Equality Logic and Uninterpreted Functions

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1 Equality Logic and Uninterpreted Functions Seminar: Decision Procedures Michaela Tießler

2 Agenda 1. Definitions 2. Use of Uninterpreted Functions 3. Decision Procedures

3 formula: atom: term: Equality Logic (= Theory of Equality) formula formula formula (formula) atom term = term identifier constant Identifiers are variables defined over a single infinite domain Constants are elements from the same domain as the identifier Example: u1 = x1 + y1 u2 = x2 +y2 z = u1 u2

4 Interpretation of = in Equality Logic Reflexivity x x = x Symmetry x, y x = y y = x Transitivity x, y, z x = y y = z x = z

5 Equality Logic and Uninterpreted Functions (EUF) formula: atom: term: formula formula formula (formula) atom term = term predicate symbol (list of terms) identifier function symbol (list of terms) Identifiers are variables defined over a single infinite domain Example: u1=f(x1, y1) u2=f(x2, y2) z = G (u1, u2)

6 Interpretation of = in EUF Reflexivity x x = x Symmetry x, y x = y y = x Transitivity x, y, z x = y y = z x = z Congruence t,,, t., t, /,, t. / 0 t 0 = t 0 / F t,,, t. = F(t, /,, t. / )

7 Negation Normal Form (NNF) Negations are only applied to atoms Each formula can easily be transformed to NNF by applying three rules A A (A B) A B (A B) A B

8 Agenda 1. Definitions 2. Use of Uninterpreted Functions 3. Decision Procedures

9 Why do we use uninterpreted functions? Used for abstracting, or generalizing, theorems Easier to reason about Example use: Proving Equivalence of Programs

10 Use Case: Verifying a Compilation Process with Translation Validation Suppose that a source program contains the statement z = Outcome of the compiling process: x, x 8 8 y, + y 8 u, = < = ; u < 8 = u, u, ; = y, + y 8 ; z = u 8 > Ø New auxiliary variables u,, u 8 and have been added Verification condition: u, = x, u x 8 = u, u, = y, + y 8 z = u 8 8 z = x, x 8 8 y, + y 8 Abstracted version: (u, = F(x,, x 8 ) u 8 = G(u,, u, ) = H(y,, y 8 ) z = G(u 8, ) z = G G F x,, x 8, F x,, x 8, H y,, y 8

11 Agenda 1. Definitions 2. Use of Uninterpreted Functions 3. Decision Procedures Simplify Equality Formula Ackermann`s Reduction Congruence Closure

12 Simplify Equality Formula Equality Graphs x, x 8 x C x D Dashed edges represent equalities Solid edges represent disequalities

13 Simplify Equality Formula Algorithm Input: Equality formula φ F 1. Let φ F/ = φ F 2. Construct the equality graph G F (φ F ) 3. Replace each pure literal "x = y" or "x y" in φ F with TRUE, if (x, y) not in a simple contradictory cycle 4. If anything changed in the last two steps go to step 2 5. Return φ F

14 Simplify Equality Formula Example φ F x, x 8 f, f 8 x, = f, y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8 z Equality Graph x, x 8 y, z f, f 8 y 8 y D g, g 8

15 Simplify Equality Formula Example φ F x, x 8 f, f 8 x, = f, y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8 z Replacing literals TRUE TRUE TRUE φ F y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8 TRUE

16 Simplify Equality Formula Example φ F x, x 8 f, f 8 x, = f, y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8 z Simplification φ F y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8

17 Simplify Equality Formula Example φ F y, = y 8 = y D y, = y D y 8 y D z = g, g, = g 8 z g 8 Reconstructing Equality Graph y, y D z y 8 g, g 8 à no further changes possible

18 Ackermann`s Reduction Transforms EUF formula φ ST to an equality logic formula φ F of the form: φ F FC F flat F Where FC F is a conjunction of functional consistency constraints and flat F is a flattening of φ ST Functional consistency: Instances of the same function return the same value if given equal arguments φ F is valid if and only if φ ST is valid

19 Ackermann`s Reduction Algorithm 1. Indexing the functions, beginning with the inner functions 2. Take the original expression φ ST and transform it to its flat flat F 3. Compute conjunction of constraints FC F 4. Let φ F FC F flat F

20 Ackermann`s Reduction Example x, = x 8 y, = y 8 F G G x,, y, = F G G x 8, y 8 g 8 g D g, f, f 8 1. Indexing the functions, beginning with the inner functions g, G x, G x 8 g 8 G g, g D G f, F g 8 f 8 F g D

21 Ackermann`s Reduction Example x, = x 8 y, = y 8 F G G x,, y, = F G G x 8, y 8 g 8 g D g, f, f 8 2. Take the original expression φ ST and transform it to its flat flat F flat F : = x, = x 8 y, = y 8 f, = f 8

22 Ackermann`s Reduction Example x, = x 8 y, = y 8 F G G x,, y, = F G G x 8, y 8 g 8 g D g, f, f 8 3. Compute conjunction of constraints FC F : = x, = x 8 g, = g, = g 8 = g D g 8 = g D y, = y 8 f, = f 8 x, = g, g, = g 8 x 8 = = g D g, = g 8 = g D

23 Ackermann`s Reduction Example x, = x 8 y, = y 8 F G G x,, y, = F G G x 8, y 8 g 8 g D g, 4. Let φ F FC F flat F f, f 8 φ F := (x, = x 8 g, = g, = g 8 = g D g 8 = g D y, = y 8 f, = f 8 x, = g, g, = g 8 x 8 = = g D g, = g 8 = g D ) (x, = x 8 y, = y 8 f, = f 8 )

24 Congruence Closure Algorithm to decide a conjunction of equalities and uninterpreted functions Input: A conjunction φ ST of equality predicates over variables and uninterpreted functions in negation normal form Output: equivalence classes Satisfiability of φ ST

25 Congruence Closure Algorithm 1. Put two terms t,, t 8 in their own equivalence class if (t, = t 8 ) is a predicate in φ ST 2. Merge equivalence classes with a shared term until there are no more classes to be merged 3. Merge terms F(t 0 ) and F(t \ ) if t 0 and t \ are in the same equivalence class 4. φ ST is Satisfiable if there doesn t exist any unequality t 0 t \ with t 0 and t \ of the same equivalence class

26 Congruence Closure Example φ ST : = x, = x 8 x 8 = x D = x C x C x, F x,, x C F D G x, = ) 1. Put two terms t,, t 8 in their own equivalence class if (t, = t 8 ) is a predicate in φ ST x, F(x,,x C ) G(x, ) x D x 8 x 8 x C D ) )

Congruence Closure Example φ ST : = x, = x 8 x 8 = x @ x D = x C x C x, F x,, x C F x @,x D G x, = G(x @ ) 2.

27 Congruence Closure Example φ ST : = x, = x 8 x 8 = x D = x C x C x, F x,, x C F D G x, = ) 2. Merge equivalence classes with a shared term until there are no more classes to be merged x D x 8 F(x,,x 8 ) x C G(x, ) x, D ) )

28 Congruence Closure Example φ ST : = x, = x 8 x 8 = x D = x C x C x, F x,, x C F D G x, = ) 3. Merge terms F(t 0 ) and F(t \ ) if t 0 and t \ are in the same equivalence class F(x,,x C ) G(x, ) x, x D x 8 x C D ) )

29 Congruence Closure Example φ ST : = x, = x 8 x 8 = x D = x C x C x, F x,, x C F D G x, = ) 4. φ ST is Satisfiable if there doesn t exist any unequality t 0 t \ with t 0 and t \ of the same equivalence class x,, x 8, x D,x C, F x,, x 8, F x D, G x, }, {G x C F x,, x C F x D Unsatisfiable

30 Congruence Closure based on Graphs G = (V, E) u, v directed Graph vertices λ(v) R label Relation on V u, v congruent under R if λ u = λ v && u 0,v 0 with V u = u,, u., V v = v,, v. u i, v i ε R {u, v} R u, v closed under congruences R if u, v with u, v congruent under R R / minimal extention of R; with R / equivalence Relation && R / closed under R

31 Congruence Closure based on Graphs Example F V1 x, V2 x 8 V3 F V4 F V5 V6 x D V7 x C V8 R Relation V1, V4 V1, V4 congruent under R Ø Congruence closure of R must include at least V2,V5 and V3,V6 Ø Minimal equivalence Relation: V1,V4, V2, V5, V3,V6

32 Congruence Closure Example for Implementation Based on the operations union and find: find(u): returns the unique name of the equivalence class u belongs to union(u, v): combines equivalence classes of vertices u and v Other operations: congruent(u, v): returns true if u and v are congruent

33 Congruence Closure Example for Implementation Compute congruence closure merge (u, v) if find(u)=find(v) then return P = set of predecessors of all vertices equivalent to u P = set of predecessors of all vertices equivalent to v union(u, v) For each pair (x, y) with x P and y P if find (u) find (v) && congruent(u,v) = TRUE then merge(x, y)

34 Congruence Closure Example for Implementation Decision Procedure t, = u, t = u r, = s, r = s, Construct Graph G which corresponds to the set of all terms appearing in the conjunction let τ t be the vertex in G representing t let R be the identityrelation on the vertices of G for 1 i p merge τ t 0, τ u 0 for 1 i q if τ r, τ s 0 equivalent then return UNSATISFIABLE return SATISFIABLE

35 Congruence Closure Complexity Graph based CC: O nm with n number of vertices of G and m number of edges of G Average time of fastest CC algorithm: O n log n with n length of conjunction

36 Further approaches Graph-Based Reduction to Propositional Logic Solving equality logic formulas by relying on the small-model property of this logic

SMT BASICS WS 2017/2018 ( ) LOGIC SATISFIABILITY MODULO THEORIES. Institute for Formal Models and Verification Johannes Kepler Universität Linz

SMT BASICS WS 2017/2018 ( ) LOGIC SATISFIABILITY MODULO THEORIES. Institute for Formal Models and Verification Johannes Kepler Universität Linz LOGIC SATISFIABILITY MODULO THEORIES SMT BASICS WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität