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Sound and Complete Calculus for Predicate Specialization in Program Verification

This joint work introduces a calculus supporting efficient reasoning with inductive predicates in program analysis. The proposed method prunes infeasible disjuncts, discards unsatisfiable branches, and abstracts states for improved verification performance.

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Sound and Complete Calculus for Predicate Specialization in Program Verification

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  1. A specialization calculus for program verification CristianGherghina Joint work with: Wei-NganChin, RazvanVoicu, Quang LocLe Florin Craciun, ShengchaoQin TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

  2. Focus • Logics with inductive predicates provide an expressive abstraction mechanism • Becoming popular in the field of program analysis • Tricky to efficiently reason with

  3. Folding/Unfolding • Given a predicate definition • Unfolding performance loss • Unfolded states are costlier due to disjunctions Unfolding Folding

  4. Proposal • We introduce a sound and complete calculus to support pruning of infeasible disjuncts • Use predicate specialization. • Benefits: • eagerly discards unsatisfiabledisjuncts • state in abstractedform

  5. Overview • Motivating example • Informal description of the calculus • Predicate Specialization • Annotation inference • Experiments

  6. Motivating Example

  7. Motivation • Consider the entailment: • The LHS unfolds to:

  8. Performance penalties • Unfold operations are followed by costly satisfiabilitychecks • The remaining satisfiabledisjuncts expose considerable information • Detailed information not always needed • Reasoning with larger formulas is inherently costly

  9. Overview • Motivating example • Informal description of the calculus • Predicate Specialization • Correctness • Experiments

  10. Predicate definition changes • Invariant family • Pruning conditions

  11. Entailment - revisited The previous entailment with annotations • Predicate specialization, for list x • Pruning • Invariant enrichment

  12. Entailment - revisited • Predicate specialization, for list y • Pruning • Invariant enrichment

  13. Overview • Motivating example • Informal description of the calculus • Predicate Specialization • Annotation inference • Experiments

  14. Predicate Specialization • Convention: • We will use the term context ( C ) to denote the pure part of the formula • The rationale is that C will be the context in which predicate specialization takes place

  15. Predicate Specialization • Predicate specialization • Aims for • fewer viable branches : L2L1 • fewer possible pruning conditions : R2R1 • stronger context : C1 C2

  16. Predicate Specialization • Given • Pick a pruning condition • Drop the infeasible branches from L • Enrich the context • Drop irrelevant pruning conditions

  17. L={1,2} ; C : ; • From pick: • Contradicts with C : -> such checks can be syntactic • Drop infeasible branches : • Add the invariant of to C C1 : • Drop irrelevant pruning conditions

  18. Irrelevant pruning conditions • Given: • C : • L : {1} • Result:

  19. Predicate specialization gains • Simple implication checks (mostly syntactic) • Considerable drop in formula size after an unfold • Increase in formula information without an unfold

  20. Overview • Motivating example • Informal description of the calculus • Predicate Specialization • Annotation inference • Experiments

  21. Annotation inference • We need a mechanism for computing • Invariant family • Pruning conditions

  22. Inferring the invariant family • Given a predicate definition • Compute fixpoint for the predicate definition • For each possible set of branches compute a conjunctive invariant

  23. Inferring the invariant family (for dll) • Replace recursive points with, the fixpoint of • For each possible subset of the branches:

  24. Inferring the pruning conditions • Given a predicate definition and the invariant families • Compute an approximation of the closure of branch invariants • For each atomic constraint in all closures construct the list of branches in which it appears (by which it is implied)

  25. Inferring the pruning conditions • Compute an approximation of the transitive closure of each branch invariant • Group all branches that imply an atomic constraint

  26. Overview • Motivating example • Informal description of the calculus • Predicate Specialization • Annotation inference • Experiments

  27. Experiments • Added the calculus to a program verifier (HIP) • Verified functional correctness for small and medium-sized programs with moderate complexity. • A benchmark of 17 small programs (7% faster) Singly, doubly, sorted and circular linked lists, selection-sort, insertion- sort, methods for handling heaps an perfect trees • Complex shapes and invariants (12-90% faster) • Red black trees, balanced binary trees, quick sort, merge sort

  28. Conclusions • Presented an effective, sound and complete calculus for predicate specialization • Application of the calculus benefits in two ways: • Keep abstraction, where possible • Improve verification performance by • Pruning unsatisfiabledisjuncts • Propagate invariant constraints • Various optimization techniques (details in paper).

  29. Questions?Thank you!

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