SMELS: Sat Modulo Equality with Lazy Superposition

1. SMELS: Sat Modulo Equality with Lazy Superposition Christopher Lynch – Clarkson Duc-Khanh Tran - MPI

2. Interest Verification problems often reduce to formulas containing • mostly ground equations and • quantified equations representing properties or theories

3. Goal Efficient inference system for deciding satisfiability of sets of equational clauses, mostly ground

4. Assumptions • DPLL(cc) most efficient way of solving ground equational clauses • Superposition most efficient way of solving nonground equational clauses • Develop complete implementable combination of the two methods • DPLL(cc(Sup))

5. Contents of Talk • DPLL(cc) • Superposition • SMELS: DPLL(cc) with Lazy Superposition • Completeness • Implementation plans

6. DPLL(cc) • DPLL: Given set of clauses S, tries to build model of S by adding literals one by one • DPLL(cc): Given set of equational clauses, tries to build model by adding literals one by one, and checking consistency in background theory (Cong. Closure)

7. Responsibility of cc • Receives set M of (dis)equations • Notifies DPLL procedure if M inconsistent • Returns J µ M, justification of inconsistency • Clause : J (or alternative) can be added as lemma

8. Using cc for implication • Given M find L where M ² L • And find small J µ M where J ² L • DPLL adds : J Ç L (or alternative) as lemma

9. Example • f(a)=b Ç d!=e • a=c Ç i!=j • d=e Ç g!=h • i=j DPLL generates {i=j, a=c, g=h, d=e, f(a)=b} g=h is justification for f(c)=b (not only one) Then g!=h Ç f(c)=b added as lemma

10. Definition of Justification • Let S be set of clauses, M (partial) model • Model is set of (dis)equations • Let L 2 M • j is a function where • j(L) µ M and • S [ j(L) ² L

11. Summary so far • DPLL sends partial model M to cc • cc determines consistency of M • If M ² L, there 9 just. j(L) • It is sound to add : j(L) Ç L • Note: We can always have j(L) = {L} • Self-justification

12. Superposition ¡Ç u[s’] = v ¢Ç s=t -------------------------------- (¡Ç¢Ç u[t] = v)¾ • ¾ = mgu(s,s’) and s’ not variable • s !· t, u[s’] !· v, s=t max, u[s’] = v max Also for u[s’] != v

13. Orderings are crucial • Without orderings, no hope of termination • Example: • : gt(x,0) Ç gt(s(x),0) • gt(c,0) • With orderings it immediately halts

14. SMELS • Let S be set of clauses, g(S): ground clauses in S, v(S): nonground clauses in S • DPLL receives g(S) and passes M to cc • cc passes reduced implied (dis)equations T to Sup • Sup performs inferences between T and v(S), justified ground clauses sent to DPLL

15. Superposition in DPLL(cc(Sup) • There are two kinds of Superposition • Superposition among nonground clauses • Superposition among nonground clause and implied (dis)equation from cc (Justified Sup) • No Superposition between ground clauses

16. Nonground Superposition • We modify Superposition so that inferences involve maximal literals of nonground part of clause (as opposed to max of entire clause) • Equational Factoring and Equation Resolution also involve maximal nonground literal

17. Example of Nonground Sup • Premises • f(g(a))=b Ç g(x)=x Çf(g(x))=x • f(f(a))=c Ç g(a)=c Çg(y)=y • Conclusion • f(g(a))=b Ç f(f(a))=c Ç g(a)=c Ç g(x)=x Çf(x)=x

18. Justified Superposition • Between nonground clause and literal L from cc, After Superposition, we add negation of justification • Equivalently, a Superposition inference between nonground clause and : j(L) Ç L

19. Examples of Justified Sup • Suppose j(f(a)=b) = {d=e, f(b)=e} • Let g(f(c))=c Ç f(x)=x Çf(x)=g(x)2 v(S) • Then Justified Superposition gives d!=e Ç f(b)!=e Ç g(f(c))=c Ç f(a)=a Çb=g(a) • This is ground, so passed back to DPLL

20. Example of DPLL(cc(Sup)) • p(a,b) = p1 • p(c,d) = p2 • p(e,f) = p3 • p1 = p2 Ç p1 = p3 • a != c • a != e • p(x1,y1) != p(x2,y2) Ç x1= x2

21. DPLL • Input: g(S) = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2 Ç p1=p3, a != c, a != e} • Output: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2, a!=c, a!=e} • j(p1=p2) = {p1=p2} • For all other L 2 M, j(L) = ;

22. cc • Input: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2, a!=c, a!=e} • Output: T = {p(a,b)=p2, p(c,d)=p2, p(e,f)= p3, p1=p2, a!=c, a!=e} • j(p(a,b)=p2) = {p1=p2}

23. Sup • Input: T = {p(a,b)=p2, p(c,d)=p2, p(e,f)= p3, p1=p2, a!=c, a!=e} • v(S) = {p(x1,y1) != p(x2,y2) Ç x1= x2} • Justified Superposition gives {p1!=p2 Çp2!=p(x2,y2) Ç a=x2, p2!=p(x2,y2) Ç c=x2, p3!=p(x2,y2) Ç e=x2} • Also: p1!=p2 Ç a=c

24. DPLL • Input: g(S) = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2 Ç p1=p3, a != c, a != e, p1!=p2 Ç a=c} • Output: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a!=e} • j(p1=p3) = ;

25. cc • Input: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a!=e} • Output: T = {p(a,b)=p3, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a!=e} • j(p(a,b)=p3) = ;

26. Sup • Input: T = {p(a,b)=p3, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a!=e} • v(S) = {p(x1,y1) != p(x2,y2) Ç x1= x2} • Justified Superposition gives a=e

27. DPLL • Input: g(S) = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2 Ç p1=p3, a != c, a != e, p1!=p2 Ç a=c, a=e} • Output: UNSAT

28. Example 2 • Repeat example, suppose that original set did not contain a!=e • Then everything is the same up until the last DPLL step

29. DPLL • Input: g(S) = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p2 Ç p1=p3, a != c, p1!=p2 Ç a=c, a=e} • Output: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a=e}

30. cc • Input: M = {p(a,b)=p1, p(c,d)=p2, p(e,f)= p3, p1=p3, a!=c, a=e} • Output: T = {p(e,b)=p3, p(c,d)=p2, p(e,f)= p3, p1=p3, c!=e, a=e} • All justifications empty

31. Sup • Input: T = {p(e,b)=p3, p(c,d)=p2, p(e,f)= p3, p1=p3, c!=e, a=e} • v(S) = {p(x1,y1) != p(x2,y2) Ç x1= x2} • Justified Superposition gives nothing new • Therefore T is a model modulo v(S)

32. Schematic Saturation • Example theory v(S) is decidable • We could use Schematic Saturation to prove the decidability • We could also use Schematic Saturation to compile nonground theory and efficiently perform Justified Superposition

33. Instantiation • Resolution + self-justification =Instantiation • j(p(a)) = {p(a)} • Nonground clause q(x) Ç ~p(x) • Justified Resolution gives q(a) Ç ~p(a) • As far as we know, first combination of instantiation with ordered resolution

34. Completeness • Suppose S is saturated by SMELS • Let M be model of g(S) • Then M is v(S) model of g(S)

35. Completeness Proof • Modifed version of BG model generation • May have implications for selection rules and goal-directed Superposition • Justifications are key

36. Completeness implies • S is SAT implies • Ground model M (modulo v(S)) is generated in finite time, or • M (modulo v(S)) is generated in infinite time • S is UNSAT implies • Unsatisfiable ground g(S) is found

37. Comparisons • BE: Uses Eager Superposition (works for some theories) • SPASS+T: FOL theorem prover is driver, which calls SMT, not complete • InstGen: Instantiates clauses but no orderings • Simplify: Instantiates terms but is not complete

38. Conclusions • SMELS = DPLL(cc(Sup)) • DPLL sends partial model to cc • cc passes reduced implications to Sup • Sup handles nonground part using powerful orderings

39. Future Work • Implement using compilation of Justification Superposition using Schematic Saturation • Combine with other theories like Linear Arithmetic