Some Observations on the Computational Complexity of Longley’s H Functional Preliminary Report

The sequentially realizable functionals (denoted SR) is a class of “sequentially computable” higher-type functionals that are a strictly larger class than the PCF-computable functionals and that have very strong and natural mathematical properties. Longley [Lon98b] exhibited a functional H in SR with the property that if one adds H to the language PCF, then the resulting language, PCF+H, computes exactly SR. Longley hesitated to recommend PCF+H as the basis of a practical programming language as the only known ways of computing H seemed to involve high computational complexity. In this simple note we show that if P 6= NP, then the computational complexity of H (and of similar functionals in SR) is inherently infeasible. §1. Background Our presentation of the SR functionals is based on the work of van Oosten [vO99] (who was the first to formalize SR along the lines sketched below) and Longley [Lon98b]. Van Oosten and Longley study both the full and effective versions of SR. The focus here is on the effective version of SR, and to avoid decorating everything in sight with an “eff ” subor superscript, we have skewed van Oosten’s and Longley’s notation to reflect our focus. The SR functionals are based on a simple game between two partial functions f, g:N ⇀ N. In this game f successively asks g what its value is at particular arguments and the game terminates when, if ever, f decides it has enough information about g and outputs an answer instead of a question. The game can fail to terminate either because f queries g on an argument on which g is undefined, or because f , on receiving a certain response from g, goes undefined, or because f and g go on chatting forever. In playing one of these ∗Dept. of Elec. Eng. & Computer Science; Syracuse University; Syracuse, NY 13244 USA. Email: [email protected].