A logical account of pspace

Gaboardi, Marco; Marion, Jean-Yves; Rocca, Simona Ronchi Della

doi:10.1145/1328897.1328456

Cited by 17 publications

(16 citation statements)

References 27 publications

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“…but instead by restricting the primitives or the features of the languages. Various approaches have been used for that, primarily in logic and in functional programming languages: restrictions of the comprehension scheme in second-order logic [24,26]; ramification in logic or in recursion theory [25,5]; read-only functional programs [21]; variants of linear logic [15,22,16]. .…”

Section: Introductionmentioning

confidence: 99%

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Baillot

2015

Information and Computation

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Elementary linear logic is a simple variant of linear logic due to Girard and which characterizes in the proofs-as-programs approach the class of elementary functions, that is to say functions computable in time bounded by a tower of exponentials of fixed height. Other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions, but at the price of either a more complicated syntax or of more sophisticated encodings. These logical systems can serve as the basis of type systems for λ-calculus ensuring polynomial time complexity bounds on well-typed terms. This paper aims at reviving interest in elementary linear logic by showing that, despite its simplicity, it can capture smaller complexity classes than that of elementary functions. For that we carry a detailed analysis of its normalization procedure, and study the complexity of functions represented by a given type. We then show that by considering a slight variant of this system, with type fixpoints and free weakening (elementary affine logic with fixpoints) we can characterize the complexity of functions of type !W ! k+2 B, where W and B are respectively types for binary words and booleans. The key point is a sharper study of the normalization bounds. We characterize in this way the class P of polynomial time predicates, and more generally the hierarchy of classes k-EXP, for k ≥ 0, where k-EXP is the union of DTIME(2 n i k ), for i ≥ 1.

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Section: Introductionmentioning

confidence: 99%

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Baillot

2015

Information and Computation

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“…In [Gir92] characterisation of PTIME is given in terms of bounded linear logic. In [GabMarRon08] one proposes a characterization of PSPACE by means of an extension of (soft affine) typed lambda calculus. For this extension, the authors design a call-by-name evaluation machine in order to compute programs in polynomial space.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Size Analysis of First-Order Shapely Functions

Shkaravska¹,

Eekelen²,

Kesteren³

2009

Logical Methods in Computer Science

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Abstract. We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be implementations of matrix multiplication and the Cartesian product of two lists.The type system is proved to be sound w.r.t. the operational semantics of the language. The type checking problem is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic.Furthermore, we have shown that the type-inference problem is at least semi-decidable (under this restriction). We have implemented a procedure that combines run-time testing and type-checking to automatically obtain size dependencies. It terminates on total typable function definitions.

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“…Several papers have studied programming languages with implicit computational complexity properties [3,8]. This line of research is motivated both by the perspective of automated complexity analysis and providing natural characterisations of complexity classes like PTIME or PSPACE.…”

Section: Related Workmentioning

confidence: 99%

Collected Size Semantics for Strict Functional Programs over General Polymorphic Lists

Shkaravska

Eekelen

Tamalet³

2014

Foundational and Practical Aspects of Resource Analysis

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Abstract. Size analysis can be an important part of heap consumption analysis. This paper is a part of ongoing work about typing support for checking output-on-input size dependencies for function definitions in a strict functional language. A significant restriction for our earlier results is that inner data structures (e.g. in a list of lists) all must have the same size. Here, we make a big step forwards by overcoming this limitation via the introduction of higher-order size annotations such that variate sizes of inner data structures can be expressed. In this way the analysis becomes applicable for general, polymorphic nested lists.

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A logical account of pspace

Cited by 17 publications

References 27 publications

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Polynomial Size Analysis of First-Order Shapely Functions

Collected Size Semantics for Strict Functional Programs over General Polymorphic Lists

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