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The role of polymorphism in the characterisation of complexity by soft types

INFORMATION AND COMPUTATION 2016年 248卷 130-149页

作者： Chrzaszcz, Jacek Schubert, Aleksy Univ Warsaw Inst Informat Ul Banacha 2 PL-02097 Warsaw Poland

Soft type assignment systems STA, STA(+), and STA(B) characterise by means of reduction of terms computation in complexity classes PTIME, NP, and PSPACE, respectively. All these systems are inspired by linear logic and include polymorphism similar to the one of System F. We show that the presence of polymorphism gives the undecidability of typechecking and type inference. We also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in away sufficient for the traditional complexity theory. The translations we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input. (C) 2016 Elsevier Inc. All rights reserved.

关键词： implicit computational complexity lambda-Calculus Polymorphism Undecidability

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BUILD YOUR OWN CLARITHMETIC I: SETUP AND COMPLETENESS

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LOGICAL METHODS IN COMPUTER SCIENCE 2016年第3期12卷

作者： Japaridze, Giorgi Villanova Univ Dept Comp Sci 800 Lancaster Ave Villanova PA 19085 USA

Clarithmetics are number theories based on computability logic. Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized/schematic version CLA11(P4)(P1,P2,P3). By tuning the three parameters P-1, P-2, P-3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a, wide range of target tricomplexity classes, i.e., combinations of time (set by P-3), space (set by P-2) and so called amplitude (set by P-1 complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a, solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P-4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a solution from the given tricomplexity class is a theorem of the system. This article is published in two parts. The present Part T introduces the system and proves its completeness, while the forthcoming Part II is devoted to proving soundness.

关键词： Computability logic Interactive computation implicit computational complexity Game semantics Peano arithmetic Bounded arithmetic

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Bounded Combinatory Logic and lower complexity

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INFORMATION AND COMPUTATION 2016年 248卷 215-226页

作者： Redmond, Brian F. Grande Prairie Reg Coll Dept Comp Math & Stat Sci 10726-106 Ave Grande Prairie AB T8V 4C4 Canada

We introduce a stratified version of Combinatory Logic(1) in which there are two classes of terms called player and opponent such that the class of player terms is strictly contained in the class of opponent terms. We show that the system characterizes Polynomial Time in a similar way to Soft Linear Logic. With the addition of explicit "lazy" products to the player terms and various notions of distributivity, we obtain a characterization of Polynomial Space. This paper is an expanded version of the abstract presented at DICE 2013. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Combinatory Logic Stratification implicit computational complexity Polynomial time and space Soft linear logic

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BUILD YOUR OWN CLARITHMETIC II: SOUNDNESS

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LOGICAL METHODS IN COMPUTER SCIENCE 2016年第3期12卷

作者： Japaridze, Giorgi Villanova Univ Dept Comp Sci 800 Lancaster Ave Villanova PA 19085 USA

Clarithmetics are number theories based on computability logic. Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized/schematic version CLA11(P4)(P1, P2, P3). By tuning the three parameters P-1, P-2, P-3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a wide range of target tricomplexity classes, i.e., combinations of time (set by P-3), space (set by P-2) and so called amplitude (set by P-1) complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P-4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a soli ion from the given tricomplexity class is a theorem of the system. This article is published in two parts. The previous Part I has introduced the system and proved its completeness, while the present Part II is devoted to proving soundness.

关键词： Computability logic Interactive computation implicit computational complexity Game semantics Peano arithmetic Bounded arithmetic

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Computation by interaction for space-bounded functional programming

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INFORMATION AND COMPUTATION 2016年 248卷 150-194页

作者： Dal Lago, Ugo Schoepp, Ulrich Univ Bologna I-40126 Bologna Italy INRIA Sophia Antipolis Nice France Univ Munich Marchioninistr 15 D-81377 Munich Germany

When programming with sublinear space constraints one often needs to use special implementation techniques even for simple tasks, such as function composition. In this paper, we study how such implementation techniques can be supported in a functional programming language. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We apply this construction to a term model of a first-order programming language and use the resulting structure to derive the functional programming language INTML. INTML can be understood as a programming language simplification of Stratified Bounded Affine Logic. We formulate INTML by means of a type system inspired by Baillot & Terui's Dual Light Affine Logic. We show that it captures the complexity classes FLOGSPACE and NFLOGSPACE. We illustrate its expressiveness by showing how typical graph algorithms, such a test for acyclicity in undirected graphs, can be represented. (C) 2016 Elsevier Inc. All rights reserved.

关键词： implicit computational complexity Logarithmic space Type system Geometry of interaction Functional programming

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Introduction to clarithmetic II

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INFORMATION AND COMPUTATION 2016年 247卷 290-312页

作者： Japaridze, Giorgi Villanova Univ Dept Comp Sci 800 Lancaster Ave Villanova PA 19085 USA

The earlier paper "Introduction to clarithmetic I" constructed an axiomatic system of arithmetic based on computability logic, and proved its soundness and extensional completeness with respect to polynomial time computability. The present paper elaborates three additional sound and complete systems in the same style and sense: one for polynomial space computability, one for elementary recursive time (and/or space) computability, and one for primitive recursive time (and/or space) computability (C) 2016 Elsevier Inc. All rights reserved.

关键词： Computability logic Game semantics Peano arithmetic Constructive theories Interactive computation Bounded arithmetic implicit computational complexity

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Higher-order interpretations and program complexity

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INFORMATION AND COMPUTATION 2016年 248卷 56-81页

作者： Baillot, Patrick Dal Lago, Ugo Univ Lyon 1 CNRS INRIA Ecole Normale Super LyonLab Informat Parallelism F-69622 Villeurbanne France Univ Bologna I-40126 Bologna Italy INRIA Sophia Antipolis Nice France

Polynomial interpretations and their generalizations like quasi-interpretations have been used in the setting of first-order functional languages to design criteria ensuring statically some complexity bounds on programs[10]. This fits in the area of implicit computational complexity, which aims at giving machine-free characterizations of complexity classes. In this paper, we extend this approach to the higher-order setting. For that we consider simply-typed term rewriting systems[35], we define higher-order polynomial interpretations for them, and we give a criterion ensuring that a program can be executed in polynomial time. In order to obtain a criterion flexible enough to validate interesting programs using higher-order primitives, we introduce a notion of polynomial quasi-interpretations, coupled with a simple termination criterion based on linear types and path-like orders. (C) 2016 Elsevier Inc. All rights reserved.

关键词： implicit computational complexity Term rewriting systems Type systems Lambda-calculus

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Light combinators for finite fields arithmetic

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SCIENCE OF COMPUTER PROGRAMMING 2015年第Part3期111卷 365-394页

作者： Canavese, D. Cesena, E. Ouchary, R. Pedicini, M. Roversi, L. Politecn Torino Dipartimento Automat & Informat I-10129 Turin Italy Theneeds Inc San Francisco CA USA Univ Rome Tre Dipartimento Matemat & Fis I-00146 Rome Italy Univ Turin Dipartimento Informat I-10149 Turin Italy

This work completes the definition of a library which provides the basic arithmetic operations in binary finite fields as a set of functional terms with very specific features. Such a functional terms have type in Typeable Functional Assembly (TFA). TFA is an extension of Dual Light Affine Logic (DLAL). DIAL is a type assignment designed under the prescriptions of implicit computational complexity (ICC), which characterises polynomial time costing computations. We plan to exploit the functional programming patterns of the terms in the library to implement cryptographic primitives whose running-time efficiency can be obtained by means of the least hand-made tuning as possible. We propose the library as a benchmark. It fixes a kind of lower bound on the difficulty of writing potentially interesting low cost programs inside languages that can express only computations with predetermined complexity. In principle, every known and future ICC compliant programming language for polynomially costing computations should supply a simplification over the encoding of the library we present, or some set of combinators of comparable interest and difficulty. We finally report on the applicative outcome that our library has and which is a reward we get by programming in the very restrictive scenario that TFA provides. The term of TFA which encodes the inversion in binary fields suggested us a variant of a known and efficient imperative implementation of the inversion itself given by Fong. Our variant, can outperform Fong's implementation of inversion on specific hardware architectures. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Lambda calculus Finite fields arithmetic Type assignments implicit computational complexity

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On bounding space usage of streams using interpretation analysis

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SCIENCE OF COMPUTER PROGRAMMING 2015年第Part3期111卷 395-425页

作者： Gaboardi, Marco Pechoux, Romain Harvard Univ Cambridge MA 02138 USA Univ Dundee Dundee DD1 4HN Scotland Univ Lorraine INRIA Project Carte Loria UMR 7503 Nancy France

Interpretation methods are important tools in implicit computational complexity. They have been proved particularly useful to statically analyze and to limit the complexity of programs. However, most of these studies have been so far applied in the context of term rewriting systems over finite data. In this paper, we show how interpretations can also be used to study properties of lazy first-order functional programs over streams. In particular, we provide some interpretation criteria useful to ensure two kinds of stream properties: space upper bounds and input/output upper bounds. Our space upper bounds criteria ensure global and local upper bounds on the size of each output stream element expressed in terms of the maximal size of the input stream elements. The input/output upper bounds criteria consider instead the relations between the number of elements read from the input stream and the number of elements produced on the output stream. This contribution can be seen as a first step in the development of a methodology aiming at using interpretation properties to ensure space safety properties of programs working on streams. (C) 2015 Elsevier BM. All rights reserved.

关键词： Stream programs Interpretations Program space usage Lazy languages implicit computational complexity

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An abstract approach to stratification in linear logic

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INFORMATION AND COMPUTATION 2015年 241卷 32-61页

作者： Boudes, Pierre Mazza, Damiano de Falco, Lorenzo Tortora Univ Paris 13 CNRS LIPN Sorbonne Paris Cite F-93430 Villetaneuse France Univ Paris 13 CNRS UMR 7030 LIPNSorbonne Paris Cite F-93430 Villetaneuse France Univ Roma Tre Dipartimento Filosofia Rome Italy

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called "light" subsystems), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities;when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic. (C) 2014 Elsevier Inc. All rights reserved.

关键词： implicit computational complexity Light linear logics Denotational semantics Categorical semantics

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