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Advanced Features of Z3: Quantifiers, Fixed Points, and Non-Linear Arithmetic

Explore the advanced capabilities of the Z3 theorem prover, focusing on quantifier elimination procedures, support for non-linear arithmetic, and fixed-point computations. Discover tools and methodologies for analyzing satisfiability in complex logical frameworks. This overview includes topics like linear and integer arithmetic (LRA, LIA), algebraic data types, and the efficient handling of Boolean and bit-vector operations. Learn about virtual substitutions for polynomials, the efficient Datalog engine, and how Z3 can be utilized for symbolic model checking with state-of-the-art algorithms.

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Advanced Features of Z3: Quantifiers, Fixed Points, and Non-Linear Arithmetic

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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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