Solvers: Equality and Arrays Lecture 4 2012

1. Solvers: Equality and Arrays Lecture 4 2012 Nikolaj Bjørner Microsoft Research DTU Winter course January 5th2012

2. Plan • The theory of Un-interpreted functions • The Theory of Arrays

3. Key Takeaways: • Theory of equality, un-interpreted functions and arrays • Saturation-based decision procedures • A common pattern • Connections with algorithms used in program analysis • Inference rule formulation • Datalog formulation • A reduction approach to decision procedures • The theory of array reduces to theory of equalities/functions

4. Deciding Equality

5. E - The theory of equality Reflexivity: t = t Symmetry: t = s  s = t Transitivity: t = s  s= u t = u Congruence: t1= s1 ..  tn =sn  f(t1, …, tn) = f(s1, …, sn) E – the (infinite) conjunction of these axioms

6. Congruence Closure • E-satisfiability can be decided with a simple algorithm known as congruence closure. • Congruence closure creates a finite quotient for DC(E + L). • E – Equality axioms • L – Literals: extra equalities in input

7. Deciding Equality a = b, b = c, d = e, b = s, d = t, a e, a s a b c d e s t

8. Deciding Equality a = b, b = c, d = e, b = s, d = t, a e, a s a b c d e s t

9. Deciding Equality a = b, b = c, d = e, b = s, d = t, a e, a s a,b c d e s t

10. Deciding Equality a = b, b = c, d = e, b = s, d = t, a e, a s a,b c d e s t

11. Deciding Equality a = b, b = c, d = e, b = s, d = t, a e, a s a,b,c d e s t

12. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c d e s t

13. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c d,e s t

14. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c d,e s t

15. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c,s d,e t

16. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c,s d,e t

17. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c,s d,e,t

18. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c,s d,e,t

19. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e, a s a,b,c,s d,e,t Unsatisfiable

20. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e a,b,c,s d,e,t Model construction

21. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e a,b,c,s 1 2 d,e,t Model construction • |M| = {1 ,2} (universe, aka domain)

22. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e a,b,c,s 1 2 d,e,t Model construction • |M| = {1 ,2} (universe, aka domain) • M(a) = 1 (assignment)

23. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e a,b,c,s 1 2 d,e,t Alternative notation: • aM = 1 Model construction • |M| = {1 ,2} (universe, aka domain) • M(a) = 1 (assignment)

24. Deciding Equality a = b,b = c, d = e, b = s, d = t, a e a,b,c,s 1 2 d,e,t Model construction • |M| = {1 ,2} (universe, aka domain) • M(a) = M(b) = M(c) = M(s) = 1 • M(d) = M(e) = M(t) = 2

25. Deciding Equality:Termination, Soundness, Completeness • Termination: easy • Soundness • Invariant: all constants in a “ball” are known to be equal. • The “ball” merge operation is justified by: • Transitivity and Symmetry rules. • Completeness • We can build a model if an inconsistency was not detected. • Proof template (by contradiction): • Build a candidate model. • Assume a literal was not satisfied. • Find contradiction.

26. Deciding Equality:Termination, Soundness, Completeness • Completeness • We can build a model if an inconsistency was not detected. • Instantiating the template for our procedure: • Assume some literal c = d is not satisfied by our model. • That is, M(c) ≠ M(d). • This is impossible, c and d must be in the same “ball”. c,d,… i M(c) = M(d) = i

27. Deciding Equality:Termination, Soundness, Completeness • Completeness • We can build a model if an inconsistency was not detected. • Instantiating the template for our procedure: • Assume some literal c ≠ d is not satisfied by our model. • That is, M(c) = M(d). • Key property: we only check the disequalities after we processed all equalities. • This is impossible, c and d must be in the different “balls” c,… d,… i j M(c) = i M(d) = j

28. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, g(d))  f(b, g(e)) Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

29. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, g(d))  f(b, g(e)) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

30. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, v1)  f(b, g(e)) v1  g(d) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

31. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, v1)  f(b, g(e)) v1  g(d) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

32. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, v1)  f(b, v2) v1  g(d), v2  g(e) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

33. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, f(a, v1)  f(b, v2) v1  g(d), v2  g(e) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

34. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3  f(b, v2) v1  g(d), v2  g(e), v3 f(a, v1) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

35. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 f(b, v2) v1  g(d), v2  g(e), v3  f(a, v1) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

36. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) First Step: “Naming” subterms Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

37. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s d,e,t v2 v4 v1 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

38. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s d,e,t v2 v4 v1 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • d = e implies g(d) = g(e)

39. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s d,e,t v2 v4 v1 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • d = e implies v1= v2

40. Deciding Equality + (uninterpreted) Functions We say: • v1 and v2 are congruent. a = b,b = c, d = e, b = s, d = t, v3 v4 v1 g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 d,e,t v4 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • d = e implies v1= v2

41. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 d,e,t v4 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • a = b, v1= v2implies f(a, v1) = f(b, v2)

42. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4 f(b, v2) a,b,c,s v1,v2 d,e,t v4 v3 Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • a = b, v1= v2implies v3 = v4

43. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3 v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4 f(b, v2) a,b,c,s v1,v2 v3,v4 d,e,t Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn) • a = b, v1= v2implies v3 = v4

44. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, v3  v4 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 v3,v4 d,e,t Unsatisfiable Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

45. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, a v4, v2  v3 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) Changing the problem a,b,c,s v1,v2 v3,v4 d,e,t Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

46. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, a v4, v2  v3 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 v3,v4 d,e,t Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

47. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, a v4, v2  v3 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 v3,v4 d,e,t Congruence Rule: • x1 = y1, …, xn = yn implies f(x1, …, xn) = f(y1, …, yn)

48. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, a v4, v2  v3 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 1 2 3 4 v3,v4 d,e,t Model construction: • |M| = {1 ,2 ,3 ,4} • M(a) = M(b) = M(c) = M(s) = 1 • M(d) = M(e) = M(t) = 2 • M(v1) = M(v2) = 3 • M(v3) = M(v4) = 4

49. Deciding Equality + (uninterpreted) Functions a = b,b = c, d = e, b = s, d = t, a v4, v2  v3 v1  g(d), v2  g(e), v3  f(a, v1) , v4  f(b, v2) a,b,c,s v1,v2 1 2 3 4 v3,v4 d,e,t Model construction: • |M| = {1 ,2 ,3 ,4} • M(a) = M(b) = M(c) = M(s) = 1 • M(d) = M(e) = M(t) = 2 • M(v1) = M(v2) = 3 • M(v3) = M(v4) = 4 Missing: Interpretation for f and g.

50. Deciding Equality + (uninterpreted) Functions • Building the interpretation for function symbols • M(g) is a mapping from |M| to |M| • Defined as: M(g)(i) = j if there is v  g(a) s.t. M(a) = i M(v) = j = k, otherwise (k is an arbitrary element) • Is M(g) well-defined?