Local Linear Logic for Locality Consciousness in Multiset Transformation

We use Girard's linear logic (LL) to produce a semantics for Gamma, a multiset transformation language. The semantics improves on the existing structured operational semantics (SOS) of the language by highlighting Gamma's inefficiencies, which were hidden by the SOS. We propose a new logic called local linear logic (Local LL), which adds locality-consciousness to the resource-consciousness of linear logic. As a case study, we use this logic to propose a new semantics for Gamma. The new semantics suggests an annotation of Gamma which increases its efficiency without compromising its programming style. We show how the new semantics also gives us a better understanding of parallel Gamma and its implementation, and offers insight into the nature of chemicalreaction based computational models in general.

[1]  V. Michele Abrusci.  Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic , 1991, J. Symb. Log..

[2]  Daniel Le Métayer,et al.  Programming by multiset transformation , 1993, CACM.

[3]  Andre Scedrov,et al.  Bounded Linear Logic: A Modular Approach to Polynomial-Time Computability , 1992, Theor. Comput. Sci..

[4]  Yves Lafont,et al.  Interaction nets , 1989, POPL '90.

[5]  Gérard Berry,et al.  The chemical abstract machine , 1989, POPL '90.

[6]  JEAN-MARC ANDREOLI,et al.  Logic Programming with Focusing Proofs in Linear Logic , 1992, J. Log. Comput..

[7]  K. Mani Chandy,et al.  Parallel program design - a foundation , 1988 .

[8]  David N. Yetter,et al.  Quantales and (noncommutative) linear logic , 1990, Journal of Symbolic Logic.

[9]  José Meseguer,et al.  Conditioned Rewriting Logic as a United Model of Concurrency , 1992, Theor. Comput. Sci..

[10]  Tommaso Toffoli,et al.  Cellular Automata Machines , 1987, Complex Syst..

[11]  Stephen Wolfram,et al.  Cellular Automata And Complexity , 1994 .

[12]  Christian Creveuil,et al.  Dérivation systématique d'un algorithme de segmentation d'images . Un exemple d'application du formalisme GAMMA , 1989 .

[13]  Chris Hankin,et al.  A Calculus of Gamma Programs , 1992, LCPC.

[14]  Vladimir Alexiev,et al.  Applications of Linear Logic to Computation: An Overview , 1994, Log. J. IGPL.

[15]  Ralph-Johan Back,et al.  Distributed cooperation with action systems , 1988, TOPL.

[16]  Samson Abramsky,et al.  Computational Interpretations of Linear Logic , 1993, Theor. Comput. Sci..

[17]  David Gelernter,et al.  Generative communication in Linda , 1985, TOPL.

[18]  Herbert Kuchen,et al.  Shared Memory Implementation of the Gamma-Operation , 1996, J. Symb. Comput..

[19]  Philip Wadler,et al.  Linear Types can Change the World! , 1990, Programming Concepts and Methods.

[20]  Chris Hankin,et al.  Transformation of Gamma Programs , 1992, WSA.

[21]  A. Troelstra.  Lectures on linear logic , 1992 .

[22]  Hugh McEvoy.  Gamma, chromatic typing and vegetation , 1996 .

[23]  Ralph-Johan Back,et al.  Decentralization of Process Nets with Centralized Control , 1983, PODC.

[24]  Patrick Lincoln,et al.  Constant-Only Multiplicative Linear Logic is NP-Complete , 1992, Theor. Comput. Sci..

[25]  Yves Lafont,et al.  The Linear Abstract Machine , 1988, Theor. Comput. Sci..

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

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

[28]  Daniel Le Métayer,et al.  Parallel Machines for Multiset Transformation and their Programming Style / Parallele Maschinen für die Multimengen-Transformation und deren Programmierstil , 1988, it Inf. Technol..

[29]  V. Michele Abrusci.  Non-commutative intuitionistic linear logic , 1990, Math. Log. Q..

[30]  Henry G. Baker,et al.  Lively linear Lisp: “look ma, no garbage!” , 1992, SIGP.

[31]  David Gelernter,et al.  Distributed communication via global buffer , 1982, PODC '82.

[32]  G BakerHenry.  Lively linear Lisp , 1992 .

[33]  Chris Hankin,et al.  A Congruence for Gamma Programs , 1993, WSA.

[34]  Pieter H. Hartel,et al.  A parallel functional implementation of range queries , 1989 .