Computability Logic გამოთვლადობის ლოგიკა Логика Вычислимости 可计算性逻辑 (Col)
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Infinite Computation, Co-Induction and Computational Logic
Infinite Computation, Co-induction and Computational Logic Gopal Gupta Neda Saeedloei, Brian DeVries, Richard Min, Kyle Marple, Feliks Klu´zniak Department of Computer Science, University of Texas at Dallas, Richardson, TX 75080. Abstract. We give an overview of the coinductive logic programming paradigm. We discuss its applications to modeling ω-automata, model checking, verification, non-monotonic reasoning, developing SAT solvers, etc. We also discuss future research directions. 1 Introduction Coinduction is a technique for reasoning about unfounded sets [12], behavioral properties of programs [2], and proving liveness properties in model checking [16]. Coinduction also provides the foundation for lazy evaluation [9] and type inference [21] in functional programming as well as for interactive computing [33]. Coinduction is the dual of induction. Induction corresponds to well-founded structures that start from a basis which serves as the foundation: e.g., natural numbers are inductively defined via the base element zero and the successor function. Inductive definitions have 3 components: initiality, iteration and mini- mality. For example, the inductive definition of lists of numbers is as follows: (i) [] (empty list) is a list (initiality); (ii) [H|T] is a a list if T is a list and H is some number (iteration); and, (iii) the set of lists is the smallest set satisfying (i) and (ii) (minimality). Minimality implies that infinite-length lists of numbers are not members of the inductively defined set of lists of numbers. Inductive definitions correspond to least fixed point (LFP) interpretations of recursive definitions. Coinduction eliminates the initiality condition and replaces the minimality condition with maximality. Thus, the coinductive definition of a list of numbers is: (i) [H|T] is as a list if T is a list and H is some number (iteration); and, (ii) the set of lists is the largest set of lists satisfying (i) (maximality). -
Resources for Further Study
Resources for Further Study A number of valuable resources are available for further study of philosophical logic. In addition to the books and articles cited in the references at the end of each chapter included in this volume, there are four general categories of resources that can be consulted for information about the history and current research developments in philosophical logic. Additional materials can be found by soliciting advice from logicians, philosophers, and mathematicians at local colleges and universities. Logic Handbooks Many university and independent presses publish books on or related to mathematical and philo- sophical logic. There are also several special series of original monographs in logic that are worth investigating. The literature is too vast to justify a selection of the individual books that have con- tributed to the development of logic. We can nevertheless identify special categories of texts of special interest, beginning with handbooks and book series dedicated to logic and philosophical logic. Here are some recent relevant publications: Boyer, Robert S. (1988) A Computational Logic Handbook. Boston, MA: Academic Press. Sherwood, John C. (1960) Discourse of Reason: A Brief Handbook of Semantics and Logic. New York: Harper & Row. Handbook of Fuzzy Computation, ed. Enrique H. Ruspini, Piero P. Bonissone and Witold Pedrycz. Philadelphia, PA: Institute of Physics Publications, 1998. Handbook of Logic and Language, ed. Johan van Benthem and Alice ter Meulen. Cambridge, MA: MIT Press, 1997. Handbook of Logic in Artificial Intelligence and Logic Programming, ed. Dov Gabbay, C. J. Hogger, and J. A. Robinson. Oxford: Clarendon Press, 1993–8. Emmet, E. R. (1984) Handbook of Logic. -
Giorgi Japaridze
Giorgi Japaridze Curriculum Vitae CONTACT INFORMATION Address: Computing Sciences Department, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA. Email: [email protected] Telephone: (1) 610 519 7332 Fax: (1) 610 519 7889 Home page: http://www.csc.villanova.edu/~japaridz/ EDUCATION 1998 Ph.D. Computer Science, University of Pennsylvania Dissertation: The Logic of Resources and Tasks 1987 Ph.D. Logic, Moscow State University Dissertation: Modal-Logical Means of Studying Provability 1983 M.S. Philosophy, Tbilisi State University (Georgia, former USSR) Thesis: The Notion of Truth in Formalized Languages LANGUAGES: Georgian, Russian, English, Chinese (Mandarin), German. EMPLOYMENT HISTORY 2008 - present Full Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 2010 - 2013 Chair Professor School of Computer Science and Technology, Shandong University, Jinan, China 2004-2008 Associate Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 2007 Visiting Professor School of Information Science and Technology, Institute of Artificial Intelligence, Xiamen University, Fujian, Xiamen, China 1998 -2003 Assistant Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 1995-1998 Research Assistant Dept. of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA 1993-1994 Visiting Associate Professor Philosophy Department, University of Notre Dame, Notre Dame, IN, USA 1992-1993 Postdoctoral Fellow Dept. of Mathematics and Computer Science, University of Amsterdam, Amsterdam, The Netherlands 1987-1992 Senior Researcher Institute of Philosophy, Georgian Academy of Sciences, Tbilisi, Georgia (former USSR) Main Contributions to Science Provability and interpretability logics (1985-1998) While a student, introduced polymodal provability logic GLP, and proved its arithmetical completeness. This contained a solution of an open problem on the logic of provability raised by George Boolos a decade earlier (1985-1988). -
Propositional Computability Logic I∗
Propositional computability logic I∗ Giorgi Japaridze† Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA. Email: [email protected] URL: http://www.csc.villanova.edu/∼ japaridz/ Abstract In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational prob- lems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of “always computable” problems. Computability logic has been introduced semantically, and now among its main technical goals is to axiomatize the set of valid formu- las or various natural fragments of that set. The present contribution signifies a first step towards this goal. It gives a detailed exposition of a soundness and completeness proof for the rather new type of a deductive propositional system CL1, the logical vocabulary of which contains operators for the so called parallel and choice operations, and the atoms of which represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper “Introduction to computability logic” [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, — the context that determines the value of the present results. -
Types in Logic, Mathematics, and Programming from the Handbook of Proof Theory
CHAPTER X Types in Logic, Mathematics and Programming Robert L. Constable Computer Science Department, Cornell University Ithaca, New York 1~853, USA Contents 1. Introduction ..................................... 684 2. Typed logic ..................................... 692 3. Type theory ..................................... 726 4. Typed programming languages ........................... 754 5. Conclusion ...................................... 766 6. Appendix ...................................... 768 References ........................................ 773 HANDBOOK OF PROOF THEORY Edited by S. R. Buss 1998 Elsevier Science B.V. All rights reserved 684 R. Constable 1. Introduction Proof theory and computer science are jointly engaged in a remarkable enter- prise. Together they provide the practical means to formalize vast amounts of mathematical knowledge. They have created the subject of automated reasoning and a digital computer based proof technology; these enable a diverse community of mathematicians, computer scientists, and educators to build a new artifact a globally distributed digital library of formalized mathematics. I think that this artifact signals the emergence of a new branch of mathematics, perhaps to be called Formal Mathematics. The theorems of this mathematics are completely formal and are processed digitally. They can be displayed as beautifully and legibly as journal quality mathematical text. At the heart of this library are completely formal proofs created with computer assistance. Their correctness is based on the axioms and rules of various foundational theories; this formal accounting of correctness supports the highest known standards of rigor and truth. The need to formally relate results in different foundational theories opens a new topic in proof theory and foundations of mathematics. Formal proofs of interesting theorems in current foundational theories are very large rigid objects. Creating them requires the speed and memory capacities of modern computer hardware and the expressiveness of modern software. -
Contribution of Warsaw Logicians to Computational Logic
axioms Article Contribution of Warsaw Logicians to Computational Logic Damian Niwi ´nski Institute of Informatics, University of Warsaw, 02-097 Warsaw, Poland; [email protected]; Tel.: +48-22-554-4460 Academic Editor: Urszula Wybraniec-Skardowska Received: 22 April 2016; Accepted: 31 May 2016; Published: 3 June 2016 Abstract: The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal m-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the m-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. -
Towards a Theory of Proofs of Classical Logic Lutz Straßburger
Towards a Theory of Proofs of Classical Logic Lutz Straßburger To cite this version: Lutz Straßburger. Towards a Theory of Proofs of Classical Logic. Logic in Computer Science [cs.LO]. Université Paris-Diderot - Paris VII, 2011. tel-00772590 HAL Id: tel-00772590 https://tel.archives-ouvertes.fr/tel-00772590 Submitted on 10 Jan 2013 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. Towards a Theory of Proofs of Classical Logic Habilitation a` diriger des recherches Universit´eDenis Diderot – Paris 7 Lutz Straßburger Jury: Richard Blute (rapporteur) Pierre-Louis Curien (rapporteur) Gilles Dowek Martin Hyland (rapporteur) Delia Kesner Christian Retor´e Alex Simpson (rapporteur) Soutenance : 7 janvier 2011 Table of Contents Table of Contents iii 0 Vers une th´eorie des preuves pour la logique classique v 0.1 Cat´egoriesdespreuves . vi 0.2 Notationssyntaxiquepourlespreuves . xv 0.3 Tailledespreuves................................ xx 1 Introduction 1 1.1 CategoriesofProofs .............................. 1 1.2 SyntacticDenotationsforProofs . 3 1.3 SizeofProofs................................... 5 2 On the Algebra of Proofs in Classical Logic 7 2.1 WhatisaBooleanCategory?. 7 2.2 Star-AutonomousCategories . 9 2.3 Someremarksonmix .............................. 12 2.4 ∨-Monoids and ∧-comonoids........................... 16 2.5 Orderenrichment................................ -
Baltic International Yearbook of Cognition, Logic and Communication
Baltic International Yearbook of Cognition, Logic and Communication Volume 8 Games, Game Theory and Game Semantics Article 5 2013 Ptarithmetic Giorgi Japaridze School of Computer Science and Technology, Shandong University; Department of Computing Sciences, Villanova University Follow this and additional works at: https://newprairiepress.org/biyclc Part of the Theory and Algorithms Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Japaridze, Giorgi (2013) "Ptarithmetic," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 8. https://doi.org/10.4148/1944-3676.1074 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. Ptarithmetic 2 The Baltic International Yearbook of and developed (Japaridze 2011, unpublished, forth.). In retrospect, Cognition, Logic and Communication however, the author finds that “Ptarithmetic” contained a number of potentially useful ideas that have not been subsequently adopted by November 2013 Volume 8: Games, Game Theory the “clarithmetics” line of research (at least not yet), and that, for this and Game Semantics reason, it would be a pity to let this material remain unpublished. pages 1-186 DOI: 10.4148/1944-3676.1074 Probably the most important of such ideas is the “Polynomial Time Induction” (PTI) rule of Ptarithmetic, with no close or distant relatives elsewhere in the literature. -
Computational Logic: Memories of the Past and Challenges for the Future
Computational Logic: Memories of the Past and Challenges for the Future John Alan Robinson Highland Institute, 96 Highland Avenue, Greenfield, MA 01301, USA. Abstract. The development of computational logic since the introduction of Frege's modern logic in 1879 is presented in some detail. The rapid growth of the field and its proliferation into a wide variety of subfields is noted and is attributed to a proliferation of subject matter rather than to a proliferation of logic itself. Logic is stable and universal, and is identified with classical first order logic. Other logics are here considered to be first order theories, syntactically sugared in notationally convenient forms. From this point of view higher order logic is essentially first order set theory. The paper ends by presenting several challenging problems which the computational logic community now faces and whose solution will shape the future of the field. 1 Introduction Although logic and computing are each very old subjects, going back very much more than a mere hundred years, it is only during the past century that they have merged and become essentially one subject. As the logican Quine wrote in 1960, "The basic notions of proof theory converge with those of machine computing. ... The utterly pure theory of mathematical proof and the utterly technological theory of machine computation are thus at bottom one, and the basic insights of each are henceforth insights of the other" ([1], p. 41). The aptly-named subject of computational logic - no pun intended - subsumes a wide variety of interests and research activities. All have something to do with both logic and computation. -
Games and Logic
Baltic International Yearbook of Cognition, Logic and Communication Volume 8 Games, Game Theory and Game Semantics Article 8 2013 Games and Logic Gabriel Sandu University of Helsinki Follow this and additional works at: https://newprairiepress.org/biyclc Part of the Logic and Foundations Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Sandu, Gabriel (2013) "Games and Logic," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 8. https://doi.org/10.4148/1944-3676.1072 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. Games and Logic 2 The Baltic International Yearbook of x”. Their scopal dependencies and independencies in a first-order sen- Cognition, Logic and Communication tence are recast in terms of the strategic interaction of two players in a game of perfect information. For instance, the first-order sentence November 2013 Volume 8: Games, Game Theory ∀x∃yB(x, y) is analyzed in terms of a game between the universal and Game Semantics player (Abelard) and the existential player (Eloise). They choose indi- pages 1-30 DOI: 10.4148/1944-3676.1072 viduals from the underlying universe of discourse to be the values of the variables x and y, respectively. The order of the choices and the information sets of the players are indicated by the syntax of the sen- GABRIEL SANDU tence. -
ON the SYSTEM CL12 of COMPUTABILITY LOGIC Contents
Logical Methods in Computer Science Vol. 11(3:1)2015, pp. 1–71 Submitted Jan. 13, 2013 www.lmcs-online.org Published Jul. 28, 2015 ON THE SYSTEM CL12 OF COMPUTABILITY LOGIC GIORGI JAPARIDZE Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA URL: http://www.csc.villanova.edu/∼japaridz/ e-mail address: [email protected] Abstract. Computability logic (CoL) is a long-term project for redeveloping logic on the basis of a constructive game semantics, with games seen as abstract models of interactive computational problems. Among the fragments of CoL successfully axiomatized so far is CL12 — a conservative extension of classical first-order logic, whose language augments that of classical logic with the so called choice (“constructive”) sorts of quantifiers and connectives. This system has already found fruitful applications as a logical basis for constructive and complexity-bound versions of Peano arithmetic, such as arithmetics for polynomial time computability, polynomial space computability, and beyond. The present paper introduces a third, indispensable complexity measure for interactive computations termed amplitude complexity, and establishes the adequacy (soundness/completeness) of CL12 and the associated Logical Consequence mechanism with respect to (simultaneously) A amplitude, S space and T time computability under certain minimal conditions on the triples (A,S,T ) of function classes. This result very substantially broadens the potential application areas of CL12, even when time and/or space complexity is the only concern. It would be sufficient to point out that, for instance, now CL12 can be reliably used as a logical basis of systems for logarithmic space or exponential time computabilities — something that the earlier-known crude adequacy results for CL12 were too weak to allow us to do. -
Finding Unity in Computational Logic
Finding Unity in Computational Logic Dale Miller INRIA Saclay and LIX/Ecole´ polytechnique Route de Saclay, 91128 PALAISEAU Cedex, France [email protected] While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in many, diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around recent developments in the theory of sequent calculus proofs. We outline how new tools and methodologies can be developed around a boarder approach to computational logic. Computational logic, unity of logic, proof theory 1. SOFTWARE AND HARDWARE CORRECTNESS Our ability to provide at least some formal guarantees IS CRITICALLY IMPORTANT about software systems will be directly related to our ability to deploy new functionality and services. The de- Computer systems are everywhere in our societies manding requirements in safety critical systems matched and their integration with all parts of our lives is with our inability to formally establish their correct behav- constantly increasing. There are a host of computer ior means that developments and deployments of, say, systems—such as those in cars, airplanes, missiles, avionic systems or high-speed train switches, moves at hospital equipment—where correctness of software is a glacial pace and clings to old technological solutions paramount. In the area of consumer electronics, big that have no history of failing. If we cannot distinguish changes in the attitude towards correctness has taken applets from viruses, we cannot expect people to really place.