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COMPUTABILITY LOGIC Homepage
Giorgi Japaridze'sComputability Logic Homepage** This material is based upon work supported by the NationalScience Foundation under Grant No. 0208816. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.What is computability logic?Lecture notes on computability logicPotential applications of computability logicPapers on computability logicLinks to related websitesLaTeX macros for the operators of computability logicWhat is computabilitylogic? Computability is certainly one of the most interesting and fundamental concepts in mathematics and computer science, and it would be more thannatural to ask what logic it induces. Let us faceit: this question has not only never been answered, but never even beenaskedwithin a reasonably coherent and comprehensive formal framework.This is where Computability Logic comes in.It is a formal theory of computability in the samesense as classical logic is a formal theory of truth. In a broader and more proper sense, computability logic is not just a particular theory but an ambitious and challenging program for redeveloping logic following the scheme "from truth to computability". It was introduced in 2003 and, at present, still remains in its infancy stage, with open problems prevailing over answered questions. It is largely a virgin soil offering plenty of research opportunities, with good chances of interesting findings, for those with interests in logic and its applications in computer science.Computation and computational problems in computability logic are understood in their most general, interactive sense, and are precisely seen as games played by amachine (computer, agent, robot) against its environment (user, nature, or the devil himself). Computability of such problems means existence of a machine that always wins the game. Logical operators stand for operations on computational problems, and validity of a logical formula means being a scheme of "always computable" problems.Remarkably, classical, intuitionistic and linear (in a broad sense)logics turn out to be three natural fragments of computability logic. This is no accident. The classical concept of truth is nothing but a special case of computability -- computability restrictedto problems of zero interactivity degree. Correspondingly, classicallogic is nothing but a special fragment of computability logic. One of the main -- so farrather abstract -- intuitions associated with intuitionistic logic isthat it must be a logic of problems (Kolmogorov 1932); this is exactly what computability logic is, only in a much more expressive language than intuitionistic logic. And one of the main -- again, so far rather abstract -- claims of linear logic is that it is a logic of resources. Reversing of the roles of the machine and its environment turns computational problems into computational resources, which makes computability logic a logic of resources, only, again, in a more expressive language than that of linear logic, and based on an intuitively convincing and mathematically strict resource semantics rather than some naive philosophy and unreliable syntactic arguments. For more about computability logic vs. linear logic, visit GameSemantics or Linear Logic?. For an extended online tutorial introduction to the subject please see Lecture Notes on Computability Logic.Potential applications of computability logic: The computability logic paradigm is not only about "what can be computed", butequally about "how can be computed". This opens applicationareas far beyond pure computation theory, such as: Knowledgebase systems.Computability logic is an appealing alternative to the traditional knowledge base,knowledge representation and query logics. Its advantages include: Computability logic is a logic of interaction, and this is what a good knowledgebase logic needs to be. After all, most of the real knowledgebase andinformation systems are interactive. Computability logic is resource-conscious: the potential knowledge provided by a disposable pregnancy test device is knowledge of only one (even though an arbitrary one) woman'spregnancy status, but not two. Computability logic naturally captures this sort of extremely relevant differences whilethe traditional systems fail to account for them.Computability logic naturally differentiates between truth and an agent's actualability to find/compute/tell what is true. To achieve similar effects,traditional knowledgebase logics have to appeal to the controversial,messy and troublemaking epistemic constructs. Systems for planning and action.Computability logic might be a reasonable alternative to the traditional AIplanning logics. Its advantages include: Computability logic is a logic of interaction, and planning systemsshould be well able to account for an agent's interaction with theenvironment. A good planning logic should be able to naturally account for the fact that with one ballistic missile one can destroy only one (even though an arbitrary one) target but not two. The resource-conscious computability logic fits the bill. Computability logic is a logic of knowledge and informational resources, whichautomatically takes care of the knowledge preconditions problem. Computability-logic-based planning systems are immune to the notorious frameproblem. Constructive applied theories.Computability logic is a conservative extension of classical logic, which makes it a reasonable alternative to the latter in its most traditional and unchallengedapplication areas. In particular, it makes perfect sense to base applied theories (such as, say, Peano arithmetic) on computability logic instead ofclassical logic. The advantages of doing so include that computability-logic-based applied theories,compared with their classical-logic-based counterparts, are: Expressive: the language of classical logic is only a modest fraction of the language of computability logic. Computationally meaningful: every theorem of a computability-logic-basedtheory represents a computable problem rather than just a truestatement. Constructive: any proof of a formula F in a computability-logic-based theory effectivelyencodes a solution to the computational problem represented by F.Selected papers on computability logic: The official journal versions of the following papers may be slightly different from the corresponding online preprints. It is recommended that you try to use the journal version first. Access to some journals may be restricted though, in which case you may download the online preprint.Recommendation: For the first acquaintance with the subject, most recommended is reading the tutorial-style Sections 1-10 of "In the beginning was game semantics". [1]G.Japaridze, Introduction to computability logic.Annals of Pure andApplied Logic 123 (2003), pp.1-99. #3 "HOTTEST" ARTICLE of the journal. Official
journal version
Online preprintProvides a fundamentalintroduction to the subject. It is long but easy to read. If you decideto read it, please make sure you check out the Erratum.¡¡ [2]G.Japaridze,Propositional computability logic I. ACM Transactions on Computational Logic 7 (2006), No. 2, pp. 302-330.
Official journal version
Online preprintContains a detailed exposition of a soundness and completeness proof forone of the most basic fragments of computability logic. ¡¡ [3]G.Japaridze,Propositional computability logic II. ACM Transactions on Computational Logic 7 (2006), No. 2, pp. 331-362. Official journal version Online preprintExtends the soundness/completeness result of itspredecessor to a much more expressive fragment of computability logic. ¡¡ [4] G.Japaridze,Introduction to cirquent calculus and abstract resourcesemantics. Journal of Logic and Computation 16 (2006), No.4, pp. 489-532. Official journal version Online preprint Cirquent calculus is a new syntactic approach, which can be seen as a refinement of sequent calculus. Unlike the latter, it is flexible enough to be used as a deductive framework for computability logic. ¡¡ [5]G.Japaridze,Computability logic: a formal theory of interaction.In: InteractiveComputation: The New Paradigm. D.Goldin, S.Smolka and P.Wegner, eds. Springer Verlag, Berlin 2006, pp. 183-223. Official book version Online preprint Presents a relatively compact and nontechnical introduction to computability logic and anoverview of the newestresults. Most recommended for those whose primary interests are incomputer science. Writtenin a semitutorial style with non-expert audience in mind. ¡¡[6]G.Japaridze,From truth to computability I. Theoretical Computer Science 357 (2006), pp. 100-135. Official journal version Online preprint Proves soundness and completeness for the first-order fragment CL3of computability logic, which is the predicate version of logic CL1studied in [2].¡¡[7] G.Japaridze,From truth to computability II. Theoretical Computer Science379 (2007), pp. 20-52. Official journal version Online preprint Extends the results of its predecessor in the same sense onthe first-order level as [3] extends [2] on the propositional level.¡¡[8] G.Japaridze,Intuitionistic computability logic.ActaCybernetica 18 (2007), No.1, pp.77-113. Official journal version Online preprint Proves soundness for Heyting's intuitionistic predicate calculus with respect to the computability-logic semantics. ¡¡[9]G.Japaridze,The logic of interactive Turing reduction.Journal of SymbolicLogic 72 (2007), No.1, pp. 243-276. Official journal version Online preprint A proof of the completeness of the imlicative fragment ofintuitionistic logic with respect to the semantics of computability logic. ¡¡[10]G.Japaridze,The intuitionistic fragment of computability logic at thepropositional level.Annals of Pure andApplied Logic 147 (2007), No.3, pp.187-227. #1 "HOTTEST" ARTICLE of the journal. Official journal version Online preprint Proof of the (soundness and) completeness of the full propositional intuitionistic logic with respect to the semantics of computability logic.¡¡[11] G.Japaridze,Cirquent calculus deepened. Journal of Logic and Computation Advance Access published online on July 21, 2008; doi: 10.1093/logcom/exn019 Official journal version Online preprint Elaborates a deep version (as opposed to the shallow version of [4]) of cirquent calculus for the (Ø,Ú,Ù)-fragment of computability logic and classical propositional logic.¡¡[12]G.Japaridze,In the beginning was game semantics.In: Games: Unifying Logic, Language and Philosophy.O. Majer, A.-V. Pietarinen and T. Tulenheimo, eds. Springer Verlag,Berlin (to appear). Official book version Online preprint A comprehensive overview of the philosophy, motivations andtechniques of computability logic. Supported by ample illustrations and examples.Contains a proof of the soundness of affine logic, claimed but neverofficially proven before. ¡¡[13]G.Japaridze,Many concepts and two logics of algorithmic reduction.Studia Logica (to appear in the Spring of 2009). Official journal version Online preprint The main result here is a proof of the fact that implicative binary tautologies and their instances are precisely captured by the implicative fragment of intuitionistic calculus without contraction. This is the logic of the basic (linear) sort of reduction.¡¡[14] G.Japaridze, Sequential operators in computability logic. Information and Computation (to appear). Official journal version Online preprintIntroduces the new, sequential group (conjunction/disjunction, quantifiers and recurrences) of game operations. Constructs sound and complete axiomatizations.¡¡[15] G.Japaridze,Towards applied theories based on computability logic.Journal of SymbolicLogic (under review). Official journal version Online preprint Basing Peano arithmetic on computability logic instead of classical logic. Built on the new sound and complete deductive system CL12 for computability logic, which substantially strengthens the systemCL3 introduced in [6]. ¡¡Other websites on computability logic:GameSemantics or Linear Logic?Course on Computability Logic --- taught in 2007. Includes lecture notes in PowerPoint. Some other web pages linked to this site: Logic and Games (Stanford Encyclopedia of Philosophy) DeepInference and the Calculus of StructuresProofs as games Computability logic-1 Lambda theUltimate: Introduction to computability logic Lambda theUltimate: In the beginning was game semantics Mathematical Logicaround the WorldKeywords: გამოთვლადობის ლოგიკა, §§à§Ô§Ú§Ü§Ñ §Ó§í§é§Ú§ã§Ý§Ú§Þ§à§ã§ä§Ú,¿É¼ÆËãÐÔÂß¼, Game semantics, Resource semantics, Linear logic, Intuitionistic logic, Constructive logics, Interaction, Interactive computation, Knowledgerepresentation, Knowledgebase systems, Resourcebase systems, Systems for planning and action, Computational logic, Calculus of structures  var sc_project=258215;  |
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