Characterizing complexity classes by higher type primitive recursive definitions
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We study the expressive power of 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(2p(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, 7].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.