Concrete Abstractions: Formalizing and Analyzing Discrete Theories and Algorithms with the RISCAL Model Checker

ISBN-13: 9783031249334 / Angielski / Twarda / 2023 / 280 str.

Wolfgang Schreiner

Opis

This book demonstrates how to formally model various mathematical domains (including algorithms operating in these domains) in a way that makes them amenable to a fully automatic analysis by computer software.

The presented domains are typically investigated in discrete mathematics, logic, algebra, and computer science; they are modeled in a formal language based on first-order logic which is sufficiently rich to express the core entities in whose correctness we are interested: mathematical theorems and algorithmic specifications. This formal language is the language of RISCAL, a “mathematical model checker” by which the validity of all formulas and the correctness of all algorithms can be automatically decided. The RISCAL software is freely available; all formal contents presented in the book are given in the form of specification files by which the reader may interact with the software while studying the corresponding book material.

Spis treści

Theories and Algorithms

1.1 Theories and Models

1.2 Theorems

1.3 Problems

1.4 Algorithms

1.5 Summary

Further Reading

2 Searching and Sorting

2.1 Searching in Arrays

2.2 Loop Invariants and Termination Measures

2.3 Generating and Checking Verification Conditions

2.4 Constructing Invariants

2.5 Binary Search

2.6 Sorting Arrays

Further Reading

3 Sets, Relations, and Graphs

3.1 Sets

3.2 Relations and Directed Graphs

3.3 The Reachability of Nodes

3.4 Shortest Paths Between All Pairs of Nodes

3.5 Shortest Paths Between Two Nodes

Further Reading

4 Propositional Logic

4.1 Boolean Algebras

4.2 Propositional Formulas

4.3 Clausal Normal Forms

4.4 Translating Propositional Formulas to Normal Forms

4.5 The Minimization of Disjunctive Normal Forms

4.6 The Satisfiability of Conjunctive Normal Forms

Further Reading

5 Big Number and Polynomial Arithmetic

5.1 Arbitrary Precision Numbers

5.2 The Karatsuba Algorithm

5.3 Modular Arithmetic

5.4 Univariate Polynomials

5.5 Multivariate Polynomials

Further Reading

6 Puzzles and Games

6.1 Chess Puzzles

6.2 Mathematical Puzzles

6.3 River-Crossing Puzzles

6.4 Games for Two Players

Further Reading

7 Concurrent Systems

7.1 Peterson’s Algorithm

7.2 The Alternating Bit Protocol

7.3 A Resource Allocator

Further Reading

8 Further Topics

8.1 Checking Temporal Logic Specifications

8.2 Proving First-Order Formulas

Further Reading

A The RISCAL Software

B RISCAL Models

B.1 Theories and Algorithms

B.2 Searching and Sorting

B.3 Sets, Relations, and Graphs

B.4 Propositional Logic

B.5 Big Number and Polynomial Arithmetic

B.6 Puzzles and Games

B.7 Concurrent Systems

References

Index

O autorze

Wolfgang Schreiner is an associate professor at the Research Institute for Symbolic Computation (RISC) of the Johannes Kepler University Linz, Austria. He is working in formal methods of computer science (with previous research in parallel computing and functional programming), and has produced various software packages related to formal semantics, specification, and verification, in particular the RISC ProofNavigator, the RISC ProgramExplorer, and the RISC Algorithm Language (RISCAL). Formerly he directed a degree programme on Computer-based Learning at the Upper Austria University of Applied Sciences in Hagenberg.