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# Topic: Computability logic

 Computability logic Computability logic is a formal theory of computability, introduced by Giorgi Japaridze in 2003. Computability logic is a conservative extension of classical logic and, being more expressive, constructive and computationally meaningful, it has a wide range of application areas. Computability logic is an attempt to lay foundations for a comprehensive and systematic theory of interactive computation. www.arikah.com /encyclopedia/Computability_logic   (334 words)

 Computability logic - Wikipedia, the free encyclopedia Introduced by Giorgi Japaridze in 2003, Computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. Being a conservative extension of the former, computability logic is, at the same time, by an order of magnitude more expressive, constructive and computationally meaningful. Besides classical logic, linear logic (understood in a relaxed sense) and intuitionistic logic also turn out to be natural fragments of computability logic. en.wikipedia.org /wiki/Computability_logic   (315 words)

 Computability logic Computability logic is a formal theory of computability, introduced by Japaridze in 2003. While the central semantic concept in classical logic is (Tarski an) truth, computability logic is based on a concept of computability. Besides classical logic, intuitionistic and linear logic s (understood in a relaxed sense) also turn out to be natural fragments of computability logic. www.nebulasearch.com /encyclopedia/article/Computability_logic.html   (496 words)

 COMPUTABILITY LOGIC: a theory of interactive computation HOMEPAGE Computability logic is a formal theory of (interactive) computability in the same sense as classical logic is a formal theory of truth. Computability logic has been introduced semantically, and now among its main technical goals is to axiomatize the set of valid formulas or various natural fragments of that set. Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. www.cis.upenn.edu /~giorgi/cl.html   (3951 words)

 Computability theory - Wikipedia, the free encyclopedia Computability theory is the part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. A computable number is a real number which can be approximated to any arbitrary degree of accuracy by an algorithm. Electronic computers, and even quantum computers, are exactly equivalent to Turing machines, if they have access to an unbounded supply of memory. en.wikipedia.org /wiki/Computability_theory   (750 words)

 Game Semantics or Linear Logic? Logical operators are understood as operations on such tasks/resources/games, atoms as variables ranging over tasks/resources/games, and validity of a logical formula as existence of a machine that always (under every particular intretpretation of atoms and against any possible behavior by the environment) successfully accomplishes/provides/wins the task/resource/game represented by the formula. With this semantics, computability logic is a formal theory of computability in the same sense as classical logic is a formal theory of truth. Computability logic starts with a mathematically strict and intuitively convincing semantics, and only after that, as a natural second step, asks the question about what the corresponding logic and its axiomatizations (syntax) are. www.csc.villanova.edu /~japaridz/CL/gsoll.html   (641 words)

 Computability logic - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-11-05) Computability, Complexity, and Languages : Fundamentals of Theoretical Computer Science (Computer Science and Scientific Computing) Computability: Computable Functions, Logic, and the Foundations of Mathematics, with Computability: A Timeline Computability in Analysis and Physics (Perspectives in Mathematical Logic) encyclopedia.worldsearch.com /computability_logic.htm   (372 words)

 Workshop on Computability and Logic, Abstracts   (Site not responding. Last check: 2007-11-05) A logical learning paradigm is defined as a class W of structures over some vocabulary, and a set E of first-order formulas that represent data. This notion is a basic notion for the lattice structure of the computably enumerable sets and was further investigated in several papers under different points of view, e.g., their T-degrees (by Jockusch and Lerman), isomorphisms between them (by Maass and Stob), index sets (by Lempp), e-dominance (by Robinson) and generalizations of major subset (by Herrmann). Abstract: Fuzzy logic (in the narrow sense) is a logic with comparative notion of truth; main systems of (truth-functional) fuzzy logic are based on the notion of a continues t-norm as the truth function of conjunction and its residuum as the truth function of implication; this is the standard semantics. math.uni-heidelberg.de /logic/colo2003/abstracts.html   (4316 words)

 Computer Science Colloquium - April 29 Computability Logic (CL) is a logic of computational (and/or informational) resources and tasks, with these entities understood in their most general - interactive - sense. CL is a formal theory of computability in the same sense as classical logic is a formal theory of truth. With logical operators representing certain most basic and natural operations on games, computability logic provides a powerful formal tool for studying interactive computational problems in a systematic way. web.gc.cuny.edu /ComputerScience/cs_cllqm/talks/2004_05_06.html   (245 words)

 Logic and computability (from logic, philosophy of) --  Encyclopædia Britannica These findings of Gödel and Montague are closely related to the general study of computability, which is usually known as recursive function theory (see mathematics, foundations of: The crisis in foundations following 1900: Logicism, formalism, and the metamathematical method) and which is one of the most important branches of contemporary logic. fundamental principles of logic: (1) law of contradiction—something cannot exist and not exist at the same time; (2) law of excluded middle—something either exists or it does not, no middle condition is possible; (3) law of identity—something is always identical with itself; 20th-century philosophers have criticized, even rejected, the laws, which derive from ancient... Discusses the Aristotelean logic, predicate calculus, geometry of Euclid, formal theories of mathematics, and Plato and Aristotle’s philosophy of mathematics. www.britannica.com /eb/article-36296   (895 words)

 Computability Theory Post in the 1930's, and includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, subrecursive hierarchy classifications, computable structures, and complexity theory relating to the preceding. There is also a Computability and Complexity in Analysis network, and one for Foundations of Mathematics, some of whose discussions relate to computability theory. Logic Colloquium '97 (the 1997 European Meeting of the Association of Symbolic Logic) was held in Leeds, and a substantial part of the programme was devoted to computability theory and related topics. www.amsta.leeds.ac.uk /pure/staff/cooper/research.html   (553 words)

 Science: Mathematics: Logic: Computability Logic - Open Site It would be accurate to say that computability logic, introduced by Giorgi Japaridze in 2003, is a formal theory of computability in the same sense as classical logic is a formal theory of truth. Remarkably, the classical concept of truth turns out to be a special case of computability (as it is understood in computability logic), and hence classical logic a special fragment of computability logic. Computability logic is a serious attempt to lay foundations for a comprehensive and systematic theory of interactive computation. open-site.org /Science/Mathematics/Logic/Computability_Logic   (264 words)

 Robot Soul: Aw, where was I...oh yes...Computability Logic Computability Logic is a logic that's based on semantics. Semantics-based logics are the bread-and-butter logics for writers, of course, but I've always assumed that no card-carrying mathematician would take such "writerly" logic as serious. Having a formal logic at my disposal that is based on semantics is likely to help me bridge the gap between the knowledge that is encoded in my application right now and the theoretical knowledge that is currently available on this application. twomorrow.twoday.net /stories/966626   (423 words)

 Prof. David Harel - Books In Dynamic Logic, such programs are first-class objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. The treatment of some topics in Part I is therefore necessarily abbreviated, and a passing familiarity with the basic concepts of mathematical logic, computability, formal languages and automata, and program verification is a desirable prerequisite. Apart from the obvious heavy reliance on classical logic, computability theory and programming, the subject has its roots in the work of Thiele [198] and Engeler [42] in the late 1960's, who were the first to advance the idea of formulating and investigating formal systems dealing with properties of programs in an abstract setting. www.wisdom.weizmann.ac.il /~dharel/dynamic_logic.html#preface   (848 words)

 Computability Complexity Logic Book To the towering achievement of the mathematics of the last one hundred years belongs the formulation of a precise, embracing concept of formal language and a general concept of algorithm. The efficiency of a programming language and the degree of correctness of the programs (describing algorithmic processes) formulated in it then depends essentially on the quality of the specification language as a vehicle of description and on the reliability of the methods by which programs are constructed from the specifications. The theory of algorithms (also known as computability theory) in its modern form is, above all, the theory of the extent and complexity of classes of algorithms and the automata and machines that realise them. www.di.unipi.it /~boerger/cclbookintro.html   (590 words)

 Villanova CSC Technical Reports   (Site not responding. Last check: 2007-11-05) 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. Its formulas represent interactive computational problems, logical operators stand for operations on computational problems, and validity of a formula is understood as being a scheme of problems that always have algorithmic solutions. A need in substantially new deductive tools came with the birth of computability logic (see this http URL) - the semantically constructed formal theory of computational resources, which has stubbornly resisted any axiomatization attempts within the framework of traditional syntactic approaches. hawk.csc.villanova.edu /~TechRep   (1439 words)

 Introduction to computability logic | Lambda the Ultimate One of the distinctions is that environment in former is not limited to computable behavior, it may be "a capricious user or the blind forces of nature". Computability logic is not it, I think, but I'm interested to see how it is different from linear logic. As a sequent calculus, I think CL is somewhat similar to Guglielmi's calculus of structures because of deep inference (and yes, that was mentioned in the OP, but I made the whole circle and found it out the hard way). lambda-the-ultimate.org /node/view/204   (2614 words)

 Computability Science, Directory   (Site not responding. Last check: 2007-11-05) ECCC - Electronic Colloquium on Computational Complexity The Electronic Colloquium on Computational Complexity is a new forum for the rapid and widespread interchange of ideas, techniques, and research in computational complexity. Computability Theory Directory of researchers working in computability theory, and list of open problems. Computability Logic - Wictionary A dictionary definition of the subject. www.wacofdn.org /d2RjXzI2OTIw.aspx   (175 words)

 Logic and Computability, University of Oslo   (Site not responding. Last check: 2007-11-05) The Oslo Logic Group consists of scholars working in logic at four different departments at the University of Oslo and at the Oslo University College. In the recent years members of the Oslo Logic Group have published inside the following areas: computability theory, complexity theory, theoretical computer science, automated theorem proving, philosophical logic, computational linguistics (the list is not exhaustive). HUMIT1750 - Logikk og beregninger (Logic and Computation), 10 ECTS, Department of Linguistic. www.math.uio.no /avda/logikk   (279 words)

 Amazon.com: Books: Computability and Logic   (Site not responding. Last check: 2007-11-05) Like other good logic texts-Jeffrey's Formal Logic or Pollock's Technical Method's (out of print, but available in PDF on his website)-there is very little commentary in the brief chapters, so it is useful if you are already very familiar with the material or if you have a very worthy guide. This was meant as an intermediate logic text for philosophy and math students, but it would try the patience of a math major. People who are veterans with logic and logicians may easily spot typos, but for a first time student of the subject, I was confused as hell at some parts simply because there was a typo. www.amazon.com /exec/obidos/tg/detail/-/0521007585?v=glance   (1916 words)

 COMPUTABILITY LOGIC: a theory of interactive computation HOMEPAGE   (Site not responding. Last check: 2007-11-05) And one of the main -- again, so far rather abstract -- ambitions of linear logic is that it is a logic of resources. Constructive: any proof of a formula F in a CL-based theory effectively encodes a solution to the computational problem represented by F. Papers on CL: Currently CL is at its very first stages of developmnent, with open problems and unverified conjectures prevailing over answered questions. It is expected to constitute part 1 of a two-piece series on the intuitionistic fragment of CL, with part 2 containing an anticipated completeness proof. www.luvfeed.org /cache/3176   (3414 words)

 Harvey Mudd College An introduction to some of the mathematical foundations of computer science, particularly logic, automata, and computability theory. It is assumed that you’ve seen truth-tabular forms of logic and are already familiar with propositional connectives and quantifiers. Having introduced the necessary theory, we will return to logic and expose a fundamental limitation of logic: that there is no algorithm for determining, in any sufficiently-rich logic framework, whether a statement is or is not true. www.cs.hmc.edu /courses/2005/spring/cs81   (436 words)

 Local Realizability Toposes and a Modal Logic for Computability   (Site not responding. Last check: 2007-11-05) This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. www.cs.cmu.edu /Groups/LTC/abstracts/lrtmlc.html   (110 words)

 Computability and Logic - Cambridge University Press The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel’s Incompleteness Theorems, but also a large number of optional topics from Turing’s theory of computability to Ramsey’s theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521007585   (309 words)

 Logic, Computability and Randomness 2004   (Site not responding. Last check: 2007-11-05) The conference is sponsored by the Association for Symbolic Logic. Demuth was motivated by the study of computable aspects of concepts from (constructive) mathematical analysis. As perhaps the last mathematical innovation of an extraordinary scientific career, Kolmogorov in 1974 proposed to found statistical theory on finite combinatorial and computability principles independent of probabilistic assumptions, as the relation between the individual data and its explanation (model), expressed by Kolmogorov's structure function. www.dc.uba.ar /people/logic2004   (1714 words)

 BU CLA MA 531: Computability and Logic ---Home Page   (Site not responding. Last check: 2007-11-05) The course begins with a treatment of first-order logic as the basis for mathematical logic and an underlying language for mathematics. After describing the class of computable functions and Church's Thesis, the theory is developed through the Enumeration and Parametrization Theorems to Kleene's Recursion Theorem. The last 2 (minimum) to 4 (maximum) weeks of the semester will cover material on Intuitionistic Logic, which is of special interest to computer scientists and a prelude to a seminar I will teach in the Spring 96 semester. www.cs.bu.edu /faculty/kfoury/MA531/syllabus.html   (438 words)

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