Introduction to Languages, Machines and Logic

Transcription

1 Introduction to Languages, Machines and Logic

2 Springer-Verlag London Ltd.

3 Alan P. Parkes Introduction to Languages, Machines and Logic Computable Languages, Abstract Machines and Formal Logic Springer

4 Alan P. Parkes, BSc, PhD Distributed Multimedia Research Group, Computing Department, Lancaster University, Lancaster, LAI 4YR British Library Cataloguing in Publication Data Parkes, Alan Introduction to languages, machines and logic: computable languages, abstract machines and formal logic 1. Machine theory I. Title ISBN Library of Congress Cataloging-in-Publication Data Parkes, Alan Introduction to languages, machines and logic: computable languages, abstract machines and formal logic/ Alan P. Parkes. p.cm. Includes index. ISBN ISBN (ebook) DOI / Formallanguages 2. Machine theory. I. Title. QA267.3.P dc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. ISBN Springer-Verlag London Limited 2002 Originally published by Springer-Verlag London in 2002 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Gray Publishing, Tunbridge Wells, Kent 34/ Printed on acid-free paper SPIN

5 Contents 1 Introduction Overview... 1 What This Book Is About What This Book Tries to Do... 5 What This Book Tries Notto Do... 6 The Exercises Further Reading... 7 Some Advice... 7 Part 1 Languages and Machines 2 Elements of Formal Languages Overview Alphabets Strings Functions that Apply to Strings Useful Notation for Describing Strings Formal Languages Methods for Defining Formal Languages Set Definitions of Languages Decision Programs for Languages Rules for Generating Languages Formal Grammars Grammars, Derivations, and Languages The Relationship Between Grammars and Languages Phrase Structure Grammars and the Chomsky Hierarchy Formal Definition of PSGs Derivations, Sentential Forms, Sentences, and "L( G)" The Chomsky Hierarchy A Type 0 Grammar: Computation as Symbol Manipulation " 33 Exercises v

6 vi Contents 3 Syntax, Semantics, and Ambiguity Overview Syntax Vs. Semantics Derivation Trees Parsing Ambiguity Exercises Regular Languages and Finite State Recognisers Overview Regular Grammars Some Problems with Grammars Finite State Recognisers and Finite State Generators 51 Creating an FSR The Behaviour of the FSR The FSR as Equivalent to the Regular Grammar Non-determinism in FSRs Constructing Deterministic FSRs The DFSR as Equivalent to the Non-DFSR 63 A Simple Deterministic Decision Program Minimal FSRs Constructing a Minimal FSR Why Minimisation Works The General Equivalence of Regular Languages and FSRs.. 76 Observations on Regular Grammars and Languages Exercises Context Free Languages and ~shdown Recognisers Overview Context Free Grammars and Context Free Languages 81 Changing G Without Changing L( G) The Empty String (8) Chomsky Normal Form Pushdown Recognisers The Stack Constructing a Non-deterministic PDR Example NPDRs, M3 and MlO Deterministic PDRs M~, a Deterministic Version of M More DPDRs Deterministic and Non-deterministic CFLs Every Regular Language Is a Deterministic CFL The Non-deterministic CFLs A Refinement to the Chomsky Hierarchy in the Case of CFLs The Equivalence of the CFLs and the PDRs 103

7 Contents vii Observations on CFGs and CFLs 104 Exercises Important Features of Regular and Context Free Languages Overview Closure Properties of Languages Closure Properties of the Regular Languages Complement Union Intersection Concatenation Closure Properties of the Context Free Languages Union Concatenation Intersection Complement.... Chomsky's Hierarchy Is Indeed a Proper Hierarchy The "Repeat State Theorem" A Language that Is Context Free but Not Regular The" uvwxy" Theorem for CFLs.... {aibici: i ;;:" I} Is Not Context Free.... The "Multiplication Language" Is Not Context Free... Preliminary Observations on the Scope of the Chomsky Hierarchy Exercises Phrase Structure Languages and Thring Machines.... Overview.... The Architecture of the Turing Machine.... "Tapes" and the "Read/Write Head".... Blank Squares TM "Instructions".... TMs Defined The Behaviour of a TM TMs as Language Recognisers Regular Languages Context Free Languages TMs Are More Powerful than PDRs Introduction to (TM) Computable Languages The TM as the Recogniser for the Context Sensitive Languages Constructing a Non-deterministic TM for Reduction Parsing of a Context Sensitive Language The Generality of the Construction The TM as the Recogniser for the Type 0 Languages

8 viii Contents Amending the Reduction Parsing TM to Deal with Type 0 Productions Dealing with the Empty String The TM as the Recogniser for All Types in the Chomsky Hierarchy Decidability: A Preliminary Discussion Deciding a Language Accepting a Language End of Part Exercises Part 2 Machines and (omputation 8 Finite State Transducers Overview Finite State Transducers FSTs and Language Recognition FSTs and Memory FSTs and Computation Simple Multiplication Addition and Subtraction Simple Division and Modular Arithmetic The Limitations of the FST Restricted FST Multiplication FSTs and Unlimited Multiplication FSTs as Unsuitable Models for Real Computers Exercises Thring Machines as Computers Overview Thring Machines and Computation TMs and Arbitrary Binary Multiplication Some Basic TM Operations The''ADD''TM The"MULT"TM TMs and Arbitrary Integer Division The"SUBTRACT"TM The "DIY" TM Logical Operations TMs and the Simulation of Computer Operations Exercises Turing's Thesis and the Universality of the Thring Machine Overview

9 Contents ix Turing's Thesis Coding a TM and Its Tape as a Binary Number 205 Coding Any TM Coding the Tape The Universal Turing Machine 208 UTMs Tapes The Operation of UTM Some Implications of UTM Non-deterministic TMs Converting a Non-deterministic TM into a 4-tape Deterministic TM The Four Tapes ofthe Deterministic Machine, D 214 The Systematic Generation of the Strings of Quintuple Labels The Operation of D The Equivalence of Non-deterministic and Four-tape Deterministic TMs Converting a Multi-tape TM into a Single-tape TM 222 Example: Representing Three Tapes as One 223 The Operation of the Single-tape Machine, S. 226 The Equivalence of Deterministic Multi-tape and Deterministic Single-tape TMs The Linguistic Implications of the Equivalence of Non-deterministic and Deterministic TMs 228 Exercises Computability, Solvability, and the Halting Problem 231 Overview The Relationship Between Functions, Problems, Solvability, and Decidability 231 Functions and Computability Problems and Solvability Decision Problems and Decidability The Halting Problem UTM H Partially Solves the Halting Problem. 235 Reductio ad Absurdum Applied to the Halting Problem The Halting Problem Shown to Be Unsolvable 239 Some Implications of the Unsolvability of the Halting Problem Computable Languages An Unacceptable (Non-computable) Language 243 An Acceptable, but Undecidable, Language. 245 Languages and Machines Exercises

10 x Contents 12 Dimensions of Computation Overview Aspects of Computation: Space, Time, and Complexity Non-deterministic TMs Viewed as Parallel Processors 251 Parallel Computations and Time A Brief Look at an Unsolved Problem of Complexity 254 A Beginner's Guide to the "Big 0" Predicting the Running Time of Algorithms. 255 Linear Time Logarithmic Time 257 Polynomial Time 260 Exponential Time 267 The Implications of Exponential Time Processes 268 Is P Equal to NP? Observations on the Efficiency of Algorithms 270 End of Part Exercises Part 3 Computation and Logic 13 Boolean Logic and Propositional Logic Overview.... Boolean Logic.... Boolean Logic Operators Boolean Logic for Problem Solving. Boolean Logic and Computing Propositional Logic.... Propositions.... Implication and Equivalence Rules of Inference Problem Solving and Reasoning in Propositional Logic. 285 Using Truth Tables to Prove Things in PropositionalLogic Observations on Propositional Logic 288 Exercises First Order Predicate Logic. 291 Overview Predicate Logic Predicates 292 Functions 293 "Sentences" Revisited: Well-formed Formulae. 294 The "First Orderness" of First Order Logic

11 Contents xi Quantifiers The Existential Quantifier The Universal Quantifier A "Blocks World" Example of FOPL Representation The Semantics of FOPL: Interpretation Problem Solving and Inference in FOPL Rules ofinference for FOPL Solving Problems by Classical FOPL Reasoning The Nature of FOPL Conclusions 305 Exercises Logic and Computation Overview A Computational Form of FOPL Getting Rid of V Getting Rid of Conjunctive Normal Form Databases Resolution The Role of Unification in Resolution How to Do Resolution The Efficiency of Resolution Why Resolution Works Logic in Action Languages, Machines, and Logic Exercises Solutions to Selected Exercises Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Further Reading 341 Index