atp-haskell-1.7: src/Data/Logic/ATP/Prolog.hs
-- | Backchaining procedure for Horn clauses, and toy Prolog implementation.
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# OPTIONS_GHC -Wall #-}
module Data.Logic.ATP.Prolog where
import Data.List as List (map)
import Data.Logic.ATP.Apply (HasApply(TermOf))
import Data.Logic.ATP.FOL (var, lsubst)
import Data.Logic.ATP.Formulas (IsFormula(AtomOf))
-- import Data.Logic.ATP.Lib (deepen)
import Data.Logic.ATP.Lit (IsLiteral, JustLiteral)
import Data.Logic.ATP.Term (IsTerm(TVarOf), vt)
import Data.Map.Strict as Map
import Data.Set as Set
import Data.String (fromString)
import Test.HUnit
data PrologRule lit = Prolog (Set lit) lit deriving (Eq, Ord)
-- -------------------------------------------------------------------------
-- Rename a rule.
-- -------------------------------------------------------------------------
renamerule :: forall lit atom term v.
(IsLiteral lit, JustLiteral lit, Ord lit, HasApply atom, IsTerm term,
atom ~ AtomOf lit, term ~ TermOf atom, v ~ TVarOf term) =>
Int -> PrologRule lit -> (PrologRule lit, Int)
renamerule k (Prolog asm c) =
(Prolog (Set.map inst asm) (inst c), k + Set.size fvs)
where
fvs = Set.fold (Set.union . var) (Set.empty :: Set v) (Set.insert c asm)
vvs = Map.fromList (List.map (\(v, i) -> (v, vt (fromString ("_" ++ show i)))) (zip (Set.toList fvs) [k..]))
inst = lsubst vvs
{-
(* ------------------------------------------------------------------------- *)
(* Basic prover for Horn clauses based on backchaining with unification. *)
(* ------------------------------------------------------------------------- *)
let rec backchain rules n k env goals =
match goals with
[] -> env
| g::gs ->
if n = 0 then failwith "Too deep" else
tryfind (fun rule ->
let (a,c),k' = renamerule k rule in
backchain rules (n - 1) k' (unify_literals env (c,g)) (a @ gs))
rules;;
let hornify cls =
let pos,neg = partition positive cls in
if length pos > 1 then failwith "non-Horn clause"
else (map negate neg,if pos = [] then False else hd pos);;
let hornprove fm =
let rules = map hornify (simpcnf(skolemize(Not(generalize fm)))) in
deepen (fun n -> backchain rules n 0 undefined [False],n) 0;;
(* ------------------------------------------------------------------------- *)
(* A Horn example. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
let p32 = hornprove
<<(forall x. P(x) /\ (G(x) \/ H(x)) ==> Q(x)) /\
(forall x. Q(x) /\ H(x) ==> J(x)) /\
(forall x. R(x) ==> H(x))
==> (forall x. P(x) /\ R(x) ==> J(x))>>;;
(* ------------------------------------------------------------------------- *)
(* A non-Horn example. *)
(* ------------------------------------------------------------------------- *)
(****************
hornprove <<(p \/ q) /\ (~p \/ q) /\ (p \/ ~q) ==> ~(~q \/ ~q)>>;;
**********)
END_INTERACTIVE;;
(* ------------------------------------------------------------------------- *)
(* Parsing rules in a Prolog-like syntax. *)
(* ------------------------------------------------------------------------- *)
let parserule s =
let c,rest =
parse_formula (parse_infix_atom,parse_atom) [] (lex(explode s)) in
let asm,rest1 =
if rest <> [] & hd rest = ":-"
then parse_list ","
(parse_formula (parse_infix_atom,parse_atom) []) (tl rest)
else [],rest in
if rest1 = [] then (asm,c) else failwith "Extra material after rule";;
(* ------------------------------------------------------------------------- *)
(* Prolog interpreter: just use depth-first search not iterative deepening. *)
(* ------------------------------------------------------------------------- *)
let simpleprolog rules gl =
backchain (map parserule rules) (-1) 0 undefined [parse gl];;
(* ------------------------------------------------------------------------- *)
(* Ordering example. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
let lerules = ["0 <= X"; "S(X) <= S(Y) :- X <= Y"];;
simpleprolog lerules "S(S(0)) <= S(S(S(0)))";;
(*** simpleprolog lerules "S(S(0)) <= S(0)";;
***)
let env = simpleprolog lerules "S(S(0)) <= X";;
apply env "X";;
END_INTERACTIVE;;
(* ------------------------------------------------------------------------- *)
(* With instantiation collection to produce a more readable result. *)
(* ------------------------------------------------------------------------- *)
let prolog rules gl =
let i = solve(simpleprolog rules gl) in
mapfilter (fun x -> Atom(R("=",[Var x; apply i x]))) (fv(parse gl));;
(* ------------------------------------------------------------------------- *)
(* Example again. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
prolog lerules "S(S(0)) <= X";;
(* ------------------------------------------------------------------------- *)
(* Append example, showing symmetry between inputs and outputs. *)
(* ------------------------------------------------------------------------- *)
let appendrules =
["append(nil,L,L)"; "append(H::T,L,H::A) :- append(T,L,A)"];;
prolog appendrules "append(1::2::nil,3::4::nil,Z)";;
prolog appendrules "append(1::2::nil,Y,1::2::3::4::nil)";;
prolog appendrules "append(X,3::4::nil,1::2::3::4::nil)";;
prolog appendrules "append(X,Y,1::2::3::4::nil)";;
(* ------------------------------------------------------------------------- *)
(* However this way round doesn't work. *)
(* ------------------------------------------------------------------------- *)
(***
*** prolog appendrules "append(X,3::4::nil,X)";;
***)
(* ------------------------------------------------------------------------- *)
(* A sorting example (from Lloyd's "Foundations of Logic Programming"). *)
(* ------------------------------------------------------------------------- *)
let sortrules =
["sort(X,Y) :- perm(X,Y),sorted(Y)";
"sorted(nil)";
"sorted(X::nil)";
"sorted(X::Y::Z) :- X <= Y, sorted(Y::Z)";
"perm(nil,nil)";
"perm(X::Y,U::V) :- delete(U,X::Y,Z), perm(Z,V)";
"delete(X,X::Y,Y)";
"delete(X,Y::Z,Y::W) :- delete(X,Z,W)";
"0 <= X";
"S(X) <= S(Y) :- X <= Y"];;
prolog sortrules
"sort(S(S(S(S(0))))::S(0)::0::S(S(0))::S(0)::nil,X)";;
(* ------------------------------------------------------------------------- *)
(* Yet with a simple swap of the first two predicates... *)
(* ------------------------------------------------------------------------- *)
let badrules =
["sort(X,Y) :- sorted(Y), perm(X,Y)";
"sorted(nil)";
"sorted(X::nil)";
"sorted(X::Y::Z) :- X <= Y, sorted(Y::Z)";
"perm(nil,nil)";
"perm(X::Y,U::V) :- delete(U,X::Y,Z), perm(Z,V)";
"delete(X,X::Y,Y)";
"delete(X,Y::Z,Y::W) :- delete(X,Z,W)";
"0 <= X";
"S(X) <= S(Y) :- X <= Y"];;
(*** This no longer works
prolog badrules
"sort(S(S(S(S(0))))::S(0)::0::S(S(0))::S(0)::nil,X)";;
***)
END_INTERACTIVE;;
-}
testProlog :: Test
testProlog = TestLabel "Prolog" (TestList [])