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The SMT solver Z3 Lecture 3, 2012

The SMT solver Z3 Lecture 3, 2012. Nikolaj Bjørner Microsoft Research DTU Winter course January 4 th 2012 Organized by Hanne Riis Nielson, Flemming Nielson. Plan. Overview and architecture of Z3 What is Z3 How to use Z3. Takeaways :.

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The SMT solver Z3 Lecture 3, 2012

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  1. The SMT solver Z3 Lecture 3, 2012 Nikolaj Bjørner Microsoft Research DTU Winter course January 4th 2012 Organized by Hanne Riis Nielson, Flemming Nielson

  2. Plan • Overview and architecture of Z3 • What is Z3 • How to use Z3

  3. Takeaways: • You will have an idea of what Z3 is and ways of using it

  4. Follow-on questions

  5. What is Z3? Theories Simplify OCaml Arrays Bit-Vectors .NET SMT-LIB Lin-arithmetic Groebner basis C Recursive Datatypes Comb. Array Logic Native Free (uninterpreted) functions F# quote Quantifiers: E-matching Model Generation: Finite Models Quantifiers: Super-position Proof objects Assumption tracking Parallel Z3 By Leonardo de Moura & Nikolaj Bjørner http://research.microsoft.com/projects/z3

  6. Z3: Little Engines of Proof Freely available from http://research.microsoft.com/projects/z3

  7. Input Formats

  8. Input Formats • Text: • SMT-LIB2 - main exchange format for SMT solvers • Simplify - legacy format by Simplify Theorem Prover • Native Z3 - low-level for storing formulas (and replay) • Log - low-level log for replay • TPTP - format used for first-order theorem provers • Programmatic: • C - API functions exposed for C • Ocaml - Ocaml wrapper around C API • .NET - .NET wrapper around C API • Scala, Python - by Phillip Suter and Sascha Böhme

  9. A Primer on SMT-LIB2 • See online Interactive tutorial • http://rise4fun.com/z3tutorial

  10. LINQ/F#: Sample layer on top of API Create Quoted Expression open Microsoft.Z3 open Microsoft.Z3.Quotations do Solver.prove <@ Logic.declare (fun t11 t12 t21 t22 t31 t32 -> not ((t11 >= 0I) && (t12 >= t11 + 2I) && (t12 + 1I <= 8I) && (t21 >= 0I) && (t22 >= t21 + 3I) && (t32 + 1I <= 8I) && (t31 >= 0I) && (t32 >= t31 + 2I) && (t32 + 3I <= 8I) && (t11 >= t21 + 3I || t21 >= t11 + 2I) && (t11 >= t31 + 2I || t31 >= t11 + 2I) && (t21 >= t31 + 2I || t31 >= t21 + 3I) && (t12 >= t22 + 1I || t22 >= t12 + 1I) && (t12 >= t32 + 3I || t32 >= t12 + 1I) && (t22 >= t32 + 3I || t32 >= t22 + 1I) ) ) @> SMT@Microsoft

  11. Theories

  12. Theories • Uninterpretedfunctions • Arithmetic (linear) • Bit-vectors • Algebraic data-types • Arrays • User-defined

  13. Theories • Uninterpreted functions • Arithmetic (linear) • Bit-vectors • Algebraic data-types • Arrays • User-defined

  14. Theories • Uninterpreted functions • Arithmetic (linear) • Bit-vectors • Algebraic data-types • Arrays • User-defined

  15. Theories • Uninterpreted functions • Arithmetic (linear) • Bit-vectors • Algebraic data-types • Arrays • User-defined

  16. Theories • Uninterpreted functions • Arithmetic (linear) • Bit-vectors • Algebraic data-types • Arrays • User-defined

  17. User-interaction and Guidance

  18. Interaction models • Text: SMT-LIB, SMT-LIB2, Native Yices (high-level), Native Z3 (low-level), Simplify • Programmatic APIs: C, Ocaml, .NET, LINQ,

  19. Interaction Logical Formula Sat/Model

  20. Interaction Logical Formula Unsat/Proof

  21. Interaction Logical Formula Simplify

  22. Interaction Logical Formula • x and y are equal • z + y and x + z are equal Implied Equalities

  23. Interaction Logical Formula Quantifier Elimination

  24. Interaction Logical Formula Unsat. Core

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