The Nature of Computation: Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Proceedings

Promises are a standard way to formalize partial algorithms; and advice quantifies nonuniformity. For decision problems, the latter is captured in common complexity classes such as P/poly, that is, with advice growing in size with that of the input. We advertise constant-size advice and explore its theoretical impact on the complexity of classification problems – a natural generalization of promise problems – and on real functions and operators. Specifically we exhibit problems that, without any advice, are decidable/computable but of high complexity while, with (each increase in the permitted size of) advice, (gradually) drop down to polynomial-time.

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