DOCSLIB.ORG

0 Least Upper Bounds on the Size of Confluence and Church-Rosser

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
0 Least Upper Bounds on the Size of Confluence and Church-Rosser 0 Least Upper Bounds on the Size of Confluence and Church-Rosser Diagrams in Term Rewriting and λ-Calculus1 JEROEN KETEMA, Imperial College London JAKOB GRUE SIMONSEN, University of Copenhagen We study confluence and the Church-Rosser property in term rewriting and λ-calculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t !∗ s0 ∗ t0 expressed as a function: ∗ ∗ 0 — for confluence a function vsR(m; n) where the valleys are for peaks t s ! t with s of size at most m and the reductions of maximum length n, and ∗ 0 — for the Church-Rosser property a function cvsR(m; n) where the valleys are for conversions t $ t with t and t0 of size at most m and the conversion of maximum length n. For confluent term rewriting systems (TRSs), we prove that vsR is a total computable function, and for linear such systems that cvsR is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vsR(m; n) and cvsR(m; n) for some TRS R: In particular, we show that for every total computable function ' : N −! N there is a TRS R with a single term s such that vsR(jsj; n) ≥ '(n) and cvsR(n; n) ≥ '(n) for all n. For orthogonal TRSs R we prove that there is a constant k such that (a) vsR(m; n) is bounded from above by a function exponential in k and (b) cvsR(m; n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λ-calculus, we show that vsR(m; n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—Lambda calculus and related systems General Terms: Theory Additional Key Words and Phrases: Term rewriting, lambda calculus, Church-Rosser property, confluence, upper bounds ACM Reference Format: Ketema, J., Simonsen, J.G. 2012. Least Upper Bounds on the Size of Confluence and Church-Rosser Dia- grams in Term Rewriting and λ-Calculus. ACM Trans. Comput. Logic 0, 0, Article 0 (December 2012), 28 pages. DOI:http://dx.doi.org/10.1145/0000000.0000000 1. INTRODUCTION Confluence is the property of some rewriting systems that any peak t ∗ s !∗ t0 has a corresponding valley t !∗ s0 ∗ t0. The valley and the term s0 are said to complete the diagram. 1This work is an extended and revised version of the conference paper [Ketema and Simonsen 2010]. Authors’ addresses: J. Ketema, Department of Computing, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, United Kingdom; Jakob Grue Simonsen, Department of Computer Science, University of Copenhagen (DIKU), Njalsgade 126–128, 2300 Copenhagen S, Denmark Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is per- mitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or [email protected]. c 2012 ACM 1529-3785/2012/12-ART0 $15.00 DOI:http://dx.doi.org/10.1145/0000000.0000000 ACM Transactions on Computational Logic, Vol. 0, No. 0, Article 0, Publication date: December 2012. 0:2 J. Ketema and J.G. Simonsen a ≤n ≤n ≤n b c b o / c ≤l ≤l ≤l ≤l Ð Ð d d (a) Confluence (b) Church-Rosser property Fig. 1. Confluence and the Church-Rosser property of a rewrite system R with bounds on the lengths of the reductions. This paper is concerned with finding least upper bounds of l as a function of n. A rewriting system is confluent iff it has the so-called Church-Rosser property; the property that any conversion t $∗ t0 has a corresponding valley t !∗ s0 ∗ t0. As the two notions are equivalent, the terms ‘confluence’ and ‘Church-Rosser property’ are often used interchangeably in the literature; to the authors’ knowledge, all contem- porary research in rewriting systems considers confluence, rather than the classical Church-Rosser property, probably due to the fact that the former seems better suited to the range of standard proof methods available. In functional programming, the Church-Rosser property ensures that different ways of evaluating a program always yield the same result (modulo non-termination): The result of running a program will be independent of the evaluation order or reduction strategy. In logic, if the rewrite system induced by the derivation relation of a deductive system has the Church-Rosser property and is strongly normalizing, the system will be consistent: No statement can both hold and not hold. While the Church-Rosser property has been shown to hold for a variety of rewrite systems, there has, to the authors’ knowledge, never been an investigation into the number of reduction steps in a valley that completes a peak or a conversion of a given size (see Figure 1). Succinctly: The question “How large is the valley as a function of the peak or conversion?” has apparently not been asked before. We find the above question to be intrinsically interesting from a theoretical point of view as Church-Rosser-type results are ubiquitous. However, we also believe the practical implications in mainstream functional programming to be limited: Standard functional languages like ML and HASKELL employ a fixed evaluation strategy such as call-by-value or call-by-need, and there seems to be little interest in performing optimizations by switching strategies (modulo non-termination). However, for more specialized languages, like declarative DSLs where the evaluation order may not be fixed, there may be practical implications: If, for small peaks, the size of the smallest corresponding valley is so large that a term completing the Church-Rosser diagram cannot be computed using realistic resources, then it matters very much which reduc- tion strategy is employed: Choosing the ‘wrong’ evaluation strategy (say, call-by-value) and performing just a few steps of a computation could result in a very long reduction before a result is reached—better to backtrack to the original term and try another strategy. In this paper, we perform the first fundamental study of valley sizes for systems with the Church-Rosser property; specifically we study how the size of a peak or con- version affects the size of the smallest corresponding valley. We consider three very general settings: that of (arbitrary) first-order term rewriting systems, of orthogonal term rewriting systems (roughly corresponding to first-order functional programs that have no fixed evaluation order), and of untyped λ-calculus. We believe that these three ACM Transactions on Computational Logic, Vol. 0, No. 0, Article 0, Publication date: December 2012. Least Upper Bounds on the Size of Confluence and Church-Rosser Diagrams 0:3 Table I. The bounds derived for the valley sizes and their tightness Confluence (vs) Church-Rosser (cvs) Bound Tight? Bound Tight? TRSs Unknown (Section 4) Computable Yes Linear TRSs (Theorem 4.2) (Theorem 4.12) Computable Yes (Section 4) (Theorem 4.5) (Theorem 4.14) Orthogonal TRSs Exponential Yes E 4 Unknown (Section 5) (Theorem 5.5) (Remark 5.6) (Theorem 5.9) λ-calculus E 4 Unknown Unknown (Section 6) (Theorem 6.5) areas cover most of the non-specialized areas where the Church-Rosser property oc- curs. The most significant area omitted is that of general higher-order rewrite systems (including higher-order functional programs and logics with bound variables)—we ex- pect general upper bounds in that case to be very difficult to derive, as is foreshadowed by the difficulties we encounter in our treatment of λ-calculus in Section 6. Valley sizes are measured by two functions, vs : N2 −! N for confluence, and cvs : N2 −! N for the Church-Rosser property. Roughly, vs(m; n) is the least number of steps required to complete a valley for a peak starting from a term of size at most m and with reductions of length at most n. Similarly, cvs(m; n) is the least number of steps required to complete a valley corresponding to a conversion involving at most n steps between two terms of size at most m. Our results on upper bounds are summarized in Table I. In the table, the exponential upper bound for orthogonal TRSs depends on a constant dependent on the specific rewrite system and is thus tractable in some cases. 1.1. Related Work For all the kinds of rewriting considered in this paper, the question of valley sizes in confluence and Church-Rosser diagrams does not appear to have been investigated be- fore. However, in research on the ‘efficiency’ in λ-calculus, some related phenomena have been under scrutiny. Most pertinently, investigations have been performed con- cerning upper bounds on the length of developments by de Vrijer [1985] and concerning standard reductions by Xi [1999]. In typed systems, lower bounds for normalizing re- ductions have been studied by Statman [1979], upper bounds are treated Schwichten- berg [1982; 1991] and Beckmann [2001], and lower and upper bounds are discussed by Springintveld [1993].
Load more

Recommended publications
  • Hyperoperations and Nopt Structures
    Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles.
    [Show full text]
  • An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
    STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical
    [Show full text]
  • A Survey of Recursive Analysis and Moore's Notion of Real Computation
    A Survey of Recursive Analysis and Moore’s Notion of Real Computation Walid Gomaa INRIA Nancy Grand-Est Research Centre, France, Faculty of Engineering, Alexandria University, Egypt [email protected] Abstract The theory of analog computation aims at modeling computational systems that evolve in a continuous manner. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero- finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them. 1 Introduction Analog computation is a computational paradigm that attempts to model systems whose internal states are continuous rather than discrete. Unlike the case of discrete computation, which has enjoyed a kind of uniformity and conceptual universality, there have been several approaches to analog computations some of which are not even comparable.
    [Show full text]
  • 0.1 Axt, Loop Program, and Grzegorczyk Hier- Archies
    1 0.1 Axt, Loop Program, and Grzegorczyk Hier- archies Computable functions can have some quite complex definitions. For example, a loop programmable function might be given via a loop program that has depth of nesting of the loop-end pair, say, equal to 200. Now this is complex! Or a function might be given via an arbitrarily complex sequence of primitive recursions, with the restriction that the computed function is majorized by some known function, for all values of the input (for the concept of majorization see Subsection on the Ackermann function.). But does such definitional|and therefore, \static"|complexity have any bearing on the computational|dynamic|complexity of the function? We will see that it does, and we will connect definitional and computational complexities quantitatively. Our study will be restricted to the class PR that we will subdivide into an infinite sequence of increasingly more inclusive subclasses, Si. A so-called hierarchy of classes of functions. 0.1.0.1 Definition. A sequence (Si)i≥0 of subsets of PR is a primitive recur- sive hierarchy provided all of the following hold: (1) Si ⊆ Si+1, for all i ≥ 0 S (2) PR = i≥0 Si. The hierarchy is proper or nontrivial iff Si 6= Si+1, for all but finitely many i. If f 2 Si then we say that its level in the hierarchy is ≤ i. If f 2 Si+1 − Si, then its level is equal to i + 1. The first hierarchy that we will define is due to Axt and Heinermann [[5] and [1]].
    [Show full text]
  • Hierarchies of Recursive Functions
    Leibniz Universitat¨ Hannover Institut f¨urtheoretische Informatik Bachelor Thesis Hierarchies of recursive functions Maurice Chandoo February 21, 2013 Erkl¨arung Hiermit versichere ich, dass ich diese Arbeit selbst¨andigverfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. i Dedicated to my dear grandmother Anna-Maria Kniejska ii Contents 1 Preface 1 2 Class of primitve recursive functions PR 2 2.1 Primitive Recursion class PR ..................2 2.2 Course-of-value Recursion class PRcov .............4 2.3 Nested Recursion class PRnes ..................5 2.4 Recursive Depth . .6 2.5 Recursive Relations . .6 2.6 Equivalence of PR and PRcov .................. 10 2.7 Equivalence of PR and PRnes .................. 12 2.8 Loop programs . 14 2.9 Grzegorczyk Hierarchy . 19 2.10 Recursive depth Hierarchy . 22 2.11 Turing Machine Simulation . 24 3 Multiple and µ-recursion 27 3.1 Multiple Recursion class MR .................. 27 3.2 µ-Recursion class µR ....................... 32 3.3 Synopsis . 34 List of literature 35 iii 1 Preface I would like to give a more or less exhaustive overview on different kinds of recursive functions and hierarchies which characterize these functions by their computational power. For instance, it will become apparent that primitve recursion is more than sufficiently powerful for expressing algorithms for the majority of known decidable ”real world” problems. This comes along with the hunch that it is quite difficult to come up with a suitable problem of which the characteristic function is not primitive recursive therefore being a possible candidate for exploiting the more powerful multiple recursion. At the end the Turing complete µ-recursion is introduced.
    [Show full text]
  • Fractional Mathematical Operators and Their Computational Approximation
    Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 4356371, 11 pages http://dx.doi.org/10.1155/2016/4356371 Research Article Fractional Mathematical Operators and Their Computational Approximation José Crespo and Francisco Javier Montáns Escuela Tecnica´ Superior de Ingenier´ıa Aeronautica´ y del Espacio, Universidad Politecnica´ de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Correspondence should be addressed to Francisco Javier Montans;´ [email protected] Received 27 January 2016; Revised 30 July 2016; Accepted 9 August 2016 Academic Editor: Manuel Doblare´ Copyright © 2016 J. Crespo and F. J. Montans.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Usual applied mathematics employs three fundamental arithmetical operators: addition, multiplication, and exponentiation. However, for example, transcendental numbers are said not to be attainable via algebraic combination with these fundamental operators. At the same time, simulation and modelling frequently have to rely on expensive numerical approximations of the exact solution. The main purpose of this article is to analyze new fractional arithmetical operators, explore some of their properties, and devise ways of computing them. These new operators may bring new possibilities, for example, in approximation theory and in obtaining closed forms of those approximations and solutions. We show some simple demonstrative examples. 1. Introduction parameters [2, 3] rendering linear equations. We will also see possible applications of the presented operators in constitu- Science mostly hinges on mathematics which has been built tive modelling. on three fundamental operations: the addition, the multipli- Consider the classification described in Table 1.
    [Show full text]
  • The Grzegorczyk Hierarchy
    Chapter 4 The Grzegorczyk Hierarchy 4.1 Reprise The primitive recursive functions are defined as the closure of R and C over a basic stock of functions. Both the logician’s diagonal argument and the double- recursion diagonal arguments involve functions that can “simulate” n nested primitive recursions when given an argument n. Intuitively, the situation is that each primitive recursive function is allowed only finitely many nested primitive recursions, so functions that require arbitrarily many (without bound) are not primitive recursive. This situation calls out for greater attention to which functions can be de- fined with a given, fixed number of primitive recursion operations. In other words, it makes sense to look at the hierarchy of increasing classes En defined in terms of increasing numbers of primitive recursions, where each class is closed under composition. Every primitive recursive function will be at some finite level of the hierarchy: [ P rim = Ex x A natural idea would be to start with the basic functions closed under com- position and then to add in successively more complex functions like addition, multiplication, exponentiation, etc., closing under composition each time. But composition is a little weak. Recall that some slow-growing functions like cutoff subtraction required the R schema for their definitions even though they don’t really use it to generate faster growth. We would like to distinguish growth- generating applications of R from applications that don’t generate growth. We can get around the difficulty by closing each class under bounded re- cursion as well as composition. An application of R is bounded in a class of functions if the function resulting from the application is bounded everywhere by some other function already established to be in the class.
    [Show full text]
  • Solving for the Analytic Piecewise Extension of Tetration and the Super-Logarithm
    Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm by Andrew Robbins Abstract An overview of previous extensions of tetration is presented. Specific conditions for differentiability and piecewise continuity are shown. This leads to a way of generating approximations of the super-logarithm. These approximations are shown to converge to a function that satisfies two basic properties of extensions of tetration. Table of Contents Introduction 2 Background 4 Extensions 6 Results 10 Generalization 20 Conclusion 21 Appendices 22 Copyright 2005 Andrew Robbins 1 Introduction Of the operators in the sequence: addition, multiplication, exponentiation, and tetration, only the first three are well-defined analytic operations. Tetration, also known as the hyper4 operator, power towers, and iterated exponentiation, has only been “nicely” defined for integer y in y x . Although extensions of tetration to real y have been made, those extensions have not followed simply from the properties of hyper-operations. The particular form of this property as it pertains to tetration is: Property 1. Iterated exponential property y−1 y x = x( x ) for all real y. Tetration has two inverses, the super-root, and the super-logarithm: = y = = z x tet x ( y ) twr y ( x ) = = −1 x srt y ( z ) twr y ( z ) = = −1 y slogx ( z ) tet x ( z ) When y is fixed, the function twr y ( x ) is called an order y power tower of x [14]. When x is fixed, the function tet x ( y ) is called a base x tetrational function of y. In general, it is pronounced: x tetra y. Inverting a power tower gives the super-root, and inverting a tetrational function gives the super-logarithm, More detail about these inverses can be found in [12], [14] and [15].
    [Show full text]
  • Lecture 15 CS 6110 Spring 2015 Wed
    Advanced Progamming Languages Lecture 15 CS 6110 Spring 2015 Wed. Feb. 25, 2015 Lecture 15 Midterm reminder - Friday, March 6 Topics 1. Comments on Abstract State Machine in Dr. Rahli's lectures { The heart of Theory B { with a Theory A analogue 2. Partial recursive numerical functions (Kleene 1936, Turing 1937) and General recursive functions, Herbrand-G¨odel1932 3. Kleene Normal Form theorem (Kleene T-predicate) using primitve recursion, only the elementary subset. Related to acceptable indexing, Rogers 1967 4. Textbook material Church-Rosser Theorem p.163 Typed λ-calculus, section 2.6 p.42 Strong normalization theorem p.45 5. Deep problems remain lurking here on the Theory B side in G¨odelnumbering versus meta language insights. Need to ask the right question. 1. Abstract State Machines for the λ-calculus These machines are not only conceptually interesting, they are practical and provide the basis for functional language compilers and bootstrapping. This is Theory B semantics in practice. You can study the Krevine and Landin machines in detail in the BRICS 2003 article along with other state machines. 2. Partial recursive functions and General recursive functions Please read the Kleene material included with the lecture. He describes G¨odel'sinsight to simplify Herbrand's idea, which is this (p.274 Kleene): (i) Write down any mutually recursive equations that define a total function ' re- cursively (ii) Show that they define the intended function ' by finitary means, say intuitionistic methods. 1 G¨odel's\bold generalization" - only require (i), since showing they define a function will likely require a proof by induction, but any fintary proof will suffice.
    [Show full text]
  • Fifty Years of the Spectrum Problem: Survey and New Results
    FIFTY YEARS OF THE SPECTRUM PROBLEM: SURVEY AND NEW RESULTS A. DURAND, N. D. JONES, J. A. MAKOWSKY, AND M. MORE Abstract. In 1952, Heinrich Scholz published a question in the Journal of Sym- bolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. G¨unter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual devel- opments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz’ question was answered in various ways, Asser’s question remains open. One appendix paraphrases the contents of several early and not easily accesible papers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendix contains a compendium of questions and conjectures which remain open. To be submitted to the Bulletin of Symbolic Logic. January 23, 2018 (version 13.2) Contents 1. Introduction 3 2. The Emergence of the Spectrum Problem 4 2.1. Scholz’s problem 4 2.2. Basic facts and questions 5 arXiv:0907.5495v1 [math.LO] 31 Jul 2009 2.3. Immediate responses to H. Scholz’s problem 6 2.4.
    [Show full text]
  • Non Primitive Recursive Complexity Classes
    Non Primitive Recursive Complexity Classes Simon Halfon, ENS Cachan August 22, 2014 supervised by Philippe Schnoebelen and Sylvain Schmitz Team INFINI, LSV, ENS Cachan 1 Introduction / Summary File The general context The introduction of Well Structured Transition Systems (WSTS) in 1987 [8], i.e. transition systems that satisfies a monotony property with respect to some well-quasi-ordering (wqo), has led to an important number of decidability results of verification problems for several natural models: Petri Nets and VASS and a large number of their extensions, lossy channel systems, string rewrite systems, process algebra, communicating automaton, and so on. Surveys of results and applications obtained with this theory can be found in [9, 3, 1, 2]. The main idea behind these decidability results is a generic algorithm that explores a tree that must be finite by the wqo property: every infinite sequence of configurations of the system (xi)i2N has an increasing pair, that is a pair i < j such that xi ≤ xj. Moreover, wqo theory has provided upper bound to these algorithms by bounding the length of so called bad sequences, (finite) sequences that do not have an increasing pair. The bounds obtained are non-primitive-recursive, which is unusual in verification. In addition, matching lower bounds has been proved for several models [19, 6]. The research problem Considering the growing number of new problems with a non-primitive-recursive complexity, it is natural to ask for a finer classification, and therefore the formalization of non-elementary complexity classes. This has been achieved by the introduction of a hierarchy of ordinal-indexed fast-growing complexity classes (Fα)α [18].
    [Show full text]
  • Classes of Sub-Recursive Function Selected Bibliography
    Classes of sub-recursive function Selected bibliography Armando B. Matos1 November 7, 2012 1Email: [email protected] This is a personal bibliography on classes of sub-recursive functions. The references in the abstracts have been omitted. 1 Bibliography [AAV10] Philippe Andary, Bruno Patrou Andary, and Pierre Valarcher. A rep- resentation theorem for primitive recursive algorithms. Fundamenta Informaticae, XX:1{18, 2010. Abstract. We formalize the algorithms computing primitive recursive (PR) functions as the abstract state machines (ASMs) whose running length is computable by a PR function. Then we show that there exists a programming language (implementing only PR functions) by which it is possible to implement any one of the previously defined algorithms for the PR functions in such a way that their complexity is preserved. [Axt59] Paul Axt. On a subrecursive hierarchy and primitive recursive degrees. Transactions of the American Mathematical Society, 92:85{105, 1959. (From the Introduction) We shall investigate some problems connected with these classifications [of general and primitive recursive degrees]. First a uniqueness property of classes Cy associated with notations for the same ordinal is described and shown to hold at ordinals less than !2 and to fail at !2. The k-recursive functions of Peter are located in the hierarchy below the !! level. Although it is not yet settled whether all recursive functions are obtained, it is clear that [y2OCy is a large and interesting class. Finally primitive recursive degrees are studied, and certain similarities to and differences from the theory of general recursive degrees of [Kleene and Post reference] are obtained. [Axt63] Paul Axt.
    [Show full text]