Bounded Linear Logic: A Modular Approach to Polynomial-Time Computability

Abstract Usual typed lambda-calculi yield input/output specifications; in this paper the authors show how to extend this paradigm to complexity specifications. This is achieved by means of a restricted version of linear logic in which the use of exponential connectives is bounded in advance. This bounded linear logic naturally involves polynomials in its syntax and dynamics. It is then proved that any functional term of appropriate type actually encodes a polynomial-time algorithm and that conversely any polynomial-time function can be obtained in this way.

[1]  Andre Scedrov,et al.  Bounded Linear Logic: A Modular Approach to Polynomial Time Computability: Extended Abstract , 1990 .

[2]  Jean-Yves Girard,et al.  Geometry of Interaction 1: Interpretation of System F , 1989 .

[3]  Jean-Louis Krivine,et al.  Opérateurs de mise en mémoire et traduction de Gödel , 1990, Arch. Math. Log..

[4]  Stephen A. Cook,et al.  Feasibly constructive proofs and the propositional calculus (Preliminary Version) , 1975, STOC.

[5]  Jean-Yves Girard,et al.  Π12-logic, Part 1: Dilators , 1981 .

[6]  Jeffrey D. Ullman,et al.  Principles of Database Systems , 1980 .

[7]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[8]  Stephen A. Cook,et al.  Functional interpretations of feasibly constructive arithmetic , 1989, STOC '89.

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

[10]  Yuri Gurevich,et al.  Toward logic tailored for computational complexity , 1984 .

[11]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

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

[13]  J. Girard.  Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des Types , 1971 .

[14]  Jean-Yves Girard,et al.  Geometry of interaction 2: deadlock-free algorithms , 1990, Conference on Computer Logic.

[15]  Anil Nerode,et al.  Complexity-Theoretic Algebra II: Boolean Algebras , 1989, Ann. Pure Appl. Log..

[16]  Bruce M. Kapron,et al.  Characterizations of the basic feasible functionals of finite type , 1989, 30th Annual Symposium on Foundations of Computer Science.

[17]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[18]  Samuel R. Buss,et al.  The Polynomial Hierarchy and Intuitionistic Bounded Arithmetic , 1986, SCT.

[19]  Andre Scedrov,et al.  A brief guide to linear logic , 1990, Bull. EATCS.

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

[21]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[22]  Andre Scedrov,et al.  Some Semantic Aspects of Polymorphic Lambda Calculus , 1987, LICS.

[23]  Phokion G. Kolaitis,et al.  On the expressive power of datalog: tools and a case study , 1990, J. Comput. Syst. Sci..

[24]  Anil Nerode,et al.  Polynomially graded logic I. A graded version of system T , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[25]  Vincent Danos,et al.  The structure of multiplicatives , 1989, Arch. Math. Log..