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 (sPOP⁎ 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.