Decision Procedures in First Order Logic Decision Procedures

Decision Procedures in First Order Logic Decision Procedures for Equality Logic Daniel Kroening and Ofer Strichman 1

Outline n Introduction ¨ Definition, n complexity ¨ Reducing Uninterpreted Functions to Equality Logic ¨ Using Uninterpreted Functions in proofs ¨ Simplifications Introduction to the decision procedures ¨ The framework: assumptions and Normal Forms ¨ General terms and notions ¨ Solving a conjunction of equalities ¨ Simplifications Decision Procedures An algorithmic point of view 2

Basic assumptions and notations n Input formulas are in NNF n Input formulas are checked for satisfiability n Formula with Uninterpreted Functions: UF n Equality formula: E Decision Procedures An algorithmic point of view 3

First: conjunction of equalities n Input: A conjunction of equalities and disequalities 1. Define an equivalence class for each variable. For each equality x = y unite the equivalence classes of x and y. Repeat until convergence. 2. For each disequality u v if u is in the same equivalence class as v return 'UNSAT'. 3. Return 'SAT'. Decision Procedures An algorithmic point of view 4

Example n x 1 = x 2 Æ x 2 = x 3 Æ x 4=x 5 Æ x 5 x 1 , x 2 , x 3 Equivalence class , x 5 x 4 Equivalence class Is there a disequality between members of the same class ? Decision Procedures An algorithmic point of view 5

Next: add Uninterpreted Functions n x 1 = x 2 Æ x 2 = x 3 Æ x 4=x 5 Æ x 5 x 1 Æ F(x 1) F(x 2) F( x 1) x 1 , x 2 , x 3 , x 5 x 4 Equivalence class F( x 2) Equivalence class Decision Procedures An algorithmic point of view 6

Next: Compute the Congruence Closure n x 1 = x 2 Æ x 2 = x 3 Æ x 4=x 5 Æ x 5 x 1 Æ F(x 1) F(x 2) x 1 , x 2 , x 3 F( x 1 ), F( , x 5 x x 4 2) Equivalence class Now - is there a disequality between members of the same class ? This is called the Congruence Closure Decision Procedures An algorithmic point of view 7

And now: consider a Boolean structure n x 1 = x 2 Ç (x 2 = x 3 Æ x 4=x 5 Æ x 5 x 1 Æ F(x 1) F(x 2)) x 1 , x 2 Equivalence class x , x 3 2 x , x 5 4 ) x 1 ( F ) x 2 ( F Equivalence classes case 1 case 2 Syntactic case splitting: this is what we want to avoid! Decision Procedures An algorithmic point of view 8

Deciding Equality Logic with UFs n Input: Equality Logic formula UF n Convert UF to DNF n For each clause: ¨ Define an equivalence class for each variable and each function instance. ¨ For each equality x = y unite the equivalence classes of x and y. For each function symbol F, unite the classes of F(x) and F(y). Repeat until convergence. ¨ If all disequalities are between terms from different equivalence classes, return 'SAT'. n Return 'UNSAT'. Decision Procedures An algorithmic point of view 9

Basic notions E: x = y Æ y = z Æ z x n The Equality predicates: {x = y, y = z, z x} which we can break to two sets: E= ={x = y, y = z}, E = {z x} n The Equality Graph GE( E) = h. V, E=, E i (a. k. a “E-graph”) y x z Decision Procedures An algorithmic point of view 10

Basic notions 1 E: x = y Æ y = z Æ z x unsatisfiable 2 E : x = y Æ y = z Ç z x satisfiable y x z The graph GE( E) represents an abstraction of E It ignores the Boolean structure of E Decision Procedures An algorithmic point of view 11

Basic notions y x z n Dfn: a path made of E= edges is an Equality Path. we write x =*z. n Dfn: a path made of E= edges + exactly one edge from E is a Disequality Path. We write x *y. Decision Procedures An algorithmic point of view 12

Basic notions y x n n z Dfn. A cycle with one disequality edge is a Contradictory Cycle. In a Contradictory Cycle, for every two nodes x, y it holds that x =* y and x * y. Decision Procedures An algorithmic point of view 13

Basic notions y x z n Dfn: A subgraph is called satisfiable iff the conjunction of the predicates represented by its edges is satisfiable. n Thm: A subgraph is unsatisfiable iff it contains a Contradictory cycle Decision Procedures An algorithmic point of view 14

Basic notions n Thm: Every Contradictory Cycle is either simple or contains a simple contradictory cycle Decision Procedures An algorithmic point of view 15

Simplifications, again n n Let S be the set of edges that are not part of any Contradictory Cycle Thm: replacing all solid edges in S with False, and all dashed edges in S with True, preserves satisfiability Decision Procedures An algorithmic point of view 16

Simplification: example x 3 Tru x 4 Tru e False e x 2 x 1 n (x 1 = x 2 Ç x 1 = x 4) Æ (x 1 x 3 Ç x 2 = x 3) n (x 1 = x 2 Ç True) Æ (x 1 x 3 Ç x 2 = x 3) n (: False Ç True) = True n Satisfiable! Decision Procedures An algorithmic point of view 17

Syntactic vs. Semantic splits n So far we saw how to handle disjunctions through syntactic case-splitting. n There are much better ways to do it than simply transforming it to DNF: ¨ Semantic Tableaux, ¨ SAT-based splitting, ¨ others… n We will investigate some of these methods later in the course. Decision Procedures An algorithmic point of view 18

Syntactic vs. Semantic splits n Now we start looking at methods that split the search space instead. This is called semantic splitting. n SAT is a very good engine for performing semantic splitting, due to its ability to guide the search, prune the search-space etc. Decision Procedures An algorithmic point of view 19