DOCSLIB.ORG

Decidability: Church-Turing Thesis • While There Are a Countably Infinite

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Decidability: Church-Turing Thesis • While There Are a Countably Infinite Decidability: Church-Turing Thesis • While there are a countably infinite number of languages that are described by TMs over some alphabet Σ∗, there are an uncountably infinite number that are not • Are there machines that are more powerful than TMs that can describe some of these other languages? { Note: Since there can only be a countably infinite number of machines that are described by some finite representation, there will always be some languages that cannot be described • Theory of computation based on the concept of algorithms • Important concepts relating to theory of computation introduced in the early 20th century 1. Is it possible to axiomatize all mathematical structures of interest so that every true statement becomes a theorem? { The set of axioms may be infinite, but they must be decidable - there must be an algorithm that can determine whether a string is an axiom 2. Given a set of axioms, does there exist a machine that can decide whether an axiom is a theorem? • These questions were proved to be unanswerable 1. Godel's Incompleteness Theorem { Statement: Given a potentially infinite set of true statements in number theory (Peano arithmetic), some cannot be proven 2. Entscheidungsproblem (decision problem) of Hilbert and Achermann { Statement: (a) Does there exist an algorithm for deciding, for an arbitrary sentence w in first order logic, whether w is valid? (b) Given a set of axioms A and a sentence w, does there exist an algo- rithm for deciding whether w is entailed by A? (c) Given a set of axioms A and a sentence w, does there exist an algo- rithm for deciding whether w can be proved from A? 1 Decidability: Church-Turing Thesis (2) • Concept of algorithm is important for proof of the Entscheidungsproblem { Turing and Church independently formalized what is meant by algorithm 1. Turing's approach was the Turing machine 2. Church's approach was the lambda calculus • Church-Turing Thesis { Statement: Lambda calculus and Turing machines are equivalent in power { Generalization: Any formalisms capable of describing computational algo- rithms are equivalent in power { This has not been proven, hence it is a thesis, not a theorem { Implications are that 1. There is no computational model that is more powerful 2. There are problems (languages) that have no computational solutions { Thus, there are limitations to what computers can do 2 Decidability: The Halting Problem • The theory of computation is concerned with the classes D, SD, SD=D = SD − D, and :SD • The halting problem is concerned with the language H = f< M; w >: TM M halts on input string wg • Our concern here - and throughout the rest of this section - is whether a language is 2 D; 2 SD=D; or 2 :SD? • While correct syntax of strings input to decision procedures (i.e., < M; w >) is important, that is not our concern here { Our concern is whether machine M halts when it runs on input w • We also do not consider general strings from Σ∗ { Thus, :H = f< M; w >: TM M does not halt on wg • The halting problem is: { Is H decidable, or semidecidable? • Theorem 19.1 { Statement: Language H = f< M; w >: TM M halts on input string wg 2 SD { Proof: By construction Boolean MSH (<M, w>) f M(w); return TRUE; g 3 Decidability: The Halting Problem (2) • Theorem 19.2 { Statement: Language H = f< M; w >: TM M halts on input string wg 2= D { Proof: ∗ If H were decidable, there would be a TM MH that decides H ∗ It could be implemented as Boolean MH (string <M>, string <w>) f if (<M> represents a TM that halts on w) return TRUE; else return FALSE; g ∗ Now consider TM C: Boolean C (<M>) f if (MH (<M>, <M>)) loop; else halt; g ∗ The crux of the proof is the answer to the question · What is the result of C(< C >)? 1. If C were decidable, C would never halt 2. If C were not decidable, C would halt ∗ The premise on which C is based - the existence of MH - must be incor- rect ∗ Therefore, MH does not exist 4 Decidability: The Halting Problem (3) { Another way of looking at things: ∗ Consider the table below, which indicates encodings which a TM accepts < M1 > < M2 > < M3 > < M4 > ... M1 accept accept ... M2 accept ... M3 accept accept accept ... M4 accept ... ... ... ∗ Now consider this table, which indicates the result of running MH on each (row; col) pair < M1 > < M2 > < M3 > < M4 > ... M1 accept accept ... M2 reject reject accept ... M3 accept accept accept ... M4 accept reject ... ... ... ∗ Since C is a TM, it must appear in some row, and its encoding in some column ∗ The values in C's row are opposite the values of the diagonal 1. If [1; 1] is accept 2. Then [c; 1] is reject < M1 > < M2 > < M3 > < M4 > ... M1 accept accept ... M2 reject reject accept ... M3 accept accept accept ... M4 accept reject ... ... ... C reject accept reject accept ... ... ... ∗ What value should be stored in [c; c]? · If [c; c] is reject, it should be accept, and vice-versa! 5 Decidability: The Halting Problem (4) • Theorem 19.3 { Statement: If H were in D, then every language in SD would be in D too { Discussion: ∗ Let L 2 SD ∗ Then there exists a TM ML that semidecides L ∗ Suppose H were in D ∗ Then it would be decidable by some machine O ∗ To decide whether some w 2 L, we could use O to determine whether ML halts on w ∗ If the answer were yes, then we could run ML on w to see if it accepts { Proof: By construction of M 0 to decide above ∗ Construction Boolean M' (string w) f result = O(<ML, w>); if (result == accept) f result2 = ML(w); if (result2 == accept) accept(); else reject(); g else reject(); g ∗ O always halts, so M 0 always halts ∗ Therefore, if H were in D, all languages in SD would be too 6 Decidability: The Relationship Between D and SD • The following theorems explore the relations among the languages D, SD, and :SD • Theorem 20.1 { Statement: The Context Free Grammars are a proper subset of D { Proof: ∗ By Theorem 14.1, there is a decision procedure to determine whether a string w 2 L, where L is a CFL ∗ But AnBnCn is not a CFL, and it is decidable ∗ Therefore CF Ls ⊂ D • Theorem 20.2 { Statement: Every language L 2 D is also in SD { Proof: ∗ If L 2 D, L is decided by some TM M ∗ This means that M accepts all strings in L and only strings in L ∗ This meets the definition of semidecidability, so L 2 SD 7 Decidability: The Relationship Between D and SD (2) • Theorem 20.3 { Statement: There are languages that are not in SD { Lemma: There are a countably infinite number of semidecidable languages over an alphabet Σ { Proof (of lemma): ∗ Every language L 2 SD is semidecided by some TM with alphabet Σ ∗ The TMs with alphabet Σ can be lexicographically ordered ∗ By Theorem A.1, there are a countably infinite number of semideciding TMs ∗ Therefore, there are at most a countably infinite number of languages in SD, since there cannot be more semidecidable languages than TMs ∗ There is not a 1:1 correspondence between TMs and semidecidable lan- guages, since there is more than 1 TM to semidecide a language ∗ The number of semidecidable languages is at least countably infinite because D ⊂ SD, and D is countably infinite ∗ Therefore, there are at least and at most a countably infinite number of semidecidable languages { Proof (of theorem): ∗ By Theorem 2.2, there are an uncountably infinite number of languages over an alphabet. ∗ So there must be more languages over an alphabet than there are in SD ∗ So there is at least 1 language in :SD • Theorem 20.4 { Statement: D is closed under complement { Proof: By construction ∗ Let L 2 D ∗ There must be a deterministic TM that decides L ∗ Construct M 0 as follows 1. Initially, let M 0 = M 2. Make the non-accepting states accepting 3. Make the accepting states non-accepting ∗ M 0 now rejects what M accepts, and vice-versa ∗ Since M always halts, M 0 always halts: M 0 decides :L 8 Decidability: The Relationship Between D and SD (3) • Theorem 20.5 { Statement: SD is not closed under complement { Proof: By contradiction ∗ Suppose SD were closed under complementation ∗ Then :L 2 SD would mean that there exists a TM M 0 that semidecides :L ∗ Construct M 00 as follows 1. M 00 simulates M and M 0 in parallel on input w 2. Since w 2 L or w 2 :L, w is accepted by either M or M 0 ∗ This means that M 00 will halt on w ∗ So if all L 2 SD were closed under complement, all would also be decidable ∗ But this is not the case, so SD is not closed under complement • Theorem 20.6 { Statement: Language L 2 D iff L 2 SD and :L 2 SD { Proof(if part): ∗ If L 2 D, then L 2 SD ∗ By Theorem 20.4, :L 2 D ∗ Therefore, :L 2 SD { Proof(only-if part): By construction ∗ Since L 2 SD and :L 2 SD, both are semidecided by some TM ∗ Call these TMs M and M 0 ∗ Construct M 00 as follows 1. M 00 simulates M and M 0 in parallel on input w ∗ Since w is accepted by one of M and M 0, M 00 must halt ∗ Since M 00 decides L, L 2 D • Theorem 20.7 { Statement: :H2 = SD { Proof: ∗ H 2 SD by Theorem 19.1 ∗ If :H 2 SD, then H 2 D by Theorem 20.6 ∗ But H2 = D by Theorem 19.2 ∗ Therefore, :H2 = SD 9 Decidability: And Enumeration • TM M enumerates a language L if 1.
Load more

Recommended publications
  • “The Church-Turing “Thesis” As a Special Corollary of Gödel's
    “The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known.
    [Show full text]
  • Church's Thesis and the Conceptual Analysis of Computability
    Church’s Thesis and the Conceptual Analysis of Computability Michael Rescorla Abstract: Church’s thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing’s work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church’s thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing’s work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular.1 §1. Turing machines and number-theoretic functions A Turing machine manipulates syntactic entities: strings consisting of strokes and blanks. I restrict attention to Turing machines that possess two key properties. First, the machine eventually halts when supplied with an input of finitely many adjacent strokes. Second, when the 1 I am greatly indebted to helpful feedback from two anonymous referees from this journal, as well as from: C. Anthony Anderson, Adam Elga, Kevin Falvey, Warren Goldfarb, Richard Heck, Peter Koellner, Oystein Linnebo, Charles Parsons, Gualtiero Piccinini, and Stewart Shapiro. I received extremely helpful comments when I presented earlier versions of this paper at the UCLA Philosophy of Mathematics Workshop, especially from Joseph Almog, D.
    [Show full text]
  • Bundled Fragments of First-Order Modal Logic: (Un)Decidability
    Bundled Fragments of First-Order Modal Logic: (Un)Decidability Anantha Padmanabha Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] https://orcid.org/0000-0002-4265-5772 R Ramanujam Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] Yanjing Wang Department of Philosophy, Peking University, Beijing, China [email protected] Abstract Quantified modal logic is notorious for being undecidable, with very few known decidable frag- ments such as the monodic ones. For instance, even the two-variable fragment over unary predic- ates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [15] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle ∀ alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the ∃ bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both ∀ and ∃ bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the ∃ bundle cannot distinguish between constant domain and variable domain interpretations. 2012 ACM Subject Classification Theory of computation → Modal and temporal logics Keywords and phrases First-order modal logic, decidability, bundled fragments Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2018.43 Acknowledgements We thank the reviewers for the insightful and helpful comments.
    [Show full text]
  • Soundness of Workflow Nets: Classification
    DOI 10.1007/s00165-010-0161-4 The Author(s) © 2010. This article is published with open access at Springerlink.com Formal Aspects Formal Aspects of Computing (2011) 23: 333–363 of Computing Soundness of workflow nets: classification, decidability, and analysis W. M. P. van der Aalst1,2,K.M.vanHee1, A. H. M. ter Hofstede1,2,N.Sidorova1, H. M. W. Verbeek1, M. Voorhoeve1 andM.T.Wynn2 1 Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected] 2 Queensland University of Technology, Brisbane, Australia Abstract. Workflow nets, a particular class of Petri nets, have become one of the standard ways to model and analyze workflows. Typically, they are used as an abstraction of the workflow that is used to check the so-called soundness property. This property guarantees the absence of livelocks, deadlocks, and other anomalies that can be detected without domain knowledge. Several authors have proposed alternative notions of soundness and have suggested to use more expressive languages, e.g., models with cancellations or priorities. This paper provides an overview of the different notions of soundness and investigates these in the presence of different extensions of workflow nets. We will show that the eight soundness notions described in the literature are decidable for workflow nets. However, most extensions will make all of these notions undecidable. These new results show the theoretical limits of workflow verification. Moreover, we discuss some of the analysis approaches described in the literature. Keywords: Petri nets, Decidability, Workflow nets, Reset nets, Soundness, Verification 1.
    [Show full text]
  • Undecidable Problems: a Sampler
    UNDECIDABLE PROBLEMS: A SAMPLER BJORN POONEN Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathemat- ics. 1. Introduction The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. The criterion for selection of a problem in this survey is simply that the author finds it entertaining! We do not pretend that our list of undecidable problems is complete in any sense. And some of the problems we consider turn out to be decidable or to have unknown decidability status. For another survey of undecidable problems, see [Dav77]. 2. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number @0 strictly between @0 and 2 , is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [G¨od40,Coh63, Coh64].) The first examples of statements independent of a \natural" axiom system were constructed by K. G¨odel[G¨od31]. 2. Decision problem: A family of problems with YES/NO answers is called unde- cidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert's tenth problem, to decide whether a mul- tivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 2.1.
    [Show full text]
  • The ∀∃ Theory of D(≤,∨, ) Is Undecidable
    The ∀∃ theory of D(≤, ∨,0 ) is undecidable Richard A. Shore∗ Theodore A. Slaman† Department of Mathematics Department of Mathematics Cornell University University of California, Berkeley Ithaca NY 14853 Berkeley CA 94720 May 16, 2005 Abstract We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. 1 Introduction Our interest here is in the structure D of all the Turing degrees but much of the motivation and setting is shared by work on other degree structures as well. In particular, Miller, Nies and Shore [2004] provides an undecidability result for the r.e. degrees R at the two quantifier level with added function symbols as we do here for D. To set the stage, we begin then with an adaptation of the introduction from that paper including statements of some of its results and so discuss R, the r.e. degrees and D(≤ 00), the degrees below 00 as well. A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. It also serves as a guide and organizational principle for the development of construction techniques and algebraic information about the structures. A decision procedure implies (and requires) a full understanding and control of the first order properties of a structure. Undecidability results typically require and imply some measure of complexity and coding in the struc- ture. Once a structure has been proven undecidable, it is natural to try to determine both the extent and source of the complexity.
    [Show full text]
  • Warren Goldfarb, Notes on Metamathematics
    Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines .
    [Show full text]
  • Classical Metalogic an Introduction to Classical Model Theory & Computability
    Classical Metalogic An Introduction to Classical Model Theory & Computability Notes by R.J. Buehler January 5, 2015 2 Preface What follows are my personal notes created in preparation for the UC-Berkeley Group in Logic preliminary exam. I am not a computability theorist, nor a model theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slight hint of a guarantee that everything contained within is factual and correct (indeed, some areas are entirely unchanged from the moment I copied them off the blackboard). This said, if you find a serious error, I would greatly appreciate it if you would let me know so that it can be corrected. The material for these notes derives from a wide variety of sources: Lectures by Wes Holliday Lectures by Antonio Montalban Lectures by John Steel Kevin Kelly’s computability theory notes David Marker’s “Model Theory: An Introduction” Wilfrid Hodge’s “A Shorter Model Theory” Robert Soare’s “Recursively Enumerable Sets and Degrees" Richard Kaye’s “Models of Peano Arithmetic” Chang and Keisler’s “Model Theory" While I certainly hope my notes are beneficial, if you’re attempting to learn the contained material for the first time, I would highly suggest picking up (at least) a copy of Marker and Soare’s texts in addition. To those Group in Logic students who may be using these notes to help themselves prepare for their preliminary exam, chapters 1-5, 7, and 9-18 contain relevant material, as well as chapter 8, section 3.
    [Show full text]
  • METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
    METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence.
    [Show full text]
  • Chapter 7: Proof Systems: Soundness and Completeness
    Chapter 7: Proof Systems: Soundness and Completeness Proof systems are built to prove statements. Proof systems are an inference machine with special statements, called provable state- ments being its final products. The starting points of the inference are called axioms of the system. We distinguish two kinds of axioms: logi- calAL and specific SP . 1 Semantical link : we usually build a proof system for a given language and its seman- tics i.e. for a logic defined semantically. First step : we choose as a set of logical ax- ioms AL some subset of tautologies, i.e. statements always true. A proof system with only logical axioms AL is called a logic proof system. Building a proof system for which there is no known semantics we think about the logi- cal axioms as statements universally true. 2 We choose as axioms a finite set the state- ments we for sure want to be universally true, and whatever semantics follows they must be tautologies with respect to it. Logical axioms are hence not only tautolo- gies under an established semantics, but they also guide us how to establish a se- mantics, when it is yet unknown. The specific axioms SP are these formulas of the language that describe our knowl- edge of a universe we want to prove facts about. Specific axioms are not universally true, they are true only in the universe we are inter- ested to describe and investigate. 3 A proof system with logical axioms AL and specific axioms SP is called a formal the- ory. The inference machine is defined by a finite set of rules, called inference rules.
    [Show full text]
  • G22.2390-001 Logic in Computer Science Fall 2007 Lecture 3
    Review Last week • Propositional Logic: Semantics • Satisfiability and Tautologies • Propositional Connectives and Boolean Functions G22.2390-001 Logic in Computer Science Fall 2007 • Compactness Lecture 3 1 2 Outline Sources • Computability and Decidability Enderton: 1.6, 1.7. • Boolean Circuits N. J. Cutland. Computability. • Boolean Satisfiability (SAT) W. Hodges. A Shorter Model Theory. Cambridge Press, 1997. • Binary Decision Diagrams (BDD’s) S. Malik. “The Quest for Efficient Boolean Satisfiability Solvers.” Available from http://www.princeton.edu/∼sharad/. Used with permission. E. Hirsch, A. Kojevnikov. “UnitWalk: A New SAT Solver that Uses Local Search Guided by Unit Clause Elimination.” PDMI Preprint. Sept. 2001. R. E. Bryant. Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers, C-35(8), 1986. E. Clarke. Bug Catching: Automated Program Verification and Testing. Lecture notes, Fall 2002. http://www-2.cs.cmu.edu/∼emc/15-398/. D. Dill. Automatic Formal Verification Techniques. Lecture notes, Fall 2001. http://verify.stanford.edu/cs356/. 3 4 Computability Computability and Decidability The important notion of computability relies on a formal model of computation. All of these models are equivalent to what can be achieved by a computer with any standard programming language, given arbitrary (but finite) time and memory. Many formal models have been proposed: Church’s Thesis 1. General recursive functions defined by means of an equation calculus (Godel-Herbrand-Kleene)¨ A notion known as Church’s thesis states that all models of computation are either equivalent to or less powerful than those just described. 2. λ-definable functions (Church) We will accept Church’s thesis and thus define a function to be computable if we 3.
    [Show full text]
  • Source Theories the Signature of Arithmetic Decidability
    Source Theories A signature is a (finite or infinite) set of predicate and function symbols. We fix a signature S. “Formula” means now “formula over the signature S”. A theory is a set of formulas T closed under consequence, i.e., if F ,...,F T and F ,...,F = G then G T . G.S. Boolos, J.P. Burgess, R.C Jeffrey: 1 n ∈ { 1 n} | ∈ Computability and Logic. Cambridge University Press 2002. Fact: Let be a structure suitable for S. The set F of formulas such A that (F )=1 is a theory. A We call them model-based theories. Fact: Let be a set of closed formulas. The set F of formulas such F that = F is a theory. F | 1 2 The signature of arithmetic Decidability, consistency, completeness, . The signature SA of arithmetic contains: A set of formulas is decidable if there is an algorithm that decides F a constant 0, for every formula F whether F holds. • ∈ F a unary function symbol s, • Let T be a theory. two binary function symbols + and , and • · T is decidable if it is decidable as a set of formulas. a binary predicate symbol <. • T is consistent if for every closed formula F either F / T or F / T . ∈ ¬ ∈ (slight change over previous definitions) T is complete if for every closed formula F either F T or F T . ∈ ¬ ∈ Arith is the theory containing the set of closed formulas over SA T is (finitely) axiomatizable if there is a (finite) decidable set T X ⊆ that are true in the canonical structure.
    [Show full text]