Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concurrent Strategies on Games with Algebras

Sacha Huriot-Tattegrain, Glynn Winskel

TL;DR

The paper tackles concurrent games with algebra under imperfect information, proposing a bijection between strategies on a game with algebra and strategies on a larger expanded game, and showing this correspondence persists when neutral events are included. It develops the formalism of $\\\ ext{A}$-games and $\\\ ext{A}$-strategies, defines free-logic winning conditions, and establishes how algebraic structure propagates to expansions and to compositions. It further investigates extensions such as neutral moves and copycat strategies, and provides concrete examples (A-homomorphism and Ehrenfeucht–Fraïssé games) to illustrate how algebraic reasoning maps to classical model-theoretic notions like homomorphisms and isomorphisms. The work also introduces $\\\\\\\\\\\\\\\\\\\\\\Lambda$-games to model imperfect information and discusses reasoning techniques for these strategies. Overall, the results advance the use of games with algebras in modeling concurrent computation, while leaving open questions about bicategorical structure and optimal strategy composition under imperfect information.

Abstract

Probabilistic concurrent/distributed strategies have so far not been investigated thoroughly in the context of imperfect information, where the Player has only partial knowledge of the moves made by the Opponent. In a situation where the Player and Opponent can make concurrent moves according to the game, and the Player cannot see the move of the Opponent, the move of the Player should be probabilistically independent of the move of the Opponent. What has been achieved is showing a bijection between strategies on a game with algebra and strategies on a regular (albeit more complex) game. We also succeeded in showing the results holds with neutral events. However it is still unclear if a well-formed bicategory of concurrent games with algebras can be defined. Our attempts to compose these strategies while managing the added structure didn't pan out. Concerning the other classic extensions of concurrent games the first results we presented show promise of a more general usage of games with algebra.

Concurrent Strategies on Games with Algebras

TL;DR

The paper tackles concurrent games with algebra under imperfect information, proposing a bijection between strategies on a game with algebra and strategies on a larger expanded game, and showing this correspondence persists when neutral events are included. It develops the formalism of -games and -strategies, defines free-logic winning conditions, and establishes how algebraic structure propagates to expansions and to compositions. It further investigates extensions such as neutral moves and copycat strategies, and provides concrete examples (A-homomorphism and Ehrenfeucht–Fraïssé games) to illustrate how algebraic reasoning maps to classical model-theoretic notions like homomorphisms and isomorphisms. The work also introduces -games to model imperfect information and discusses reasoning techniques for these strategies. Overall, the results advance the use of games with algebras in modeling concurrent computation, while leaving open questions about bicategorical structure and optimal strategy composition under imperfect information.

Abstract

Probabilistic concurrent/distributed strategies have so far not been investigated thoroughly in the context of imperfect information, where the Player has only partial knowledge of the moves made by the Opponent. In a situation where the Player and Opponent can make concurrent moves according to the game, and the Player cannot see the move of the Opponent, the move of the Player should be probabilistically independent of the move of the Opponent. What has been achieved is showing a bijection between strategies on a game with algebra and strategies on a regular (albeit more complex) game. We also succeeded in showing the results holds with neutral events. However it is still unclear if a well-formed bicategory of concurrent games with algebras can be defined. Our attempts to compose these strategies while managing the added structure didn't pan out. Concerning the other classic extensions of concurrent games the first results we presented show promise of a more general usage of games with algebra.
Paper Structure (17 sections, 9 theorems, 36 equations)

This paper contains 17 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Context of the subject
  3. Preliminary definitions
  4. Games and Strategies
  5. Adjoining an algebra to a game
  6. Relation between games with and without algebra
  7. Further results on games
  8. An approach with neutral events
  9. Concurrent copycat strategies
  10. $\mathcal{A}$-algebras from $\mathcal{A}$-strategies and games
  11. Example of games
  12. $A$-homomorphism game
  13. Ehrenfencht-Fraissé games
  14. An extension to imperfect information
  15. Definition of access levels
  16. ...and 2 more sections

Key Result

Lemma 2.13

$x\models_\sigma W_E\vee W_F$ iff ($x_E\models_{\sigma_E}W_E$ or $x_F\models_{\sigma_F}W_F$)

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.8
  • ...and 26 more