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Soundness and Completeness of the Cirquent Calculus System CL6 for Computability Logic

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Soundness and Completeness of the Cirquent Calculus System CL6 for Computability Logic Soundness and completeness of the Cirquent calculus system CL6 for computability logic WENYAN XU and SANYANG LIU, Department of Mathematics, Xidian University, Xi’an, 710071, PR China. E-mail: [email protected]; [email protected] Downloaded from Abstract Computability logic is a formal theory of computability. The earlier article ‘Introduction to cirquent calculus and abstract resource semantics’ by Japaridze proved soundness and completeness for the basic fragment CL5 of computability logic. The present article extends that result to the more expressive cirquent calculus system CL6, which is a conservative extension of http://jigpal.oxfordjournals.org/ both CL5 and classical propositional logic. Keywords: Cirquent calculus, computability logic. 1 Introduction Computability logic (CoL), introduced by G. Japaridze [1, 2, 6], is a semantical and mathematical platform for redeveloping logic as a formal theory of computability. Formulas in CoL represent interactive computational problems, understood as games between a machine and its environment at CSU Fresno on July 29, 2015 (symbolically named as and ⊥, respectively); logical operators stand for operations on such problems; ‘truth’ of a problem/game means existence of an algorithmic solution, i.e. ’s effective winning strategy; and validity of a logical formula is understood as such truth under every particular interpretation of atoms. The approach induces a rich collection of (old or new) logical operators. Among those, relevant to this article are ¬ (negation), ∨ (parallel disjunction) and ∧ (parallel conjunction). Intuitively, ¬ is a role switch operator: ¬A is the game A with the roles of and ⊥ interchanged (’s legal moves and wins become those of ⊥, and vice versa). Both A∧B and A∨B are games playing which means playing the two components A and B simultaneously (in parallel). In A∧B, is the winner if it wins in both components, while in A∨B winning in just one component is sufficient. The symbols and ⊥, together with denoting the two players, are also used to denote two special (the simplest) sorts of games. Namely, is a moveless (‘elementary’) game automatically won by the player , and ⊥ is a moveless game automatically won by ⊥. Cirquent calculus is a refinement of sequent calculus. Unlike the more traditional proof theories that manipulate tree-like objects (formulas, sequents, hypersequents, etc.), cirquent calculus deals with graph-style structures termed cirquents, with its main characteristic feature thus being allowing to explicitly account for sharing subcomponents between different components. The approach was introduced by Japaridze [4] as a new deductive tool for CoL, and was further developed in [5, 7, 8]. The article [4] constructed a cirquent calculus system CL5 for the basic (¬,∧,∨)-fragment of CoL, and proved its soundness and completeness with respect to the semantics of CoL. Vol. 20 No. 1, © The Author 2011. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/jigpal/jzr043 Advance Access published 29 October 2011 [17:18 13/1/2012 jzr043.tex] Paper Size: a4 paper Job: JIGPAL Page: 317 317–330 318 Cirquent calculus system The atoms of CL5 represent computational problems in general, and are said to be general atoms. The so-called elementary atoms, representing computational problems of zero degree of interactivity (such as the earlier-mentioned games and ⊥) and studied in other pieces of literature on CoL, are not among them. Thus, CL5 only describes valid computability principles for general problems. This is a significant limitation of expressive power. For example, the formula A→A∧A is not valid in CoL when A is a general atom, but becomes valid (as any classical tautology for that matter) when A is elementary.1 So the language of CL5 naturally calls for an extension. Japaridze [4] claimed without a proof that the soundness and completeness result for CL5 could be extended to the more expressive cirquent calculus system CL6 (reproduced later), which is a conservative extension of both CL5 and classical propositional logic. This article is devoted to a soundness and completeness proof for system CL6, thus contributing to the task of extending the cirquent-calculus approach so as to accommodate incrementally expressive fragments of CoL. Downloaded from 2 Preliminaries This article primarily targets readers already familiar with Japaridze [4], and can essentially be treated as a technical appendix to the latter. However, in order to make it reasonably self-contained, http://jigpal.oxfordjournals.org/ in this section we reproduce the basic concepts from [4, 6]. An interested reader may consult (the extremely well written) [4] and [6] for additional explanations, illustrations, examples and insights. 2.1 Games and strategies The letter ℘ is used as a variable ranging over {,⊥}, with ¬℘ meaning ℘’s adversary. A move is a finite string over the standard keyboard alphabet. A labmove is a move prefixed (‘labeled’) with or ⊥. A run is a finite or infinite sequence of labmoves, and a position is a finite run. Runs are at CSU Fresno on July 29, 2015 usually delimited by ‘’ and ‘’, with thus denoting the empty run. For any run , ¬ is the same as , with the only difference that every label ℘ is changed to ¬℘. A game2 is a pair A=(LrA,WnA), where: (i) LrA is a set of runs satisfying the condition that a finite or infinite run is in LrA iff so are all of ’s non-empty finite initial segments.3 If ∈LrA, then is said to be a legal run of A; otherwise is an illegal run of A. A move α is a legal move for a player ℘ in a position of A iff ,℘α∈LrA; otherwise α is an illegal move. When the last move of the shortest illegal initial segment of is ℘-labeled, is said to be a ℘-illegal run of A. (ii) WnA is a function that sends every run to one of the players or ⊥, satisfying the condition that if is a ℘-illegal run of A, then WnA=¬℘. When WnA=℘, is said to be a ℘-won run of A. Let A be a game. The depth of A is the smallest integer d such that the length of every legal run of A is ≤d. A is said to be elementary iff the depth of it is 0. There are exactly two elementary games, for which the same symbols and ⊥ are used as for the two players. We identify these with the two propositions of classical logic: (true) and ⊥ (false). ‘Snow is white’ is thus a moveless game automatically won by the machine, while ‘Snow is black’ is automatically lost. This is exactly what eventually makes classical propositional logic a natural fragment of CoL. 1This will be explained later. 2The concept of a game considered in CoL is more general than the one defined here, with games in the present sense called constant games. For simplicity, in this article we only consider constant games, omitting the word ‘constant’ and just saying ‘game’. 3This condition can be seen to imply that the empty run is always in LrA. [17:18 13/1/2012 jzr043.tex] Paper Size: a4 paper Job: JIGPAL Page: 318 317–330 Cirquent calculus system 319 Let be a run and α be a move. The notation α will be used to indicate the result of deleting from all moves (together with their labels) except those that look like αβ for some move β, and then further deleting the prefix ‘α’ from such moves. For instance, 1.α,⊥2.β,1.γ,⊥2.δ1. =α,γ. The game operations outlined intuitively in the preceding section are defined below, where A, A1, A2 are arbitrary games, α ranges over moves, i ∈{1,2}, is an arbitrary run, and is any legal run of the game that is being defined. (1) ¬A (negation) is defined by: Downloaded from (i) ∈Lr¬A iff ¬∈LrA. (ii) Wn¬A = iff WnA¬ =⊥. (2) A1 ∧A2 (parallel conjunction) is defined by: ∧ . (i) ∈LrA1 A2 iff every move of is i.α for some i,α and, for both i, i ∈LrAi . ∧ . (ii) WnA1 A2 = iff, for both i, WnAi i =. http://jigpal.oxfordjournals.org/ (3) A1 ∨A2 (parallel disjunction) is defined by: ∨ . (i) ∈LrA1 A2 iff every move of is i.α for some i,α and, for both i, i ∈LrAi . ∨ . (ii) WnA1 A2 = iff, for some i, WnAi i =. Intuitively, making a move i.α in A1 ∧A2 or A1 ∨A2 means making the move α in the Ai component of the game; and i. (respectively i.) is the subrun of (respectively ) that took place in that component. In what follows, we explain—formally or informally—several additional basic concepts of the CoL semantics. at CSU Fresno on July 29, 2015 (1) Static games: CoL restricts its attention to a special yet very wide subclass of games termed ‘static’. Intuitively, static games are interactive tasks where the relative speeds of the players are irrelevant, as it never hurts a player to postpone making moves. A formal definition of this concept can be found in [6], which we will not reproduce here as nothing in this article relies on it. The only relevant for us fact is that all elementary games are static, and that, as proven in [1, 6], the class of static games is closed under the operations ¬,∧,∨ (as well as any other game operations studied in CoL). (2) HPMs: CoL understands ’s effective strategies as interactive machines. Several sorts of such machines have been proposed and studied in CoL, all of them turning out to be equivalent in computing power once we exclusively consider static games. The most basic sort of such machines is called the hard-play machine (HPM). This is a kind of a Turing machine with the additional capability of making moves, and has two tapes4: the ordinary read/write work tape and the read-only run tape.
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