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A game-semantic model of computation

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Download games semantic system

Postby Vosar В» 08.04.2019

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Consequently, we have given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers , e. As immediate corollaries, some of the well-known theorems in computability theory such as the smn theorem and the first recursion theorem are generalized. This work is intended to be a stepping stone toward a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.

This result leads to a novel mathematical foundation of computation beyond classical computation , e. In mathematics, however, there are various kinds of non-classical computation , where by classical computation we mean what merely implements a function on natural numbers, since there are a variety of mathematical objects other than natural numbers, for which TMs have certain limitations. As an example of non-classical computation, consider higher-order computation [ 50 ], i.

However, TMs cannot capture higher-order computation in a natural or systematic fashion. In fact, although TMs may compute on symbols that encode other TMs, e. For this point, one may argue that oracle TMs [ 47 , 64 ] may treat an input computation as an oracle , a black-box-like computation that does not have to be recursive; however, it is like a function rather than a computational process that computes just in a single step, which appears conceptually mysterious and technically ad hoc.

Another approach is to give an input computation as a potentially infinite sequence of symbols on the input tape [ 69 ], but it may be criticized in a similar manner. Also, unlike classical computability, a confluence between different notions of higher-order computability has been rarely established [ 50 ]. Perhaps more crucially than the limitation for non-classical computation mentioned above, one may argue that TMs are not appropriate as mathematics of computational processes since computational steps of TMs are often too low-level to see what they are supposed to compute.

Footnote 1 Also, what TMs formulate is essentially symbol manipulations ; however, the content of computation on mathematical, semantic, non-symbolic objects seems completely independent of its symbolic representation, e. Therefore, it would be rather appropriate, at least from the conceptual and the mathematical points of view, to formulate such high-level computational processes in a more abstract, in particular syntax-independent , manner, in order to explain low-level computational processes, and then regard the latter as executable symbolic implementations of the former.

In fact, this or similar perspective is nothing new and shared with various prominent researchers; for instance, Robin Milner stated:. We address this problem in the present paper. However, since there are so many kinds of computation, e. Game semantics of computation [ 3 , 6 , 37 ] is a particular kind of denotational semantics of programming languages [ 8 , 32 , 70 ], in which types and terms are modeled as games and strategies whose definitions are given in Sect. An advantage of game semantics is this flexibility: It models a wide range of programming languages by simply varying constraints on strategies [ 6 ], which enables one to systematically compare and relate different languages ignoring syntactic details.

Also, as full completeness and full abstraction results [ 15 ] in the literature have demonstrated, game semantics in general has an appropriate degree of abstraction and thus it has a good potential to be mathematics of high-level computational processes. Informally, one can imagine that games provide a high-level description of interactive computation between a TM and an oracle, and therefore, they seem appropriate as an approach to the research problem defined in Sect.

Note that such an intensional nature stands in sharp contrast to the traditional domain-theoretic denotational semantics [ 8 ] which, e. In the following, let us give a brief, informal introduction to games and strategies as defined in [ 6 ] in order to sketch the main idea of the present paper. Moves of a game are nodes of the game, where some moves are distinguished and called initial ; only initial moves can be the first element or occurrence of a position of the game.

A strategy on a game, on the other hand, is what tells P which move together with a pointer she should make at each of her turns in the game. Let us consider a simple example. The game N of natural numbers is the following rooted tree which is infinite in width :. In the following, the pointers of most strategies are obvious, and thus, we often omit them. B such that the change of AB -parity i. Next, a fundamental construction!

Each occurrence of an initial move called an initial occurrence of A points to an initial occurrence of B. Girard translation makes explicit the point that some functions need to refer to an input more than once to produce an output, i. As seen in the examples given above, games and strategies capture higher-order computation in an abstract, conceptually natural fashion, where O plays the role of an oracle as part of the formalization.

Thus, one may expect that games and strategies would be appropriate as mathematics of high-level computational processes, solving the research problem of Sect. However, conventional games and strategies have never been formulated as a mathematical model of computation in the sense of TMs ; rather, the primary focus of the field has been full abstraction [ 8 , 15 ], i.

This situation is in a sense frustrating since games and strategies seem to have a good potential to give a semantic , intrinsic i. For the potential, we have decided to employ games and strategies as our basic mathematical framework and extend them to give mathematics of computational processes in the sense described in Sect. However, these variants of games and strategies are just conventional ones, and consequently, such step-by-step processes have no official status in their categories.

The problem lies in the point that in conventional game semantics composition of strategies is executed as parallel composition plus hiding [ 3 ], where hiding is the matter. Let us illustrate this point by a simple, informal example as follows. Their computations can be described by the following diagrams:. N has been arbitrarily chosen i. The resulting play is to be read as follows:. Nevertheless, the present author and Samson Abramsky have introduced a novel, dynamic variant of games and strategies that systematically model dynamics and intensionality of computation, and also studied their algebraic structures [ 71 ].

In contrast to the previous work mentioned above, dynamic strategies themselves embody step-by-step processes in computation by retaining intermediate occurrences of moves, and composition of them is parallel composition without hiding. Moreover, to obtain a powerful model of computation, they should be at least Turing complete , i. This sets up, in addition to the conceptual quest so far, an intriguing mathematical question in its own right:.

Not surprisingly, perhaps, this problem has turned out to be challenging, and the main technical achievement of the present paper is to give a positive answer to it. This is achieved roughly as follows. The concepts introduced below make sense for conventional i. As we see in Sect. This sets up a finitary representation of game-semantic computation on natural numbers. Thus, it may seem at first glance that finitary dynamic strategies in the following sense suffice: A strategy is finitary if its representation by a partial function [ 38 , 51 ] that assigns the next P-move to previous moves, called its table , is finite.

As described in Fig. First, its partial function representation can be given by the following infinitary table n. This example illustrates why we need viable not only finitary dynamic strategies for Turing completeness, where recall that minimization in the general form or an equivalent construction is vital to construct all partial recursive functions [ 16 ]. Note also that viability is defined solely in terms of games and strategies without any axiom or induction.

Also, viable dynamic strategies solve the problem defined in Sect. First, as we have seen via examples, games and strategies give an abstract, syntax-independent formulation of high-level computational processes, e. Moreover, an instruction strategy for a viable dynamic strategy describes a low-level computational process that implements the dynamic strategy. We hope that these technical results would convince the reader that viability of dynamic strategies is a natural, reasonable generalization of Church—Turing computability.

Another, more conceptual contribution of the present work is to establish a single mathematical framework for both high-level and low-level computational processes, where the former defines what computation does, while the latter describes how to execute the former. In comparison with existing mathematical models of computation, our game-semantic approach has some novel features. First, in comparison with computation by TMs or programming languages, plays of games are a more abstract concept; in particular they are not necessarily symbol manipulations, which is why they are suitable for abstract, high-level computational processes.

Next, computation in a game proceeds as an interaction between P and O, which may be seen as a generalization of computation by TMs in which just one interaction occurs i.

The present work inherits this interactive nature of game semantics. Last but not least, games are a semantic counterpart of types , where note that types do not a priori exist in TMs, and types in programming languages are syntactic entities. Hence, our approach provides a deeper clarification of types in the context of theory of computation.

Moreover, by exploiting the flexibility of game semantics, our approach would be applicable to a wide range of computation though it is left as future work. Footnote 8 Of course, we need to work out details for these developments, which is out of the scope of the present paper, but it is in principle clear how to apply our framework to existing game semantics. In this sense, the present work would serve as a stepping stone toward these extensions.

In the literature, there have been several attempts to provide a mathematical foundation of computation beyond classical or symbolic ones. We do not claim at all our game-semantic approach is best or canonical in comparison with the previous work; however, our approach certainly has some advantages. Notably, ASMs define a very general notion of computation, namely computation as structure transition.

Nevertheless, there are notable differences between computability logic and the present work. Next, our framework inherits the categorical structure of existing game semantics see [ 71 ] for this point , providing a compositional formulation of logic and computation, i. Nevertheless, it would be interesting to adopt his TMs-based approach in our framework and compare the resulting computational power with that of the present work.

Finally, let us mention some of the precursors of game semantics. To clarify the notion of higher-order computability, Stephen Cole Kleene considered a model of higher-order computation based on dialogues between computational oracles in a series of papers [ 42 , 43 , 44 ], which can be seen as the first attempt to define a mathematical notion of algorithms in a higher-order setting [ 50 ].

Moreover, Gandy and his student Giovanni Pani refined these works by Kleene to obtain a model of PCF that satisfies universality though this work was not published. These previous papers are direct ancestors of game semantics in particular the so-called HO-games [ 38 ] by Martin Hyland and Luke Ong. As another line of research motivated by the full abstraction problem for PCF [ 58 ] , Pierre-Louis Curien and Gerard Berry conceived of sequential algorithms [ 10 ] which was the first attempt to go beyond extensional functions to capture sequentiality of PCF.

Sequential algorithms preceded and became highly influential to the development of game semantics; in fact, sequential algorithms are presented in the style of game semantics in [ 50 ], and it is shown in [ 14 ] that the oracle computation developed by Kleene can be represented by sequential algorithms though the converse does not hold. The rest of the paper proceeds roughly as follows. This introduction ends with fixing some notation. Then, recalling dynamic games and strategies in Sect.

Finally, we draw a conclusion and propose future work in Sect. As already explained, we have chosen this variant since, in contrast to conventional games and strategies, dynamic games and strategies capture step-by-step processes in computation, which is essential for a TMs-like model of computation.

However, we need some modifications of dynamic games and strategies. In particular, we have to employ exponential! In addition, we slightly refine the original definition of dynamic games by requiring that an intermediate occurrence of an O-move in a position of a dynamic game must be a mere copy of the last occurrence of a P-move, which reflects the example of composition without hiding in the introduction.

Let us remark, however, that this refinement is technically trivial, and it is not our main contribution. This section presents the resulting variant of games and strategies. To make this paper essentially self-contained, we shall explain motivations and intuitions behind the definitions. On the other hand, such a bijection is necessary only for manipulating effective tags, and so we would like to avoid an involved mechanism to achieve it. Then, our solution for this problem is to simply introduce elements to denote the bijection:.

Footnote 10 We shall utilize outer tags for exponential! As already stated, our games are slightly modified dynamic games introduced in [ 71 ]. We first review the slightly modified dynamic games in the present section; see [ 71 ] for the details, and [ 3 , 6 , 37 ] for a general introduction to game semantics.

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Re: download games semantic system

Postby Zolorisar В» 08.04.2019

Levy, P. Springer A strategy on a game, on the other hand, is what http://enjoystake.site/poker-games/poker-games-kazan-play-1.php P which move together with a pointer she should make at each of her turns in the game. Springer, New York Google Scholar 7. Moreover, the subgame relation is preserved under these constructions, i.

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Postby Taushakar В» 08.04.2019

It is roughly why game semantics has been highly successful in denotational semantics. To make this paper essentially self-contained, we shall explain motivations and intuitions behind the definitions. S2 It is deterministic i. In a position of the game, O always performs the first move by a question, and then P and Seantic alternately play by alternationin which every non-initial occurrence is performed please click for source a specific previous occurrence by justification .

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Postby Dourn В» 08.04.2019

Let us begin with recalling these two concepts. Search SpringerLink Search. Next, computation in a game proceeds as an interaction between P and O, which may be seen as a generalization of computation click TMs in which just one interaction occurs i. Automata, Languages, and Programming, pp. This is achieved roughly as follows.

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Postby Kisar В» 08.04.2019

However, we need some modifications of dynamic games and system. In the following, let us give a brief, informal introduction download games and strategies as defined in [ 6 ] in order to sketch the main idea of the present paper. However, the most imminent future work is games, by exploiting the visit web page of game semantics, to enlarge the scope of the present work i. As seen in the semantic given above, games and strategies capture higher-order computation in an abstract, conceptually natural fashion, where O plays the role of an oracle as part of the formalization. Finally, let us mention some of the precursors of game semantics.

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Postby Kezilkree В» 08.04.2019

Note also gamess we shall focus on innocent strategies as a means to narrow down previous occurrences to be concerned with. Also, the occurrence becomes no longer initial as soon as the pairing is post-concatenated; thus, it does not suffice to trace the first occurrence of each position. A tag refers to an outer or inner tag. Reprints and Permissions.

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Re: download games semantic system

Postby Nalrajas В» 08.04.2019

This process is experimental and the keywords may be updated as the learning algorithm improves. HodgesR. Kozen, D.

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Postby Tejar В» 08.04.2019

Abramsky, S. ACM 21 8— It is, however, viable for any normalized game Awhich is perhaps surprising to many readers.

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Postby Fezshura В» 08.04.2019

Views Read Edit View history. This point is unsatisfactory because:. The author acknowledges the financial support from Funai Overseas Scholarship, and also he is gamex to Samson Abramsky and Robin Piedeleu for fruitful discussions. Semantic Web Semantic wiki. We have defined our games and strategies in the previous section.

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Re: download games semantic system

Postby Kigazshura В» 08.04.2019

In: Gabbay, D. We have chosen zemantic unary representation for its simplicity. Henceforth, st-algorithms refer to standard ones by default. For their simplicity, let us skip their informal definitions and just present the formal ones:.

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Postby Mooguzilkree В» 08.04.2019

Rahman and N. Pietarinen and T. However, these variants of games and strategies are just conventional ones, and consequently, such step-by-step processes have no official status in their categories.

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Postby Moogugal В» 08.04.2019

Theory 42— It is clear that pairing of strategies may be handled in a completely similar manner; currying and uncurrying are even simpler. Kozen, D. This question seems highly interesting from a source perspective.

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Postby Faera В» 08.04.2019

As already explained, we have chosen this variant since, in contrast to conventional games and strategies, dynamic games and strategies capture step-by-step processes in computation, which is essential for a TMs-like model of computation. This result leads to a novel mathematical foundation of computation beyond classical computatione. They are static legal positions that satisfy additional axioms:. HylandW.

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Postby Zulkir В» 08.04.2019

Its typical play is as follows:. However, we need some modifications of dynamic games and strategies. In: Buss, Syatem. Search SpringerLink Search. Clerbout: Immanent Reasoning or Equality in Action.

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Postby Kajijora В» 08.04.2019

Addison-Wesley, Reading Google Scholar Dal Lago, U. Blass, A. Power, J. Jaber, G.

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Postby Feramar В» 08.04.2019

Rahman and N. Springer, London This sets up, in addition to the conceptual quest so far, an intriguing mathematical question in its own http://enjoystake.site/gambling-movies/gambling-movies-runner-3-1.php. Its typical plays are as follows:.

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Postby Gukora В» 08.04.2019

Ong, C. N has been arbitrarily chosen i. This line was further gamez by Samson AbramskyRadhakrishnan JagadeesanPasquale Malacaria and independently Martin Hyland and Luke Ongwho placed special emphasis on http://enjoystake.site/gambling-near/gambling-near-me-deflate-meme-1.php, i. In: Arge, L.

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Postby Mejinn В» 08.04.2019

Lorenzen, K. Games are played between two agents: a machine and its environment, where the machine is required to follow only effective strategies. Analysis Computational Lexical lexis lexicology Statistical Structural Prototype theory Force dynamics Ssemantic linguistics problems Theory of descriptions.

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Postby JoJorg В» 08.04.2019

Of course, one may consider ganes TMs that may interact with O, but then it is no longer TMs in the gambling sense; in fact, this idea naturally leads to the game-semantic model of computation developed in the present paper. We hope that these technical results would convince the reader that click the following article of dynamic strategies is a natural, reasonable generalization of Game computability. Formally, we define promotion of games as follows:. Informally, one can imagine that games provide a high-level description of interactive computation between a TM and an oracle, and game, they seem appropriate as an approach to the research problem defined crossword Sect. Recall that here static legal position defined in [ card ] is a j-sequence that satisfies alternation and visibilityi.

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Postby Tojajinn В» 08.04.2019

Abramsky, S. ENW EndNote. The predecessor strategy is defined by:. Cambridge University Press, Cambridge In the following, let us give a brief, informal introduction to games and strategies as defined in [ 6 ] in order to sketch the main idea of the present paper.

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