Yves Lafont and Thomas Streicher. Games semantics for linear logic. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, July 1991. IEEE Computer Society Press, Los Amitos, California.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....lattices, and categories. This is because the closed sets of a comonoid are the open sets of another comonoid on the same set of points. The participants in this coalgebra workshop would recognize it most readily as a comonoid (A, #, #) in chu, the monoidal category of (bi)extensional Chu spaces [1,4,3,8], where is such a Chu space and # : A# # : A are Chu morphisms satisfying the coassociativity and two counit equations. Compare this with the notion of monoid in a monoidal category , I) as a triple (A, #) where is an object of C and : are morphisms of C satisfying the ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

....and Par is expressed thus in terms of the trip condition ( Gir87] Introduction, Section III.4.3) ffl In the case of Omega there is no cooperation: if we start with A , then we come back through A before entering B after which we come back through B . This example is taken from [LS91], but the same idea can be found in [Con76] 9 ffl in the case of there is cooperation: if we start again with A , then we are expected through B , from which we go to B and eventually come back through A . Thus we get the following possible transitions in trips: B: A A B B ....

....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 function, we see some connection ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. Sixth Annual Symposium on Logic in Computer Science, pages 43--51. Computer Society Press, 1991.

....# A # B A # A B # B A, A # B # B (#, left) #,A # B (cut) 2. 1 Categorical Interpretation of Linear logic Having already introduced the syntax and the proof theory of linear logic, we are now ready to present its linear categorical semantics which has been mostly developed by Lafont [12] and Seely[17] with contributions of several other people ( 15] 14] 1] The linear logic model which is adopted in this paper is a simple axiomatization of linear category known as a closed symmetrical monoidal category with the notion of a dualizing object. We recall here the notion of a ....

Y. Lafont, T. Streicher, Games Semantics for Linear Logic, in Proc. Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, July 91, pp. 43-50.

....data specialization. 1 Introduction This paper describes an attempt to mix parallelism and partial evaluation. This is motivated by the increasing interest for parallelism based on linear logic [Gir87] The geometry of interaction [Lam90, GAL92, Abr90] interaction nets [Laf90] games semantics [Bla, LS91] are promising theories for understanding concurrency and relation between processes. We have already worked on partial evaluation of interaction nets [Bec92] However, here, we are interesting to know if classical partial evaluation (i.e. based on binding time analysis [J#r92, Con88, JSS89] is ....

Y. Lafont and T. Streicher. Game semantics for linear logic. In Sixth IEEE Symposium on Logic in Computer Science. IEEE, 1991.

....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 linear logic was given in [Lam94] while the ones in [LafStr91] or [Mey94] the latter for predicate logic without contractions) are not intensional. Subsequently this lead in 1993 to the development of intensional game theoretical models in the semantics of programming languages independently by [AbrJagMal94, HylOng1, Ni96] These models proved to be very ....

Y. Lafont, T. Streicher, Game Semantics for Linear Logic, in: Proceedings of the Sixth International Symposium on Logic in Computer Science, Computer Society Press of the IEEE, 1991, p. 43--50

.... to coherent domains, 3 are now understood to yield mathematical models for linear logic proofs, more precisely, for the relation t is a linear logic proof of a formula A [26, 88, 18, 19] Other versions of game semantics are given by Abramsky and Jagadeesan [2] and by Lafont and Streicher [65]. Event spaces, which come about from Pratt s work in semantics of concurrency, also provide models for certain linear logic proofs [82] Models investigated by de Paiva [39] are motivated by important proof theoretic transformations. A mathematical model for the linear logic provability relation ....

Y. Lafont and T. Streicher. Game semantics for linear logic. In Proc. 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 43--50. IEEE Computer Society Press, Los Alamitos, California, July 1991.

....one by: A ( B A A B B A; A ( B B ( left) A B (cut) 2. 1 Categorical Interpretation of Linear logic Having already introduced the syntax and the proof theory of linear logic, we are now ready to present its linear categorical semantics which has been mostly developed by Lafont [12] and Seely[17] with contributions of several other people ( 15] 14] 1] The linear logic model which is adopted in this paper is a simple axiomatization of linear category known as a closed symmetrical monoidal category with the notion of a dualizing object. We recall here the notion of a ....

Y. Lafont, T. Streicher, Games Semantics for Linear Logic, in Proc. Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, July 91, pp. 43-50.

....moves, and the game acquires the usual von Neumann Morgenstern form, with the client trying to minimize and the programmer to maximize this gain. The intuitive and logical meaning of the pairs of arrows in opposite directions, like in the adaptation morphisms, has been analyzed in this context in [6], connecting games, linear logic and the Chu construction. In any case, semantics of speci cation carrying programs must draw ideas and structures from sources as varied as institutions and game theory, although the goals and methods in each case di er essentially. On the level of abstract ....

Y. Lafont and T. Streicher, Games semantics for linear logic. Proc 6 th LICS Conf. (IEEE 1991) 43-49

....coherence spaces and of perp and tensor of coherence spaces see e.g. Girard [Gir87] For definiteness we define a coherence space to be its web (V; R) a set V of vertices or tokens together with a symmetric reflexive binary relation R V 2 constituting an undirected graph. Lafont and Streicher [LS91] exhibit the following full embedding of Coh in Chu(Set; 2) 3 Definition 3 The Coh embedding in Chu is a functor G : Coh Chu representing (V; R) as the Chu space (A; r; X) where A is the set of cliques a V of the web, X the set of anticliques x V , and r(a; x) ja xj 1 for each ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

....Chu spaces while broadening the applicability of linear logic. In outline, our proof begins with semisimple (par of tensors) MLL formulas A, and associates a MIX proof net with every Chu logical transformation j of A. We pull j back along the Lafont Streicher embedding (LS) of coherence spaces [LS91] to yield a dinatural b j, then appeal to full completeness for MLL with MIX [Tan97] to obtain (Section 3.1) Next we show that determines j not only in the LS image but also beyond, by using logical relations to tie down its behaviour at an arbitrary Chu space via the LS image of its ....

....(u; v) 2 l. Tensor product: jU Omega V j = jU j Theta jV j with (u; v) U Omega V (u 0 ; v 0 ) iff u U u 0 and v V v. Tensor unit: jI j = f g, with (necessarily) I . Linear maps I X correspond to cliques of X . 2.5. 1 The Lafont Streicher embedding Lafont and Streicher [LS91] exhibit a full and faithful functor LS : Coh Chu. Points are cliques, states are anticliques, and matrix entries are given by intersection: LS(U) U ffl ; u ; U ffi ) where a u x = ja xj. Note that clique and an anticlique can intersect in at most one point. On linear maps l : U ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

....of PTIME as the functions definable in certain subsystems of SLR. The proofs of these results are only briefly sketched in this paper; the full version will appear elsewhere [11] The technique to obtain such results is an interpretation of SLR in various models based on presheaves and Chu spaces [18, 12]. These models can serve as a justification of the system in their own right. 2 Syntax In this section we define types, contexts, and expressions and give the rules for typing and subtyping. 2.1 Types and subtyping The type expressions of the calculus SLR are given by the following grammar. A; ....

Yves Lafont and Thomas Streicher. Game semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, editor, Logic in Computer Science (LICS), pages 43--49, 1991.

....Chu transforms is a full embedding of the category of the former in that of the latter. Furthermore the embedding is concrete, meaning that the representing Chu spaces have the same carriers as the algebras they represent and transform via the same functions. As pointed out by Lafont and Streicher [14], topological spaces can be represented as Chu spaces in the obvious way, a full embedding of Top in Chu(Set; 2) Combining that embedding with the above, and generalizing to structures with multiple relations and multiple sorts as per [23] demonstrates that Chu spaces can model all relational ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

.... and in turn inspired related systems by Chirimar, Gunter, and Riecke [CGR92] Lincoln and Mitchell [LM92] Mackie [Mac91] Troelstra [Tro92] and Wadler [Wad90, Wad91] Seely provided a categorical model, that subsumes other models such as coherence spaces [Gir87] event spaces [Pra91] games [LS91], and the Geometry of Interaction [AJ92] Unfortunately, Abramsky s syntax is incoherent with Seely s semantics: different derivations of the same term may yield different semantics. The basic problem is that Promotion does not commute with substitution. All of the above syntaxes suffer from a ....

Y. Lafont and T. Streicher. Game semantics for linear logic. In 6'th Symposium on Logic in Computer Science, IEEE Press, Amsterdam, July 1991.

....to the usual operations for concurrent processes. Finally we connect them with event structures, showing how any event structure can be represented as an eks. These structures have previously been studied by Barr[Bar79] as instances of Chu s construction on sets, and by Lafont and Streicher[LS91] as games over 2. Brown and Gurr[BG90] have used similar structures to study Petri nets. 1.1 Theory of an eks An eks (E; Q) is a set Q of subsets of 2 E . We can therefore write it out as a formula in Disjunctive Normal Form in propositional logic, with variables E and clauses Q, with each ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

....(2) this implies that the proof structure ( Gamma; OE oe ) is in fact a proof net. 5 Further Directions Extension to Infinite Games [Preliminary calculations suggest that the obvious extension of the definition works ] Big gaps here. Related Work Old work: Con76] Joy77] Hyl90] [LS91] Lamarche: Lam93] Concrete Data Structure and Sequential Algorithms: Cur93] Acknowledgements We would like to thank members of the Cambridge Imperial Joint Seminars on Game Semantics for helpful discussions. ....

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. Sixth Annual IEEE Symp. Logic in Computer Science, pages 42--51, Computer Society Press, 1991.

....functions. This is most easily seen using the form rowA (g(y) rowB (y) ffi f of the adjointness condition, with composition (of functions B 2 as the open sets of B) with f (yielding functions A 2) being exactly the inverse image function f Gamma1 : 2 B 2 A . Lafont and Streicher [LS91] mention in passing this realization along with that of Girard s coherent spaces [Gir87] also in Chu 2 , and the realization of vector spaces over the field K in Chu U(K) Locales. A spatial locale [Isb72, Joh82] is a column maximal T 0 topological space, one to which no point can be added ....

.... a wide range of quality of palettes, from much worse than vector spaces (e.g. sets) to much better (e.g. Boolean algebras) Vector spaces fall exactly in the middle of this spectrum; indeed the n dimensional vector space over GF (2) is representable as a square (whence in the middle ) Chu space [LS91] with 2 n points and 2 n states (the dual points or functionals in the usual sense of vector spaces) with Chu transforms between vector spaces so represented being exactly their linear transformations. The canonical choice of nontrivial palette B is the two point one state Boolean algebra = ....

[Article contains additional citation context not shown here]

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

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Yves Lafont and Thomas Streicher. Games semantics for linear logic. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, July 1991. IEEE Computer Society Press, Los Amitos, California.

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Yves Lafont and Thomas Streicher. Game semantics for linear logic. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science (LICS 91), pages 43--50, 1991.

No context found.

Y. Lafont and T. Streicher. Game semantics for linear logic. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science (LICS 91), pages 43--50, 1991.

No context found.

Yves Lafont and Thomas Streicher. Game semantics for linear logic. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science (LICS 91), pages 43--50, 1991.

No context found.

Y. Lafont and T. Streicher. Game semantics for linear logic. In Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science (LICS 91), pages 43--50, 1991.

No context found.

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

No context found.

Y. Lafont,T.Streicher, Game Semantics for Linear Logic, in: Proceedings of the Sixth International Symposium on Logic in Computer Science, Computer Society Press of the IEEE, 1991, p. 43--50

No context found.

Y. Lafont and T. Streicher. Games semantics for linear logic. In Proc. 6th Annual IEEE Symp. on Logic in Computer Science, pages 43--49, Amsterdam, July 1991.

No context found.

Yves Lafont and Thomas Streicher. Games semantics for linear logic. In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, July 1991. IEEE Computer Society Press, Los Amitos, California.

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