1 / 9

Decision Procedures in First Order Logic

Decision Procedures in First Order Logic. Decision Procedures for Equality Logic. Outline. Introduction Definition, complexity Reducing Uninterpreted Functions to Equality Logic Using Uninterpreted Functions in proofs Simplifications Introduction to the decision procedures

peigi
Download Presentation

Decision Procedures in First Order Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Decision Procedures in First Order Logic Decision Procedures forEquality Logic Daniel Kroening and Ofer Strichman

  2. Outline • Introduction • Definition, 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

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

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

  5. x4,x5 Example • x1 = x2Æx2 = x3Æx4=x5Æx5x1 x1,x2,x3 Equivalence class Equivalence class Is there a disequality between members of the same class ? Decision Procedures An algorithmic point of view

  6. x4,x5 Next: add Uninterpreted Functions • x1 = x2Æx2 = x3Æx4=x5Æx5x1 ÆF(x1)F(x2) F(x1) x1,x2,x3 Equivalence class F(x2) Equivalence class Equivalence class Equivalence class Decision Procedures An algorithmic point of view

  7. x4,x5 Next: Compute the Congruence Closure • x1 = x2Æx2 = x3Æx4=x5Æx5x1 ÆF(x1)F(x2) x1,x2,x3 F(x1),F(x2) Equivalence class Equivalence class Now - is there a disequality between members of the same class ? This is called theCongruence Closure Decision Procedures An algorithmic point of view

  8. And now: consider a Boolean structure • x1 = x2Ç (x2 = x3Æx4=x5Æx5x1 ÆF(x1) F(x2)) x1,x2 F(x1) F(x2) x4,x5 x2,x3 Equivalence class Equivalence classes case 1 case 2 Syntactic case splitting: this is what we want to avoid! Decision Procedures An algorithmic point of view

  9. Deciding Equality Logic with UFs • Input: Equality Logic formula UF • Convert UF to DNF • 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'. • Return 'UNSAT'. Decision Procedures An algorithmic point of view

More Related