## Sequentiality vs. Concurrency in Games and Logic (2001)

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Venue: | Math. Structures Comput. Sci |

Citations: | 13 - 0 self |

### BibTeX

@ARTICLE{Abramsky01sequentialityvs.,

author = {Samson Abramsky},

title = {Sequentiality vs. Concurrency in Games and Logic},

journal = {Math. Structures Comput. Sci},

year = {2001},

volume = {13},

pages = {2003}

}

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### Abstract

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

### Citations

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Citation Context ...rministic synchronization trees inductively, as given by expressions of the form X i2I a i :P i (i 6= j ) a i 6= a j ) i.e. as the least set closed under the operation of disjointly guarded summation =-=[Mil89-=-]. It is understood that the summation, as in CCS, is associative and commutative (idempotence does not arise because of the disjointness condition). The basic case of the inductive denition is given ... |

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Citation Context ...mains unclear: is there true concurrency lurking somewhere in the computational interpretation of classical logic? It would also be interesting to look at bounded systems such as Light Linear Logic [=-=Gir9-=-8], and the connections to complexity classes. Appendix: a brief review of MALL The formulas of the system are built from literals, i.e. propositional atoms ,s, . . . and their negations ? ,s? , . . ... |

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Citation Context ...x ! P ) ! P [v=x]: This property of persistence of outputs (which corresponds to the information increasingness in the domain-theoretic presentation) is more akin to concurrent constraint programming =-=[Sar93-=-]. We can connect this operational view very directly to our view of strategies as closure operators by dening a denotational semantics for terms of our process calculus as closure operators, and prov... |

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320 |
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Citation Context ...choices, and hence restoring con uence. This approach is typied by systems such as Girard's LC [Gir91], the systems LKT and LKQ of Danos, Joinet and Schellinx [DJS96], and the -calculus of Parigot [Pa=-=r92-=-]. 2. Study the full, `untamed' non-con uent calculus, andsnd the computational structure which is there. This is typied by work such as [BB96, Urb00, Lai01]. In a similar fashion, there seem to be tw... |

262 | Models for concurrency
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Citation Context ... and ba, taking account of the order in which the `events' a and b occur. In a true concurrency model, if the events a and b are causally idependent of each other, we can identify these two histories =-=[WN95]-=-. The purpose of proof nets [Gir87, Gir95a] and Geometry of Interaction [Gir89, Gir90, Gir95, DR93, DR95] was tosnd a more intrinsic representation of proofs in which the spurious ordering of rules im... |

240 |
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Citation Context ...polarized games due to Blass [Bla72, Bla92], which pregured 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, prexing, summation, the Expansion theorem). We analyze the `shocking' fact that this very clear and intuitive idea leads to a non-associative composition; a kind of incompati... |

209 | and full completeness for multiplicative linear logic
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Citation Context ...tice of referring to the two players as P (`Proponent') and O (`Opponent'). In process terms, we can think of P as representing the System currently under consideration, and O as its Environment (see =-=[AJ92b-=-] for a more detailed account of the correspondence between notions of game theory and process theory). Formally, we take games to be inductively (or, if preferred, coinductively) dened to be either a... |

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Citation Context ...tion. One can go beyond the scope of Intuitionistic Linear Logic, and study a more expressive, yet still `tamed' system of Linear Logic. This is Linear Logic with focussing, as introduced by Andreoli =-=[AP91]-=-, and used in the current work by Girard on Ludics [Gir01]. We shall study this approach in the next section. 2. Alternatively, we can extend the semantics to embrace true concurrency, and hence overc... |

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160 |
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Citation Context ...tax, by restricting the permitted reductions, or by adding information to resolve the non-deterministic choices, and hence restoring con uence. This approach is typied by systems such as Girard's LC [=-=Gir9-=-1], the systems LKT and LKQ of Danos, Joinet and Schellinx [DJS96], and the -calculus of Parigot [Par92]. 2. Study the full, `untamed' non-con uent calculus, andsnd the computational structure which i... |

131 |
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Citation Context ...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 v... |

103 | A new deconstructive logic: Linear logic
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Citation Context ...mation to resolve the non-deterministic choices, and hence restoring con uence. This approach is typied by systems such as Girard's LC [Gir91], the systems LKT and LKQ of Danos, Joinet and Schellinx [=-=DJS-=-96], and the -calculus of Parigot [Par92]. 2. Study the full, `untamed' non-con uent calculus, andsnd the computational structure which is there. This is typied by work such as [BB96, Urb00, Lai01]. I... |

99 |
Geometry of interaction I: Interpretation of system f
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Citation Context ...ations. Note that in any symmetric monoidal category, the symmetric group S(n) acts in a canonical way on the tensor power n O A = A : : : A | {z } n : In the original form of Geometry of Interaction =-=[Gir89-=-] (dubbed `particle style GoI' in [Abr96]) the monoidal structure used for the representation is coproduct (disjoint union). In our setting, we can use products to exactly the same eect. Thus in our e... |

88 | Proof-nets : the parallel syntax for proof theory. Logic and Algebra
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Citation Context ...lly complete model for Linear Logic in its original form (i.e. not `hyper-sequentialized'), and indeed the correspondence is with proof-nets, the `parallel syntax for proof theory' in Girard's phrase =-=[Gir95a]-=-. In this way, the distinction between sequentiality and concurrency is re ected at a fundamental level, in the analysis of the `space of proofs'. The main aim of the present paper is to expose some o... |

75 |
Categorical combinators, sequential algorithms, and functional programming
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Citation Context ...l space of strategies. Various sub-classes of S are important. In particular, there are constraints on strategies which correspond to important computational properties. One of these is sequentiality =-=[C-=-ur86]. A closure operator 2 S has an output function f : G ! G if 8x 2 dom(): (x) = f(x) _ x 25 where dom() = fx 2 G j (x) 6= >g: We recall the domain-theoretic conditions of stability and sequential... |

71 | New foundations for the Geometry of Interaction
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- 1994
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Citation Context ...hould be noted that mathematically, the concurrent games model is merely a rephrasing of the `New Foundations for the Geometry of Interaction' introduced by the present author and Radha Jagadeesan in =-=[AJ92a]-=-. As demonstrated in [Abr96, AHS00], this and the original form of geometry of interaction in [Gir89] are indeed both instances of a single general scheme, as far as the interpretation of the multipli... |

67 | A symmetric lambda calculus for classical program extraction - Barbanera, Berardi - 1996 |

65 | Retracing some paths in process algebra
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Citation Context ...dal category, the symmetric group S(n) acts in a canonical way on the tensor power n O A = A : : : A | {z } n : In the original form of Geometry of Interaction [Gir89] (dubbed `particle style GoI' in =-=[A-=-br96]) the monoidal structure used for the representation is coproduct (disjoint union). In our setting, we can use products to exactly the same eect. Thus in our example of the sequent ? O ? ; the... |

59 | Classical Logic and Computation - Urban - 2000 |

49 | Game semantics - Abramksy, McCusker - 1998 |

49 | Concurrent games and full completeness - Abramsky, Melliès - 1999 |

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38 |
A.: Categories, Allegories
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Citation Context ...d to date fall into two broad classes: Proofs using decomposition arguments 2 There were a number of signicant precursors, as noted in [AJ92b], including representation theorems in category theory [F=-=S9-=-1], full abstraction results in programming language semantics [Mil75, Plo77], Plotkin's characterization of denability in the -calculus using logical relations [Plo80], studies of parametric polymorp... |

36 | Semantics of Interaction
- Abramsky
- 1997
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Citation Context ...eed to dene the notion of winning strategy in the usual fashion. For non-well-founded game trees, innite plays are possible, and we must add information to say who wins these. For more on this, see [A=-=br97]-=-, and for further development of the connection between processes, games and Linear Logic, see [Abr00]. We have already developed these ideas more than enough for our purposes in this paper. 3 Composi... |

35 | Concrete domains
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Citation Context ... conditions of stability and sequentiality [AC98]. These make sense because our underlying domain G is simple and concrete|it is in fact a `concrete domain' in the technical sense of Kahn and Plotkin =-=[KP78-=-]. Indeed, because it is simply a (countable) product ofsat domains (in fact, G = N ? ! ), we can use Vuillemin's original denition of sequentiality [Vui73]: a function f : G ! G is sequential if: 8c ... |

32 | A semantic analysis of control - Laird - 1999 |

30 | Degrees of indeterminacy of games - Blass - 1972 |

30 | A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations (preliminary extended abstract - Ong - 1996 |

29 | Geometry of interaction III : accommodating the additives
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Citation Context ...nteraction can be faithfully represented. By being a little more abstract, an elegant account meeting all these desiderata can be found. 1 The extension of Geometry of Interaction to the additives in =-=[Gir95-=-], while containing much of interest, is generally agreed not to oer a fully satisfactory treatment. 17 7.2 A domain-theoretic formalization of games and strategies Let us begin by revisiting Blass ga... |

25 | Finitary PCF is not decidable
- Loader
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Citation Context ...e such as PCF (the most elegant way of doing this uses realizability ideas [Abr00]), then it is fully abstract, and even universal if we restrict to eective strategies. Moreover, by Loader's results [=-=Loa96]-=-, this is about as good a characterization of sequentiality as we are likely to get. Let us give asrst indication of how we can use our simple universal (type-free) space of strategies to recover the ... |

21 | Full completeness of the multiplicative linear logic of chu spaces - Devarajan, Hughes, et al. - 1999 |

19 | Linear Lauchli semantics - Blute, Scott - 1996 |

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16 |
Domain Theory. Handbook of Logic
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Citation Context ...me board, into which any other (subject to countability) can be embedded. The positions are then partial functions on natural numbers, which under inclusion form one of the basic examples of a domain =-=[AJ94]-=-. Moreover, as play progresses, and more moves are made, the position increases in the natural information ordering: f(c 1 ; v 1 ); (c 2 ; v 2 )g (c 3 ;v 3 ) ! f(c 1 ; v 1 ); (c 2 ; v 2 ); (c 3 ; v 3 ... |

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15 |
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Citation Context ...' in the technical sense of Kahn and Plotkin [KP78]. Indeed, because it is simply a (countable) product ofsat domains (in fact, G = N ? ! ), we can use Vuillemin's original denition of sequentiality [=-=Vui73]: a function -=-f : G ! G is sequential if: 8c 2 C: 8x 2 G : f(x)(c)" ) 8y x: f(y)(c)" _ (9c 0 : x(c 0 )" ^ 8y x: f(y)(c)# ) y(c 0 )#): There says that there is some particular cell in the input whic... |

12 | Local and asynchronous beta-reduction (an analysis of Girard’s execution formula - Danos, Regnier - 1993 |

12 | Linear Logic, Totality and Full Completeness - Loader |

11 | Models of lambda calculi and linear logic: structural, equational and proof-theoretic characterisations - Loader - 1994 |

11 | Exhausting strategies, joker games and full completeness for IMLL with Unit. Theoretical Computer Science - Murawski, Ong - 2003 |

11 |
Concrete domains. Theoretical Computer Science, Böhm Festschrift special issue
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Citation Context ... conditions of stability and sequentiality [AC98]. These make sense because our underlying domain G is simple and concrete|it is in fact a `concrete domain' in the technical sense of Kahn and Plotkin =-=[KP78-=-]. Indeed, because it is simply a (countable) product ofsat domains (in fact, G = N ? ! ), we can use Vuillemin's original denition of sequentiality [Vui73]: a function f : G ! G is sequential if: 8c ... |

10 |
Non-sequential processes
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Citation Context ... seen are essentially concerned with choice and causality? 1 We can in fact characterize the behavioural features of the various levels of connectives in Linear Logic using the concepts of Net Theory =-=[Pet7-=-7]: the multiplicatives express concurrency (i.e. pure causal independence) the additives express con ict and causality (i.e. choice and sequencing). (This is of course just the `true concurrency' p... |

9 | Process realizability
- Abramsky
- 2000
(Show Context)
Citation Context ...nite plays are possible, and we must add information to say who wins these. For more on this, see [Abr97], and for further development of the connection between processes, games and Linear Logic, see =-=[Abr00]-=-. We have already developed these ideas more than enough for our purposes in this paper. 3 Composition We now turn to the crucial issue of composition of strategies. It seems that, with our simple but... |

9 | Algebraic types in PER models - Hyland, Robinson, et al. |

6 |
Rendiconti del Seminario Matematico dell' Universit'a e Politecnico Torino
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Citation Context ...ve proofs are represented by permutations acting on structures of some kind. 7.1 Background The original version of the Geometry of Interaction was developed by Girard for the multiplicative fragment =-=[Gir88]-=-. This is still the best setting in which to explain the basic ideas on which the interpretation is based. Consider then the multiplicative fragment, with the restriction that the Axiom is only used f... |

5 |
What is the logic of parallel processing
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Citation Context ...ame theory nor concurrency theory oers ready made solutions to this problem. While some suggestive remarks connecting games of imperfect information with concurrent `teams' of players have been made [=-=HS95]-=-, no precise, let alone tractable formalization has been proposed so far as we are aware. As we have seen, Blass games give primacy to the additive connectives of Linear Logic. They give a direct and ... |

4 | A deconstruction of non-deterministic classical cut elimination - Laird - 2001 |