## A Compositional Game Semantics for Multi-Agent Logics of Partial Information

Citations: | 2 - 0 self |

### BibTeX

@MISC{Abramsky_acompositional,

author = {Samson Abramsky},

title = {A Compositional Game Semantics for Multi-Agent Logics of Partial Information},

year = {}

}

### OpenURL

### Abstract

We consider the following questions: What kind of logic has a natural semantics in multi-player (rather than 2-player) games? How can we express branching quantifiers, and other partial-information constructs, with a properly compositional syntax and semantics? We develop a logic in answer to these questions, with a formal semantics based on multiple concurrent strategies, formalized as closure operators on Kahn-Plotkin concrete domains. Partial information constraints are represented as co-closure operators. We address the syntactic issues by treating syntactic constituents, including quantifiers, as arrows in a category, with arities and co-arities. This enables a fully compositional account of a wide

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Citation Context ...ake a decision at cell c in state s if it would have made the same decision in the smaller (less information) state γM(c)(s). 3 This is a direct analogue of the use of views in Hyland-Ong style games =-=[HO00]-=- to define ‘innocent strategies’. However, apart from the more general format of our games, there is greater flexibility in the provision of the function γM as a separate component of the game structu... |

136 | A Modal Logic for Coalitional Power in Games - Pauly |

73 | New foundations for the geometry of interaction - Abramsky, Jagadeesan - 1994 |

68 |
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Citation Context ...nto the partial information constraint (5) to yield exactly the desired interpretation. 255 Compositionality Reconsidered The issue of compositionality for IF-logic has attracted some attention; see =-=[Hin98]-=-. In [Hod97], Hodges gives a compositional Tarski-style semantics for IF logic. The key idea is to work at the level of sets of sets of assignments, rather than the sets of assignments used in the sta... |

57 |
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Citation Context ...than 2-player) games? Another topic which has been studied extensively in recent years has been the logical aspects of games of imperfect information, starting with Henkin-style branching quantifiers =-=[Hen61]-=-, and Hintikka’s game-theoretical interpretation of these, and continuing with the IF-logic of Hintikka and Sandu [HS89, HS95, HS96]. The issue of whether and how a compositional semantics for IF-logi... |

50 | Compositional Semantics for a Language of Imperfect Information
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Citation Context ...he IF-logic of Hintikka and Sandu [HS89, HS95, HS96]. The issue of whether and how a compositional semantics for IF-logic can be given has been studied by several authors, particularly Wilfrid Hodges =-=[Hod97]-=-. However, there is an even more basic question which does not seem to have received much, if any, attention: namely, how to give a properly compositional syntax for such constructs. For example, how ... |

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

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Citation Context ... convenient, and indeed very natural, to use as our formal representations of games were introduced by Gilles Kahn and Gordon Plotkin in 1975 (although their paper only appeared in a journal in 1993) =-=[KP78]-=-. They arose for Kahn and Plotkin in providing a representation theory for their notion of concrete domains. The term used by Kahn and Plotkin for these structures was information matrices; subsequent... |

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Citation Context ...work coming from Theoretical Computer Science which is suggestive: namely the π-calculus [MPW92, MPW92a], with its scope restriction and extrusion. The action calculi subsequently developed by Milner =-=[Mil93]-=- are even more suggestive, although only certain features are relevant here. We shall reformulate our view of logical syntax as follows. Each syntactic constituent will have an arity and a co-arity. C... |

19 | From chaotic iteration to constraint propagation
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Citation Context ...semantics of dataflow [JPP89] and concurrent constraint programming [SRP]. They have also been applied extensively to constraint programming and constraint propagation algorithms over the past decade =-=[Apt97]-=-. Our own previous development of these ideas appears in a number of papers [AM99, Abr00a, Abr03]. 3.4 Outcomes and Valuations One ingredient which has been missing thus far in our account of the sema... |

13 |
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Citation Context ...y first appear. Consider an idea which may seem strange at first, although it has also arisen in Dynamic Game Logic [vBen03] and implicitly in the semantics of non-deterministic programming languages =-=[AP81]-=-. Instead of treating quantifiers as prefixing operators Qαx.ϕ in the usual way, we can consider them as stand-alone particles Qαx ≡ Qαx.1. We should expect to have (Qαx) · ϕ ≡ (Qαx.1) · ϕ ≡ Qαx.(1 · ... |

11 | Game-Theoretical Semantics', in J. van Benthem & A. ter Meulen (eds) Handbook of Logic and Language - Hintikka, Sandu - 1997 |

9 | A game semantics for generic polymorphism
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Citation Context ...ould be continuous. For example, we could take Vα to be the interval domain I[0,1] on the unit interval. An extension to full second-order logic, although technically more demanding, is also possible =-=[AJ05]-=-. • What is the full spectrum of possible connectives which can be used to explore the resources of our semantics? The logic LA we have introduced is quite natural, but this question remains wide open... |

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Citation Context ...as, and π ∈ S(A). Quantifiers as Particles The syntax of LA is more powerful than may first appear. Consider an idea which may seem strange at first, although it has also arisen in Dynamic Game Logic =-=[vBen03]-=- and implicitly in the semantics of non-deterministic programming languages [AP81]. Instead of treating quantifiers as prefixing operators Qαx.ϕ in the usual way, we can consider them as stand-alone p... |

3 | finite observations and strong bisimulation - Quantales - 2001 |

2 |
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Citation Context ... . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Compositionality Reconsidered 26 6 Further Directions 27 1 Introduction We begin with the following quote from the manifesto of a recent Workshop =-=[AdM7]-=-: “Traditionally, logic has dealt with the zero-agent notion of truth and the oneagent notion of reasoning. In the last decades, research focus in logic shifted from these topics to the vast field of ... |

2 | Abramsky: Sequentiality vs. Concurrency - Samson - 2003 |

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2 |
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Citation Context ...e. Our use of closure operators as strategies, and the definition of the evaluation of strategy profiles as least common fixpoints, builds on ideas which arose originally in the semantics of dataflow =-=[JPP89]-=- and concurrent constraint programming [SRP]. They have also been applied extensively to constraint programming and constraint propagation algorithms over the past decade [Apt97]. Our own previous dev... |