The SMT solver Z3

Lecture 3, 2012

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

Plan

Overview and architecture of Z3

What is Z3

How to use Z3

Takeaways:

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

Follow-on questions

What is Z3?

TheoriesBit-Vectors

Lin-arithmetic Groebner basis

Free (uninterpreted) functions

Arrays

Quantifiers:E-matching

OCaml

.NET

CNative

SMT-LIB

Model Generation:Finite Models

Simplify

Comb. Array LogicRecursive Datatypes

Quantifiers:Super-position

Proof objects

Parallel Z3Assumption

tracking

By Leonardo de Moura & Nikolaj Bjørner http://research.microsoft.com/projects/z3

F# quote

Z3: Little Engines of Proof

CongruenceClosur

e

SAT Solve

r

Simplifier

Quant.Instan

ces

Simplex

Bit-Arith

Super-positio

n

Arrays

Datatypes

User-Theori

es

MBQI

Q- Elim

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

Input Formats

Input FormatsText:

SMT-LIB2 - main exchange format for SMT solversSimplify - legacy format by Simplify Theorem ProverNative Z3 - low-level for storing formulas (and replay)Log - low-level log for replayTPTP - format used for first-order theorem provers

Programmatic:C - API functions exposed for COcaml - Ocaml wrapper around C API.NET - .NET wrapper around C APIScala, Python - by Phillip Suter and Sascha Böhme

A Primer on SMT-LIB2See online Interactive tutorial

http://rise4fun.com/z3tutorial

LINQ/F#: Sample layer on top of API

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

Theories

Theories

Uninterpreted functionsArithmetic (linear)Bit-vectorsAlgebraic data-typesArraysUser-defined

Uninterpreted functionsArithmetic (linear)Bit-vectorsAlgebraic data-typesArraysUser-defined

Theories

Theories

Uninterpreted functionsArithmetic (linear)Bit-vectorsAlgebraic data-typesArraysUser-defined

Theories

Uninterpreted functionsArithmetic (linear)Bit-vectorsAlgebraic data-typesArraysUser-defined

Theories

Uninterpreted functionsArithmetic (linear)Bit-vectorsAlgebraic data-typesArraysUser-defined

User-interaction

and Guidance

Interaction modelsText: SMT-LIB, SMT-LIB2, Native Yices (high-level), Native Z3 (low-level), SimplifyProgrammatic APIs: C, Ocaml, .NET, LINQ,

Interaction

Logical Formula

Sat/Model

Interaction

Logical Formula

Unsat/Proof

Interaction

Simplify

Logical Formula

Interaction

ImpliedEqualities

- x and y are equal- z + y and x + z are equal

Logical Formula

Interaction

QuantifierEliminatio

n

Logical Formula

Interaction

Logical Formula

Unsat. Core