Tag Archives: Formal Methods

Notes on Synthesis and Equation Proving for Catlab.jl

By: Philip Zucker

Re-posted from: https://www.philipzucker.com/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
        head: Symbol call
        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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol compose
              2: Symbol f
              3: Symbol g
          3: Symbol h
      right: Expr
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          6: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol B
              3: Symbol C
          7: Expr
            head: Symbol call
            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.

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
        head: Symbol call
        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
        head: Symbol call
        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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          5: Expr
            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol compose
              2: Symbol f
              3: Symbol g
          3: Symbol h
      right: Expr
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          6: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol B
              3: Symbol C
          7: Expr
            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            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
            head: Symbol call
            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

Notes on Synthesis and Equation Proving for Catlab.jl

By: philzook58

Re-posted from: https:/www.philipzucker.com/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
        head: Symbol call
        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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol compose
              2: Symbol f
              3: Symbol g
          3: Symbol h
      right: Expr
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          6: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol B
              3: Symbol C
          7: Expr
            head: Symbol call
            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
        head: Symbol call
        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
        head: Symbol call
        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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          5: Expr
            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol compose
              2: Symbol f
              3: Symbol g
          3: Symbol h
      right: Expr
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
              3: Symbol B
          6: Expr
            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol B
              3: Symbol C
          7: Expr
            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Symbol f
          3: Expr
            head: Symbol call
            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
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            head: Symbol call
            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
        head: Symbol call
        args: Array{Any}((3,))
          1: Symbol compose
          2: Expr
            head: Symbol call
            args: Array{Any}((2,))
              1: Symbol id
              2: Symbol A
          3: Symbol f
      right: Symbol f
      context: OrderedCollections.OrderedDict{Symbol,Union{Expr, Symbol}}
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          1: Symbol A
          2: Symbol B
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        vals: Array{Union{Expr, Symbol}}((3,))
          1: Symbol Ob
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            head: Symbol call
            args: Array{Any}((3,))
              1: Symbol Hom
              2: Symbol A
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      doc: Nothing nothing
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    keys: Array{Symbol}((16,))
      1: Symbol ⋅
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      16: #undef
    vals: Array{Symbol}((16,))
      1: Symbol compose
      2: #undef
      3: #undef
      4: #undef
      5: #undef
      ...
      12: #undef
      13: #undef
      14: #undef
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Yet Another MicroKanren in Julia

By: philzook58

Re-posted from: https://www.philipzucker.com/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 embedable 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) 
    while isa(u,Var) && haskey(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 
        @assert u.head === :call && v.head === :call 
        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

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))

Thoughts & Links

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

https://github.com/ekmett/guanxi https://www.youtube.com/watch?v=D7rlJWc3474&ab_channel=MonadicWarsaw

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