Intuitionistic Light Affine Logic

This article is a structured introduction to Intuitionistic Light Affine Logic (ILAL). ILAL has a polynomially costing normalization, and it is expressive enough to encode, and simulate, all PolyTime Turing machines. The bound on the normalization cost is proved by introducing the proof-nets for ILAL. The bound follows from a suitable normalization strategy that exploits structural properties of the proof-nets. This allows us to have a good understanding of the meaning of the § modality, which is a peculiarity of light logics. The expressive power of ILAL is demonstrated in full detail. Such a proof gives a hint of the nontrivial task of programming with resource limitations, using ILAL derivations as programs.

[1]  J. Girard,et al.  Proofs and types , 1989 .

[2]  John C. Mitchell,et al.  Polymorphic Type Inference and Containment , 1988, Inf. Comput..

[3]  Lorenzo Tortora de Fal.  Additives of linear logic and normalization -- Part II: the additive standardization theorem , 2000 .

[4]  J. Girard.  PROOF-NETS : THE PARALLEL SYNTAX FOR PROOF-THEORY , 1996 .

[5]  Andrea Asperti.  Light affine logic , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[6]  Ron Dinishak.  The optimal implementation of functional programming languages , 2000, SOEN.

[7]  Daniel Leivant,et al.  Lambda Calculus Characterizations of Poly-Time , 1993, Fundam. Informaticae.

[8]  Lorenzo Tortora de Falco.  Reseaux, coherence et experiences obsessionnelles , 2000 .

[9]  Lorenzo Tortora de Falco,et al.  Additives of linear logic and normalization - Part I: a (restricted) Church-Rosser property , 2003, Theor. Comput. Sci..

[10]  Luca Roversi.  A Polymorphic Language Which Is Typable and Poly-step , 1998, ASIAN.

[11]  Andrea Asperti,et al.  The optimal implementation of functional programming languages , 1998, Cambridge tracts in theoretical computer science.

[12]  Jean-Yves Girard,et al.  Light Linear Logic , 1998, Inf. Comput..

[13]  Vincent Danos,et al.  Linear Logic & Elementary Time , 1999 .

[14]  Luca Roversi.  Light Affine Logic as a Programming Language: A First Contribution , 2000, Int. J. Found. Comput. Sci..

[15]  Daniel Leivant,et al.  A Foundational Delineation of Poly-time , 1994, Inf. Comput..

[16]  Luca Roversi.  A P-Time Completeness Proof for Light Logics , 1999, CSL.