Ground Interpolation for the Theory of Equality

Ground Interpolation for the Theory of Equality A. Fuchs1, A. Goel2, J. Grundy2, S. Krstic2, C. Tinelli1 1 The University of Iowa 2 Intel Corporation

Logical Interpolation in Formal Methods Logical interpolants are useful in model checking, e.g., to • accelerate the computation of reachability relations • improve predicate abstraction We will focus on ground interpolants

Ground Interpolation in First-order Theories A theory T admits ground interpolation iff every two ground formulas A and B inconsistent in T have a ground T -interpolant, a ground formula I s.t. • I’s symbols are shared by A and B • A |=T I • I, B are inconsistent in T ( I, B |=T false )

Contribution of This Work A new ground interpolation procedure for EUF Highlights: • Interpolants are extracted from colored congruence graphs (CCGs) • A CG represents compactly a proof of inconsistency for sets of ground literals • CGs are easily produced by usual congruence closure algorithms for deciding ground satisfiability in EUF

Contribution of This Work A new ground interpolation procedure for EUF Highlights: • Our interpolants are: • conjunctions of ground Horn clauses • in simplest possible form for EUF • smaller and simpler than in previous method by McMillan [McM05]

Simplifying Assumptions We consider only conjunctions of literals • Any interpolation procedure for such formulas, in any theory, can be uniformly extended to arbitrary ground formulas [e.g., McM05, CGS08]

(Only?) Previous Work • Interpolation procedure for EUF by McMillan [MCM03] • Based on a inference system for EUF with 6 rules (for reflexivity, symmetry, etc. of = ) • Rules extended with annotations [u, v, , ] for premises and conclusions, and increased to 11 • If A, B derives false[u, v, , ] then    is an interpolant of A, B

Our view: Interpolation as a Cooperative Game u1 = f(x, v0) B = v1 = f(x, u0) u = h(v), u2 v2 u0 = v0 A = u2= g(u1, u) v2=g(v1, h(v))

Ground Interpolation as a Cooperative Game u1 = f(x, v0) B = v1 = f(x, u0) u = h(v), u2 v2 u0 = v0 A = u2= g(u1, u) v2=g(v1, h(v))

Ground Interpolation as a Cooperative Game u1 = f(x, v0) B = v1 = f(x, u0) u = h(v), u2 v2 u0 = v0 A = u2= g(u1, u) v2=g(v1, h(v)) Interpolant: u0 = v0  (u1 = v1 u = h(v)  u2 = v2)

Ground Interpolation as a Cooperative Game Concrete Result for EUF: A procedure to retrofit the interpolation game to congruence graphs

Basic edge Derived edge Congruence Graph: Example L = {x1 = z1, z1 = z2, z2 = x2, z3 = f(x1), f(x2) = z4, x3 = z5, z5 = f(z3), f(z4) = z6, z6 = x4, y1 = z7, z7 = f(x3), f(x4) = z8, z8 = y2} T= {terms in L} z1 x1 z2 x2 z4 z3 f(x1) f(x2) z6 x4 x3 z5 f(z3) f(z4) z8 y2 y1 z7 f(x3) f(x4)

Congruence Graphs and EUF Fact: decision procedures for EUF essentially compute congruence graphs Prop. LetL = {equalities and disequalities}, T = {all terms in L}. L is inconsistent in EUF iff there is a CG (T, ) and s  t  L s.t. s * t

Congruence Graphs and EUF • Let G be any CG showing that L is inconsistent in EUF • Let L = A B We can extract an interpolant of A, Bfrom G by first suitably coloring G with{A, B} The interpolant can be seen as generated from a run of the interpolation game between an A-prover and a B-prover

z1 x1 z2 x2 z4 z3 f(x1) f(x2) z6 x4 x3 z5 f(z3) f(z4) z8 y2 y1 z7 f(x3) f(x4) Colored Congruence Graph: Example A = {x1= z1, z2 = x2, z3 = f(x1), f(x2) = z4, x3= z5, z6 = x4, z7 = f(x3), f(x4) = z8} B = { z1 = z2, z5 = f(z3) , f(z4) = z6, y1= z7, z8 = y2 } Coloring scheme: • Nodes in A \B colored A • ” ” B \ A ” B • ” ” A B ” AB • Basic edgesin A colored A • ” ” ” B ” B • Derived edges colored A (B) if both endpoints are A (B)

Fixing Uncolorable Graphs • It is possible (and easy) to modify the graph to remove uncolorable edges • Reason: EUF is equality interpolating Lemma. [YM05]If A, B |= s = t one can compute a AB-term u s.t. A, B |= s = u  u = t

s5 v6 u6 r5 u5 s6 v5 r2 s4 v3 u3 v4 u4 r4 r2 s2 u1 s1 v1 r1 s3 u2 v2 r3 s7 v7 u7 r7 u v s r Extracting Interpolants from Colored Congruence Graphs CCG for A, B with s  r B : Notation: let xy denote a path from node x to node y

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = I(su)  I(uv)  I(vr) = I(s1r1)  I(uv)  = I(s1u1)  I(u1v1)  I(v1r1)  I(uv) =  {u1=v1}   I(uv)

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = {u1=v1}  I(uv) = {u1=v1}  { v3 =u3  v6 =u6  v4 =u4  u2 =v2  u = v}  I(v3 =u3)  I(v6 =u6)  I(v4 =u4)  I(u2 =v2)

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = {u1=v1}  I(uv) = {u1=v1}  { v3 =u3  v6 =u6  v4 =u4  u2 =v2  u = v}      I(u2 =v2)

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = {u1=v1}  { v3 =u3  v6 =u6  v4 =u4  u2 =v2  u = v}  I(u2 =v2)

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = {u1=v1}  { v3 =u3  v6 =u6  v4 =u4  u2 =v2  u = v}  I(s7 =r7)

s5 v6 u6 r5 s6 u5 v5 r2 s7 v7 u7 r7 s4 v3 u3 v4 u4 r4 s2 r2 s1 u1 v1 r1 s3 u2 v2 r3 u s v r I(sr) = {u1=v1}  { v3 =u3  v6 =u6  v4 =u4  u2 =v2  u = v}  {u5 =v5  u7 = v7} Note:A |= I(sr) andB, I(sr) |= s = r but s  r  B

Interpolation Function:Formal Definition {I() |  is a factor of st} if st has ≥ 2 factors I(st) = {I() |  is a parent of a link in st} if st is a B-path {I() |   P(st)}{J(st)} if st is a A-path {P() |  is a factor of st} if st has ≥ 2 factors P(st) = {st} if st is a B-path {P() |  is a parent of a link in st} if st is a A-path J(st) = {u = v | uv  P(st)}  s = t

Main Theoretical Result Lemma. Function I is well defined and computable over any CCG, and returns a set of ground Horn clauses. Theorem. Let G be a CCG for A, B. If sr is a path in G s.t. s  r  B, then I(sr) is an EUF-interpolant ofAandB. Note: The paper also defines an I’ for when s  r  A.

Interpolation Procedure Given a literal set L inconsistent in EUF and a partition A, B of L • run CC to find a CG G over L connecting s, r for some s  r  L • modify G as needed to make it colorable and color it (in any allowed way) • If s  r B return I(sr) else returnI’(sr)

Main Differences with McMillan’s Procedure • CGs condense inferences by reflexivity, symmetry and transitivity into paths (big step vs. small step proof) Ex:z1= x1 = z2 = x2 = f(z3) = x3 = z4z1 Our interpolant: z1 = z4 McMillan’s: z1 = z2 z2 = f(z3)  f(z3) = z4

Main Differences with McMillan’s Procedure • Interpolants with simple Boolean structure Ex. 7,10 in our paper: Our interpolant: (z1 = z2 z3 = z4)  (z5 = z6 z7 = z8) McMillan’s: (z1 = z2 (z3 = z4 z5 = z6))  z3 = z4  z7 = z8

Main Differences with McMillan’s Procedure • Minimal number of new, auxiliary terms vs. many new terms produced on-the-fly • Non-deterministic coloring step (2) vs. fully specified annotation mechanism • Overall smaller and simpler interpolants

Experimental Results • Interpolation procedure implemented in SMT-solver DPT • Compared with state-of-the-art implementation of McMillan’s procedure in MathSAT [Cim08] • Both systems extend interpolation to general ground EUF formulas in the same way (relying on similar DPLL-style SAT engines) • Resolution proofs from the two DPLL engines are comparable in size • Same benchset as in [Cim08]

Experimental Results DTP vs. MathSAT on 45 benchmarks derived from SMT-LIB Runtimes: Comparable Interpolant size: DPT’s 3.8 times smaller on average

Conclusion • New interpolation procedure for EUF • Easy to implement on top of CC procedures within SMT solvers • Generates smaller and simpler interpolants • Provides basis for further refinements and implementations • Its flexibility could be useful when the notion of interpolant quality is better understood

Theories with Ground Interpolation • Equality over uninterpreted function symbols (EUF) • Real arithmetic • Linear Integer Arithmetic with divisibility operator • … • Any FOL theory admitting quantifier elimination

Coloring Congruence Graph Let A, B be disjoint sets of literals Every symbol of A (B) is A-colorable (B-colorable) A term is A-colorable (B-colorable) if all of its symbols are To color a CG for A B, color • a node withA (resp., B) if it occurs inA (resp., B) • a basic edge with A (resp.,B) if it occurs inA (resp.,B) • a derived edge with A(alternatively, with B) if its end-points are both colored with A (with B)

Congruence Graph for L Any undirected graph G built during this procedure Input:L= {ground literals}, T= {ground terms} Let G := (T, ) with  :=  Repeat as long as possible For each(s, t)  TT \ * such that s = t  L or t = s  L or s is f(s1,…,sn), t is f(t1,…,tn) and s1* t1, …, sn* tn do add (s, t) to 

Ground Interpolation for the Theory of Equality