DLOGTIME

Abstract: We introduce a weak uniformity condition for families of P systems, DLOGTIME uniformity, inspired by Boolean circuit complexity. We then prove that DLOGTIME-uniform families of P systems with active membranes working in logarithmic space (not counting their input) can simulate logarithmic-space deterministic Turing machines.

Abstract: We study the complexity of building pseudorandom generators (PRGs) from hard functions. We show that, starting from a function f : f0;1g l ! f0;1g that is mildly hard on average, i.e. every circuit of size 2 ›(l) fails to compute f on at least a 1=poly(l) fraction of inputs, we can build a PRG : f0;1g O(log n) ! f0;1g n computable in ATIME(O(1);logn) = alternating time O(logn) with O(1) alternations. Such a PRG implies BP ¢ AC0 = AC0 under DLOGTIME -uniformity. On the negative side, we prove a tight time-alternations tradeoff for black-box PRG constructions that are based on worst-case hard functions. We also prove a tight time-alternations tradeoff for black-box worst-case hardness amplification, which is the problem of producing an average-case hard function starting from a worstcase hard one. In particular, we obtain that there is no black-box worst-case hardness amplification within the polynomial time hierarchy. These lower bounds are obtained by showing that constant depth circuits cannot compute extractors and list-decodable codes.

Abstract: We define DLOGTIME proof systems, DLTPS, which generalize NC0 proof systems. It is known that functions such as Exact_k and Majority do not have NC0 proof systems. Here, we give a DLTPS for Exact_k (and therefore for Majority) and also for other natural functions such as Reach and Cliquek. Though many interesting functions have DLTPS, we show that there are languages in NP which do not have DLTPS. We consider the closure properties of DLTPS and prove that they are closed under union and concatenation but are not closed under intersection and complement. Finally, we consider a hierarchy of polylogarithmic time proof systems and show that the hierarchy is strict.