Abstract
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 language | English (US) |
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Article number | 9 |
Journal | Logical Methods in Computer Science |
Volume | 3 |
Issue number | 1 |
DOIs | |
State | Published - Mar 13 2007 |
Keywords
- Basic feasible functional
- Compositional semantics
- Higher-type computation
- Implicit computational complexity
- Type systems
ASJC Scopus subject areas
- Theoretical Computer Science
- General Computer Science