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Two Algorithms in Search of a Type-System

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Abstract

The authors’ \(\mathsf{ATR}\) programming formalism is a version of call-by-value \(\mathsf{PCF}\) under a complexity-theoretically motivated type system. \(\mathsf{ATR}\) programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are \(\mathsf{ATR}\) -definable ( \(\mathsf{ATR}\) types are confined to levels 0, 1, and 2). A limitation of the original version of \(\mathsf{ATR}\) is that the only directly expressible recursions are tail-recursions. Here we extend \(\mathsf{ATR}\) so that a broad range of affine recursions are directly expressible. In particular, the revised \(\mathsf{ATR}\) can fairly naturally express the classic insertion- and selection-sort algorithms, thus overcoming a sticking point of most prior implicit-complexity-based formalisms. The paper’s main work is in refining the original time-complexity semantics for \(\mathsf{ATR}\) to show that these new recursion schemes do not lead out of the realm of feasibility.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT, 06459, USA

    Norman Danner

  2. Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, 13210, USA

    James S. Royer

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  1. Norman Danner
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  2. James S. Royer
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Correspondence to Norman Danner.

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Danner, N., Royer, J.S. Two Algorithms in Search of a Type-System. Theory Comput Syst 45, 787–821 (2009). https://doi.org/10.1007/s00224-009-9181-y

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