Higher type recursion, ramification and polynomial time

Abstract It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial-time computable functions. The restrictions are obtained by using a ramified type structure, and by adding linear concepts to the lambda calculus.

[1]  Daniel Leivant,et al.  Subrecursion and lambda representation over free algebras , 1990 .

[2]  Lev Gordeev,et al.  Basic proof theory , 1998 .

[3]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[4]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

[5]  Karl-Heinz Niggl,et al.  The $\mu$-measure as a tool for classifying computational complexity , 2000, Arch. Math. Log..

[6]  Elaine J. Weyuker,et al.  Computability, complexity, and languages - fundamentals of theoretical computer science , 2014, Computer science and applied mathematics.

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

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

[9]  Karl-Heinz Niggl,et al.  Ranking Primitive Recursions: The Low Grzegorczyk Classes Revisited , 1999, SIAM J. Comput..

[10]  Kenneth W. Regan,et al.  Computability , 2022, Algorithms and Theory of Computation Handbook.

[11]  Daniel Leivant,et al.  Ramified Recurrence and Computational Complexity IV : Predicative Functionals and Poly-Space , 2000 .

[12]  D. Hilbert.  Über das Unendliche , 1926 .

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

[14]  Martin Hofmann,et al.  A Mixed Modal/Linear Lambda Calculus with Applications to Bellantoni-Cook Safe Recursion , 1997, CSL.

[15]  Daniel Leivant.  Predicative Recurrence in Finite Types , 1994, LFCS.

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

[17]  Harold Simmons,et al.  The realm of primitive recursion , 1988, Arch. Math. Log..

[18]  D. Leivant.  Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time , 1995 .