Comparing hierarchies of types in models of linear logic

  • Authors:

  • Paul-André Melliès

  • Affiliations:

  • Equipe PPS, Université Paris VII, 2 place Jussieu, Case 7014, 75251 Paris Cedex 05, France

  • Venue:
  • Information and Computation
  • Year:
  • 2004

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Abstract

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F: C →← D : G and transformations IdC ⇒ GF and IdD = ⇒ FG, and (2) their exponentials !M and !N are related by distributive laws ρ : !NF ⇒ F!M and η : !MG ⇒ G!N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. nonuniform.