Model Based Theory Combination

Transcription

1 Model Based Theory Combination SMT 2007 Leonardo de Moura and Nikolaj Bjørner {leonardo, Microsoft Research Model Based Theory Combination p.1/20

2 Combination of Theories In practice, we need a combination of theories. Examples: Given x+2 = y f(read(write(a,x, 3),y 2)) = f(y x+1) f(f(x) f(y)) f(z),x + z y x z < 0 Σ = Σ 1 Σ 2 T 1, T 2 : theories over Σ 1, Σ 2 T = DC(T 1 T 2 ) Is T consistent? Given satisfiability procedures for conjunction of literals of T 1 and T 2, how to decide the satisfiability of T? Model Based Theory Combination p.2/20

3 Nelson-Oppen Combination Let T 1 and T 2 be consistent, stably infinite theories over disjoint (countable) signatures. Assume satisfiability of conjunction of literals can decided in O(T 1 (n)) and O(T 2 (n)) time respectively. Then, 1. The combined theory T is consistent and stably infinite. 2. Non-deterministic NO: Satisfiability of quantifier free conjunction of literals in T can be decided in O(2 n2 (T 1 (n) + T 2 (n)). 3. Deterministic NO: If T 1 and T 2 are convex, then so is T and satisfiability in T is in O(n 4 (T 1 (n) + T 2 (n))). Model Based Theory Combination p.3/20

4 Nelson-Oppen Combination Procedure The combination procedure: Initial State: φ is a conjunction of literals over Σ 1 Σ 2. Purification: Preserving satisfiability transform φ into φ 1 φ 2, such that, φ i Σ i. Interaction: Guess a partition of V(φ 1 ) V(φ 2 ) into disjoint subsets. Express it as conjunction of literals ψ. Example. The partition {x 1 }, {x 2,x 3 }, {x 4 } is represented as x 1 x 2,x 1 x 4,x 2 x 4,x 2 = x 3. Component Procedures : Use individual procedures to decide whether φ i ψ is satisfiable. Return: If both return yes, return yes. No, otherwise. Model Based Theory Combination p.4/20

5 Convexity A theory T is convex iff for all finite sets Γ of literals and for all non-empty disjunctions i I x i = y i of variables, Γ = T i I x i = y i iff Γ = T x i = y i for some i I. For convex theories, instead of guessing, one can deduce the equalities to be shared. Key idea: propagate x y to Γ 2 whenever T 1 Γ 1 = x y, and vice-versa. Sharing equalities is sufficient: Theory T 1 can assume that x M2 y M 2 whenever x y is not implied by T 2 and vice versa. Model Based Theory Combination p.5/20

6 Combining theories in practice Propagate all implied equalities. Deterministic Nelson-Oppen. Complete only for convex theories. It may be expensive for some theories. Delayed Theory Combination. Nondeterministic Nelson-Oppen. Create set of interface equalities (x = y) between shared variables. Use SAT solver to guess the partition. Disadvantage: the number of additional equality literals is quadratic in the number of shared variables. Model Based Theory Combination p.6/20

7 Model based theory combination Common to these methods is that they are pessimistic about which equalities are propagated. Model-based Theory Combination Optimistic approach. Use a candidate model M i for one of the theories T i and propagate all equalities implied by the candidate model, hedging that other theories will agree. if M i = T i Γ i {u = v} then propagate u = v. If not, use backtracking to fix the model. It is cheaper to enumerate equalities that are implied in a particular model than of all models. Model Based Theory Combination p.7/20

8 Model based theory combination: Example Purifying x = f(y 1),f(x) f(y), 0 x 1, 0 y 1 Model Based Theory Combination p.8/20

9 Model based theory combination: Example x = f(z),f(x) f(y), 0 x 1, 0 y 1,z = y 1 Model Based Theory Combination p.8/20

10 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,f(z)} x E = 1 0 x 1 x A = 0 f(x) f(y) {y} y E = 2 0 y 1 y A = 0 {z} z E = 3 z = y 1 z A = 1 {f(x)} f E = { 1 4, {f(y)} 2 5, 3 1, else 6 } Assume x = y Model Based Theory Combination p.8/20

11 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,y,f(z)} x E = 1 0 x 1 x A = 0 f (x) f (y) {z} y E = 1 0 y 1 y A = 0 x = y {f (x),f (y)} z E = 2 z = y 1 z A = 1 f E = { 1 3, x = y 2 1, else 4 } Unsatisfiable Model Based Theory Combination p.8/20

12 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,f(z)} x E = 1 0 x 1 x A = 0 f(x) f(y) {y} y E = 2 0 y 1 y A = 0 x y {z} z E = 3 z = y 1 z A = 1 {f(x)} f E = { 1 4, x y {f(y)} 2 5, 3 1, else 6 } Backtrack, and assert x y. T A model need to be fixed. Model Based Theory Combination p.8/20

13 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,f(z)} x E = 1 0 x 1 x A = 0 f(x) f(y) {y} y E = 2 0 y 1 y A = 1 x y {z} z E = 3 z = y 1 z A = 0 {f(x)} f E = { 1 4, x y {f(y)} 2 5, 3 1, else 6 } Assume x = z Model Based Theory Combination p.8/20

14 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,z,f(x),f(z)} x E = 1 0 x 1 x A = 0 f(x) f(y) {y} y E = 2 0 y 1 y A = 1 x y {f(y)} z E = 1 z = y 1 z A = 0 x = z f E = { 1 1, x y 2 3, x = z else 4 } Satisfiable Model Based Theory Combination p.8/20

15 Model based theory combination: Example T E T A Literals Eq. Classes Model Literals Model x = f(z) {x,z,f(x),f(z)} x E = 1 0 x 1 x A = 0 f(x) f(y) {y} y E = 2 0 y 1 y A = 1 x y {f(y)} z E = 1 z = y 1 z A = 0 x = z f E = { 1 1, x y 2 3, x = z else 4 } Let h be the bijection between S E and S A. h = { 1 0, 2 1, 3 1, 4 2,...} Model Based Theory Combination p.8/20

16 Model based theory combination: Example T E T A Literals Model Literals Model x = f(z) x E = 1 0 x 1 x A = 0 f(x) f(y) y E = 2 0 y 1 y A = 1 x y z E = 1 z = y 1 z A = 0 x = z f E = { 1 1, x y f A = { , x = z 1 1 else 4 } else 2} Extending A using h. h = { 1 0, 2 1, 3 1, 4 2,...} Model Based Theory Combination p.8/20

17 Simplex: a model base theory solver Tableau: B and N denote the set of basic and nonbasic variables. x i = x j N a ij x j x i B, Solver stores upper and lower bounds l i and u i, and a mapping β that assigns a value β(x i ) to every variable. The bounds on nonbasic variables are always satisfied by β, that is, the following invariant is maintained x j N, l j β(x j ) u j. Bounds constraints for basic variables are not necessarily satisfied by β, but pivoting steps can be used to fix bounds violations. Model Based Theory Combination p.9/20

18 Simplex: a model based theory solver The current model for the simplex solver is given by β. Bound propagation Equations + Bounds can be used to derive new bounds. Example: x = y z, y 2, z 3 x 1. Model Based Theory Combination p.10/20

19 Opportunistic equality propagation Efficient (and incomplete) methods for propagating equalities. Notation A variable x i is fixed iff l i = u i. A linear polynomial x j V a ijx j is fixed iff x j is fixed or a ij = 0. Given a linear polynomial P = x j V a ijx j : β(p) denotes x j V a ijβ(x j ). Model Based Theory Combination p.11/20

20 Opportunistic equality propagation Equality propagation in arithmetic: FixedEq l i x i u i, l j x j u j = x i = x j if l i = u i = l j = u j EqRow x i = x j + P = x i = x j if P is fixed, and β(p) = 0 EqOffsetRows x i = x k + P 1 = x i = x j if x j = x k + P 2 P 1 and P 2 are fixed, and β(p 1 ) = β(p 2 ) EqRows x i = P + P 1 x j = P + P 2 = x i = x j if P 1 and P 2 are fixed, and β(p 1 ) = β(p 2 ) Model Based Theory Combination p.12/20

21 Opportunistic theory/equality propagation These rules can miss some implied equalities. Example: z = w is detected, but x = y is not because w is not a fixed variable. x = y + w + s z = w + s 0 z w 0 0 s 0 Remark: bound propagation can be used imply the bound 0 w, making w a fixed variable. Model Based Theory Combination p.13/20

22 Model mutation Sometimes x M = y M by accident. Model mutation: diversify the current model. For a Simplex based procedure, freedom intervals model mutation without pivoting. Model Based Theory Combination p.14/20

23 Ackermann s reduction Ackermann s reduction is used to remove uninterpreted functions. For each application f( a) in φ create a fresh variable f a. For each pair of applications f( a), f( c) in φ add the clause a c f a = f c. Replace f( a) with f a in φ. It is used in some SMT solvers to reduce T LA T E to T LA. Main problem: quadratic number of new clauses. It is also problematic to use this approach in the context of several theories and when combining SMT solvers with quantifier instantiation. Model Based Theory Combination p.15/20

24 Ackermann s reduction Congruence closure based algorithms miss the following inference rule f(n) f(m) = n i m i Following simple formula takes O(2 N ) time to be solved using SAT + Congruence closure. N (p i x i = v 0 ), ( p i x i = v 1 ), (p i y i = v 0 ), ( p i y i = v 1 ), i=1 f(x N,...,f(x 2,x 1 )...) f(y N,...,f(y 2,y 1 )...) It can be solved in polynomial time with Ackermann s reduction. A similar behavior is also observed in several pipeline verification problems. Model Based Theory Combination p.16/20

25 Dynamic Ackermann s reduction This performance problem reflects a limitation in the current congruence closure algorithms used in SMT solvers. It is not related with the theory combination problem. Dynamic Ackermannization: clauses corresponding to Ackermann s reduction are added when a congruence rule participates in a conflict. CC Ack Dyn Ack conflicts time (s) conflicts time (s) conflicts time (s) c10bi f10id > > Model Based Theory Combination p.17/20

26 Experimental Results # MathSAT MathSAT-dtc Yices Z3 EufLaArithmetic (11) (1) Hash Wisa (1) RandomCoupled (51) RandomDecoupled (1) (51) Simple (29) (53) 1.00 Ackermann (82) (82) Total (95) (112) (155) Model Based Theory Combination p.18/20

27 Experimental Results (cont.) Z3-dtc Z3-dtc* Z3-ack Z3-neq Z3-ndack Z3 EufLaArithmetic (11) (4) (1) Hash Wisa RandomCoupled (166) (103) RandomDecoupled (85) (71) Simple Ackermann (77) (77) (77) 1.72 Total (339) (255) (1) (77) Model Based Theory Combination p.19/20

28 Conclusion New theory combination method. Solves a number of practical deficiencies with other known solutions to integrating theories. Optimizations: Opportunistic equality propagation. Model mutation. Dynamic Ackermannization copes with limitations in the current congruence closure algorithms. We are experimenting with a model-based array theory. Model Based Theory Combination p.20/20