Mastering Z3: A Comprehensive Tutorial

The Z3 theorem prover, developed by Lawrence C. Paulson and Basil Miller, is a powerful tool for automated deduction and verification. It's widely used in academia, industry, and software engineering for various applications. This Z3 tutorial will guide you through understanding and using this versatile tool.

Before diving into the tutorial, ensure you have Z3 installed. You can download it from the Z3 GitHub page. This tutorial assumes you're using the command line interface. Let's start with the basics.

Understanding Z3's Architecture

Z3 is an efficient SMT ( Satisfiability Modulo Theories) solver. It uses a novel method called 'superposition' to display impressive performance. It supports various logics and theories, including arithmetic, arrays, bit-vectors, and uninterpreted functions.

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Z3 has two main components: the parser and the solver. The parser translates formulas into an intermediate representation called 'AST' (Abstract Syntax Tree). The solver then processes this AST to find solutions, simplify expressions, or check satisfiability.

Installation and Setup

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Install Z3 by following the instructions in the official guide. After installation, you can verify by opening a terminal and typing:

z3 --version

This command should display the installed Z3 version.

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Running Z3

Z3 can be run from the command line with various options. Here's a simple example:

z3 example.smt2

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Z3 reads formulas from the 'example.smt2' file, which should contain well-formed SMTlib2 formulas. The solver outputs results based on the given commands.

Z3 Syntax and Semantics

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Z3 uses an SMTlib2-like syntax. Here are some basic syntax rules:

(assert ) - Asserts an expression as a fact.
(check-sat) - Checks if the asserted formulas are satisfiable.
(get-model) - Retrieves a model satisfying the formula if it's satisfiable.

Let's create a simple 'example.smt2' file:

(assert (= (+ 1 2) 3)) (check-sat)

This formula asserts that 1 + 2 equals 3. Running Z3 with this file will output sat since the formula is true.

Arithmetic and Comparison

Z3 supports various arithmetic and comparison operations. You can negotiate these using standard mathematical symbols like '+', '-', '*', '/', and comparisons using '<', '>', '<=', '>=', and '='.

For example:

(assert (<= (* 2 3) (+ 1 4)))

Z3 will assert that 6 is less than or equal to 5. Running this will output unsat as the formula is false.

Boolean Logic

Z3 allows logical operations using 'and', 'or', 'not', and 'xor'. Boolean expressions can be combined using these operations. Here's an example:

(assert (or (not (= 1 2)) (= 2 2)))