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(* ========================================================================= *) (* Backchaining procedure for Horn clauses, and toy Prolog implementation. *) (* ========================================================================= *) (* ------------------------------------------------------------------------- *) (* Rename a rule. *) (* ------------------------------------------------------------------------- *) let renamerule k (asm,c) = let fvs = fv(list_conj(c::asm)) in let n = length fvs in let vvs = map (fun i -> "_" ^ string_of_int i) (k -- (k+n-1)) in let inst = subst(fpf fvs (map (fun x -> Var x) vvs)) in (map inst asm,inst c),k+n;; (* ------------------------------------------------------------------------- *) (* 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;;