(* ========================================================================= *) (* Model elimination procedure (MESON version, based on Stickel's PTTP). *) (* *) (* Copyright (c) 2003-2007, John Harrison. (See "LICENSE.txt" for details.) *) (* ========================================================================= *) (* ------------------------------------------------------------------------- *) (* Example of naivety of tableau prover. *) (* ------------------------------------------------------------------------- *) START_INTERACTIVE;; tab <>;; tab <>;; (* ------------------------------------------------------------------------- *) (* The interesting example where tableaux connections make the proof longer. *) (* Unfortuntely this gets hammered by normalization first... *) (* ------------------------------------------------------------------------- *) tab <<~p /\ (p \/ q) /\ (r \/ s) /\ (~q \/ t \/ u) /\ (~r \/ ~t) /\ (~r \/ ~u) /\ (~q \/ v \/ w) /\ (~s \/ ~v) /\ (~s \/ ~w) ==> false>>;; END_INTERACTIVE;; (* ------------------------------------------------------------------------- *) (* Generation of contrapositives. *) (* ------------------------------------------------------------------------- *) let contrapositives cls = let base = map (fun c -> map negate (subtract cls [c]),c) cls in if forall negative cls then (map negate cls,False)::base else base;; (* ------------------------------------------------------------------------- *) (* The core of MESON: ancestor unification or Prolog-style extension. *) (* ------------------------------------------------------------------------- *) let rec mexpand rules ancestors g cont (env,n,k) = if n < 0 then failwith "Too deep" else try tryfind (fun a -> cont (unify_literals env (g,negate a),n,k)) ancestors with Failure _ -> tryfind (fun rule -> let (asm,c),k' = renamerule k rule in itlist (mexpand rules (g::ancestors)) asm cont (unify_literals env (g,c),n-length asm,k')) rules;; (* ------------------------------------------------------------------------- *) (* Full MESON procedure. *) (* ------------------------------------------------------------------------- *) let puremeson fm = let cls = simpcnf(specialize(pnf fm)) in let rules = itlist ((@) ** contrapositives) cls [] in deepen (fun n -> mexpand rules [] False (fun x -> x) (undefined,n,0); n) 0;; let meson fm = let fm1 = askolemize(Not(generalize fm)) in map (puremeson ** list_conj) (simpdnf fm1);; (* ------------------------------------------------------------------------- *) (* Example. *) (* ------------------------------------------------------------------------- *) START_INTERACTIVE;; let davis_putnam_example = meson < (F(y,z) /\ F(z,z))) /\ ((F(x,y) /\ G(x,y)) ==> (G(x,z) /\ G(z,z)))>>;; END_INTERACTIVE;; (* ------------------------------------------------------------------------- *) (* With repetition checking and divide-and-conquer search. *) (* ------------------------------------------------------------------------- *) let rec equal env fm1 fm2 = try unify_literals env (fm1,fm2) == env with Failure _ -> false;; let expand2 expfn goals1 n1 goals2 n2 n3 cont env k = expfn goals1 (fun (e1,r1,k1) -> expfn goals2 (fun (e2,r2,k2) -> if n2 + r1 <= n3 + r2 then failwith "pair" else cont(e2,r2,k2)) (e1,n2+r1,k1)) (env,n1,k);; let rec mexpand rules ancestors g cont (env,n,k) = if n < 0 then failwith "Too deep" else if exists (equal env g) ancestors then failwith "repetition" else try tryfind (fun a -> cont (unify_literals env (g,negate a),n,k)) ancestors with Failure _ -> tryfind (fun r -> let (asm,c),k' = renamerule k r in mexpands rules (g::ancestors) asm cont (unify_literals env (g,c),n-length asm,k')) rules and mexpands rules ancestors gs cont (env,n,k) = if n < 0 then failwith "Too deep" else let m = length gs in if m <= 1 then itlist (mexpand rules ancestors) gs cont (env,n,k) else let n1 = n / 2 in let n2 = n - n1 in let goals1,goals2 = chop_list (m / 2) gs in let expfn = expand2 (mexpands rules ancestors) in try expfn goals1 n1 goals2 n2 (-1) cont env k with Failure _ -> expfn goals2 n1 goals1 n2 n1 cont env k;; let puremeson fm = let cls = simpcnf(specialize(pnf fm)) in let rules = itlist ((@) ** contrapositives) cls [] in deepen (fun n -> mexpand rules [] False (fun x -> x) (undefined,n,0); n) 0;; let meson fm = let fm1 = askolemize(Not(generalize fm)) in map (puremeson ** list_conj) (simpdnf fm1);; (* ------------------------------------------------------------------------- *) (* The Los problem (depth 20) and the Steamroller (depth 53) --- lengthier. *) (* ------------------------------------------------------------------------- *) START_INTERACTIVE;; (*********** let los = meson <<(forall x y z. P(x,y) ==> P(y,z) ==> P(x,z)) /\ (forall x y z. Q(x,y) ==> Q(y,z) ==> Q(x,z)) /\ (forall x y. Q(x,y) ==> Q(y,x)) /\ (forall x y. P(x,y) \/ Q(x,y)) ==> (forall x y. P(x,y)) \/ (forall x y. Q(x,y))>>;; let steamroller = meson <<((forall x. P1(x) ==> P0(x)) /\ (exists x. P1(x))) /\ ((forall x. P2(x) ==> P0(x)) /\ (exists x. P2(x))) /\ ((forall x. P3(x) ==> P0(x)) /\ (exists x. P3(x))) /\ ((forall x. P4(x) ==> P0(x)) /\ (exists x. P4(x))) /\ ((forall x. P5(x) ==> P0(x)) /\ (exists x. P5(x))) /\ ((exists x. Q1(x)) /\ (forall x. Q1(x) ==> Q0(x))) /\ (forall x. P0(x) ==> (forall y. Q0(y) ==> R(x,y)) \/ ((forall y. P0(y) /\ S0(y,x) /\ (exists z. Q0(z) /\ R(y,z)) ==> R(x,y)))) /\ (forall x y. P3(y) /\ (P5(x) \/ P4(x)) ==> S0(x,y)) /\ (forall x y. P3(x) /\ P2(y) ==> S0(x,y)) /\ (forall x y. P2(x) /\ P1(y) ==> S0(x,y)) /\ (forall x y. P1(x) /\ (P2(y) \/ Q1(y)) ==> ~(R(x,y))) /\ (forall x y. P3(x) /\ P4(y) ==> R(x,y)) /\ (forall x y. P3(x) /\ P5(y) ==> ~(R(x,y))) /\ (forall x. (P4(x) \/ P5(x)) ==> exists y. Q0(y) /\ R(x,y)) ==> exists x y. P0(x) /\ P0(y) /\ exists z. Q1(z) /\ R(y,z) /\ R(x,y)>>;; ****************) (* ------------------------------------------------------------------------- *) (* Test it. *) (* ------------------------------------------------------------------------- *) let prop_1 = time meson <

q <=> ~q ==> ~p>>;; let prop_2 = time meson <<~ ~p <=> p>>;; let prop_3 = time meson <<~(p ==> q) ==> q ==> p>>;; let prop_4 = time meson <<~p ==> q <=> ~q ==> p>>;; let prop_5 = time meson <<(p \/ q ==> p \/ r) ==> p \/ (q ==> r)>>;; let prop_6 = time meson <

>;; let prop_7 = time meson <

>;; let prop_8 = time meson <<((p ==> q) ==> p) ==> p>>;; let prop_9 = time meson <<(p \/ q) /\ (~p \/ q) /\ (p \/ ~q) ==> ~(~q \/ ~q)>>;; let prop_10 = time meson <<(q ==> r) /\ (r ==> p /\ q) /\ (p ==> q /\ r) ==> (p <=> q)>>;; let prop_11 = time meson <

p>>;; let prop_12 = time meson <<((p <=> q) <=> r) <=> (p <=> (q <=> r))>>;; let prop_13 = time meson <

(p \/ q) /\ (p \/ r)>>;; let prop_14 = time meson <<(p <=> q) <=> (q \/ ~p) /\ (~q \/ p)>>;; let prop_15 = time meson <

q <=> ~p \/ q>>;; let prop_16 = time meson <<(p ==> q) \/ (q ==> p)>>;; let prop_17 = time meson <

r) ==> s <=> (~p \/ q \/ s) /\ (~p \/ ~r \/ s)>>;; (* ------------------------------------------------------------------------- *) (* Monadic Predicate Logic. *) (* ------------------------------------------------------------------------- *) let p18 = time meson < P(x)>>;; let p19 = time meson < Q(z)) ==> P(x) ==> Q(x)>>;; let p20 = time meson <<(forall x y. exists z. forall w. P(x) /\ Q(y) ==> R(z) /\ U(w)) ==> (exists x y. P(x) /\ Q(y)) ==> (exists z. R(z))>>;; let p21 = time meson <<(exists x. P ==> Q(x)) /\ (exists x. Q(x) ==> P) ==> (exists x. P <=> Q(x))>>;; let p22 = time meson <<(forall x. P <=> Q(x)) ==> (P <=> (forall x. Q(x)))>>;; let p23 = time meson <<(forall x. P \/ Q(x)) <=> P \/ (forall x. Q(x))>>;; let p24 = time meson <<~(exists x. U(x) /\ Q(x)) /\ (forall x. P(x) ==> Q(x) \/ R(x)) /\ ~(exists x. P(x) ==> (exists x. Q(x))) /\ (forall x. Q(x) /\ R(x) ==> U(x)) ==> (exists x. P(x) /\ R(x))>>;; let p25 = time meson <<(exists x. P(x)) /\ (forall x. U(x) ==> ~G(x) /\ R(x)) /\ (forall x. P(x) ==> G(x) /\ U(x)) /\ ((forall x. P(x) ==> Q(x)) \/ (exists x. Q(x) /\ P(x))) ==> (exists x. Q(x) /\ P(x))>>;; let p26 = time meson <<((exists x. P(x)) <=> (exists x. Q(x))) /\ (forall x y. P(x) /\ Q(y) ==> (R(x) <=> U(y))) ==> ((forall x. P(x) ==> R(x)) <=> (forall x. Q(x) ==> U(x)))>>;; let p27 = time meson <<(exists x. P(x) /\ ~Q(x)) /\ (forall x. P(x) ==> R(x)) /\ (forall x. U(x) /\ V(x) ==> P(x)) /\ (exists x. R(x) /\ ~Q(x)) ==> (forall x. V(x) ==> ~R(x)) ==> (forall x. U(x) ==> ~V(x))>>;; let p28 = time meson <<(forall x. P(x) ==> (forall x. Q(x))) /\ ((forall x. Q(x) \/ R(x)) ==> (exists x. Q(x) /\ R(x))) /\ ((exists x. R(x)) ==> (forall x. L(x) ==> M(x))) ==> (forall x. P(x) /\ L(x) ==> M(x))>>;; let p29 = time meson <<(exists x. P(x)) /\ (exists x. G(x)) ==> ((forall x. P(x) ==> H(x)) /\ (forall x. G(x) ==> J(x)) <=> (forall x y. P(x) /\ G(y) ==> H(x) /\ J(y)))>>;; let p30 = time meson <<(forall x. P(x) \/ G(x) ==> ~H(x)) /\ (forall x. (G(x) ==> ~U(x)) ==> P(x) /\ H(x)) ==> (forall x. U(x))>>;; let p31 = time meson <<~(exists x. P(x) /\ (G(x) \/ H(x))) /\ (exists x. Q(x) /\ P(x)) /\ (forall x. ~H(x) ==> J(x)) ==> (exists x. Q(x) /\ J(x))>>;; let p32 = time meson <<(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))>>;; let p33 = time meson <<(forall x. P(a) /\ (P(x) ==> P(b)) ==> P(c)) <=> (forall x. P(a) ==> P(x) \/ P(c)) /\ (P(a) ==> P(b) ==> P(c))>>;; let p34 = time meson <<((exists x. forall y. P(x) <=> P(y)) <=> ((exists x. Q(x)) <=> (forall y. Q(y)))) <=> ((exists x. forall y. Q(x) <=> Q(y)) <=> ((exists x. P(x)) <=> (forall y. P(y))))>>;; let p35 = time meson < (forall x y. P(x,y))>>;; (* ------------------------------------------------------------------------- *) (* Full predicate logic (without Identity and Functions) *) (* ------------------------------------------------------------------------- *) let p36 = time meson <<(forall x. exists y. P(x,y)) /\ (forall x. exists y. G(x,y)) /\ (forall x y. P(x,y) \/ G(x,y) ==> (forall z. P(y,z) \/ G(y,z) ==> H(x,z))) ==> (forall x. exists y. H(x,y))>>;; let p37 = time meson <<(forall z. exists w. forall x. exists y. (P(x,z) ==> P(y,w)) /\ P(y,z) /\ (P(y,w) ==> (exists u. Q(u,w)))) /\ (forall x z. ~P(x,z) ==> (exists y. Q(y,z))) /\ ((exists x y. Q(x,y)) ==> (forall x. R(x,x))) ==> (forall x. exists y. R(x,y))>>;; let p38 = time meson <<(forall x. P(a) /\ (P(x) ==> (exists y. P(y) /\ R(x,y))) ==> (exists z w. P(z) /\ R(x,w) /\ R(w,z))) <=> (forall x. (~P(a) \/ P(x) \/ (exists z w. P(z) /\ R(x,w) /\ R(w,z))) /\ (~P(a) \/ ~(exists y. P(y) /\ R(x,y)) \/ (exists z w. P(z) /\ R(x,w) /\ R(w,z))))>>;; let p39 = time meson <<~(exists x. forall y. P(y,x) <=> ~P(y,y))>>;; let p40 = time meson <<(exists y. forall x. P(x,y) <=> P(x,x)) ==> ~(forall x. exists y. forall z. P(z,y) <=> ~P(z,x))>>;; let p41 = time meson <<(forall z. exists y. forall x. P(x,y) <=> P(x,z) /\ ~P(x,x)) ==> ~(exists z. forall x. P(x,z))>>;; let p42 = time meson <<~(exists y. forall x. P(x,y) <=> ~(exists z. P(x,z) /\ P(z,x)))>>;; let p43 = time meson <<(forall x y. Q(x,y) <=> forall z. P(z,x) <=> P(z,y)) ==> forall x y. Q(x,y) <=> Q(y,x)>>;; let p44 = time meson <<(forall x. P(x) ==> (exists y. G(y) /\ H(x,y)) /\ (exists y. G(y) /\ ~H(x,y))) /\ (exists x. J(x) /\ (forall y. G(y) ==> H(x,y))) ==> (exists x. J(x) /\ ~P(x))>>;; let p45 = time meson <<(forall x. P(x) /\ (forall y. G(y) /\ H(x,y) ==> J(x,y)) ==> (forall y. G(y) /\ H(x,y) ==> R(y))) /\ ~(exists y. L(y) /\ R(y)) /\ (exists x. P(x) /\ (forall y. H(x,y) ==> L(y)) /\ (forall y. G(y) /\ H(x,y) ==> J(x,y))) ==> (exists x. P(x) /\ ~(exists y. G(y) /\ H(x,y)))>>;; let p46 = time meson <<(forall x. P(x) /\ (forall y. P(y) /\ H(y,x) ==> G(y)) ==> G(x)) /\ ((exists x. P(x) /\ ~G(x)) ==> (exists x. P(x) /\ ~G(x) /\ (forall y. P(y) /\ ~G(y) ==> J(x,y)))) /\ (forall x y. P(x) /\ P(y) /\ H(x,y) ==> ~J(y,x)) ==> (forall x. P(x) ==> G(x))>>;; (* ------------------------------------------------------------------------- *) (* Example from Manthey and Bry, CADE-9. *) (* ------------------------------------------------------------------------- *) let p55 = time meson < hates(x,y) /\ ~richer(x,y)) /\ (forall x. hates(agatha,x) ==> ~hates(charles,x)) /\ (hates(agatha,agatha) /\ hates(agatha,charles)) /\ (forall x. lives(x) /\ ~richer(x,agatha) ==> hates(butler,x)) /\ (forall x. hates(agatha,x) ==> hates(butler,x)) /\ (forall x. ~hates(x,agatha) \/ ~hates(x,butler) \/ ~hates(x,charles)) ==> killed(agatha,agatha) /\ ~killed(butler,agatha) /\ ~killed(charles,agatha)>>;; let p57 = time meson < P(x,z)) ==> P(f(a,b),f(a,c))>>;; (* ------------------------------------------------------------------------- *) (* See info-hol, circa 1500. *) (* ------------------------------------------------------------------------- *) let p58 = time meson < ((P(v) \/ R(w)) /\ (R(z) ==> Q(v))))>>;; let p59 = time meson <<(forall x. P(x) <=> ~P(f(x))) ==> (exists x. P(x) /\ ~P(f(x)))>>;; let p60 = time meson < exists y. (forall z. P(z,y) ==> P(z,f(x))) /\ P(x,y)>>;; (* ------------------------------------------------------------------------- *) (* From Gilmore's classic paper. *) (* ------------------------------------------------------------------------- *) (*** Amazingly, this still seems non-trivial... in HOL it works at depth 45! let gilmore_1 = time meson < G(y)) <=> F(x)) /\ ((F(y) ==> H(y)) <=> G(x)) /\ (((F(y) ==> G(y)) ==> H(y)) <=> H(x)) ==> F(z) /\ G(z) /\ H(z)>>;; ***) (*** This is not valid, according to Gilmore let gilmore_2 = time meson < F(z,y)) /\ (F(z,y) <=> F(z,z)) /\ (F(x,y) <=> F(y,x)) ==> (F(x,y) <=> F(x,z))>>;; ***) let gilmore_3 = time meson < (G(y) ==> H(x))) ==> F(x,x)) /\ ((F(z,x) ==> G(x)) ==> H(z)) /\ F(x,y) ==> F(z,z)>>;; let gilmore_4 = time meson < F(y,z) /\ F(z,z)) /\ (F(x,y) /\ G(x,y) ==> G(x,z) /\ G(z,z))>>;; let gilmore_5 = time meson <<(forall x. exists y. F(x,y) \/ F(y,x)) /\ (forall x y. F(y,x) ==> F(y,y)) ==> exists z. F(z,z)>>;; let gilmore_6 = time meson < G(v,u) /\ G(u,x)) ==> (exists u. forall v. F(u,y) ==> G(v,u) /\ G(u,y)) \/ (forall u v. exists w. G(v,u) \/ H(w,y,u) ==> G(u,w))>>;; let gilmore_7 = time meson <<(forall x. K(x) ==> exists y. L(y) /\ (F(x,y) ==> G(x,y))) /\ (exists z. K(z) /\ forall u. L(u) ==> F(z,u)) ==> exists v w. K(v) /\ L(w) /\ G(v,w)>>;; let gilmore_8 = time meson < (G(y) ==> (forall u. exists v. H(u,v,x)))) ==> F(x,x)) /\ ((F(z,x) ==> G(x)) ==> (forall u. exists v. H(u,v,z))) /\ F(x,y) ==> F(z,z)>>;; (*** This is still a very hard problem let gilmore_9 = time meson < (forall u. exists v. F(x,u,v) /\ G(z,u) /\ ~H(x,z)) ==> (forall u. exists v. F(x,u,v) /\ G(y,u) /\ ~H(x,y))) /\ ((forall u. exists v. F(x,u,v) /\ G(y,u) /\ ~H(x,y)) ==> ~(forall u. exists v. F(x,u,v) /\ G(z,u) /\ ~H(x,z)) ==> (forall u. exists v. F(y,u,v) /\ G(y,u) /\ ~H(y,x)) /\ (forall u. exists v. F(z,u,v) /\ G(y,u) /\ ~H(z,y)))>>;; ***) (* ------------------------------------------------------------------------- *) (* Translation of Gilmore procedure using separate definitions. *) (* ------------------------------------------------------------------------- *) let gilmore_9a = time meson <<(forall x y. P(x,y) <=> forall u. exists v. F(x,u,v) /\ G(y,u) /\ ~H(x,y)) ==> forall x. exists y. forall z. (P(y,x) ==> (P(x,z) ==> P(x,y))) /\ (P(x,y) ==> (~P(x,z) ==> P(y,x) /\ P(z,y)))>>;; (* ------------------------------------------------------------------------- *) (* Example from Davis-Putnam papers where Gilmore procedure is poor. *) (* ------------------------------------------------------------------------- *) let davis_putnam_example = time meson < (F(y,z) /\ F(z,z))) /\ ((F(x,y) /\ G(x,y)) ==> (G(x,z) /\ G(z,z)))>>;; (* ------------------------------------------------------------------------- *) (* The "connections make things worse" example once again. *) (* ------------------------------------------------------------------------- *) meson <<~p /\ (p \/ q) /\ (r \/ s) /\ (~q \/ t \/ u) /\ (~r \/ ~t) /\ (~r \/ ~u) /\ (~q \/ v \/ w) /\ (~s \/ ~v) /\ (~s \/ ~w) ==> false>>;; END_INTERACTIVE;;