M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....net is an abstract representation of a proof: the translation of cut free proofs into proof nets identifies proofs modulo inessential commutations of rules. The identifications have since been verified as canonical from a semantic perspective, with numerous full completeness results for MLL, e.g. [AJ94, HO93, Loa94, Tan97, BS96, DHPP99]. Furthermore, the identifications correspond to coherences of free star autonomous categories [BCST96] The problem of finding a satisfactory extension of the theory of proof nets to unit free multiplicative additive linear logic (MALL) has remained open since the inception of linear logic ....

J. M. E. HYLAND & C.-H. L. ONG (1993): Fair games and full completeness for multiplicative linear logic without the MIX-Rule. On Ong's web page.

....strategies as morphisms and show full completeness of MLL with MIX. But the MIX rule, from Gamma and Delta infer Gamma; Delta, although validated by most naturally arising models of linear logic, is nevertheless omitted from pure linear logic. Seeking the real McCoy, Hyland and Ong [6] modify AJ s model to refute MIX by disallowing certain plays they characterize as unfair. This work was supported by ONR under grant number N00014 92 J 1974 This paper proves full completeness of a modest fragment of MLL without MIX interpreted over Chu spaces. As a static object a Chu ....

....MLL for Chu spaces. Here we prove the converse for a fragment of MLL: every dinatural transformation between the interpretations of terms of this fragment interprets some proof. Our modest fragment of MLL restricts to conjunctive normal form (CNF) formulas (called semisimple by Hyland and Ong [6]) those expressible as the tensor product c : Omega c n of clauses each formed as a par c i = P 1 . ....

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J.M.E. Hyland and C.-H.L. Ong. Fair games and full completeness for multiplicative linear logic without the mix-rule. Available by ftp from ftp.comlab.ox.ac.uk as fcomplete.ps.gz in /pub/Documents/techpapers/Luke.Ong, 1993.

.... for now is that this can indeed be done using suitable uniformity and preservation properties in a number of semantic settings; this is the content of the various full completeness theorems for Multiplicative Linear Logic which have appeared over the past few years, starting with [AJ92b] see e.g. [HO92, BS96, Loa94a, Hag00]) The question is, how can this view of the multiplicatives, which has no scope for expressing causal dependencies (this is exactly the sense in which it is asynchronous) be reconciled with the additives, which as we have seen are essentially concerned with choice and causality We can in ....

.... , and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule) This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic, e.g. [HO92, BS96, Loa94a, Loa94b, Tan97, DHPP99, Hag00]. The proofs of full completeness which have appeared to date fall into two broad classes: Proofs using decomposition arguments There were a number of signi cant precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in ....

M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992.

.... for now is that this can indeed be done using suitable uniformity and preservation properties in a number of semantic settings; this is the content of the various full completeness theorems for Multiplicative Linear Logic which have appeared over the past few years, starting with [AJ92b] see e.g. [HO92, BS96, Loa94a, Hag00]) The question is, how can this view of the multiplicatives, which has no scope for expressing causal dependencies (this is exactly the sense in which it is asynchronous) be reconciled with the additives, which as we have seen are essentially concerned with choice and causality 1 We can in ....

.... 2 , and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule) This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic, e.g. [HO92, BS96, Loa94a, Loa94b, Tan97, DHPP99, Hag00]. The proofs of full completeness which have appeared to date fall into two broad classes: Proofs using decomposition arguments 2 There were a number of signi cant precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in ....

M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992.

.... by Vuillemin and Milner (cf. V] and [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, ....

J.M.E. Hyland and C.-H. L. Ong. Fair games and full completeness for multiplicative linear logic without the MIX-rule. Manuscript (1993).

....More speculatively, an intensional model of computation (i.e. one that reflects some properties of the process of computation) could perhaps also be used to model computation related 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, ....

J. M. E. Hyland, C.-H. L. Ong, Fair Games and Full Completeness for Multiplicative Linear Logic without the Mix-Rule, Unpublished Manuscript, 1993

....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 can be made on both sides. The linear arrow A # B ....

J. M. E. Hyland and C.-H. L. Ong, Fair games and full completeness for Multiplicative Linear Logic without the Mix-Rule, available from ftp://ftp.comlab.ox.ac.uk/pub/ Documents/techpapers/Luke.Ong, 1993.

....lax functors B C resulting in lax functors B cChu C cChu that preserve dualization are the isomorphisms. 5 Chu cells in rel : a connection with games Interactions and games have been studied to nd models for certain fragments of linear logic. Rather than the usual trees (cf. e.g. 6] 1] 2] [10], and [11] here we wish to use bipartite labeled state transition systems (LSTSs) of the form R : x y u v r0 r1 a b with two state sets r 0 (for Opponent) and r 1 (for Player) and two labeling relations r 0 r 1 0 a and r 1 r 0 1 b to be interpreted as moves. In fact, these are ....

Hyland, J. M. E., and Ong, C.-H. L. Fair games and full completeness for multiplicative linear logic without the mix-rule. Working Draft, July 1993.

....structures do not satisfy the middle interchange law. 0 Introduction In order to model certain fragments of linear logic, games based on alternating trees have been introduced as objects of symmetric monoidal closed categories with certain strategies as morphisms, cf. e.g. 3] 1] 2] [4], 5] In [6] we showed these games can also be viewed as 1 cells in a 2 category gam with sets as objects and inclusions as 2 cells. The 1 cell composition in gam was inspired by the relation product and appeared to be in some sense orthogonal to the composition of strategies. Here we show ....

Hyland, J. M. E., and Ong, C.-H. L. Fair games and full completeness for multiplicative linear logic without the mix-rule. Working Draft, July 1993.

....cannot be functorial on gam . Nevertheless, the composition of games may be viewed as orthogonal to the familiar composition of strategies in a common framework. 0 Introduction People who study games from a mathematical perspective (e.g. Blass, Abramsky, Hyland, Ong et al. cf. 3] 1] 2] [6], and [5] tend to think of games as certain kinds of trees. In fact, they construct categories with games as objects and strategies as morphisms. In contrast to that approach, here we wish to derive games as the natural choice of morphisms in a certain bicategory with sets as objects. This ....

Hyland, J. M. E., and Ong, C.-H. L. Fair games and full completeness for multiplicative linear logic without the mix-rule. Working Draft, July 1993.

....categorical description of the structure that leads to the monoidal closed category is even more satisfying. In particular, we then obtain an explicit involution. 0 Introduction People who study games from a mathematical perspective (e.g. Blass, Abramsky, Hyland, Ong et al. cf. 3] 1] 2] [6], and [5] tend to think of games as certain kinds of trees. In fact, they construct categories with games as objects and strategies as morphisms. In contrast to that approach, here we wish to derive games as the natural choice of morphisms in a certain bicategory with sets as objects. This ....

Hyland, J. M. E., and Ong, C.-H. L. Fair games and full completeness for multiplicative linear logic without the mix-rule. Working Draft, July 1993.

....a stronger result than mere completeness. Not only does the existence of winning strategies (in their sense) imply provability in multiplicative linear logic plus MIX, but every strategy comes from a proof. This property is called full completeness. It should also be mentioned that Hyland and Ong [19] subsequently found a further modification that eliminates the need for adding MIX to the logic. The first of Abramsky s and Jagadeesan s modifications of game semantics is a uniformity requirement on strategies. Consider a sequent # and its various game interpretations, obtained by interpreting ....

J. Martin E. Hyland and C.-H. Luke Ong. Fair games and full completeness for multiplicative linear logic without the MIX-rule. preprint, 1993.

....problems for the semantics of programming languages. Full completeness is also of considerable interest for logic itself, and in fact this paper spawned a (still growing) literature on full completeness results for the same or similar fragments of Linear Logic with respect to a variety of models [HO92, Loa94, BS96]. Full Abstraction for PCF Motivated by the full completeness results, it became of compelling interest to re examine perhaps the best known open problem in the semantics of programming languages, namely the Full Abstraction problem for PCF , using the new tools provided by game semantics. 2 ....

M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992. 5

.... and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule) This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic (MLL) [HO92, BS96, Loa94a, Loa94b]. However, there have been no results for logics beyond the (very weak) multiplicative fragment of Linear Logic. In this paper, we make a first significant extension beyond the multiplicative fragment, by proving that the concurrent games model is fully complete for Multiplicative Additive Linear ....

M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992.

....parameters of the theorem, possibly subject to given axioms. More recently analogous semantic notions of abstract constructive proof have begun to appear, in particular natural and dinatural transformations [LS86, BS96] and related notions such as logical transformations [Plo80] game strategies [AJ94, HO93], and uniformity conditions [Loa94] The naturality condition expresses transformational invariance for all transformations of the parameters of the proof, again possibly subject to given axioms. The interpretation of natural transformations as constructive proofs is suggested by the the ....

J.M.E. Hyland and C.-H.L. Ong. Fair games and full completeness for multiplicative linear logic without the mix-rule. Available by ftp from ftp.comlab.ox.ac.uk as fcomplete.ps.gz in /pub/Documents/techpapers/Luke.Ong, 1993.

....but nothing of this sort seems to be implicit in the formalism or the underlying intuitions of linear logic. But see the discussion of A below. As suggested by the title of this paper, the protocols considered here can be viewed as games (or debates or dialogs) between the client and the server [1, 2, 10, 11]. In this connection, the server is usually called the proponent or player, and the client is called the opponent. The protocol specifies who is to move (see Note 2) and what moves are legal at any point during a play of the game. Our protocols, unlike some versions of games [1, 2] but like the ....

....logic than option (2) In fact, if we were to adopt the implication corresponding to option (2) then the implication corresponding to (1) would be deducible. Finally, we might give an argument from consensus for option (1) namely that this option was adopted by all authors on game semantics [1, 2, 3, 10, 11]. Additional arguments will arise when we consider duality in the next section. We turn next to the description of the data type A ( B. Recall that this is the type of linear functions from A to B. Thus the following description of its access protocol (and data delivery) seems reasonable. A ....

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J. M. E. Hyland and C.-H. L. Ong, Fair games and full completeness for multiplicative linear logic without the MIX-rule, preprint (1993).

.... the dinatural transformations between such terms in Chu(Set; 2) are in exact correspondence with the cut free proofs (understood suitably abstractly) of multiplicative linear logic without MIX, a full completeness result in the sense of Abramsky and Jagadeesan [1] and Hyland and Pratt Ong [11], but with their game semantics replaced by dinaturality semantics along the lines of Blute and Scott [4,5] This summer with Gordon Plotkin we have been able to remove the restriction on two variables. This entailed strengthening dinaturality to logicality, necessitated by the presence of at ....

J.M.E. Hyland and C.-H.L. Ong. Fair games and full completeness for multiplicative linear logic without the mix-rule. Available by ftp from ftp.comlab.ox.ac.uk as fcomplete.ps.gz in /pub/Documents/techpapers/Luke.Ong, 1993.

....explain our ideas. We believe we can extend discreet games to model multiplicative LLL without diculty: games for the classical system (i.e. extended by involutive negation) can be obtained by admitting positions that begin with P moves; weakening can be invalidated by either introducing fairness [5] at the level of positions or exhaustion [8] at the level of strategies. 2 IMLAL Intuitionistic Multiplicative Light Ane Logic (IMLAL) formulas are generated from atoms a; b; c; by the connectives , x (read whisper ) and (read shriek ) Ane here means that the weakening rule is ....

Hyland, J. M. E., Ong, C.-H. L.: Fair games and full completeness for Multiplicative Linear Logic without the MIX-rule. Preprint (1993)

....explain our ideas. We believe we can extend discreet games to model multiplicative LLL without diculty: games for the classical system (i.e. extended by involutive negation) can be obtained by admitting positions that begin with P moves; weakening can be invalidated by either introducing fairness [5] at the level of positions or exhaustion [8] at the level of strategies. 2 IMLAL Intuitionistic Multiplicative Light Ane Logic (IMLAL) formulas are generated from atoms a; b; c; by the connectives , x (read whisper ) and (read shriek ) Ane here means that the weakening rule is ....

J. M. E. Hyland and C.-H. L. Ong. Fair games and full completeness for Multiplicative Linear Logic without the MIX-rule. Preprint, 1993.

....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 ....

J. M. E. Hyland and C.-H. L. Ong. Fair games and full completeness for Multiplicative Linear Logic without the mix-rule. ftp-able at theory.doc.ic.ac.uk in directory papers/Ong, 1993.

....the model is fully complete for MLL augmented 1 andrzej comlab.ox.ac.uk 2 Homepage: http: www.comlab.ox.ac.uk oucl people luke.ong.html Email: Luke.Ong comlab.ox.ac.uk Preprint submitted to Elsevier Preprint 30 July by the Mix rule 3 . Soon after the result was announced, Hyland and Ong [5] constructed a fully complete model for MLL proper, using what they call fair games (which invalidate the Mix rule) In both models, the treatment of units is unsatisfactory. In the former, the full completeness result really only applies to the unit free fragment of the logic; the situation is ....

....criterion of essential nets. The algorithm can be extended to a linear time tautology checker for MLL. 5 Connexions with Fair Games For the purpose of comparison, we take a slight detour and briefly sketch another fully complete model for IMLL Gamma based on the notion of fair games [5]. Fairness so restricts positions on linear function space games A ( B that players can reach a maximal position in B only if a maximal position has already been reached in A in the same play so far. Definition 5.1. A fair game G is a triple h MG ; G ; FG i where MG is a non empty set of an even ....

[Article contains additional citation context not shown here]

J. M. E. Hyland and C.-H. L. Ong. Fair games and full completeness for Multiplicative Linear Logic without the MIX-rule. preprint, 1993.

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M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992.

No context found.

J. M. E. HYLAND & C.-H. L. ONG (1993): Fair games and full completeness for multiplicative linear logic without the MIX-Rule. On Ong's web page.

No context found.

J. M. E. Hyland, C.-H. L. Ong, Fair Games and Full Completeness for Multiplicative Linear Logic without the Mix-Rule, Unpublished Manuscript, 1993

No context found.

M. Hyland and L. Ong. Fair games and full completeness for Multiplicative Linear Logic without the Mix rule. ftp-able at theory.doc.ic.ac.uk in papers/Ong, 1993.

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