Productivity of stream definitions

This paper provides a criterion based on interpretation methods on term rewrite systems in order to characterize the polynomial time complexity of second order functionals. For that purpose it introduces a first order functional stream language that allows the programmer to implement second order functionals. This characterization is extended through the use of exp-poly interpretations as an attempt to capture the class of Basic Feasible Functionals (bff). Moreover, these results are adapted to provide a new characterization of polynomial time complexity in computable analysis. These characterizations give a new insight on the relations between the complexity of functional stream programs and the classes of functions computed by Oracle Turing Machine, where oracles are treated as inputs.

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“…Productive streams are defined in the literature [20] as terms weakly normalizing to infinite lists, which is in our case equivalent to:…”

Section: Lemma 4 If E Is An Expression Of a Program With An Interprementioning

confidence: 99%

Characterizing polynomial time complexity of stream programs using interpretations

Férée

Hainry

Hoyrup

et al. 2015

Theoretical Computer Science

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“…The use of lazy evaluation allows us to do away with the restriction of an 'independent data-layer' required in [11,12,9,54,55], meaning that symbols of inductive sort are disallowed having coinductive arguments. An independent data-layer excludes, e.g., the function head(x : y) → x which takes a coinductive argument (a stream) and returns an element of an inductive sort.…”

Section: Contribution and Outlinementioning

confidence: 99%

Lazy productivity via termination

Endrullis

Hendriks

2011

Theoretical Computer Science

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a b s t r a c tWe present a procedure for transforming strongly sequential constructor-based term rewriting systems (TRSs) into context-sensitive TRSs in such a way that productivity of the input system is equivalent to termination of the output system. Thereby automated termination provers become available for proving productivity. A TRS is called productive if all its finite ground terms are constructor normalizing, and all 'inductive constructor paths' through the resulting (possibly non-wellfounded) constructor normal form are finite. To our knowledge, this is the first complete transformation from productivity to termination.The transformation proceeds in two steps: (i) The strongly sequential TRS is converted into a shallow TRS, where patterns do not have nested constructors. (ii) The shallow TRS is transformed into a context-sensitive TRS, where rewriting below constructors and in arguments not 'consumed from' is disallowed.Furthermore, we show how lazy evaluation can be encoded by strong sequentiality, thus extending our transformation to, e.g., Haskell programs.Finally, we present a simple, but fruitful extension of matrix interpretations to make them applicable for proving termination of context-sensitive TRSs.

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“…Such functions were studied (e.g.) in [1,26]. Combining these methods with the method we have described here can be one of the objectives for the future work.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Inductive and Coinductive Components of Corecursive Functions in Coq

Bertot

Komendantskaya

2008

Electronic Notes in Theoretical Computer Science

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In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.

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Productivity of stream definitions

Cited by 29 publications

References 15 publications

Characterizing polynomial time complexity of stream programs using interpretations

Characterizing polynomial time complexity of stream programs using interpretations

Lazy productivity via termination

Inductive and Coinductive Components of Corecursive Functions in Coq

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