The geometry of linear higher-order recursion

Linearity and ramification constraints have been widely used to weaken higher-order (primitive) recursion in such a way that the class of representable functions equals the class of poly time functions. We show that fine-tuning these two constraints leads to different expressive strengths, some of them lying well beyond polynomial time. This is done by introducing a new semantics, called algebraic context semantics. The framework stems from Gonthier's original work and turns out to be a versatile and powerful tool for the quantitative analysis of normalization in presence of constants and higher-order recursion.

[1]  Harry G. Mairson.  From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple , 2002, FSTTCS.

[2]  Martin Hofmann.  Safe recursion with higher types and BCK-algebra , 2000, Ann. Pure Appl. Log..

[3]  Daniel Leivant,et al.  Intrinsic reasoning about functional programs I: first order theories , 2002, Ann. Pure Appl. Log..

[4]  Daniel Leivant,et al.  Stratified functional programs and computational complexity , 1993, POPL '93.

[5]  Martín Abadi,et al.  The geometry of optimal lambda reduction , 1992, POPL '92.

[6]  Daniel Leivant,et al.  Ramified Recurrence and Computational Complexity II: Substitution and Poly-Space , 1994, CSL.

[7]  Ugo Dal Lago.  Context Semantics, Linear Logic and Computational Complexity , 2005, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[8]  Ugo Dal Lago,et al.  An Invariant Cost Model for the Lambda Calculus , 2005, CiE.

[9]  Daniel Leivant.  Applicative Control and Computational Complexity , 1999, CSL.

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

[11]  Dan R. Ghica,et al.  Slot games: a quantitative model of computation , 2005, POPL '05.

[12]  Klaus Aehlig,et al.  Linear Ramified Higher Type Recursion and Parallel Complexity , 2001, Proof Theory in Computer Science.

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

[14]  Vincent Danos,et al.  Reversible, Irreversible and Optimal Lambda-machines , 1999, Linear Logic Tokyo Meeting.

[15]  Stephen A. Cook,et al.  A new recursion-theoretic characterization of the polytime functions , 1992, STOC '92.

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

[17]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[18]  Daniel Leivant,et al.  Ramified Recurrence and Computational Complexity III: Higher Type Recurrence and Elementary Complexity , 1999, Ann. Pure Appl. Log..

[19]  Vincent Danos,et al.  Game Semantics & Abstract Machines. , 1996, LICS 1996.

[20]  S. S. Wainer,et al.  Proof Theoretic Complexity , 2002 .

[21]  Vincent Danos,et al.  Game semantics and abstract machines , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

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

[23]  Daniel Leivant,et al.  Intrinsic reasoning about functional programs II: unipolar induction and primitive-recursion , 2004, Theor. Comput. Sci..

[24]  David Sands,et al.  Operational Theories of Improvement in Functional Languages (Extended Abstract) , 1991, Functional Programming.

[25]  Martin Hofmann.  Linear types and non-size-increasing polynomial time computation , 2003, Inf. Comput..

[26]  Ugo Dal Lago.  Context Semantics, Linear Logic and Computational Complexity , 2006, LICS.

[27]  R. Simmons,et al.  Complexity Analysis of , 1993 .

[28]  Vincent Danos,et al.  Reversible, Irreversible and Optimal lambda-Machines , 1999, Theor. Comput. Sci..

[29]  Harold Simmons,et al.  Tiering as a recursion technique , 2005, Bull. Symb. Log..

[30]  Guillaume Bonfante,et al.  On complexity analysis by quasi-interpretation , 2004 .

[31]  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).

[32]  Ralph Matthes,et al.  Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T , 2003, Arch. Math. Log..

[33]  Helmut Schwichtenberg,et al.  Higher type recursion, ramification and polynomial time , 2000, Ann. Pure Appl. Log..

[34]  Patrick Baillot,et al.  Elementary Complexity and Geometry of Interaction , 1999, Fundam. Informaticae.

[35]  Ugo Dal Lago,et al.  Higher-Order Linear Ramified Recurrence , 2003, TYPES.