution

Wos book chapter


Lambda

thf(easymode, conjecture, ?[P : $i > $i] : ((P @ X) = X)).
echo '
thf(f_type, type, f : $i > $i > $i).
thf(f_type, type, g : $i > $i > $i).


% thf(g_permutes_f, axiom, ![X:$i, Y:$i]:((f @ X @ Y) = (g @ Y @ X))).
thf(conj, conjecture, ?[G: $i > $i > $i]: (![X:$i, Y:$i]: ( (f @ X @ Y) = (G @ Y @ X)) ) ).
' > /tmp/example.p
eprover-ho --auto --proof-object /tmp/example.p 
echo '
thf(unifythis, conjecture, ?[P : $i > $i] : (![X : $i]: ((P @ X) = X))).
' > /tmp/example.p
eprover-ho  --auto /tmp/example.p --conjectures-are-questions #fails wth GaveUp
#eprover-ho --auto /tmp/example.p  --print-strategy > /tmp/strategy # succeeds
eprover-ho --auto /tmp/example.p  --conjectures-are-questions --parse-strategy=/tmp/strategy #succeeds
#vampire --mode portfolio /tmp/example.p
echo '
;re
(declare-sort I ())
(assert (not (exists ((P (Array I I))) (forall ((X Int)) (= (select P X) X)))))
;thf(unifythis, conjecture, ?[P : $i > $i] : (![X : $i]: ((P @ X) = X))).
' > /tmp/example.smt2
eprover-ho --auto /tmp/example.smt2 #--conjectures-are-questions #fails wth GaveUp

echo '
%re
thf(f_type, type, f : $i > $i > $i ).
thf(unifythis, conjecture, ?[P : $i > $i > $i] : (![X : $i, Y : $i]: ((P @ X @ Y) = (f @ Y @ X)))).
' > /tmp/example.p
eprover-ho --auto /tmp/example.p --conjectures-are-questions #fails wth GaveUp
eprover-ho --auto /tmp/example.p --print-strategy > /tmp/strategy # succeeds
eprover-ho --auto /tmp/example.p --conjectures-are-questions #--parse-strategy=/tmp/strategy #succeeds

echo '
%re
#thf(f_type, type, x : $i).
thf(easymode, conjecture, ?[P : $i > $i] : (![X : $i]: ((P @ X) = X))).
%thf(easymode, conjecture, ?[P : $i > $i] :((P @ x) = x)).
' > /tmp/example.p
eprover-ho --auto /tmp/example.p --conjectures-are-equations #fails
eprover-ho --auto /tmp/example.p  --print-strategy > /tmp/strategy
eprover-ho --auto /tmp/example.p  --conjectures-are-questions --parse-strategy=/tmp/strategy



# --print-strategy #--conjectures-are-questions
  • eprover-ho –auto /tmp/example.p –proof-object

Conjectures-as-questions is segfaulting. Ok fixed. Hmm. Why is --auto necessary to make this work? Kind of odd behavior even if in the readme

Lambda lifting

echo '
thf(f_type, type, f : $i > $i ).
thf(problem1, conjecture, ?[G : $i > $i]: (![X: $i]: ((f @ X) = (G @ X)))). ' > /tmp/example.p
eprover-ho --auto /tmp/example.p  --conjectures-are-questions #--proof-object
# This ought to be imposble. Ok Good.
echo '
thf(f_type, type, f : $i > $i ).
thf(problem1, conjecture, ?[G : $i > $i]: (![X: $i, Y:$i]: ((f @ X) = (G @ Y)))). ' > /tmp/example.p
eprover-ho --auto /tmp/example.p --proof-object

Structures

In coq or lean we prove somwethign has properties, then we abstract it an use only the properties

thf(monoid_def, axiom, monoid(E, Op) <=> ( ! [X,Y,Z] : ( 
    Op @ (Op @ X @ Y) @ Z = Op @ X @ (Op @ Y @ Z)
    & Op @ E @ X = X
    & Op @ X @ E = X
))).


Peano

fof(myaxioms, axiom, 
    nat(z)  % unnecessary?
    & (nat(X) => nat(s(X)))
    & (s(X) = s(Y) => X = Y) % need nat(X) nat(Y) ?
    & s(X) != z

    % induction is of course the sticky one
    % & (((P @ z) & (! [X] : (P @ X => P @ (s(X))))) => P @ Y)

).

Getting Saturated with Induction

ECTS


% don't connect posibly non well formed things
cnf(comp_assoc, axiom, comp(X,comp(Y,Z)) = comp(comp(X,Y),Z)). 
cnf(wf_comp, axiom, ~wf(F) | ~wf(G) | dom(F) != cod(G) 
                            | wf(comp(F,G))).
cnf(wf_id, axiom, wf(id(A))).
cnf(comp_id_left, axiom, comp(id(cod(F)),F) = F).
cnf(comp_id_right, axiom, comp(F,id(dom(F))) = F).

cnf(term_final, axiom, cod(F) != unit | cod(G) != unit | dom(F) != dom(G) 
                                            | F = G ).


%fof(elem_def, axiom, elem(X,A) <=> ( comp(X,A) ) )



What about doing bash + vampire as a proof system

What about universes?

Set Theory

TPTP has tons of axiom sets for this. But let’s play around

fof(mystuff, axiom,
    $true
    %~ elem(X,empty)
    & ((elem(X,A) | elem(X,B)) <=> elem(X,union(A,B)))
    %& ((elem(X,A) & elem(X,B)) <=> elem(X,inter(A,B)))
    %& (elem(X,sing(Y)) <=> (X = Y))
    %& ( (![X] : (elem(X,A) <=> elem(X,B))) <=> (A = B)) % extensionality 
    %& ( (![X] : (elem(X,A) => elem(X,B))) <=> sub(A,B)) 
    
).
%fof(sub_equal, conjecture,   (sub(X,Y) & sub(Y,X)) <=> (X = Y) ).
%fof(union_comm, conjecture, union(A,B) = union(B,A)).
fof(union_assoc, conjecture, union(A,union(B,C)) = union(union(A,B),C)).

https://lawrencecpaulson.github.io/2022/02/02/Formalising_Math_Set_theory.html dicsussed in this article also. Good access

Art Quaife has a whole paper on NBG set theory https://link.springer.com/article/10.1007/BF00263451 “Automated deduction in von Neumann-Bernays-Gödel set theory” It’s also in hs thesis. This rules. https://link.springer.com/epdf/10.1007/BF00263451?sharing_token=DMYVYRCAVaP61kk6cSO7R_e4RwlQNchNByi7wbcMAY7K-nOLlPOv92rkTo0zdkBRH1ERODIQa047sigWp8Z1p0NHwi3G_fwBKYangMYWPUABQfWSxir_V9ADxEz2QrhRhQiW5Sq5dAYb02p8l3YzVQ%3D%3D belinfante

https://link.springer.com/article/10.1007/BF02328452 Set theory in first-order logic: Clauses for Gödel’s axioms - Robert Boyer, Ewing Lusk, William McCune, Ross Overbeek, Mark Stickel & Lawrence Wos https://link.springer.com/epdf/10.1007/BF02328452?sharing_token=Lu-d29xQHlQKnUAd_Fg2BPe4RwlQNchNByi7wbcMAY6sM0fmv_8YXIaliyF3Y4UE0bQhtBZKpKWX8GiVshMOVsmncPyXxm8V9-VnEQ-vPt2gqKtSzs2g3zPMCLQeDSWgwZdnCMT0Ae3SHVYV89KFjQ%3D%3D

https://dl.acm.org/doi/10.5555/1762639.1762657 a fnite first order presentation of set theory

https://arxiv.org/pdf/cs/9311103.pdf paulson - set theory in isabelle

Simplicial Set

https://math.jhu.edu/~eriehl/ssets.pdf

face

d(J,d(I,X)) = d(I,d(J-1,X))) # J < I 
d(I,s(I,X)) = X



d1(X,d2(X)) = d1(d0(X),X)
d1(d0(X),X) = d1(X,d2(X))
d2(d1(X),X) = d2(X,d0(X))


Free Logic / Partial Functions

See the category theory CAT003 example in TPTP with comments about paper by Scott Identity and Existence in Intuitionist Logic Avigad book Free logic article

We don’t want it necessarily to assume our function symbols are total. How do we model this? We could switch t a relational character. Functions are so gosh darn nice though.

Query Containment

Queries are first order logical statements with perhaps some free variables. They ae evaluated (model checked) with respect to particular finite models. Asking if one statement implies another in all models is semantic entailment. Because |- and |= are the same thing here (? Really we are relying on soundness), We can write query containment as a simple theorem proving question

fof(q1_def, axiom, q1(X) <=> (r(X,Y) & r(Y,X))). % just implication or biimplication?

fof(q2_def, axiom, q2(X) <=> r(X,Y)). 

fof(contain, conjecture, 
q1(X) => q2(Q)
).

This also implies a method for solving CSPs.


% or really the full clark completion? Yea. I almost certainly do need it.
fof(k3_edges, axiom, k3edge(a,b) & k3edge(b,c) & k3edge(c,a) & k3edge(b,a) & k3edge(c,b) & k3edge(a,c)
    & ~k3edge(a,a) & ~k3edge(b,b) & ~k3edge(c,c)
).

fof(mygraph, axiom, edge(1,2)).

fof(verts, axiom, vert(V) <=> (edge(V,X)) | edge(X,V)).

fof(color3 , conjecture, edge(X,Y) => k3(color(X),color(Y)))

hhmmm.

Systems

Vampire

First-Order Theorem Proving and VAMPIRE vampire tutorial

vampire --show_options on

lots of intrigung little options

vampire -av off –question_answering answer_literal –show_everything –show_ordering options about induction manual_cs manually pick clauses to select… ????

fof(a,axiom,prover(vampire)).
fof(a,axiom,prover(e)).
fof(a,axiom,tutorial(vampire)).
fof(a,axiom,tutorial(probabilistic)).
fof(a,conjecture,?[X]: (prover(X) & tutorial(X))).

--sos

inference restrictions --unit_resulting_resolution

echo "
fof(ground, axiom,
    edge(a,b) & edge(b,c)
).
fof(path1, axiom,
    ![X,Y]: (edge(X,Y) => path(X,Y))
).
fof(path2, axiom,
    ![X,Y,Z]:  ((edge(X,Y) & path(Y,Z)) => path(X,Z))
).
" |  vampire  --unit_resulting_resolution on

Hmm. Vampire with the option or not. Not really sure what to take away from that. It might be recogzninig a datalog fragment If I wanted to use this like an egraph, I can’t look inside the equality. It’s entirely unobseravle. Maybe I could use the reflection trick.

E prover

enormalizer is an interesting sounding program. Give it a pile of unit equalities and it will normalize with respect to thm EPR grounder to DIMACS The e calculus is a bit puzzling. I haven’t seen the analog for vampire

2.6 manual It’s also in the github repo if you make doc

I like how –answer mode works a little better for e.

Database printing feature -S. Doesn’t print stuff I would expect though? Kind of prints everything by default right? Early stopping conditions clause size

eprover --help | less

--watchlist - “constructive” proofs that aren’t seeded by refuation training - you can train it if you have a database of indicative theorems. This might be useful if you have a sequence of increasingly hard theorems, or if you are making a tool that spits out formula. -S print saturated clause set -W literaeelection stretagory NoGeneration will inhibit all generating instances “Each of the strategies that do actually select negative literals has a corresponding counterpart starting with P that additionally allows paramodulation into maximal positive literals”

echo "
fof(ground, axiom,
    edge(a,b) & edge(b,c)
).
fof(path1, axiom,
    ![X,Y]: (edge(X,Y) => path(X,Y))
).
fof(path2, axiom,
    ![X,Y,Z]:  ((edge(X,Y) & path(Y,Z)) => path(X,Z))
).
" | eprover  --literal-selection-strategy=SelectNegativeLiterals

--generated-limit=100 Ok this basically did what i wanted. I’m not sure what it is though?

“The most natural clause representation for E is probably a literal disjunction: a=$true;b!=$true;c!=$true.”


Prover9

Prover9 and Mace4 Prover9 is intriguing. It seems to have interaction modalities that other provers don’t Mace proverx a web gui variant

Some examples I didn’t write:

echo "% Prove that a group in which all elements have order 2 is commutative.

assign(max_seconds, 60).

formulas(assumptions).

  % group axioms
  (x * y) * z = x * (y * z).   % associativity
  1 * x = x.                   % left identity
  x * 1 = x.                   % right identity
  x' * x = 1.                  % lef inverse
  x * x' = 1.                  % right inverse
  
  % special assumption
  x * x = 1.                  % all elements have order 2

end_of_list.

formulas(goals).
  x * y = y * x.
end_of_list.
" | prover9
if(Prover9).
    assign(order, kbo).
    assign(max_weight, 25).
end_if.

formulas(sos).

% lattice theory

x v y = y v x.
(x v y) v z = x v (y v z).
x ^ y = y ^ x.
(x ^ y) ^ z = x^ (y ^ z).
x ^ (x v y) = x.
x v (x ^ y) = x.

end_of_list.

formulas(sos).
(x ^ y) v (x ^ z) = x ^ ((y ^ (x v z)) v (z ^ (x v y))) # label(H82).
end_of_list.

formulas(goals).
x ^ (y v (x ^ z)) = x ^ (y v (z ^ ((x ^ (y v z)) v (y ^ z)))) # label(H2).
end_of_list.

assign(order, kbo).  % The default (lpo) al works, but it takes longer.

formulas(assumptions).

% Prover9 should produce a proof in a few seconds.

% combinators B and W

  a(a(a(B,x),y),z) = a(x,a(y,z))   # label(B).
  a(a(W,x),y) = a(a(x,y),y)        # label(W).

end_of_list.

formulas(goals).

  % existence of a fixed point combinator

  exists Q all x (a(Q,x) = a(x,a(Q,x))) # label(fixed_point_combinator).

end_of_list.

% Here is a different way of specifying the goal, which tells us
% what the fixed point combinator is.  We use an answer attribute.
% Unfortunately, answer attributes are not allowed on non-clausal
% (e.g., goal) formulas, so we deny the goal and state it as an assumption.
% Note that this introduces the Skolem function f.
%
% formulas(assumptions).
% a(x,f(x)) != a(f(x),a(x,f(x))) # answer(x).
% end_of_list.

Otter

https://www.cs.unm.edu/~mccune/otter/ otter lambda - dumping lambda terms in otter? http://www.michaelbeeson.com/research/otter-lambda/ Otter seemed to offer more control over the resolution process Prover9 is the succssor to otter

otter manual “Although OTTER has an autonomous mode, most work with OTTER involves interaction with the user”

LeanTAP and ilk

See prolog some Build Your Own First-Order Prover https://www.philipzucker.com/javascript-automated-proving/

Isabelle

https://www.tptp.org/Seminars/THF/SystemIsabelle.html

refute nitpick

cnf(x ,axiom, x).
cnf(notx ,axiom, ~x).

hmm this is not working.

echo '
%thf(unifythis, conjecture, ?[P : $i > $i] : (![X : $i]: ((P @ X) = X))).
cnf(x ,axiom, x).
cnf(notx ,axiom, ~x).
' > /tmp/example.p
#isabelle tptp_sledgehammer 10 /tmp/example.p
isabelle tptp_isabelle 10 /tmp/example.p
  tptp_graph - TPTP visualisation utility
  tptp_isabelle - Isabelle tactics for TPTP competitive division
  tptp_isabelle_hot - Isabelle tactics for TPTP demo division
  tptp_nitpick - Nitpick for TPTP
  tptp_refute - Refute for TPTP
  tptp_sledgehammer - Sledgehammer for TPTP
  tptp_translate - translate between TPTP formats

Twee

https://nick8325.github.io/twee/

stack install twee twee-lib jukebox minisat

echo '
fof(right_identity, axiom,
    ![X]: f(X, e) = X).
fof(right_inverse, axiom,
    ![X]: f(X, i(X)) = e).
fof(associativity, axiom,
    ![X, Y, Z]: f(X, f(Y, Z)) = f(f(X, Y), Z)).
% left inverse is also right inver
fof(left_inverse, conjecture,
    ![X]: f(i(X),X) = e).
    ' > /tmp/group.p
twee --explain-encoding --tstp --trace /tmp/trace mymodule /tmp/group.p

Impressivly clear proof output

twee --expert-help
echo "
%re https://github.com/nick8325/twee/blob/master/tests/deriv.p
% Axioms about arithmetic.

cnf('commutativity of +', axiom,
    '+'(X,Y) = '+'(Y,X)).
cnf('associativity of +', axiom,
    '+'(X, '+'(Y, Z)) = '+'('+'(X , Y) , Z)).
cnf('commutativity of *', axiom,
    '*'(X,Y) = '*'(Y , X)).
cnf('associativity of *', axiom,
    '*'(X , '*'(Y , Z)) = '*'('*'(X , Y) , Z)).
cnf('plus 0', axiom,
    '+'('0' , X) = X).
cnf('times 0', axiom,
    '*'('0' , X) = '0').
cnf('times 1', axiom,
    '*'('1' , X) = X).
cnf('distributivity', axiom,
    '*'(X,'+'(Y,Z)) = '+'('*'(X , Y) , '*'(X , Z))).
cnf('minus', axiom,
    '+'(X , '-'(X))= '0').
cnf('derivative of 0', axiom,
    d('0') = '0').
cnf('derivative of 1', axiom,
    d('1') = '0').
cnf('derivative of x', axiom,
    d(x) = '1').
cnf('derivative of +', axiom,
    d('+'(T,U)) = '+'(d(T) , d(U))).
cnf('derivative of *', axiom,
    d('*'(T,U)) = '+'(('*'(T,d(U))) , '*'(U,d(T)))).
cnf('derivative of sin', axiom,
    d(sin(T)) = '*'(cos(T),d(T))).
cnf('derivative of cos', axiom,
    d(cos(T)) = '-'('*'(sin(T),d(T)))).

%fof(goal, conjecture,
%    ?[T]: d(T) = '*'(d(x),'*'(x,cos(x)))).
fof(goal, conjecture,
    ?[T]: d(T) = '*'(x,cos(x))).
    " > /tmp/deriv.p
twee --tstp /tmp/deriv.p

https://github.com/nick8325/jukebox

Can I extract terms by cost? cost(T) = N is problematic if we use actual equality. cost_le(T,N) = true may be better. cost(foo(X), N) = or(cost(foo(X),N-1), cost()). cost_lt(startterm, 10) = true) or ?[T]: cost(T,10) = true and T = startterm Dump the datbase at the end and rewrite with it? Ground direct? Extend with lambda terms?

PyRes

# atomic formula first. Ground is basically the same.

# sets are unhashable, so are lists
#print({[], []})

# One canonical form is sorted, deduped pos,neg lists for clauses
def clause(pos,neg):
    return (tuple(sorted(set(pos))), tuple(sorted(set(neg))))

print(clause(["a","a"], ["c"]))
print(clause(["foo(a)","bar(a)"], ["c"]))

# we could also put a wrapper around set that does have a hash defined

def resolve(C1,C2):
    C1[0].diff(C2[1])
    C2[0].diff(C1[1])

    if len(C1[0].intersect(C2[1])) > 0:


processed = set()
unprocessed = []
while len(unprocessed) > 0:
    C = unprocessed.pop()
    for C1 in processed:
        resolve(C1,C)

# terms are sexp style lists
# ["f", ["a"], ["g", "b"]]

A sqlized given-clause loop

unproc = []
proc = sqlite3.connect(":memory:") # store processed clauses in DB.
while len(unproc) > 0:
    C.pop(unproc)
    sql = query(C)
    unproc.append(filter(cur.fetchall(), lambda: do_some_filtering))
    cur.execute("INSERT INTO", C)

Ground Subsumption queyr on clauses. Find all sets that have C as subset

CREATE Table lit (clause INTEGER, lit STRING); -- UNIQUE (clause, lit) gives redundant lit deletion
-- find all clauses c such it contains literal a,b,c.
SELECT DISTINCT clause from lit a, lit b, lit c where a.clause = b.clause and a.clause = c.clause and a.lit = ? and b.lit = ? and c.lit = ?

Find all sets that have C as superset

SELECT DISTINCT clause from lit a 
    group by a.clause 
    HAVING COUNT(CASE WHEN lit = ? THEN 1 END) +  COUNT(CASE WHEN lit = ? THEN 1 END) +  COUNT(CASE WHEN lit = ? THEN 1 END) = count(*);
    
    count(case when lit != ? or lit != ? or lit != ? then 1 end) = 0
    count(*) <= 3 -- would this help or hurt? redundant info 
    and 
    min(case when lit = ? then 1 when lit = ? when else 0 ) = 1

t1 = "fah0"
t2 = "f01"
i1 = 0
i2 = 0
while i1 < len(t1) and i2 < len(t2):
    c1 = t1[i1] 
    c2 = t2[i2]
    if t2 


grammar = '''
E : F | G | Var | A
F : "f" E E
A : "a"
Var : NUMBER
'''

grammar = '''
Term : CNAME | CNAME "("  ("," Term)* ")" | Var
'''

class Unify(Transformer):
    env = {}
    def Var(self, v):
        if v in env:
    def term

Big question is, what is the representation of formulas, literals, clauses.

  • Strings
  • Json
  • Relational Terms / hashconsed
  • Feature vectors.

Zipperposition

logtk. Huh. LPO as a constraint solving problem https://sneeuwballen.github.io/zipperposition/2.0/logtk/Logtk_solving/index.html

demo resolution prover

#use "topfind";;
#require "logtk";;
print_endline "hello world";;
open Logtk
(* https://github.com/sneeuwballen/zipperposition/blob/master/tests/testTerm.ml *)
module T = Term
let f_ = ID.make "f"
let g_ = ID.make "g"
let ty = Type.term
let ty_t = Term.of_ty ty 
let prop = Type.prop
let f_fun = (T.const ~ty:Type.([ty;ty] ==> ty) f_)
let f x y = T.app f_fun [x; y]

let prec = Precedence.default [f_;g_]
(*let ord = Logtk.Ordering.lambda_kbo prec *)
let names = Ordering.names ()

let () = List.iter (fun name -> print_endline name) names

Zenon

https://github.com/Deducteam/zenon_modulo Wired in for producing dedukti and coq proofs. related to focalize project?


Imandra

I don’t know if this really belongs in this note Imandra is like ACL2 for ocaml I think python interface

import imandra

Comparisons

Datalog vs ATP

What makes an ATP that different from a datalog? Both are saturating. If I could limit of prune clauses of size > N, that might emulate a datalog. Or perhaps protect rule-rule resolution from happening.

The given clause algorithm has some flavor of seminaive. Datalog focuss on unit clauses Not at all clear how to do extraction on a paramodulated database.

Given that ATP subsumes in some sense both prolog and datalog, what does it say about Magic Set?

Extraction from ATP. The equality is invisible. The copying trick?

Prolog vs ATP

Set of support strategy with horn clauses in passive set, goal in active set, hyperresolution as only inference rule behaves akin to prolog.

What is the difference between E and a PROLOG system?

Prolog = is unification. ATP = is rewrite/semantical.

ATP vs ITP




Methodology

Given clause Discount loop

Strategies

Relationship to focusing http://web4.ensiie.fr/~guillaume.burel/download/LFAT.pdf

-Set of Support - Only infer on clauses that trace provenance to user specified set of support.

https://www.doc.ic.ac.uk/~kb/MACTHINGS/SLIDES/2013Notes/7LControl4up13.pdf The Predicate Ordering Syntactic Refinement - order predicates. Only resolve on lowest Locking Refinement - mark each literal in clause and resolve only on lowest one atom ordering

  • Resonance Strategy
  • Weighting

Unification

  • see note on unification Two way matching. Is there a substitution for variables that solves an equation. Yes

Anti-unification is an interesting topic. Generalization.

# https://github.com/pythological/unification
from unification import var, unify
x = var("x")
y = var("y")
y1 = var("y") # same y.
print(unify((x,y), (y1,1))) # returns subst dict

Or can use Maude’s unification. Or prolog’s. Or just write it.

Resolution

Put clauses in cnf. Each clause is an \/ of positive and negative atoms. Take two clauses that have matching (unifiable) positive and negative pieces

Hyperresolution https://www.doc.ic.ac.uk/~kb/MACTHINGS/SLIDES/8LHyper4up.pdf You are seeking clauses of only positive literals. Only use a “nuecleii” if it covers all of the negative literals.

UR-resolution - unit-resulting resolution. Only perform resultion if it results in nw unit literal (all other chunks line up). UR is essentially datalog stye resolution. It does however allow universals in the unit facts.

SLD resolution is ATP view of prolog.

Factoring

# Ground resolution
from typing import Any

Form = Any
Lit = Tuple[bool,Form]
Clause = Set[Lit]

def factor(c, i1, i2):
    (pos1, fm1) = c1[i1]
    (pos2, fm2) = c2[i2]
    assert pos1 != pos2
    assert fm1 == fm2
    res = []
    for i,x in enumerate(c1) 
        if i != i1 and i != i2:
            res.append(x)
    return res

def resolve(c1,i1, c2, i2):
    (pos1, fm1) = c1[i1]
    (pos2, fm2) = c2[i2]
    assert pos1 != pos2
    assert fm1 == fm2
    res = []
    for i,x in enumerate(c1) 
        if i != i1:
            res.append(x)
    for i,x in enumerate(c2) 
        if i != i2:
            res.append(x)
    return sorted(res)

processed = []
unprocessed = []

def resolve_cand(lit):
    lit2 = (not lit[0], lit)
    for c in processed:
        if lit2 in c:
            
while len(unprocessed) != 0:
    c = unprocessed.pop()
    for lit in c:
        resolve_cand(lit)
        cnew = resolve(c,i,c, i2)
        if cnew not in processed:
            unprocessed.append(cnew)

Index = Dict[Lit,Set[Clause]]


Subsumption

Cleaning out the database.

given C(X) , C(a) is redundant (subsumed). You can remove it from clause database. A question is how to do this quickly. See Term indexing.

https://www.doc.ic.ac.uk/~kb/MACTHINGS/SLIDES/2013Notes/6LSub4up13.pdf https://www.cs.cmu.edu/~fp/courses/99-atp/lectures/lecture20.html

encoding subusmption to SAT https://fmcad.org/FMCAD22/presentations/08%20-%20SAT%20and%20SMT%202/01_rath.pdf

Narrowing

see term rewriting

Paramodulation

paramodulation is unification based equational substitution

https://math.stackexchange.com/questions/865562/a-simple-yet-non-superficial-explanation-of-what-paramodulation-means-in-the

Paramodulation is dealing with equality pieces in clauses. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.9958&rep=rep1&type=pdf

Demodulation

rewrite rules basically? destructively simplify terms using them Positive literals can be marked or inferred as demodulators (what we ordinary think of as rewrite rules) By having both paramodulation and demodulation exist, the system has both destructive and conservative rules.

Superposition

https://en.wikipedia.org/wiki/Superposition_calculus

ntroduction to Superposition calculus - Cruanes slides

New results on rewrite-based satisfiability procedures

Term orderings - see term rewriting

Implementation of Saturating Theorem Provers

Superposition 25 http://www.eprover.org/EVENTS/Superposition-25/TutorialSP_ho.pdf

backmair and ganzinger 1994

superposition can be used instead of resolution by encoding predicates as functions to true

If I run superposition on unit positive clauses, is it performing completion? Completion of guarded rewrite systems?

E a brainiac theorem prover

The discussion of the encoding of t = s is a bit goofy. Yes, = is commutative. In that sense in is reasoonable to use sets. Could the opposite convention be used? This is kind of like the set-theoretic convention of (a,b) === { {a}, {a,b} __ } (stupid liquid formatting error) which is pretty bizarre too

t = s === { __ {t}, {s} __ } t != s === { __ {t,s} __ } Multiset orderings

Tableau

leantap - see prolog

Connection

http://www.leancop.de/ leancop http://leancop.de/nanocop/ nanocop https://github.com/01mf02/cop-rs - meancop rust impl

Literal Selection

Ordered resolution - order on literals, only resolve on biggest one (you want to get rid of big stuff)

Term Indexing

Triemaps that match SPJ and Graf https://www.youtube.com/watch?v=cT8G6FS2v94&ab_channel=Konfy

type expr = Var of string | App of expr * expr
type 'a exprmap = {var : 'a String.Map.t; app : 'a exprmap exprmap } (* I thing if we play that nested data structure stuff we're gonna have a bad time  in ocaml*)

See Automated reasoning handbook chapter 18 https://www.cs.cmu.edu/~fp/courses/atp/lectures/22-indexing.html towards efficient subsumption - Tammet Give reasoning a trie Indexes are pretty good for serialized structures. You can serialize a term/tree by . Perfect indexing vs approximate indexing. You don’t necessarily need to index only the terms you need. YOu can iterate over the returned bits and just toss out the stuff you didn’t want. Depends how expensive perfect vs imperfect is

Indexing has different kinds of queries. Is the database ground or not?

  • Variants
  • Unifiers
  • Forward Subsumption
  • Backwards subsumption You may want multi-term indexing

Path Indexing

most suitable for backward subsumption) Consider paths from root to leaf of tree. Can put this in a trie. Look at all paths in a query term of interest and take set intersection from path index trie.

Generalization? Consider just paths through a graph? f(a,b,c), g(a,d,c), … . Up down and around. Kind of reminds me of code2vec

Discrimination Trees

most suitable for unit forward subsumption

A depth first traversal of the whole term

Substitution trees

Feature Vectors

Simple and Efficient Clause Subsumption with Feature Vector Indexing

Fingerprint Indexing for Paramodulation and Rewriting

You can make summary data on terms that is monotonic with respect to instantation.

count and max depth appearance of symbols. A generalization of seperating terms out by their head symbol Imperfect indexing is useful. It does mean you may have to implement algorithms more than once.

Theories

Inductiion Superposition with Structural Induction Datatypes Superposition with Datatypes and Codatatypes Arrays Arith Beagle – A Hierarchic Superposition Theorem Prover

Proof

dedukti lfsc alethe

some provers are Directly outputting

sledgehammer. Use fastest provers, get clues out, reconstruct using slower simpler system specific provers.

Semi-Automated Bash

# verify sig text
function verify() {
    openssl dgst -sha256 -verify /tmp/publickey.pem -signature $1 $2
}

function trust() {
    openssl dgst -sha256 -sign /tmp/privatekey.pem $1
}

function infer() {
    for hyp in hyps
    do
        verify $hyp $1
    done
    
}


openssl genrsa -out /tmp/privatekey.pem 2048
echo "foo" > /tmp/foo
trust /tmp/foo > /tmp/foo.sig
verify /tmp/foo.sig /tmp/foo

Higher Order

https://events.model.in.tum.de/mod23/lectures.html jasmn banchette lectures

Applicative encoding. Turn f(x,y) into app(app(f,x),y). Wasteful and inefficint but it can work. Built in appreciation of higher order structure (currying basically?) is better Higher order lambda free. HORPO - term orderings that appreciate the applicative structure superposition for lambda free higher order Mechanical Mathematicians A new generation of automatic theorem provers eliminate bugs in software and mathematics

Differentiation? antiderivative search

Inside higher order systems Meson hammers blast First-Order Proof Tactics in Higher-Order Logic Theorem Provers - Hurd Metis

Misc

https://github.com/evhub/pyprover

Uwe Waldemann - automated reasoning courses

Ordering-Based Strategies for Hor n Clauses* - Dershowitz Equational horn clause. positive is larger than negative in literal selection

33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies

Set of Support, Demodulation, Paramodulation: A Historical Perspective pfenning book

loveland book fitting book theorem proving in otter

wos summary https://ininet.org/automated-reasoning-and-resolution.html

http://wp.doc.ic.ac.uk/kb/teaching/ autmated reasoning course krysia

Set of Support prio

Resolution is bottom up, tableau is top down?

Automated Deduction in Equational Logic and Cubic Curves

  • Idea: Convert resolution proof to bus proofs tree. Is this even possible or desirable?

Pelletier Problems

Schulz teaching automated theorem proving polymorphic vampire

An Impossible Asylum - ATP checking of Smullyan puzzle found hypotheses inconsistent

Phillips & Stanovsky https://www2.karlin.mff.cuni.cz/~stanovsk/math/slides_esarm.pdf loop theory and non associative lagerbas https://www2.karlin.mff.cuni.cz/~stanovsk/math/slides_lpar08.pdf

Searching for a Fixed Point Combinators by Using Automated Theorem Proving: A Preliminary Report

Challenge Problems Focusing on Equality and Combinatory Logic: Evaluating Automated Theorem-Proving Programs

tptp bibiography. It’s insane how much work is here

geometric database horn clauses.

Waldmeister and Twee - UEQ

REDUCING HIGHER-ORDER THEOREM PROVING TO A SEQUENCE OF SAT PROBLEMS

vampire and the fool

Dedam: A Kernel of Data Structures and Algorithms for Automated Deduction with Equality Clauses

efficient full higher order unification zipperposition

Thesis on Implementation of Higher-Order Superposition

auto2 interesting system that combines egraphs and sturation proving https://github.com/bzhan/auto2 he alos mentions gowers as influence Coming to Terms with Quantified Reasoning algerbaic datatypes require an infinite number of acyclicity axioms.

https://www.ps.uni-saarland.de/Publications/documents/Treinen_tacs91.pdf FOL + datatypes is more powerful than just FOL.

from pysmt.shortcuts import get_env
from pysmt.logics import LOGICS
from pysmt.shortcuts import Symbol, get_env, Solver
env = get_env().factory.add_generic_solver("vampire", ["/home/philip/.local/bin/vampire", "--input_syntax", "smtlib2"], LOGICS)
print("opening")
with Solver(name="vampire", logic="QF_UFLRA") as s:
  print("starting")
  print(s.is_sat(Symbol("x"))) # True
from z3 import *
s = Solver()
x = Bool("x")
s.add(And(x, Not(x)))
#s.add(x)
s.check()
print(s.to_smt2())
import tempfile
import subprocess
with tempfile.NamedTemporaryFile(suffix=".smt2") as fp:
    fp.write(s.to_smt2().encode() + b"(get-model)")
    fp.flush()
    res = subprocess.run(["/home/philip/.local/bin/vampire", fp.name],  capture_output=True)
    print(res.stdout.decode())