On symmetry of information and polynomial time invertibility

Abstract

Symmetry of information states that for two strings x and y, K(xy)=K(x)+K(y ¦ x)±O(log¦xy¦). We consider the statement of whether symmetry of information holds in a polynomial time bounded environment. Intuitively, this problem is related the complexity of inverting a polynomial time computable function. We give some evidence supporting this intuition, by proving the following relations: 1. If the polynomial time symmetry of information holds, then there is a polynomial time algorithm that computes the shortest description of a string for “almost all” strings. 2. If the polynomial time symmetry of information holds, then every polynomial time computable function is probabilistic polynomial time invertible for “almost all” strings in its domain. 3. If P=NP (i.e., every polynomial time computable function is polynomial time invertible), then the polynomial time symmetry of information holds. KeywordsPolynomial TimePolynomial Time AlgorithmKolmogorov ComplexityRandom StringProbabilistic Polynomial TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.