Jörg Vogel

Theta2p-Completeness: A Classical Approach for New Results

In this paper we present an approach for proving Θ2p- completeness. There are several papers in which different problems of logic, of combinatorics, and of approximation are stated to be complete for parallel access to NP, i.e. Θ2p-complete. There is a special acceptance concept for nondeterministic Turing machines which allows a characterization of Θ2p as a polynomial-time bounded class. This characterization is the starting point of this paper. It makes a master reduction from that type of Turing machines to suitable boolean formula problems possible. From the reductions we deduce a couple of conditions that are sufficient for proving Θ2p-hardness. These new conditions are applicable in a canonical way. Thus we are able to do the following: (i) we can prove the Θ2p-completeness for different combinatorial problems (e.g. max-card-clique compare) as well as for optimization problems (e.g. the Kemeny voting scheme), (ii) we can simplify known proofs for Θ2p-completeness (e.g. for the Dodgson voting scheme), and (iii) we can transfer this technique for proving Δ2p-completeness (e.g. TSPcompare).

Restarting Automata with Rewriting

Motivated by natural language analysis we introduce restarting automata with rewriting. They are acceptors on the one hand, and (special) regulated rewriting systems on the other hand. The computation of a restarting automaton proceeds in cycles: in each cycle, a bounded substring of the input word is rewritten by a shorter string, and the computation restarts on the arising shorter word.

Different Types of Monotonicity for Restarting Automata

We consider several classes of rewriting automata with a restart operation and the monotonicity property of computations by such automata. It leads to three natural definitions of (right) monotonicity of automata. Besides the former monotonicity, two new types, namely a-monotonicity and g-monotonicity, for such automata are introduced. We provide a taxonomy of the relevant language classes, and answer the (un)decidability questions concerning these properties.

Monotonic Rewriting Automata with a Restart Operation

We introduce a hierarchy of monotonic rewriting automata with a restart operation and show that its deterministic version collapses in a sequence of characterizations of deterministic context-free languages. The nondeterministic version of it gives a proper hierarchy of classes of languages with the class of context-free languages on the top.

Two-way automata with more than one storage medium

Abstract This paper is concerned with the computational power of two-way automata with more than one subrecursive storage medium. Two-way automata with a stack (a nonerasing stack or a pushdown store, respectively) and an arbitrary number of checking stacks are of special interest. They are able to accept exactly those sets which are elementary in the sense of Kalmar . If the number of checking stacks is fixed, then the computational power of the corresponding restricted classes of automata can also be characterized in terms of time and space complexity classes.

The Operators minCh and maxCh on the Polynomial Hierarchy

In this paper we introduce a new acceptance concept for nondeterministic Turing machines with output device which allows a characterization of the complexity class Θ2p = PNP[log] as a polynomial time bounded class. Thereby the internal structure of the output is essential: it looks at output with maximal number of mind changes instead of output with maximal value which was realized for the first time by Krentel [Kre88]. Motivated by this characterization we define in a general way two operators, the so called maxCh- and minCh- operator, respectively which are special types of optimization operators. Following a paper by Hempel/Wechsung [HW96] we investigate the behaviour of these operators on the polynomial hierarchy. We prove a collection of relations regarding the interaction of operators maxCh, minCh,

Restarting Automata

, ∃, ¬, ⊕, Sig, C and U. So we get a tool to show that the maxCh- and minCh- classes are distinct under reasonable structural assumptions. Finally, our proof techniques allow to solve one of the open questions of Hempel/Wechsung.