A new order-theoretic characterisation of the polytime computable functions.

We propose a new order-theoretic characterisation of the class of polytime computable functions. To this avail we define the small polynomial path order ([Formula: see text] 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 [Formula: see text] 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.