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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 Ptarithmetic 2 The Baltic International Yearbook of and developed (Japaridze 2011, unpublished, forth.).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(at least not yet), and that, for this and Game Semantics reason, it would be a pity to let this material remain unpublished.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. GIORGI JAPARIDZE What follows is a copy of the original manuscript without any mod- School of Computer Science and Technology, Shandong University ifications, except some minor changes made in 2010 before shelving Department of Computing Sciences, Villanova University the paper. Some parts of the above manuscript were used in Japaridze (2011), which explains occasional textual overlaps between the two articles. PTARITHMETIC 1. INTRODUCTION Computability logic (CL), introduced in Japaridze (2003, 2006b, ABSTRACT: The present article introduces ptarithmetic (short 2009a), is a semantical, mathematical and philosophical platform, and for “polynomial time arithmetic”) — a formal number theory sim- an ambitious program, for redeveloping logic as a formal theory of ilar to the well known Peano arithmetic, but based on the recently computability, as opposed to the formal theory of truth which logic has born computability logic instead of classical logic. The formulas of more traditionally been. Under the approach of CL, formulas represent ptarithmetic represent interactive computational problems rather computational problems, and their “truth” is seen as algorithmic solv- than just true/false statements, and their “truth” is understood as ability. In turn, computational problems — understood in their most existence of a polynomial time solution. The system of ptarith- general, interactive sense — are defined as games played by a machine metic elaborated in this article is shown to be sound and com- against its environment, with “algorithmic solvability” meaning exis- plete. Sound in the sense that every theorem T of the system rep- resents an interactive number-theoretic computational problem tence of a machine that wins the game against any possible behavior with a polynomial time solution and, furthermore, such a solu- of the environment. And an open-ended collection of the most basic tion can be effectively extracted from a proof of T. And complete and natural operations on computational problems forms the logical in the sense that every interactive number-theoretic problem with vocabulary of the theory. With this semantics, CL provides a systematic a polynomial time solution is represented by some theorem T of answer to the fundamental question “what can be computed?”, just as the system. classical logic is a systematic tool for telling what is true. Furthermore, The paper is self-contained, and can be read without any prior as it turns out, in positive cases “what can be computed” always allows familiarity with computability logic. itself to be replaced by “how can be computed”, which makes CL of potential interest in not only theoretical computer science, but many This paper was completed in early 2009 (cf. more applied areas as well, including interactive knowledge base sys- http://arxiv.org/abs/0902.2969), and has remained unpublished tems, resource oriented systems for planning and action, or declarative since then. Among the reasons is that an alternative — in a sense more programming languages. general and elegant — approach named “clarithmetic” was introduced Vol. 8: Games, Game Theory and Game Semantics 3 Giorgi Japaridze Ptarithmetic 4 While potential applications have been repeatedly pointed out and lution. Does not this sound exactly like what the constructivists have outlined in introductory papers on CL, so far all technical efforts have been calling for? been mainly focused on finding axiomatizations for various fragments Unlike the mathematical or philosophical constructivism, however, of this semantically conceived and inordinately expressive logic. Con- and even unlike the early-day theory of computation, modern com- siderable advances have already been made in this direction (Japaridze puter science has long understood that, what really matters is not just 2006d,a,c, 2009b, 2010; Mezhirov & Vereshchagin 2010), and more re- computability, but rather efficient computability. So, the next natural sults in the same style are probably still to come. It should be, however, step on the road to revealing the importance of CL for computer science remembered that the main value of CL, or anything else claiming to be would be showing that it can be used for studying efficient computabil- a “Logic” with a capital “L”, will eventually be determined by whether ity just as successfully as for studying computability-in-principle. Any- and how it relates to the outside, to the extra-logical world. In this one familiar with the earlier work on CL could have found reasons for respect, unlike many other systems officially qualified as “logics”, the optimistic expectations here. Namely, every provable formula of any merits of classical logic are obvious; these merits are most eloquently of the known sound axiomatizations of CL happens to be a scheme of demonstrated by the fact that applied formal theories, a model exam- not only “always computable” problems, but “always efficiently com- ple of which is Peano arithmetic PA, can be and have been successfully putable” problems as well, whatever efficiency exactly mens in the con- based on it. Unlike pure logics with their meaningless symbols, such text of interactive computation that CL operates in. That is, at the level theories are direct tools for studying and navigating the real world with of pure logic, computability and efficient computability yield the same its non-man-made, meaningful objects, such as natural numbers in the classes of valid principles. The study of logic abounds with phenomena case of arithmetic. To make this point more clear to a computer scien- in this style. One example would be the well known fact about classical tist, one could compare a pure logic with a programming language, logic, according to which validity with respect to all possible models is and applied theories based on it with application programs written equivalent to validity with respect to just models with countable do- in that language. A programming language created for its own sake, mains. mathematically or aesthetically appealing but otherwise unusable as a At the level of reasonably expressive applied theories, however, general-purpose, comprehensive basis for application programs, would one should certainly expect significant differences depending on hardly be of much interest. whether the underlying concept of interest is efficient computability or So, in parallel with studying possible axiomatizations and vari- computability-in-principle. For instance, the earlier-mentioned system ous metaproperties of pure computability logic, it would certainly be CLA1 proves formulas expressing computable but not always efficiently worthwhile to devote some efforts to justifying its right to existence computable arithmetical problems. The purpose of the present paper through revealing its power and appeal as a basis for applied theo- is to construct a CL-based system for arithmetic which, unlike CLA1, ries. The first and so far the only concrete steps in this direction have proves only efficiently — specifically, polynomial time — computable been made very recently in Japaridze (2010), where a CL-based system problems. The new applied formal theory PTA (“ptarithmetic”, short CLA1 of (Peano) arithmetic was constructed.1 Unlike its classical-logic- for “polynomial time arithmetic”) presented in Section 12 achieves this based counterpart PA, CLA1 is not merely about which arithmetical purpose. facts are true, but about which arithmetical problems can be actually Just like CLA1, our present system PTA is not only a cognitive, but computed or effectively solved. More precisely, every formula of the lan- also a problem-solving tool: in order to find a solution for a given prob- guage of CLA1 expresses a number-theoretic computational problem lem, it would be sufficient to write the problem in the
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