Comparing hierarchies of types in models of linear logic

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.

[1]  Paul-André Melliès.  Categorical models of linear logic revisited , 2002 .

[2]  J. Cheney,et al.  A sequent calculus for nominal logic , 2004, LICS 2004.

[3]  Roberto M. Amadio,et al.  Domains and lambda-calculi , 1998, Cambridge tracts in theoretical computer science.

[4]  Samson Abramsky,et al.  A fully abstract game semantics for general references , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[5]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[6]  Thomas Ehrhard.  Parallel and serial hypercoherences , 2000, Theor. Comput. Sci..

[7]  John Longley.  The sequentially realizable functionals , 2002, Ann. Pure Appl. Log..

[8]  John Power,et al.  Combining a monad and a comonad , 2002, Theor. Comput. Sci..

[9]  Martin Hyland.  Semantics and Logics of Computation: Game Semantics , 1997 .

[10]  M. Barr,et al.  THE SEPARATED EXTENSIONAL CHU CATEGORY , 1998 .

[11]  Pierre-Louis Curien,et al.  Sequential Algorithms on Concrete Data Structures , 1982, Theor. Comput. Sci..

[12]  Matthias Felleisen,et al.  Fully Abstract Semantics for Observably Sequential Languages , 1994, Inf. Comput..

[13]  Hanno Nickau.  Hereditarily Sequential Functionals , 1994, LFCS.

[14]  J. V. Oosten,et al.  A combinatory algebra for sequential functionals of finite type , 1997 .

[15]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[16]  Antonio Bucciarelli.  Bi-Models: Relational Versus Domain-Theoretic Approaches , 1997, Fundam. Informaticae.

[17]  AbramskySamson,et al.  Games and full completeness for multiplicative linear logic , 1994 .

[18]  Thomas Ehrhard.  A Relative PCF-Definability Result for Strongly Stable Functions and some Corollaries , 1999, Inf. Comput..

[19]  Thomas Ehrhard.  A relative PCF-definability result for strongly stable functions and some corollaries , 2000, Electron. Notes Theor. Comput. Sci..

[20]  Pierre-Louis Curien.  On the Symmetry of Sequentiality , 1993, MFPS.

[21]  Radha Jagadeesan,et al.  Full Abstraction for PCF , 1994, Inf. Comput..

[22]  Samson Abramsky,et al.  Concurrent games and full completeness , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[23]  S. Lack,et al.  The formal theory of monads II , 2002 .

[24]  J. Beck,et al.  TRIPLES, ALGEBRAS AND COHOMOLOGY , 1967 .

[25]  Radha Jagadeesan,et al.  Games and Full Completeness for Multiplicative Linear Logic , 1994, J. Symb. Log..

[26]  Antonio Bucciarelli,et al.  Sequentiality and strong stability , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[27]  S. Abramsky.  Game Semantics , 1999 .

[28]  Gordon D. Plotkin,et al.  Towards a mathematical operational semantics , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[29]  Gavin M. Bierman.  What is a Categorical Model of Intuitionistic Linear Logic? , 1995, TLCA.

[30]  Paul-André Melliès,et al.  Sequential algorithms and strongly stable functions , 2005, Theor. Comput. Sci..

[31]  Martin Hyland,et al.  Abstract Games for Linear Logic , 1999, CTCS.

[32]  Russell Harmer.  Games and full abstraction for non-deterministic languages , 1999 .

[33]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[34]  Ulrich Berger,et al.  Computational Logic , 2001, NATO ASI Series.

[35]  Thomas Ehrhard,et al.  Anatomy of an Extensional Collapse , 1997 .

[36]  Thomas Ehrhard.  Hypercoherences: A Strongly Stable Model of Linear Logic , 1993, Math. Struct. Comput. Sci..

[37]  John R. Longley,et al.  Realizability toposes and language semantics , 1995 .