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Quantifiers, Arithmetic and Fixed-points

Quantifiers, Arithmetic and Fixed-points. Quantifier Elimination Procedures in Z3 Support for Non-linear arithmetic Fixed-points – features and a preview. Quantifier Elimination. O ption: ELIM_QUANTIFIERS=true LRA – Linear real arithmetic LIA – Linear integer arithemtic

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Quantifiers, Arithmetic and Fixed-points

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  1. Quantifiers, Arithmetic and Fixed-points • Quantifier Elimination Procedures in Z3 • Support for Non-linear arithmetic • Fixed-points – features and a preview

  2. Quantifier Elimination • Option: ELIM_QUANTIFIERS=true • LRA – Linear real arithmetic • LIA – Linear integer arithemtic • D – Algebraic Datatypes • Booleans & Bit-vectors – (All-SAT) • NRA2 – Quadratic (using virtual substitutions) • Arrays – ad hoc

  3. LRA Terms Atoms Formulas

  4. Quantifier Elimination Samples

  5. LIA Terms Atoms Formulas

  6. D – algebraic data-types • Domain Closure: • Eliminate accessors: • Solve equalities: • Virtual substitution:

  7. NRA • Virtual substitutions for second-degree polynomials • Method by Weispfenning et.al. (Redlog) • Used both as quantifier elimination (all SAT) and ground decision procedure (first SAT) • ….

  8. Analysis Tool Logic Engine Z3

  9. Tool Encodings Methodology Fixed-Point SLAyer Sep. Logic Abstract Interpretation Logic Programming GateKeeper Simulation Relation Predicate Based MC Summaries SAGE BDD MC Abstraction Refinement Datalog Havoc Houdini Interpolating MC

  10. The Z Tool • Ships with Z3 • Online demo • BDD tablesample in distribution • Mostly developed by Krystof Hoder

  11. Why fixed-points Variant for Connoisseurs: Recall the basic sausage* rule: In a nutshell: Aim of Satisfiability Modulo Fixed-points and Theories. Is valid? Is satisfiable? *“sausage” terminology by AndreyRybalchenko

  12. Portfolio approach to fix-points • Efficient Datalog Engine • Finite Tables • Symbolic Tables • ComposableAbstract Relations: • Use abstract interpretation domains. • Use SMT as a domain. • Reduced product operators for sharing • Efficient Algorithms from Symbolic MC Modulo Theories • I will give a taste of this later. Is satisfiable? BDD packages Abstract Domains Interpolation Tools

  13. Core Engine Compilation  Restarts Relational Algebra Abstract Machine

  14. Core Engine Plugin architecture: New domains added using plugins implementing Relational Algebra operations. Restarts

  15. Relation representation x 0 1 y z 0 1 Bounds Intervals + = + Pentagons =

  16. Relation representation x 0 1 y z 0 1 Bounds Intervals • Product: Table x Table • Indexed Relation: Table x Relation • Reduced Product: Relation x Relation

  17. Preview – Generalized PDR Is valid? Is satisfiable? • PDR: Property Directed ReachabilityA new Algorithm For Symbolic Model Checking of Hardware • by Aaron Bradley. • In • Lift it to proceduresmultiple operators, non-linear • Lift beyond propositional logic Theories, non-ground

  18. Simple sample demo

  19. Generalizations • PDR works for linearTransformers • Generalize to non-linear • PDR works with a singleTransformer • Work with multipletransformers. •  A Solver for Datalog/Boolean Programs • PDR is for propositionallogic • Search Modulo Theories (with McMillan’s FociZ3 and other methods)

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