Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992. 255  Home/Search   Document Not in Database   Summary   Related Articles   Check

....semantics from the study of this labelled method that emphasizes the role of requirements and assertions on labels. The requirements, respectively the assertions could be seen as questions, respectively answers that have to interact and could be related to game semantics proposals for linear logic [4]. We will study the actual impact of the choice of di erent semantics for the countermodel generation process and thus propose new appropriate resource semantics for proof search with countermodel generation facility in such substructural logics. 8.3 Properties of the MILL labelled calculus ....

A. Blass. A Game Semantics for Linear Logic. Annals of Pure and Applied Logic, 56:183220, 1992.

....supported by the Austrian Science Fund Project N Z29 INF, and the Max Kade Foundation. Most of this research has been carried out while the first author was with the University of Gieen, Germany, and the third author was with TU Wien. in [27] and subsequently studied in numerous papers, e.g. [18, 64, 5, 9, 33, 50]. Unifying the two approaches, the most prominent work from this area is the seminal paper of Lindstrom [34] whose formalization of generalized quantifiers is mostly used to date. After diminished attention for some time, generalized quantifiers gained in the 90 s increasing interest in ....

....generalized quantifiers are used in different areas. On the foundational side, they have been considered extensively in descriptive complexity theory as a tool for designing logical languages for capturing computational complexity classes. For example, restricted Henkin quantifiers capture NL [9] over ordered structures, while standard Henkin quantifiers capture Theta over ordered structures [22] transitive closure logic captures NL over ordered structures, while deterministic transitive closure logic captures L over ordered structures [30, 25] and, various extensions of first order ....

A. Blass and Y. Gurevich. Henkin Quantifiers and Complete Problems. Annals of Pure and Applied Logic, 32:1--16, 1986. 24

....cut free forms of or 0 and 0 . Thus any congruence (for instance, a denotational equivalence) generated by such a cut elimination procedure must equate and 0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction [9] and game semantics [4, 1, 5], that cut elimination is a process analogous to computation. Non determinism is both a standard property of computational processes and a key feature of many important algorithms and the physical systems on which programs are run. But can a connection between non deterministic computation and ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied logic, 56:183 -- 220, 1992.

.... [Mi] The problem of extending this notion to higher order programs, like those of Godel system T, has led to various definitions: ffl Sequential algorithms on concrete data structures (see [C1] and more recently various gametheoretic models (see [AJ, C2, HO, L] inspired by the work of Blass [Bl1, Bl2], which are quite intentional models where programs of functional type are not simply interpreted by functions, but by more complicated objects ( algorithms or strategies ) which contain detailed informations about their behaviour. ffl Strong stability, a notion which coincides with ordinary ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic 56, pp. 183-220, 1992.

....a very strong way the intensional components of sequential algorithms. This approach has been developed by Berry and Curien [C1] and has now interesting developments in the direction of models of linear logic [L, C2, AJ] which have strong analogies with the game theoretic model proposed by Blass [Bl]. This 2 approach has been very successful since it has allowed for a new characterization of the fully abstract model of PCF [AJM, HO] With Bucciarelli, we also developed an abstract theory of sequential algorithms [BE3] The other approach consists in reformulating the Kahn Plotkin condition. ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic 56(1992), pp. 183-220.

....the implementation. Although not shown in this paper, non determinism is one of the key features of AsmL. It allows designers to clearly mark those areas where an implementation must make a decision. In AsmL, non determinsm is restricted; you can choose or quantify only over bounded sets [5]. AsmL specifications are model programs: they are operational specifications of the behavior expected of any implementation. Thus, they provide a minimal model by constraining implementations as little as possible. There are three main properties of AsmL that support this. 1. The ASM notion of ....

A. Blass, Y. Gurevich, and S. Shelah. Choiceless Polynomial Time. Annals of Pure and Applied Logic, 100:141--187, 1999.

....easier to understand, but it is worth acknowledging the work that others have undertaken on the basic concepts of games. Abramsky, Jagadeesan, and Malacaria [AJP94] are the originators of the specific form of games described in this report. Other types of games have also been proposed by Blass [Bla92], and Hyland and Ong [HO94, HO95] Games naturally incorporate a notion of linearity and there is a strong correspondence with Girard s linear logic [Gir87] The syntax chosen for the calculus associated with linear logic used in this report has been particularly influenced by the work of Wadler ....

Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56, 1992.

....of our main result: First, Immerman s question about the negation TChierarchy is answered. Second, it gives a new proof of our earlier result that (FO DTC) is weaker than (FO TC) 15] Third, it implies a hierarchy theorem for narrow Henkin quantifiers, via results of Blass and Gurevich [4]. Finally, it extends our earlier results in [14] on a non collapsing hierarchy inside (FO pos TC) formed by interleaving TC operators and universal quantifiers) In section 6 we consider the implications of our main result on the fine structure of stratified Datalog. In the last section, we ....

....from (FO TC) We show that on every class that satisfies a certain criterion, called the short path criterion, FO DTC) collapses to first order logic. One of the most important classes satisfying the short path criterion is the class of all hypercubes. Henkin quantifiers. Blass and Gurevich [4] have shown that transitive closure logic is equivalent to (FO NH) the extension of first order logic by narrow Henkin quantifiers. More precisely, positive occurrences of TC operators correspond to negative occurrences of narrow Henkin quantifiers, and vice versa. Thus, our results imply that ....

A. Blass and Y. Gurevich, Henkin quantifiers and complete problems, Annals of Pure and Applied Logic 32 (1986), 1--16.

....is, a logic in which handling of resources is explicitly taken into account. Thus, propositions are interpreted as games, connectives as operation on games and validity as the existence of a winning strategy for P. In order to combine both ideas using games, the first change provided by Blass [Bla92] concerns the length of a dialogue which is now allowed to take infinitely many moves. Specifically, an atomic formula is not an end point to a debate, on the contrary, it may be debatable like any other formula. Further, Blass interprets it as a method to point out the situation when both players ....

....output as an suitable interpretation for a formula and its dual respectively. Rule 2 concerns the starting player for a tensor product game. Considering O as the starting player mimics the intuition that the onus in showing that a certain statement is false is from opponent. This idea is clear in [Bla92] and has came up naturally in practical games. Rules 3 and 4 are the kernel of the resource consciousness. In not allowing players to re attack and re defend (and stipulating that a formula may be used 15 just once) is essential for achieving the proposed idea. One is able to see that these ....

Blass, A.; Game Semantics for Linear Logic. Annals of Pure and Applied Logic, 56:183--223, 1992.

....extensions of FO. In most cases, the 0 1 law no longer holds for FO(Q) where Q is a (set of) generalized quantifier(s) This is the case for instance of partially ordered quantifiers, such as Henkin quantifiers [Hen61] which yield the same expressive power as existential second order logic [BG86]. For example, the following sentence expresses the fact that the cardinality of the domain is even. 8x9v 8y9w 0 (x = y , v = w) x = w , y = v) v 6= x) 1 A : Indeed, it states that there exist two unary functions f and g, which are equal and one one; since the domain is ....

A. Blass and Y. Gurevich. Henkin quantifiers and complete problems. Annals of Pure and Applied Logic 32 (1986) 1--16.

....states, events, and resources [15, 28, 29] A connection between linear logic and randomized interaction was investigated in [22] which introduced a game semantics of a fragment of linear logic by means of probabilistic games. Other notions of game semantics for linear logic were considered in [8, 1, 2, 17, 20, 19, 11]. Another direction is emphasized here: linear logic proof game simulations of probabilistic games from complexity theory. Such simulations are then used to show that certain problems in linear logic that are known to be hard to decide are also hard to approximate. We shall consider so called ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....of cofinite sets, denoted by Fr. The families F and F are dual in a different sense; this means that a set X containing an element of each member of F (resp. F ) must belong to F (resp. F) In particular Fr and [ are dual. From more general work of Aczel ( 1] and Blass ([3]) there is a duality between games in which a player chooses X k 2 F while the other player responds with n k 2 X k , and games in which a player chooses Y k 2 F while the other player responds with n k 2 Y k . The point is that the statements (8X 2 F) 9n 2 X)OE(n) and (9Y 2 F ) 8n 2 Y ....

....supersets (therefore Z c is closed under subsets) the best strategy for II is to play finite sets as small as possible, namely singleton since a legal move must be nonempty. The next Lemma regards the duality mentioned in the introduction and is taken from the work of Aczel [1] and Blass (see [3], Theorem 1) We include a hint of the proof for completeness. Theorem 3.2. 1] 3] For a given filter F , the game G(F ; Z) and the game G(F ; Z c ) are dual; that is a player has a winning strategy in one of these games if and only if the other player has a winning strategy in ....

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A. Blass, Selective Ultrafilters and Homogeneity, Annals of Pure and Applied Logic 38 (1988), 215-255.

.... schemes) was shown to be a proper sub class of the polynomial time solvable problems, to contain every problem de ned by the sentences of inductive xed point logic and to contain problems not in the class of problems known as Choiceless 4 Polynomial Time, due to Blass, Gurevich and Shelah [6]. In this paper, we augment the basic class of program schemes NPS(1) with a queue and an extra numeric universe. We prove that when both augmentations are present, the class of problems accepted by the resulting class of program schemes NPSQ (1) is exactly the class of recursively solvable ....

A. Blass, Y. Gurevich and S. Shelah, Choiceless polynomial time, Annals of Pure and Applied Logic 100 (1999) 141-187.

....and presented it in the talk The logic of effective truth at the Logic and Computer Science conference in Marseille (June 1992) At the same conference I met Andreas Blass and learned that he had found, earlier than I, a very similar semantics. It is described in his remarkable paper [2], where the decidability of the following two fragments of the corresponding logic is established: 1. The multiplicative propositional fragment, i.e. the fragment that uses only the connectives : 5 and 4. 2. The fragment consisting of additive, i.e. 5 and 4 free, sequents (a sequent hOE 1 ; ....

....are considerable differences between Blass s and our approaches and, especially, the consequences of these approaches: 1. Our games are finite (every play has a finite length, that is) whereas Blass s games are infinite and this fact plays a crucial role in all partial completeness proofs in [2]. 4 This infiniteness makes things only second order definable while all the theory of our games of bounded depth, including the completeness proof for the logic of effective truth, can be formalized in Peano Arithmetic. 2. At the same time, Blass s semantics does not require that Proponent s ....

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A.Blass, A game semantics for linear logic. Annals of Pure and Applied Logic, v.56 (1992), pp. 183-220.

....plays finite or infinite , the nature of the board numbers, space , the nature of gain possibility of a draw, numerical gain i.e. money , the nature of the strategies e.g. memory free , to see that the question is not that simple. All extant notions, e.g. the games of Blass [5] validate the laws of linear logic, but also other laws that are sometimes completely unacceptable, e.g. classically wrong. 4.4 The school of Lorenzen It is only now that one starts to realize that Lorenzen [26] had around 1960 his own prefiguration of this program. His school remained ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....a two person game (played between the Proponent asserting the thesis, and the Opponent seeking to refute it) and a proof of the formula as a winning strategy for Proponent. Significant further contributions were made by Joyal [Joy77] who organized Conway s games into a category, and by Blass [Bla93], who gave a game semantics for Girard s Linear Logic. 1 In joint work with Radha Jagadeesan in 1992 [AJ92] a number of key ideas were introduced: ffl A dictionary between the concepts of game semantics, and those of process models of computation, in which the Player or Proponent is ....

A. Blass. A Game Semantics for Linear Logic. Annals of Pure and Applied Logic 56, 1993.

....3 we show how MALL proof structures are constructed from strategies, and in Section 4 we outline the proofs of the correctness criteria for these proof structures. Finally, Section 5 gives the main result. 2 The concurrent games model As a convenient point of departure, we begin with Blass games [Bla92]. These have the form Y i2I A i or X i2I A i where each A i is (co)inductively a Blass game. The idea is that in Q i2I A i , Opponent starts by playing some i 2 I , and play then proceeds as in A i . P i2I A i is defined dually, with Player making the initial move i 2 I . Thus these ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183-220, North-Holland, 1992.

....of (innocent) strategies which are denotations of terms. Here we have in mind the various kinds of tit fortat strategies in which P simply copies O moves from one component of the play to the other. Strategies of such nature occur also in various game models of linear logic; see e.g. [Bla92, AJ94, HO93]. It would be very useful to have a generic calculus capable of capturing a general class of such schematic strategies. It has been suggested to us that a calculus along the lines of Sangiorgi s higher order calculus [San93] may well fit our requirements, but we have not yet investigated the ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....these languages correspond to a Henkin quantifier H Sxyzw as follows. 8x9y(8z=x; y) 9w=x; y) Sxyzw j 8x 9y 8z 9w Sxyzw: Further, each complex Henkin quantifier prefix H can be defined by a quantifier prefix H of the following bifid form (2) see [Krynicki, 1993] qualifying that of [Blass Gurevich, 1986] and [Walkoe, 1970] 8x 1 : x n 9y 8z 1 : z m 9w (for some n; m 2 ) 2) 6 AHTI PIETARINEN AND GABRIEL SANDU We call sentences whose prefixes are put into this form Krynicki normal form for Henkin quantifiers. The formulas for the Krynicki normal forms for Henkin quantifiers ....

Blass, A. and Gurevich, Y.: Henkin quantifiers and complete problems, Annals of Pure and Applied Logic 32, pp. 1--16.

....3 we show how MALL proof structures are constructed from strategies, and in Section 4 we outline the proofs of the correctness criteria for these proof structures. Finally, Section 5 gives the main result. 2 The concurrent games model As a convenient point of departure, we begin with Blass games [Bla92]. These have the form Y i2I A i or X i2I A i where each A i is (co)inductively a Blass game. The idea is that in Q i2I A i , Opponent starts by playing some i 2 I , and play then proceeds as in A i . P i2I A i is defined dually, with Player making the initial move i 2 I . Thus these ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183-220, NorthHolland, 1992.

....one can try to plug a single loudspeaker to the two outputs plugs simultaneously ; maybe it works, maybe it explodes, but anyway the behaviour of such an experimental plugging is not covered by the guarantee : 2. 4 Game semantics Recently Blass introduced a semantics of linear logic, see [5], this volume. The semantics is far from being complete (i.e. it accepts additional principles) but this direction is promising. Let us forget the state of the art and let us focus on what could be the general pattern of a convincing game semantics. 2.4.1 Plays, strategies etc. Imagine a game ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....has truth valued entailment and resembles Birkhoff and von Neumann s quantum logic [BvN36] while the other three have the set valued entailment characteristic of categorical logic: A B as the set of morphisms from A to B. A number of other models have also been proposed. Inspired by Blass [Bla92], Abramsky and Jagadeesan [AJ94] have interpreted linear logic over sequential games, further studied by Hyland and Ong [HO93] Barr has proposed fuzzy relations as a model [Bar96] while Blute [Blu96] has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the ....

Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....at the practical level, certain control operators for functional programming languages [23] To some extent Girard s linear logic [21] has also renewed interest in game theory and functional programming languages. The refined connectives of linear logic have helped shed new light on work on games [12]. The games models have proved useful for programming language semantics: the recent fully abstract models of pcf [2, 26] are good examples of this. In addition, intuitionistic linear logic (ILL) has been proposed as a resource sensitive foundation of functional programming languages. Thus it ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

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Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992. 255

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Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

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Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992. 255

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A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56 (1992), p. 183--220

....other. This is similar to the motivation for the Geometry of Interaction programme [Gir89b, Gir89a, AJ92a] indeed, we shall exhibit strong connections between our semantics and the Geometry of Interaction. 1. 1 Overview of Results Blass has recently described a Game semantics for Linear Logic [Bla92b]. This has good claims to be the most intuitively appealing semantics for Linear Logic presented so far. However, there is a considerable gap between Blass semantics and Linear Logic: 1. The semantics validates Weakening, so he is actually modelling Affine logic. 2. Blass characterises validity ....

....in this paper is adopted, then one may as well work in G as in C. 6.2 Abstract Games De Paiva has studied the Dialectica Categories DC, and Linear categories GC [dP89] These are abstract constructions, but reflect some game theoretic intuitions. Indeed, Blass applies his game semantics to DC [Bla92b]. Again, Lafont and Streicher [LS91] have developed a Game Semantics for Linear Logic . An object in the category GameK is a structure (A ; A ; e) where e : A Theta A K, for some fixed set K. If we think of A as strategies for Player, A as counterstrategies and e as the payoff ....

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A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....and the semantics of proofs are investigated, with particular reference to games and Linear Logic. 1 Introduction We use Games and Logic as a mirror to understand an aspect of the sequentiality concurrency distinction. We begin with the simple, intuitive notion of polarized games due to Blass [Bla72, Bla92], which pre gured many of the ideas in Linear Logic [Gir87] and which can be seen as a polarized version of ideas familiar from process calculi such as CCS [Mil99] synchronization trees, pre xing, summation, the Expansion theorem) We analyze the shocking fact that this very clear and ....

....if G = GOH = G i OH) GOH j ) The reader can similarly write down the de nition for the Linear Implication. This notion of game tree, and the interpretations of the multiplicative and additive connectives of Linear Logic, correspond exactly to the game semantics of Andreas Blass [Bla92] (in the relaxed form in his terminology) Note that the additives provide the operations by which all such game trees can be constructed, and the multiplicatives are eliminated by the polarized versions of the Expansion Theorem given above. Hence the additives have primacy in this form of ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183220, North-Holland, 1992.

.... Semantics: For First Order Logic, truth in a model can be defined using game theoretic semantics (see, e.g. 66] A game theoretically natural extension of this semantics has led to Independence Friendly Logic [65, 109] Likewise, game theoretic semantics have been proposed for Linear Logic [21, 2]. Dialogue Games: The relation of logical consequence has been viewed as a dialogue game in [83] where the precise rules of the game determine whether one obtains intuitionistic or classical logical consequence. Extensions of this work have yielded dialogue games for various modal logics and ....

.... is [54] RPDL (which is also known as #PDL) and a closely related system LPDL (PDL loop) are studied in [61] Concurrent Propositional Dynamic Logic was introduced in [108] and followed up by [107] For textbooks on Process Algebra, see [49, 8] Game semantics for Linear Logic is discussed in [21, 2]. Game Over In this final chapter, we take another look at the relationship between Game Logic and Coalition Logic, showing that they embody two di#erent approaches to reasoning about multi agent systems. As shall be explained, the di#erence between these two approaches is in fact well known in ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....and the semantics of proofs are investigated, with particular reference to games and Linear Logic. 1 Introduction We use Games and Logic as a mirror to understand an aspect of the sequentiality concurrency distinction. We begin with the simple, intuitive notion of polarized games due to Blass [Bla72, Bla92], which pre gured many of the ideas in Linear Logic [Gir87] and which can be seen as a polarized version of ideas familiar from process calculi such as CCS [Mil99] synchronization trees, pre xing, summation, the Expansion theorem) We analyze the shocking fact that this very clear and ....

....GOH = a i2I (G i OH) a j2J (GOH j ) The reader can similarly write down the de nition for the Linear Implication. This notion of game tree, and the interpretations of the multiplicative and additive connectives of Linear Logic, correspond exactly to the game semantics of Andreas Blass [Bla92] (in the relaxed form in his terminology) Note that the additives provide the operations by which all such game trees can be constructed, and the multiplicatives are eliminated by the polarized versions of the Expansion Theorem given above. Hence the additives have primacy in this form of ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183-220, North-Holland, 1992.

....aspects like computational complexity. These observations beginning in 1992 in [AbrJag92] and independently in [HylOng92] led to the construction of very satisfactory game semantical models for linear logic (a resource sensitive logic introduced in [Gir87] another model had been given in [Bla92], but with non associative composition) These model were intensional in nature: thus the usual completeness results, stating that provability of a formula is reflected in the model, were strengthened to full completeness results where each proof is itself represented. Another games model for ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56 (1992), p. 183--220

....states, events, and resources [15, 28, 29] A connection between linear logic and randomized interaction was investigated in [22] which introduced a game semantics of a fragment of linear logic by means of probabilistic games. Other notions of game semantics for linear logic were considered in [8, 1, 2, 17, 20, 19, 11]. Another direction is emphasized here: linear logic proof game simulations of probabilistic games from complexity theory. Such simulations are then used to show that certain problems in linear logic that are known to be hard to decide are also hard to approximate. We shall consider so called ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, vol. 56 (1992), pp. 183--220.

....is manifest in the model means that games are particularly well suited to modelling linear GAMES AND DEFINABILITY FOR FPC 349 A # (A # B) # B b 1 b 1 a 1 a 1 a 2 a 2 b 2 b 2 . Figure 1. A winning strategy for A # (A # B) # B . logic [12] This was first noticed by Blass [8], whose ideas were refined by several others [2, 16, 22] leading to the first definability results. Games make the distinction between the additive conjunction ANB and the multiplicative conjunction A# B clear: in ANB , a play consists of a single play of either A or B , while in A# B , moves ....

A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, vol. 56 (1992).

....games where only Opponent may play the rst move. However this restriction prevents us from taking advantadge of the natural duality of game semantics, namely the symmetry Opponent Player. It has proved dicult to extend the game semantics to a satisfying model of linear negation (see in particular [3]) For example, the model of saturated strategies for MELL [2] is not completely satisfactory as it uses non deterministic strategies for modeling terms (and is therefore incomplete) In this paper we build a game model of the polarized fragment LLP of LL and introduce a polarization of ....

....not bifunctorial in the full category which corresponds to the premonoidal structure of control categories of P. Selinger [19] our notion of centrality is the same as Selinger s one, see corollary 2. 1) All this has also to be linked with the problem of constructions on strategies for Blass games [3], solved here by adding the centrality constraint. De nition 12 (Lifting of a strategy) Let : A P B be a strategy, the strategy # : A B is cl P (fb as j (a; b)s 2 g) 2 Linear Logic with Polarities (linear case) 2.1 MALLP This calculus is a linear fragment (without contraction ....

Andreas Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183-220, April 1992.

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A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56, 1992, pp. 183-220.

.... Linear logic, introduced in [10] is a refinement of classical logic often described as being resource sensitive because of its intrinsic ability to reflect computational states, events, and resources [11, 27, 28] Several notions of game semantics for linear logic are investigated in [6, 1, 2, 13, 17, 15, 9]. Connections between linear logic proof search and probabilistic games considered in complexity theory are investigated in [20, 21, 22] In particular, linear logic proof search may also be seen as a game. This game, the linear logic proof game, is played on linear logic formulas, and its moves ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

.... from the start [Gir87] the categorical model of coherence spaces played a prominent role in the presentation of linear logic, and since other categorical models, arguably of a more informative character, have appeared; notable are those based on game semantics, initiated by Blass s work (see [Bla92, AJ92]) Not everyone is happy with a categorical model as the explanation of a new logic, and in his pioneering paper [Gir87] Girard also gave a phase space semantics for linear logic. There have since been investigations of several other kinds of structures as models of linear logic, quantales ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 192.

....one of a forall loop. Note that unlike the program schemes mentioned in the previous paragraph, our program schemes are deterministic (and every problem accepted by such a program scheme is in P) A related model of computation has recently been examined by Blass, Gurevich and Shelah 1 . In [3], Blass, Gurevich and Shelah introduced a model of computation CPTime, Choiceless Polynomial Time, a program ( p(n) q(n) of which is an adapted Abstract State Machine (see [11, 12] augmented with two polynomial bounds, p(n) and q(n) with p(n) bounding the length of any run of the ....

....run time and the number of parallel executions, the program would have the capacity to build a computational domain of arbitrary size; and, indeed, it is not dicult to show that such an unrestricted program can simulate an arbitrary Turing machine. The instructions, or rules in the terminology of [3], of the programs of CPTime have similarities with those of the program schemes of this paper. For example: there are dynamic function symbols and assignments via update rules, whereas our program schemes have arrays and assignment blocks; there are conditional rules, whereas our program schemes ....

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A. Blass, Y. Gurevich and S. Shelah, Choiceless polynomial time, Annals of Pure and Applied Logic 100 (1999) 141-187.

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A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56, 1992, pp. 183-220.

....1 Introduction Abramsky and Jagadeesan [AJ92] henceforth abbreviated as AJ) have recently described a Game semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted by games, and proofs by winning strategies. Their work builds on an important work of Blass [Bla92] but improves on it in a number of ways. First, unlike Blass semantics which is in essence a Game semantics for Affine Logic, Weakening is invalidated in the AJ setting. Secondly, Blass s model actually only characterises classical propositional tautology, and it does not form a category, whereas ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

....other. This is similar to the motivation for the Geometry of Interaction programme [Gir89b, Gir89a, AJ92a] indeed, we shall exhibit strong connections between our semantics and the Geometry of Interaction. 1. 1 Overview of Results Blass has recently described a Game semantics for Linear Logic [Bla92b]. This has good claims to be the most intuitively appealing semantics for Linear Logic presented so far. However, there is a considerable gap between Blass semantics and Linear Logic: 1. The semantics validates Weakening, so he is actually modelling Affine logic. 2. Blass characterises validity ....

....in this paper is adopted, then one may as well work in G as in C. 6.2 Abstract Games De Paiva has studied the Dialectica Categories DC, and Linear categories GC [dP89] These are abstract constructions, but reflect some game theoretic intuitions. Indeed, Blass applies his game semantics to DC [Bla92b]. Again, Lafont and Streicher [LS91] have developed a Game Semantics for Linear Logic . An object in the category GameK is a structure (A ; A ; e) where e : A ThetaA K, for some fixed set K. If we think of A as strategies for Player, A as counter strategies and e as the ....

[Article contains additional citation context not shown here]

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183-- 220, 1992.

....other. This is similar to the motivation for the Geometry of Interaction programme [Gir89b, Gir89a, AJ92b] indeed, we shall exhibit strong connections between our semantics and the Geometry of Interaction. 1. 1 Overview of Results Blass has recently described a Game semantics for Linear Logic [Bla92b]. This has good claims to be the most intuitively appealing semantics for Linear Logic presented so far. However, there is a considerable gap between Blass semantics and Linear Logic: 1. The semantics validates Weakening, so he is actually modelling Affine logic. 2. Blass characterises validity ....

....in this paper is adopted, then one may as well work in G as in C. 6.2 Abstract Games De Paiva has studied the Dialectica Categories DC, and Linear categories GC [dP89] These are abstract constructions, but reflect some game theoretic intuitions. Indeed, Blass applies his game semantics to DC [Bla92b]. Again, Lafont and Streicher [LS91] have developed a Game Semantics for Linear Logic . An object in the category GameK is a structure (A ; A ; e) where e : A Theta A K, for some fixed set K. If we think of A as strategies for Player, A as counter strategies and e as the ....

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A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

.... Linear logic, introduced in [12] is a refinement of classical logic often described as being resource sensitive because of its intrinsic ability to reflect computational states, events, and resources [13, 32, 33, 23] Several notions of game semantics for linear logic are investigated in [6, 1, 2, 15, 19, 17, 9]. Connections between linear logic and probabilistic games considered in complexity theory are investigated in [24, 27, 25] In particular, linear logic proof search may also be seen as a game. This game, the linear logic proof game, is played on linear logic formulas, and its moves are instances ....

A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183--220, 1992.

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A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183-220, North-Holland, 1992.

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