Time-Complexity Semantics for Feasible Affine Recursions

  • Norman Danner
  • James S. Royer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


The authors’ ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable (ATR types are confined to levels 0, 1, and 2). A limitation of the original version of ATR is that the only directly expressible recursions are tail-recursions. Here we extend ATR so that a broad range of affine recursions are directly expressible. In particular, the revised 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 extending and simplifying the original time-complexity semantics for ATR to develop a set of tools for extracting and solving the higher-type recurrences arising from feasible affine recursions.


Base Type Recursive Call Typing Rule Elimination Rule Type Context 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Norman Danner
    • 1
  • James S. Royer
    • 2
  1. 1.Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459USA
  2. 2.Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13210USA

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