DOCSLIB.ORG

9789400707139.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
9789400707139.Pdf Games, Norms and Reasons SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Editors-in-Chief: VINCENT F. HENDRICKS, University of Copenhagen, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A. Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A. Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Krako´w, Poland VOLUME 353 For further volumes: http://www.springer.com/series/6607 Games, Norms and Reasons Logic at the Crossroads Edited by Johan van Benthem ILLC, University of Amsterdam, The Netherlands and Stanford University, USA Amitabha Gupta Indian Institute of Technology Bombay, India and Eric Pacuit Tilburg University, Tilburg Institute for Logic and Philosophy of Science, The Netherlands 123 Editors Prof. Johan van Benthem Prof. Amitabha Gupta University of Amsterdam Adi Shankaracharya Marg Institute for Logic 503 Whispering Woods Language and Computation (ILLC) Powai Vihar, Bldg. 3 Science Park, P.O. Box 94242 700076 Powai, Mumbai 1090 GE Amsterdam India The Netherlands [email protected] [email protected] Asst. Prof. Eric Pacuit Tilburg University Tilburg Institute for Logic and Philosophy of Science Warandelaan 2 5037 AB Tilburg The Netherlands [email protected] ISBN 978-94-007-0713-9 e-ISBN 978-94-007-0714-6 Set ISBN 978-94-007-0920-1 DOI 10.1007/978-94-007-0714-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011923658 c Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This book continues a series called “Logic at the Crossroads” of which the first vol- ume appeared in 2007.1 The series title reflects a view of our discipline as a vantage point for seeing new developments across academic fields, as well as a catalyst in making them happen. In particular, the editors feel that the deep insights from the classical phase of mathematical logic can form a harmonious mixture with a new, more ambitious research agenda of understanding and enhancing human reasoning and intelligent interaction in their full extent, whence our title “Games, Norms and Reasons”. This broad view of logic animated the Second Indian Conference on “Logic and its Relationship with other Disciplines” held at IIT Bombay, Mumbai 2007, the origin of most of the papers in this volume, while a few additional ones have been solicited by the editors to round out coverage. But the Mumbai conference also added a very concrete focus, by honouring one person who embodies the general trend described, namely Professor Rohit Jivanlal Parikh. Rohit stands at the cross- roads of many academic disciplines, and of two major human cultures, and why and how this is so, is admirably explained in the Congratulatory Citation that was read at a special event at IIT Bombay celebrating Rohit’s 70th birthday. This document has been reprinted here, after the following brief editorial description of contents. The papers in this volume do two things. First, they paint a lively portrait of modern logic as it is expanding today with many new themes and concerns. But at the same time, they include a gallery of friends and colleagues of Rohit’s who eagerly responded to our call for honouring him. The chapters to follow highlight major dimensions of his work and influence, in particular, his “Social Software” program for studying the procedural behaviour of interactive human agents with beliefs and preferences. Horacio Arlo´ Costa and Arthur Paul Pedersen start off with a key interest of Parikh’s, viz. bridging between realistic models of human behaviour and logic based 1 Logic at the Crossroads: An Interdisciplinary View, Amitabha Gupta, Rohit Parikh and Johan van Benthem, editors, Allied Publishers, New Delhi. Recently reissued by Springer Publishers under the new title Proof, Computation and Agency: Logic at the Crossroads. v vi Preface ones. They analyze Gigerenzer’s “Take the Best” algorithm for successful decisions by bounded-rational agents, and show how it can be understood and taken further using formal models of choice and preference. Sergei Artemov continues with an account of “justification logics”, which merge ideas from proof theory and prov- ability logic with epistemic logic to arrive at rich and more realistic accounts of evidence in human reasoning that link up with formal epistemology. Next, Akeel Bilgrami discusses a key aspect of what holds communication together, viz. mean- ingful discourse, and argues that linguistic meanings are not normative in any deep sense – while still fully acknowledging Wittgenstein’s insights about the norma- tivity of human intentions. Melvin Fitting follows up on his long-standing formal studies of logics of proofs, and gives a new proof of a “realization theorem” link- ing the epistemic logic S5 to its justification counterpart. Next, Sujata Ghosh and Fernando Velazquez-Quesada´ discuss current models of belief and belief change based on networks, and relate these to the more qualitative ones found in epistemic and doxastic logic, including an account of successive dynamic steps that change a current belief network. Patrick Girard analyzes another crucial dynamic social scenario, viz. merge of individual preferences into a group-based one over a set of outcomes. He shows how preference merging in structured groups receives a per- spicuous semantic and axiomatic formulation using the tools of modal and dynamic preference logic. Amy Greenwald, Amir Jafari and Casey Marks develop new links between the fields of game theory and computational learning theory, both crucial ingredients of intelligent interaction – based on a new notion of “no-Φ-regret”. Vin- cent Hendricks returns to Parikh’s interests in the fit between formal models and our natural ways of understanding knowledge and information. He gives an insight- ful discussion of the intuitive distinction between “first person” and “third person” perspectives, linking this to a wide range of issues from philosophy to economics. Dexter Kozen and Ganesh Ramanarayanan present another innovative social use of classical ideas from proof theory, developing a universal Horn equational logic with dynamic operations of “publish”, “cite” and “forget” which describe the ac- tual construction and maintenance of mathematical knowledge. Lawrence Moss re- turns to the use of natural language in human interaction, and shows how reasoning in natural language has interesting syllogistic subsystems allowing for negations that form a hitherto hardly explored level in between the classical Syllogistic and full-fledged first-order predicate logic. Jeff Paris and Alena Vencovska´ consider an- other major aspect of our natural reasoning, namely, its use of probabilistic features, whether quantitative, or qualitative with tags like “probably”. Linking up with De Finetti’s foundations of probability, they develop a sophisticated new version of Carnap’s inductive logic for monadic predicates which handles the right inference relations between agents’ degrees of belief. They also generalize this approach to binary and higher-order predicates, and develop its mathematical theory further, us- ing a new principle of “Spectrum Exchangeability”. Finally, R. Ramanujam and S.P. Suresh look at the central role of knowledge in reasoning about security pro- tocols. The semantics of conventional epistemic logics throws up challenges, since cryptographic primitives in communication lead to problems of unboundedness and Preface vii undecidability. The authors propose an alternative epistemic logic which is expres- sive enough while its protocol verification procedure is decidable. Taken together, these papers show how logic can extend its scope without any sacrifice in rigour or focus. Moreover, their quality, and their cast of authors, speaks for itself about Rohit Parikh’s intellectual breadth and influence. It has been a plea- sure to arrange this tribute, and beyond logistics, the editors have learned a lot in the process. We end with a citation which was read during the 2007 Mumbai conference. The present volume has taken some time to appear, due to circumstances beyond our control, but the editors feel that the sentiments expressed then are as fresh and appropriate as ever. Stanford Johan van Benthem Bombay Amitabha Gupta Tilburg Eric Pacuit Dedication On the occasion of his seventieth birthday, we, the participants and organizers of the “Second Indian Conference on Logic and its Relationship with Other Disciplines”, acclaim and honour Professor Rohit Jivanlal Parikh and celebrate his many extraor- dinary achievements and outstanding contributions in his pursuit of knowledge. Born on November 20, 1936, he was educated at Mumbai, and then at Harvard University, where he obtained degrees with highest honors in Physics (A.B.) and Mathematics (Ph.D.). Parikh’s romance with Logic began with a course by Quine. Later on he was
Load more

Recommended publications
  • Bounded Arithmetic and Propositional Proof Complexity
    Bounded Arithmetic and Propositional Proof Complexity Samuel R. Buss¤ Departments of Mathematics and Computer Science University of California, San Deigo La Jolla, CA 92093-0112 Abstract This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof i i complexity. We introduce the theories S2 and T2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then de¯ne the Razborov- Rudich notion of natural proofs of P =6 NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various \feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e®ectively computable in the sense of being recursively computable. ¤Supported in part by NSF grants DMS-9503247 and DMS-9205181.
    [Show full text]
  • Bounded Arithmetic Vs. Propositional Proof Systems Vs. Complexity
    Bounded Arithmetic vs. Propositional Proof Systems vs. Complexity Classes (depth oral survey) $Id: alanodepth.tex,v 1.18 2002/04/01 04:54:38 alan Exp $ LATEX'd on January 3, 2005 Alan Skelley January 3, 2005 1 Introduction We often argue that a particular mathematical concept is important if it is natural, which means that it surfaces in many places with different origins and definitions, and robust, such that a variety of disparate formulations of it end up being equivalent or at least closely related. Likewise, the applicability, maturity, and importance of a body of results are greater when that field is found to have a strong connection to another. Three areas of study intricately connected in such a useful way are computational complexity, the proof theory of arithmetic and propositional proof complexity. Computational complexity is the study of computation and the resources required to perform it. A staggering number of different kinds of computation all fall into the domain of this field. It has practical aspects, directly impacting how real computations are done by real computers, and yet seemingly fundamental, easily explained problems remain unsolved despite a good deal of effort. A particularly glaring example is the famous P vs NP problem, which asks if those two classes of problems are equal. Starting from the NP-completeness results of Cook [13] the pressure mounted with no relief, leading even to detailed, formal analysis of known proof techniques and why they are all ineffectual at tackling such problems [30]. Many complexity classes are studied and conjectures about separations and hierarchies abound, yet results are elusive.
    [Show full text]
  • 18 May 2017 Incompleteness in the Finite Domain
    Incompleteness in the finite domain Pavel Pudl´ak ∗ May 22, 2017 Abstract Motivated by the problem of finding finite versions of classical incompleteness the- orems, we present some conjectures that go beyond NP 6= coNP. These conjectures formally connect computational complexity with the difficulty of proving some sen- tences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. In this paper we review some conjectures that we have presented earlier [26, 32, 33, 34], introduce new conjectures, systematize them and prove new connections between them and some other statements studied before. 1 Introduction G¨odel’s incompleteness theorem is undoubtedly one of the most important theorems in logic. It speaks about absolute provability, i.e., about proofs without any restriction on their length. The question whether there is a “finite” or “feasible” version of the incompleteness theorem, where the complexity of proofs is bounded, has certainly intrigued many people, but very little has been published about it. With the advent of computers and theories developed for them, in particular complexity theory, the question about a finite version of the incomplete- arXiv:1601.01487v2 [math.LO] 18 May 2017 ness theorem became even more interesting. The concept of polynomial time computations turned out to be the most important concept in complexity theory.
    [Show full text]
  • A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
    This is a repository copy of A Tight Karp-Lipton Collapse Result in Bounded Arithmetic. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74793/ Proceedings Paper: Beyersdorff, O and Mueller, S (2008) A Tight Karp-Lipton Collapse Result in Bounded Arithmetic. In: Kaminiski, M and Martini, S, (eds.) Proceedings of Computer Science Logic. 17th Annual Conference of the EACSL, 16-19 September 2008, Bertinoro, Italy. Lecture Notes in Computer Science, 5213 . Springer Verlag , 199 - 214 . ISBN 978-3-540-87530-7 https://doi.org/10.1007/978-3-540-87531-4_16 Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A Tight Karp-Lipton Collapse Result in Bounded Arithmetic Olaf Beyersdorff1 and Sebastian M¨uller2⋆ 1 Institut f¨urTheoretische Informatik, Leibniz Universit¨atHannover, Germany [email protected] 2 Institut f¨urInformatik, Humboldt-Universit¨atzu Berlin, Germany [email protected] Abstract. Cook and Kraj´ıˇcek[9] have obtained the following Karp- Lipton result in bounded arithmetic: if the theory PV proves NP ⊆ P/poly, then PH collapses to BH, and this collapse is provable in PV . Here we show the converse implication, thus answering an open ques- tion from [9]. We obtain this result by formalizing in PV a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Kraj´ıˇcek[9].
    [Show full text]
  • A Study of Width Bounded Arithmetic Circuits and the Complexity of Matroid Isomorphism
    A STUDY OF WIDTH BOUNDED ARITHMETIC CIRCUITS AND THE COMPLEXITY OF MATROID ISOMORPHISM by Raghavendra Rao B. V. The Institute of Mathematical Sciences, Chennai. A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulfillment of the requirements For the Degree of DOCTOR OF PHILOSOPHY of HOMI BHABHA NATIONAL INSTITUTE September 2009 Homi Bhabha National Institute Recommendations of the Viva Voce Board As members of the Viva Voce Board, we recommend that the dissertation prepared by Raghavendra Rao B. V. entitled “A study of width bounded arithmetic cir- cuits and the complexity of matroid isomorphism” may be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date : Chairman : V. Arvind (IMSc) Date : Convener : Meena Mahajan (IMSc) Date : Member : Venkatesh Raman (IMSc) Date : Member : K. V. Subrahmanyam (CMI) Date : Member : Sundar Vishwanathan (IIT-B) Final approval and acceptance of this dissertation is contingent upon the can- didate’s submission of the final copies of the dissertation to HBNI. I hereby certify that I have read this dissertation prepared under my direction and recommend that it may be accepted as fulfilling the dissertation requirement. Date : Guide : Meena Mahajan DECLARATION I hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and the work has not been submitted earlier as a whole or in part for a degree/diploma at this or any other Institution or University. Raghavendra Rao B. V. ACKNOWLEDGEMENTS First of all I thank my supervisor Meena Mahajan for her kind support and for providing all the motiviation, inspiration and knowledge which led to this thesis.
    [Show full text]
  • February 2021 I Am Interested in Mathematical Logic and Complexity
    RESEARCH STATEMENT JÁN PICH February 2021 I am interested in mathematical logic and complexity theory. In particular, in proof complexity of circuit lower bounds. Background. Proving lower bounds on size of circuits computing explicit Boolean functions is one of the most fundamental problems in the theory of computation, underlying questions such as the P versus NP problem. Circuit lower bounds have been, however, notoriously difficult to prove. A prominent attempt to understand their complexity was made with the discovery of the natural proofs barrier of Razborov and Rudich [28], which ruled out many potential proof methods. The possibility that circuit lower bounds could present even the limits of logical reasoning was investigated by Razborov [26] who used natural proofs to derive an unprovability of circuit lower bounds in theories of bounded arithmetic. Such unprovability results correspond to lower bounds on lengths of proofs in propositional proof systems which in turn represent the Cook-Reckhow program towards the separation of NP and coNP. Unfortunately, Razborov's unprovability result as well as other existing proof complexity lower bounds work only for very weak theories resp. proof systems. Nevertheless, Razborov's approach formed a part of the motivation for the development of a theory of proof complexity generators [1, 12, 27] which was introduced in a hope to obtain lower bounds against strong proof systems like Frege - standard textbook systems for propositional logic. Razborov's conjecture. Razborov [27] conjectured that any suitable Nisan-Wigderson (NW) generator forms a good proof complexity generator in the sense that tautologies stating the existence of elements outside its range require superpolynomial-size proofs in Frege systems assuming P=poly is hard on average for NC1.
    [Show full text]
  • Some Open Problems in Bounded Arithmetic and Propositional Proof Complexity (Research Proposal Paper)
    Some open problems in bounded arithmetic and propositional proof complexity (research proposal paper) $Id: alanorp.tex,v 1.5 2002/12/10 04:57:36 alan Exp $ LATEX'd on January 3, 2005 Alan Skelley January 3, 2005 1 Introduction The broad relevance and importance of bounded arithmetic and propositional proof com- plexity are well appreciated; these subjects are two of three which are interconnected in various and interesting ways, the third subject being computational complexity theory. This latter area is rife with old, well-established open problems and one good way to get at them is by studying the other two, whose problems are different yet connected, and many of which may be tractable. The purpose of this paper is to discuss some possible avenues of research related to bounded arithmetic and propositional proof complexity. Some of these are specific technical open questions related closely to recent research and are placed in their context. Others are more vague general directions along with some discussion of what fruit may be borne by efforts so directed. Yet others are larger issues which are not expected to be resolved but which should be kept in mind and toward which some incremental progress may be possible. Some necessary definitions and background will be presented first although this will mostly be kept to a minimum. 2 Definitions and Background 2.1 Bounded Arithmetic and Complexity The study of bounded arithmetic was initiated in 1971 by Parikh with his system I∆0, similar to Peano Arithmetic, but with the important restriction of the induction scheme to ∆0 formulas: those whose quantifiers are bounded, i.e.
    [Show full text]
  • NP Search Problems in Low Fragments of Bounded Arithmetic
    NP search problems in low fragments of bounded arithmetic Jan Kraj´ıˇcek1,2 ∗ Alan Skelley1† Neil Thapen1‡ 1Mathematical Institute Academy of Sciences, Prague and 2Faculty of Mathematics and Physics Charles University, Prague September 25, 2006 Abstract We give combinatorial and computational characterizations of the 2 NP search problems definable in the bounded arithmetic theories T2 3 and T2 . By a search problem in full generality we mean simply a binary relation R(x, y) such that ∀x∃yR(x, y) holds. The search task is: given an instance x find a solution y such that R(x, y) holds. The most important class of search problems are NP search problems. Here the relation R(x, y) is decidable in polynomial time and the length of ∗The paper was written while at the Isaac Newton Institute in Cambridge (Logic and Algorithms program), supported by EPSRC grant N09176. Also supported in part by grants A1019401, AV0Z10190503, MSM0021620839, 201/05/0124, and LC505. †Partially supported by grant LC505 (Eduard Cechˇ Center) and NSERC PDF-313650- 2005. ‡Partially supported by grants A 101 94 01 and AV0Z10190503 (AS CR). 1 y is bounded by a polynomial in the length of x. Problems in NP∩coNP, the PLS problems (Polynomial Local Search) of Johnson, Papadimitriou and Yannakakis [11] and various combinatorially defined problems as in Papadim- itriou [20, 22] and Papadimitriou and Yannakakis [21] are all examples of NP search problems. Any NP search problem can be defined by an open formula φ(x, y) in the language of the bounded arithmetic theory PV of Cook [6].
    [Show full text]
  • Foundations of Proof Complexity: Bounded Arithmetic and Propositional Translations
    Foundations of Proof Complexity: Bounded Arithmetic and Propositional Translations Stephen Cook and Phuong Nguyen c Copyright 2004, 2005, 2006 October 9, 2006 DRAFT i Preface (Preliminary Version) This book studies logical systems which use restricted reasoning based on concepts from computational complexity. The underlying motivation is to de- termine the complexity of the concepts needed to prove mathematical theo- rems. The complexity classes of interest lie mainly between the basic class AC0(characterized by polynomial-size families of bounded-depth circuits), and the polynomial hierarchy PH, and includes the sequence AC0 TC0 NC1 P PH (1) ⊂ ⊆ ⊆ ⊆ We associate with each of these classes a logical theory and a propositional proof system, where the proof system can be considered a nonuniform version of the universal (or sometimes the bounded) fragment of the theory. The functions definable in the logical theory are those associated with the complexity class, and (in some cases) the lines in a polynomial size proof in the propositional system express concepts in the complexity class. This three-way association for the above classes is depicted as follows: class AC0 TC0 NC1 P PH theory V0 VTC0 VNC1 TV0 V (2) 0 0 system AC -Frege TC -Frege Frege eFrege Gi h i Consider, for example, the class NC1. The uniform version is ALogTime, the class of problems solvable by an alternating Turing machine in time O(log n). The definable functions in the associated theory VNC1 are the NC1 functions, i.e., those functions whose bit graphs are NC1 relations. A problem in nonuni- form NC1 is defined by a polynomial-size family of log-depth Boolean circuits, or equivalently a polynomial-size family of propositional formulas.
    [Show full text]
  • On the Correspondence Between Arithmetic Theories and Propositional Proof Systems - a Survey
    This is a repository copy of On the correspondence between arithmetic theories and propositional proof systems - a survey. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74440/ Article: Beyersdorff, O (2009) On the correspondence between arithmetic theories and propositional proof systems - a survey. Mathamatical Logic Quarterly, 55 (2). 116 - 137 . ISSN 0942-5616 https://doi.org/10.1002/malq.200710069 Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ ON THE CORRESPONDENCE BETWEEN ARITHMETIC THEORIES AND PROPOSITIONAL PROOF SYSTEMS OLAF BEYERSDORFF Abstract. Bounded arithmetic is closely related to propositional proof systems, and this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Kraj´ıˇcek and Pudl´ak [41]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties. Using logical methods has a rich tradition in complexity theory. In particular, there are very close relations between computational complexity, propositional proof complexity, and bounded arithmetic, and the central tasks in these areas, i.e., separating complexity classes, proving lower bounds to the length of proposi- tional proofs, and separating arithmetic theories, can be understood as different approaches towards the same problem.
    [Show full text]