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))

For completeness, sublogarithmic space is no space

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions

Lower Bounds for Alternating Online State Complexity

The notion of Online State Complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm, which is represented by a deterministic machine scanning the input from left to right in one pass. In this paper, we extend the setting to alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976: such machines run independent passes scanning the input from left to right and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem , stating that the polynomial hierarchy of alternating online state complexity is infinite, and the second is a linear lower bound on the alternating online state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model

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

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

Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.Comment: Submitted to the Journal of Logic and Computation, Special Issue on LFCS'2016) (Logical Foundations of Computer Science). Guest Editors: S. Artemov and A. Nerode. This journal version extends two conference papers: the first published in the proceedings of LFCS'2016 and the second in the proceedings of LICS'2018. arXiv admin note: substantial text overlap with arXiv:1607.0025

Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

22. Workshop Komplexitätstheorie und effiziente Algorithmen

his publication contains abstracts of the 22nd workshop on complexity theory and efficient algorithms. The workshop was held on February 8, 1994, at the Max-Planck-Institut für Informatik, Saarbrücken, Germany