The Power of Linear Functions

The linear lambda calculus is very weak in terms of expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with Booleans, natural numbers and a linear iterator. We show properties of this linear version of Godel’s System $\mathcal{T}$ and study the class of functions that can be represented. Surprisingly, this linear calculus is extremely expressive: it is as powerful as System $\mathcal{T}$

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

[2]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[3]  Ian Mackie,et al.  Lilac: a functional programming language based on linear logic , 1994, Journal of Functional Programming.

[4]  J. Roger Hindley,et al.  To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism , 1980 .

[5]  Andrea Asperti,et al.  Intuitionistic Light Affine Logic , 2002, TOCL.

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

[7]  Christine Paulin-Mohring,et al.  Inductive Definitions in the system Coq - Rules and Properties , 1993, TLCA.

[8]  Sam Lindley,et al.  Extensional Rewriting with Sums , 2007, TLCA.

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

[10]  Ian Mackie,et al.  An internal language for autonomous categories , 1993, Theory and Formal Methods.

[11]  Robin Milner,et al.  Theories for the Global Ubiquitous Computer , 2004, FoSSaCS.

[12]  Ugo Dal Lago.  The geometry of linear higher-order recursion , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[13]  A. Pitts.  INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC (Cambridge Studies in Advanced Mathematics 7) , 1987 .

[14]  J. Roger Hindley,et al.  BCK-Combinators and Linear lambda-Terms have Types , 1989, Theor. Comput. Sci..

[15]  Jean-Yves Girard,et al.  Towards a geometry of interaction , 1989 .

[16]  Yves Lafont,et al.  Soft linear logic and polynomial time , 2004, Theor. Comput. Sci..

[17]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[18]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[19]  Kazushige Terui,et al.  Light affine lambda calculus and polytime strong normalization , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[20]  Jean-Yves Girard.  Light Linear Logic , 1994, LCC.

[21]  Michael Nedzelsky,et al.  Recursion Theory I , 2008, Arch. Formal Proofs.

[22]  Patrick Baillot,et al.  Soft lambda-Calculus: A Language for Polynomial Time Computation , 2004, FoSSaCS.

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

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

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

[26]  Martin Hofmann,et al.  Linear types and non-size-increasing polynomial time computation , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[27]  Ian Mackie,et al.  Closed reduction: explicit substitutions without alpha-conversion , 2005, Math. Struct. Comput. Sci..