Identifying polynomial-time recursive functions

We present a sound and sufficient criterion for identifying polynomial-time recursive functions over higher-order data structures generalizing the seminal results by Bellantoni and Cook [1] and Leivant [2] to complex structural data-types. The criterion, presented as a deductive system, always terminates with a response that is either yes or don't know. The criterion is complete in the sense that every polynomial-time recursive function over binary strings has at least one implementation that is identified by our criterion; whether this is also true for arbitrary higher-order data structures remains an open problem. Logic programming serves as the underlying model of computation and our results apply to the Horn fragment as well to the fragment of hereditary Harrop formulas.