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

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Presentation transcript:

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

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

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

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 F# quote

Freely available from

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

See online Interactive tutorial

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

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

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

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

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

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

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

Logical Formula Sat/Model

Logical Formula Unsat/Proof

Simplify Logical Formula

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

Quantifier Elimination Quantifier Elimination Logical Formula

Unsat. Core