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Functional Programming in Sublinear Space
"... Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot comp ..."
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Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired Baillot & Terui’s Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE. 1
Applicative theories for logarithmic complexity classes
"... We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarith ..."
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We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new twosorted algebra which is very similar to Cook and Bellantoni’s famous twosorted algebra B for polynomial time [4]. The second theory describes logarithmic space by justifying concatenation and sharply bounded recursion. All theories contain the predicates W representing words, and V representing temporary inaccessible words. They are inspired by Cantini’s theories [6] formalising B. 1.
Weak Affine Light Typing is complete with respect to Safe Recursion on Notation
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"... Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset ofλterms of System F. WALT is polytime sound: if aλterm M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. The evaluation proceeds under an ..."
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Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset ofλterms of System F. WALT is polytime sound: if aλterm M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. The evaluation proceeds under any strategy of a rewriting relation which is a mix of both callbyname and callbyvalueβreductions. WALT weakens, namely generalizes, the notion of “stratification of deductions”, common to some Light Systems — those logical systems, derived from Linear logic, to characterize the set of Polynomial functions —. A weaker stratification allows to define a compositional embedding of Safe recursion on notation (SRN) into WALT. It turns out that the expressivity of WALT is strictly stronger than the one of the known Light Systems. The embedding passes through the representation of a subsystem of SRN. It is obtained by restricting the composition scheme of SRN to one that can only use its safe variables linearly. On one side, this suggests that SRN, in fact, can be redefined in terms of more primitive constructs. On the other, the embedding of SRN into WALT enjoys the two following remarkable aspects. Every datatype, required by the embedding, is represented from scratch, showing the strong structural prooftheoretical roots of WALT. Moreover, the embedding highlights a stratification structure of the normal and safe arguments, normally hidden inside