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Martin Hofmann (2001)

The Strength of Non-size-increasing Computation (Introduction and Summary)

In: Mathematical Foundations of Computer Science 2001, 26th International Symposium, MFCS 2001 Marianske Lazne, Czech Republic, August 27-31, 2001, Proceedings, ed. by Jiri Sgall and Ales Pultr and Petr Kolman, vol. 2136, pp. 58-61, Springer. Lecture Notes in Computer Science (ISBN: 3-540-42496-2).

We study the expressive power non-size increasing recursive definitions over lists. This notion of computation is such that the size of all intermediate results will automatically be bounded by the size of the input so that the interpretation in a finite model is sound with respect to the standard semantics. Many well-known algorithms with this property such as the usual sorting algorithms are definable in the system in the natural way. The main result is that a characteristic function is definable if and only if it is computable in time O(2 p(n) ) for some polynomial p. The method used to establish the lower bound on the expressive power also shows that the complexity becomes polynomial time if we allow primitive recursion only. This settles an open question posed in [1,6]. The key tool for establishing upper bounds on the complexity of derivable functions is an interpretation in a finite relational model whose correctness with respect to the standard interpretation is shown using a semantic technique.

computational complexity, inite model, semantics, higher-order functions

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