A Sharp Separation of Sublogarithmic Space Complexity Classes

We present very sharp separation results for Turing machine sublogarithmic space complexity classes which are of the form: For any, arbitrarily slow growing, recursive nondecreasing and unbounded function s there is a k in N and an unary language L such that L in SPACE(s(n)+k) setminus SPACE(s(n-1)). For a binary L the supposition łims = infty is sufficient. The witness languages differ from each language from the lower classes on infinitely many words. We use so called demon (Turing) machines where the tape limit is given automatically without any construction. The results hold for deterministic and nondeterministic demon machines and also for alternating demon machines with a constant number of alternations, and with unlimited number of alternations. The sharpness of the results is ensured by using a very sensitive measure of space complexity of Turing computations which is defined as the amount of the tape required by the simulation (of the computation in question) on a fixed universal machine. As a proof tool we use a succint diagonalization method

Refinement of the Alternating Space Hierarchy

We refine the alternating space hierarchy by separating the classes break sgak spa(s(n)) and piak spa(s(n)) from deak spa(s(n)) as well as from break deak+1 spa(s(n)), for each s(n)in Omega(łlgn) cap o(łgn), and k geq 2. We also present unary (tally) sets separating sga2 spa(s(n)) and pia2 spa(s(n)) from break dea2 spa(s(n)) as well as from dea3 spa(s(n))

Space hierarchy theorem revised

AbstractWe show that, for an arbitrary function h(n) and each recursive function ℓ(n), that are separated by a nondeterministically fully space constructible g(n), such that h(n)∈Ω(g(n)) but ℓ(n)∉Ω(g(n)), there exists a unary language L in NSPACE(h(n)) that is not contained in NSPACE(ℓ(n)). The same holds for the deterministic case.The main contribution to the well-known Space Hierarchy Theorem is that (i) the language L separating the two space classes is unary (tally), (ii) the hierarchy is independent of whether h(n) or ℓ(n) are in Ω(logn) or in o(logn), (iii) the functions h(n) or ℓ(n) themselves need not be space constructible nor monotone increasing, (iv) the hierarchy is established both for strong and weak space complexity classes. This allows us to present unary languages in such complexity classes as, for example, NSPACE(loglogn·log∗n)⧹NSPACE(loglogn), using a plain diagonalization

Sublogarithmic bounds on space and reversals

The complexity measure under consideration is SPACE x REVERSALS for Turing machines that are able to branch both existentially and universally. We show that, for any function h(n) between log log n and log n, Pi(1) SPACE x REVERSALS(h(n)) is separated from Sigma(1)SPACE x REVERSALS(h(n)) as well as from co Sigma(1)SPACE x REVERSALS(h(n)), for middle, accept, and weak modes of this complexity measure. This also separates determinism from the higher levels of the alternating hierarchy. For "well-behaved" functions h(n) between log log n and log n, almost all of the above separations can be obtained by using unary witness languages. In addition, the construction of separating languages contributes to the research on minimal resource requirements for computational devices capable of recognizing nonregular languages. For any (arbitrarily slow growing) unbounded monotone recursive function f(n), a nonregular unary language is presented that can be accepted by a middle Pi(1) alternating Turing machine in s(n) space and i(n) input head reversals, with s(n) . i(n) is an element of O(log log n . f(n)). Thus, there is no exponential gap for the optimal lower bound on the product s(n) . i(n) between unary and general nonregular language acceptance-in sharp contrast with the one-way case

The Sublogarithmic Alternating Space World

This paper tries to fully characterize the properties and relationships of space classes defined by Turing machines that use less than logarithmic space -- may they be deterministic, nondeterministic or alternating (DTM, NTM or ATM). We provide several examples of specific languages and show that such machines are unable to accept these languages. The basic proof method is a nontrivial extension of the 1 n 7! 1 n+n! technique to alternating TMs. Let llog denote the logarithmic function log iterated twice, and \Sigma k Space(S), \Pi k Space(S) be the complexity classes defined by S--space-bounded ATMs that alternate at most k \Gamma 1 times and start in an existential, resp. universal state. Our first result shows that for each k ? 1 the sets \Sigma k Space(llog ) n \Pi k Space(o(log )) and \Pi k Space(llog ) n \Sigma k Space(o(log )) are both not empty. This implies that for each S 2\Omega\Gamma2868 ) " o(log ) the classes \Sigma 1 Space(S) ae \Sigma 2 Space(S) ae \Sigma 3 Space(..