Adventures in time and space

Norman Danner, James S. Royer

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR, has its first-order programs characterize the polynomial-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomial-time. The time-restricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the na¨ıve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine.

Original languageEnglish (US)
Article number9
JournalLogical Methods in Computer Science
Issue number1
StatePublished - Mar 13 2007


  • Basic feasible functional
  • Compositional semantics
  • Higher-type computation
  • Implicit computational complexity
  • Type systems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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