Sequential algorithms and strongly stable functions

Paul-André Melliès (1)

1 PPS - Preuves, Programmes et Systèmes

Intuitionistic proofs (or PCF programs) may be interpreted as functions between domains, or as strategies on games. The two kinds of interpretation are inherently different: static vs. dynamic, extensional vs. intentional. It is extremely instructive to compare and to connect them. In this article, we investigate the extensional content of the sequential algorithm hierarchy [-] introduced by Berry and Curien two decades ago. We equip every sequential game [T] of the hierarchy with a realizability relation between plays and extensions. In this way, the sequential game [T] becomes a directed acyclic graph, instead of a tree. This enables to define a hypergraph [[T]] on the extensions (or terminal leaves) of the game [T]. We establish that the resulting hierarchy [[T]] coincides with the strongly stable hierarchy introduced by Bucciarelli and Ehrhard. We deduce from this a game-theoretic proof of Ehrhard's collapse theorem, which states that the strongly stable hierarchy coincides with the extensional collapse of the sequential algorithm hierarchy.

Mots clés: Linear logic; sequential games; hypercoherence spaces; extensional collapse

Domaines: Logique en informatique [cs.LO]

Paul-André Melliès. Sequential algorithms and strongly stable functions. Theoretical Computer Science, 2005, 343 (1-2), pp.237-281. ⟨10.1016/j.tcs.2005.05.015⟩. ⟨hal-00154206⟩