1. Nikolaj Bjørner Microsoft Research DTU Winter course January 4 th 2012 Organized by Hanne Riis Nielson, Flemming Nielson

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

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

5. Theories Bit-Vectors Lin-arithmetic Groebner basis Free (uninterpreted) functions Arrays Quantifiers: E- matching Quantifiers: E- matching OCaml.NET C C Native SMT-LIB Model Generation: Finite Models Model Generation: Finite Models Simplify Comb. Array Logic Recursive Datatypes Quantifiers: Super-position Quantifiers: Super-position Proof objects Parallel Z3 Assumption tracking By Leonardo de Moura & Nikolaj Bjørner http://research.microsoft.com/projects/z3http://research.microsoft.com/projects/z3 F# quote

6. Freely available from http://research.microsoft.com/projects/z3http://research.microsoft.com/projects/z3

8. 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. See online Interactive tutorial http://rise4fun.com/z3tutorial

10. SMT@Microsoft 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) ) @> Create Quoted Expression Expression

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

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

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

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

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

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

19. Logical Formula Sat/Model

20. Logical Formula Unsat/Proof

21. Simplify Logical Formula

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

23. Quantifier Elimination Quantifier Elimination Logical Formula

24. Unsat. Core