Elements of Finite Model Theory
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Logic in Computer Science, 2003. Proceedings. 18Th Annual IEEE Symposium On
Spectrum Hierarchies and Subdiagonal Functions Aaron Hunter Department of Computing Science Simon Fraser University Burnaby, Canada [email protected] Abstract been done on the properties of the classes in Fagin’s pro- posed spectrum hierarchy. The spectrum of a first-order sentence is the set of cardi- Similarly, very little is known about the classes of spec- nalities of its finite models. Relatively little is known about tra obtained by varying the number of binary relation sym- the subclasses of spectra that are obtained by looking only bols, or the number of unary relation symbols allowed in at sentences with a specific signature. In this paper, we sentences. In this paper, we will use model-theoretic tech- study natural subclasses of spectra and their closure prop- niques to explore the properties of these restricted classes erties under simple subdiagonal functions. We show that of spectra. many natural closure properties turn out to be equivalent to the collapse of potential spectrum hierarchies. We prove all 1.2. Closure Properties of our results using explicit transformations on first-order structures. Given a set of natural numbers S and a function f over the natural numbers, we write f(S) as a shorthand notation for {f(n)|n ∈ S}. We say that a class of spectra S is closed 1. Introduction under f if f(S) ∈Swhenever S ∈S. Using explicit transformations on graphs, More proves 1.1. Historical Motivation that F2 is closed under every polynomial with rational co- efficients that is assymptotically greater than the identity The spectrum of a first-order sentence φ is the set of car- function [12]. -
Mathematical Logic
Copyright c 1998–2005 by Stephen G. Simpson Mathematical Logic Stephen G. Simpson December 15, 2005 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 http://www.math.psu.edu/simpson/ This is a set of lecture notes for introductory courses in mathematical logic offered at the Pennsylvania State University. Contents Contents 1 1 Propositional Calculus 3 1.1 Formulas ............................... 3 1.2 Assignments and Satisfiability . 6 1.3 LogicalEquivalence. 10 1.4 TheTableauMethod......................... 12 1.5 TheCompletenessTheorem . 18 1.6 TreesandK¨onig’sLemma . 20 1.7 TheCompactnessTheorem . 21 1.8 CombinatorialApplications . 22 2 Predicate Calculus 24 2.1 FormulasandSentences . 24 2.2 StructuresandSatisfiability . 26 2.3 TheTableauMethod......................... 31 2.4 LogicalEquivalence. 37 2.5 TheCompletenessTheorem . 40 2.6 TheCompactnessTheorem . 46 2.7 SatisfiabilityinaDomain . 47 3 Proof Systems for Predicate Calculus 50 3.1 IntroductiontoProofSystems. 50 3.2 TheCompanionTheorem . 51 3.3 Hilbert-StyleProofSystems . 56 3.4 Gentzen-StyleProofSystems . 61 3.5 TheInterpolationTheorem . 66 4 Extensions of Predicate Calculus 71 4.1 PredicateCalculuswithIdentity . 71 4.2 TheSpectrumProblem . .. .. .. .. .. .. .. .. .. 75 4.3 PredicateCalculusWithOperations . 78 4.4 Predicate Calculus with Identity and Operations . ... 82 4.5 Many-SortedPredicateCalculus . 84 1 5 Theories, Models, Definability 87 5.1 TheoriesandModels ......................... 87 5.2 MathematicalTheories. 89 5.3 DefinabilityoveraModel . 97 5.4 DefinitionalExtensionsofTheories . 100 5.5 FoundationalTheories . 103 5.6 AxiomaticSetTheory . 106 5.7 Interpretability . 111 5.8 Beth’sDefinabilityTheorem. 112 6 Arithmetization of Predicate Calculus 114 6.1 Primitive Recursive Arithmetic . 114 6.2 Interpretability of PRA in Z1 ....................114 6.3 G¨odelNumbers ............................ 114 6.4 UndefinabilityofTruth. 117 6.5 TheProvabilityPredicate . -
Generalized First-Order Spectra and Polynomial-Time Recognizable Sets
Generalized First-Order Spectra and Polynomial-Time Recognizable Sets Ronald Fagin Abstract: The spectrum of a first-order sentence σ is the set of cardi- nalities of its finite models. Jones and Selman showed that a set C of numbers (written in binary) is a spectrum if and only if C is in the complexity class NEXP (nondeterministic exponential time). An alternative viewpoint of a spectrum is to consider the spectrum of σ to be the class of finite models of the existential second-order sentence 9Qσ(Q), where Q is the similarity type (set of relational symbols) of σ.A generalized spectrum is the class of finite models of an exis- tential second-order sentence 9Qσ(P; Q), where σ is first-order with similarity type P [ Q, with P and Q disjoint. Let C be a class of finite structures with similarity type P, where C is closed under isomorphism. If P is nonempty, we show that C is a generalized spectrum if and only if the set of encodings of members of C is in NP. We unify this result with that of Jones and Selman by encoding numbers in unary rather than binary, so that C is a spectrum if and only if C is in NP. We then have that C is a generalized spectrum if and only if the set of encodings of members of C is in NP, whether or not P is empty. Using this connection between logic and complexity, we take results from complexity theory and convert them into results in logic. -
Chapter 1. Cutting out Fragments
UvA-DARE (Digital Academic Repository) Logic Engineering. The Case of Description and Hybrid Logics Areces, C.E. Publication date 2000 Link to publication Citation for published version (APA): Areces, C. E. (2000). Logic Engineering. The Case of Description and Hybrid Logics. ILLC dissertation series 2000-05. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:07 Oct 2021 Chapter 1 Cutting Out Fragments The logic is invariant, ... but the data are different. So the results are different. from "Stranger in a Strange Land," Robert Heinlein 1.1 Looking for the Right Language The rules of the game are set: we are searching for good formal languages for specific tasks, where by "formal languages" we mean languages with a precise syntax and se mantics, and by "specific tasks" we mean reasoning tasks. -
BOUNDED DEGREE and PLANAR SPECTRA 1. Introduction The
Logical Methods in Computer Science Vol. 13(4:6)2017, pp. 1–21 Submitted Sep. 08, 2016 https://lmcs.episciences.org/ Published Nov. 06, 2017 BOUNDED DEGREE AND PLANAR SPECTRA ANUJ DAWAR AND ERYK KOPCZYNSKI´ Department of Computer Science and Technology, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0FD, England e-mail address: [email protected] Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail address: [email protected] Abstract. The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra. 1. Introduction The spectrum of a sentence φ of some logic, denoted spec(φ), is the set of positive integers n such that φ has a model of cardinality n. In this paper we are solely concerned with first- order logic and we use the word spectrum to mean a set of integers that is is the spectrum of some first-order sentence. Scholz in [Sch52] posed the question of characterizing those sets of integers which are spectra. -
Model Theory on Finite Structures∗
Model Theory on Finite Structures∗ Anuj Dawar Department of Computer Science University of Wales Swansea Swansea, SA2 8PP, U.K. e-mail: [email protected] 1 Introduction In mathematical logic, the notions of mathematical structure, language and proof themselves become the subject of mathematical investigation, and are treated as first class mathematical objects in their own right. Model theory is the branch of mathematical logic that is particularly concerned with the relationship between structure and language. It seeks to establish the limits of the expressive power of our formal language (usually the first order predicate calculus) by investigating what can or cannot be expressed in the language. The kinds of questions that are asked are: What properties can or cannot be formulated in first order logic? What structures or relations can or cannot be defined in first order logic? How does the expressive power of different logical languages compare? Model Theory arose in the context of classical logic, which was concerned with resolving the paradoxes of infinity and elucidating the nature of the infinite. The main constructions of model theory yield infinite structures and the methods and results assume that structures are, in general, infinite. As we shall see later, many, if not most, of these methods fail when we confine ourselves to finite structures Many questions that arise in computer science can be seen as being model theoretic in nature, in that they investigate the relationship between a formal language and a structure. However, most structures of interest in computer science are finite. The interest in finite model theory grew out of questions in theoretical computer science (particularly, database theory and complexity theory). -
On Finite Domains in First-Order Linear Temporal Logic Denis Kuperberg, Julien Brunel, David Chemouil
On Finite Domains in First-Order Linear Temporal Logic Denis Kuperberg, Julien Brunel, David Chemouil To cite this version: Denis Kuperberg, Julien Brunel, David Chemouil. On Finite Domains in First-Order Linear Temporal Logic. 14th International Symposium on Automated Technology for Verification and Analysis, Oct 2016, Chiba, Japan. 10.1007/978-3-319-46520-3_14. hal-01343197 HAL Id: hal-01343197 https://hal.archives-ouvertes.fr/hal-01343197 Submitted on 7 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On Finite Domains in First-Order Linear Temporal Logic? Denis Kuperberg1, Julien Brunel2, and David Chemouil2 1 TU Munich, Germany 2 DTIM, Université fédérale de Toulouse, ONERA, France Abstract. We consider First-Order Linear Temporal Logic (FO-LTL) over linear time. Inspired by the success of formal approaches based upon finite-model finders, such as Alloy, we focus on finding models with finite first-order domains for FO-LTL formulas, while retaining an infinite time domain. More precisely, we investigate the complexity of the following problem: given a formula ' and an integer n, is there a model of ' with domain of cardinality at most n? We show that depending on the logic considered (FO or FO-LTL) and on the precise encoding of the problem, the problem is either NP-complete, NEXPTIME-complete, PSPACE- complete or EXPSPACE-complete. -
Reflections on Finite Model Theory
Reflections on Finite Model Theory Phokion G. Kolaitis∗ IBM Almaden Research Center San Jose, CA 95120, USA [email protected] Abstract model theory, to examine some of the obstacles that were encountered, and to discuss some open problems that have Advances in finite model theory have appeared in LICS stubbornly resisted solution. proceedings since the very beginning of the LICS Sympo- sium. The goal of this paper is to reflect on finite model 2 Early Beginnings theory by highlighting some of its successes, examining ob- stacles that were encountered, and discussing some open In the first half of the 20th Century, finite models were problems that have stubbornly resisted solution. used as a tool in the study of Hilbert’s Entscheidungsprob- lem, also known as the classical decision problem, which is the satisfiability problem for first-order logic: given a 1 Introduction first-order sentence, does it have a model? Indeed, even before this problem was shown to be undecidable by Turing During the past thirty years, finite model theory has devel- and Church, logicians had identified decidable fragments of oped from a collection of sporadic, but influential, early re- first-order logic, such as the Bernays-Schonfinkel¨ Class of sults to a mature research area characterized by technical all ∃∗∀∗ sentences and the Ackermann Class of all ∃∗∀∃∗ depth and mathematical sophistication. In this period, fi- sentences. The decidability of these two classes was estab- nite model theory has been explored not only for its con- lished by proving that the finite model property holds for nections to other areas of computer science (most notably, them: if a sentence in these classes has a model, then it has computational complexity and database theory), but also in a finite model (see [14, Chapter 6] for a modern exposition its own right as a distinct area of logic in computer science. -
A Framework for Representing and Solving NP Search Problems
A Framework for Representing and Solving NP Search Problems David G. Mitchell and Eugenia Ternovska Simon Fraser University {mitchell,ter}@cs.sfu.ca Abstract substantial effort may be required to find good encodings for some problems. NP search and decision problems occur widely in AI, and a CSP provides somewhat better modelling capabilities. number of general-purpose methods for solving them have The area has also provided a number of useful techniques. been developed. The dominant approaches include propo- sitional satisfiability (SAT), constraint satisfaction problems For example, nogood learning and backjumping methods (CSP), and answer set programming (ASP). Here, we propose developed in the CSP context are central to modern SAT a declarative constraint programming framework which we solvers (Mitchell 2005). Solvers for CSP, however, have believe combines many strengths of these approaches, while found success primarily as components in constraint logic addressing weaknesses in each of them. We formalize our ap- programming (CLP) tools. These provide rich problem solv- proach as a model extension problem, which is based on the ing environments, but are really general purpose program- classical notion of extension of a structure by new relations. ming languages and thus neither purely declarative nor tai- A parameterized version of this problem captures NP. We dis- lored to problems of moderate (in)tractability. cuss properties of the formal framework intended to support ASP is the only area of these three that took modelling effective modelling, and prospects for effective solver design. methodology seriously from the outset. ASP input lan- guages provide essentially all the modelling facilities CSP Introduction does, within a purely declarative framework. -
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. July 17, 2009 (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 2.3. Immediate responses to H. Scholz’s problem 6 2.4. Approaches and themes 8 3. Understanding Spectra: counting functions and number theory 12 3.1. -
A Finitist's Manifesto: Do We Need to Reformulate the Foundations Of
A Finitist’s Manifesto: Do we need to Reformulate the Foundations of Mathematics? Jonathan Lenchner 1 The Problem There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the “existence” of objects that contain an infinite number of elements, (ii) our ability, “in theory”, to compute with an arbitrary level of precision, or (iii) our ability, “in theory”, to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the “continuum” of real numbers, which extends in the plane to the complex numbers and gives us, among other things, “rigorous” definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra – the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over C has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of “existence” and “in theory” in (i) to (iii) above. On the one hand, we know from quantum mechanics that making arbitrarily precise measure- ments of objects is impossible. By the Heisenberg Uncertainty Principle the moment we pin down an object, typically an elementary particle, in space, thereby bringing its velocity, and hence mo- mentum, down to near 0, there is a limit to how precisely we can measure its spatial coordinates. -
Finite Model Theory and Finite Variable Logics
University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science October 1995 Finite Model Theory and Finite Variable Logics Eric Rosen University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/ircs_reports Rosen, Eric, "Finite Model Theory and Finite Variable Logics " (1995). IRCS Technical Reports Series. 139. https://repository.upenn.edu/ircs_reports/139 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-28. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/139 For more information, please contact [email protected]. Finite Model Theory and Finite Variable Logics Abstract In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including first order logic (FO), over this class. A second, related, goal is to determine which results from classical model theory remain true when relativized to the class, F, of finite structures. As it is well known that many such results become false, I also consider certain weakened generalizations of classical results. k k I prove some basic results about the languages L ∃ and L ∞ω∃, the existential fragments of the finite k k k variable logics L and L ∞ω. I show that there are finite models whose L (∃)-theories are not finitely k axiomatizable. I also establish the optimality of a normal form for L ∞ω∃, and separate certain fragments of this logic. I introduce a notion of a "generalized preservation theorem", and establish certain partial k positive results.