Generation Complexity Versus Distinction Complexity

Among the several notions of resource-bounded Kolmogorov complexity that suggest themselves, the following one due to Levin [Le] has probably received most attention in the literature. With some appropriate universal machine U understood, let the Kolmogorov complexity of a word w be the minimum of |d|+log t over all pairs of a word d and a natural number t such that U takes time t to check that d determines w. One then differentiates between generation complexity and distinction complexity [A, Sip], where the former asks for a program d such that w can actually be computed from d, whereas the latter asks for a program d that distinguishes w from other words in the sense that given d and any word u, one can effectively check whether u is equal to w. Allender et al. [A] consider a notion of solvability for nondeterministic computations that for a given resource-bounded model of computation amounts to require that for any nondeterministic machine N there is a deterministic machine that exhibits the same acceptance behavior as N on all inputs for which the number of accepting paths of N is not too large. They demonstrate that nondeterminism is solvable for computations restricted to polynomially exponential time if and only if for any word the generation complexity is at most polynomial in the distinction complexity. We extend their work and a related result by Fortnow and Kummer [FK] as follows. First, nondeterminism is solvable for linearly exponential time bounds if and only if generation complexity is at most linear in distinction complexity. Second, nondeterminism is solvable for polynomial time bounds if and only if the conditional generation complexity of a word w given a word y is at most linear in the conditional distinction complexity of w given y; hence, in particular, the latter condition implies that P is equal to UP. Finally, in the setting of space bounds it holds unconditionally that generation complexity is at most linear in distinction complexity. In general, the Kolmogorov complexity of a word w is the length |d| of a shortest program d such that d determines w effectively. In a setting of unbounded computations, this approach leads canonically to the usual notion of plain Kolmogorov complexity and its prefix-free variant. In a setting of resource-bounded computations though, there are several notions of Kolmogorov complexity that are in some sense natural – and none of them is considered canonical. M. Agrawal et al. (Eds.): TAMC 2008, LNCS 4978, pp. 463–472, 2008. c © Springer-Verlag Berlin Heidelberg 2008 464 R. Hölzl and W. Merkle A straight-forward approach is to cap the execution time and/or used space by simply not allowing descriptions that take too long or too much space for producing the word we want to describe. This notion has the disadvantage that for a fixed resource-bound there is no canonical notion of universal machine. Another approach, which has received considerable attention in the literature, was introduced by Levin [Le], where, in contrast to the notion just mentioned, arbitrarily long computations are allowed, but a large running time increases in some way the complexity value. More precisely, with some appropriate universal machine U understood, in Levin’s model the Kolmogorov complexity of a word d is the minimum of |d|+log t over all pairs of a word d and a natural number t such that U takes time t to check that w is the word determined by d. As for other notions of resource-bounded Kolmogorov complexity, here one can differentiate between generation complexity and distinction complexity [A, Sip], where the former asks for a program d such that w can actually be computed from d, whereas the latter asks for a program d that distinguishes w from other words in the sense that given d and any word u, one can effectively check whether u is equal to w. The question of how generation and distinction complexity relate to each other in the setting of Levin’s notion of resource-bounded Kolmogorov complexity has been investigated by Allender et al. [A]. They consider a notion of solvability for nondeterministic computations that — for a given resource-bounded model of computation — amounts to require that for any nondeterministic machine N there is a deterministic machine that exhibits the same acceptance behavior as N on all inputs for which the number of accepting paths of N is not too large, e.g., is at most logarithmic in the number of all possible paths. Their main result then asserts that nondeterminism is solvable for computations restricted to polynomially exponential time if and only if for any word the generation complexity is at most polynomial in the distinction complexity. We extend the work of Allender et al. [A] and a related result by Fortnow and Kummer [FK] as follows. First, nondeterminism is solvable for linearly exponential time bounds if and only if generation complexity is at most linear in distinction complexity. Second, nondeterminism is solvable for polynomial time bounds if and only if the conditional generation complexity of a word w given a word y is at most linear in the conditional distinction complexity of w given y; as a consequence, the latter condition implies in particular that P is equal to UP. Combining the result on polynomial time bounds with a result by Fortnow and Kummer [FK] about Kolmogorov complexity defined in terms of fixed polynomial time bounds, one obtains that in the model just mentioned conditional generation and distinction complexity are close if and only if they are close in Levin’s model. Finally, in the setting of space bounds, more precisely, for complexity measures Ks and KDs that logarithmically count the used space instead of the running time used on a program, it holds unconditionally that generation complexity is at most linear in distinction complexity. The notion of generation complexity considered below differs from Levin’s original notion insofar as one has to generate only single bits of the word to be Generation Complexity Versus Distinction Complexity 465 generated but not the word as a whole. This variant has already been used by Allender et al. [A]; their results mentioned above, as well as the results demonstrated below extend to Levin’s original model by almost identical proofs. For a complexity class C, we will refer by C-machine to any machine M that uses a model of computation and obeys a timeor space-bound such that M witnesses L(M) ∈ C with respect to the standard definition of C. For example, an NE-machine is a nondeterministic machine that runs in linearly exponential time. The individual bits of a word x will be denoted by x1 to x|x|. We fix an appropriate universal machine U that receives as input encoded tuples of words and e.g. (x, y, z) will be encoded by x̃01ỹ01z̃ where the word ũ is obtained by doubling every symbol in u, i.e., ũ = u1u1u2u2 . . . u|u|u|u|. Logarithms to base 2 are denoted by log, and often a term of the form log t will indeed denote the least natural number s such that t ≤ 2.