We propose a new order-theoretic characterisation of the class of polytime computable functions. To this avail we define the small polynomial path order (italicsPOP⁎ for short). This termination order entails a new syntactic method to analyse the innermost runtime complexity of term rewrite systems fully automatically: for any rewrite system compatible with sPOP⁎ that employs recursion up to depth d, the (innermost) runtime complexity is polynomially bounded of degree d. This bound is tight. Thus we obtain a direct correspondence between a syntactic (and easily verifiable) condition of a program and the asymptotic worst-case complexity of the program.
Bellantoni and Cook have given a function-algebra characterization of the polynomial-time computable functions via an unbounded recursion scheme which is called safe recursion. Inspired by their work, we characterize the exponential-time computable functions with the use of a safe variant of nested recursion.
Isabel Oitavem has introduced a term rewriting system (TRS) which captures the class FPS of polynomial-space computable functions. We propose an alternative TRS for FPS. As a consequence, it is obtained that FPS is the smallest class containing certain initial functions and closed under specific operations. It turns out that our characterization is relatively simple and suggests an uniform approach to the space-complexity.
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