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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 upto depth d, the (innermost) runtime complexity is polynomially bounded of degree d. This bound is tight.

A New Order-Theoretic Characterisation of the Polytime Computable Functions