Klaus-Jörn Lange

Reversible space equals deterministic space

This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by C. Bennett (1989) and refutes the conjecture, made by M. Li and P. Vitanyi (1996), that any reversible simulation of an irreversible computation must obey Bennetts reversible pebble game rules.

Unambiguity and Fewness for Logarithmic Space

We consider various types of unambiguity and fewness for log space bounded Turing machines and polynomial time bounded log space auxiliary pushdown automata. In particular, we introduce the notions of (general), reach, and strong unambiguity and fewness. We demonstrate that closure under complement of unambiguous classes implies equivalence of unambiguity and “unambiguous fewness”. This, as we will show, applies in the cases of reach and strong unambiguity for log space. Among the many relations we exhibit, we show that the unambiguous linear contextfree languages, which are not known to be contained in deterministic log space, nevertheless are contained in strongly unambiguous log space, and, consequently, are log space reducible to deterministic contextfree languages.

On the Complexities of Linear LL(1) and LR(1) Grammars

Several notions of deterministic linear languages are considered and compared with respect to their complexities and to the families of formal languages they generate. We exhibit close relationships between simple linear languages and the deterministic linear languages both according to Nasu and Honda and to Ibarra, Jiang, and Ravikumar. Deterministic linear languages turn out to be special cases of languages generated by linear grammars restricted to LL(1) conditions, which have a membership problem solvable in NC1. In contrast to that, deterministic linear languages defined via automata models turn out to have a DSPACE(logn)-complete membership problem. Moreover, they coincide with languages generated by linear grammars subject to LR(1) conditions.

An Unambiguous Class Possessing a Complete Set

In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a ‘promise’ or ‘semantic’ concept. These usually lead to classes apparently without complete problems.

The Emptiness Problem for Intersections of Regular Languages

Given m finite automata, the emptiness of intersection problem is to determine whether there exists a string which is accepted by all m automata. In the following we consider the case, when m is bounded by a function in the input length, i.e., in the size and number of the automata. In this way we get complete problems for nondeterministic space-bounded and timespace-bounded complexity classes. Further on, we get close relations to nondeterministic sublinear time classes and to classes which are defined by bounding the number of nondeterministic steps.

String grammars with disconnecting or a basic root of the difficulty in graph grammar parsing

The complexity of languages generated by so-called context-free string grammars with disconnecting is investigated. The result is then applied to a number of graph grammar models with finite Church Rosser property. In particular, it is shown that these graph grammars can generate NP-complete languages.

Unambiguity of circuits

Abstract The concept of unambiguity of circuits is considered. Several classes of unambiguous circuit families within the NC-hierarchy are introduced and related to unambiguous automata and to PRAMs with exclusive write access. In particular, we show CREW-TIME(logk n) = UnambACk for each positive integer k.

On the Complexity of Some Problems on Groups Input as Multiplication Tables

The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investigate CGM for particular classes of groups. The problem for general groups is in SL (symmetric log space), but any kind of hardness result seems difficult because proving it would require constructing the entire multiplication table of a group. We introduce the complexity class FOLL, or FO(loglogn), of problems solvable by uniform poly-size circuit families of unbounded fan-in and depth O(loglogn). No problem in FOLL can be hard for L or for any other class containing parity, but FOLL is not known to be contained even in SL. We show that CGM for cyclic groups is in FOLL?L and that CGM for abelian groups is in FOLL. We then examine the case of some solvable groups, showing in particular that CGM for nilpotent groups is also provably not hard for any class containing parity. We also consider the problem of testing for various properties of a group input as a table: we prove that cyclicity and nilpotency can each be tested in FOLL?L. Finally, we examine the implications of our results for the complexity of iterated multiplication, powering, and division of integers in the context of the recent results of Chiu, Davida, and Litow and of Hesse.

Characterizing unambiguous augmented pushdown automata by circuits

The notions of weak and strong unambiguity of augmented push-down automata are considered and related to unambiguities of circuits. In particular we exhibit circuit classes exactly characterizing polynomially time bounded unambiguous augmented push-down automata.

Characterizing TC 0 in Terms of Infinite Groups

We characterize the languages in TC0 = L(Maj[<,Bit]) and L(Maj[<]) as inverse morphic images of certain groups. Necessarily these are infinite, since nonregular sets are concerned. To limit the power of these infinite algebraic objects, we equip them with a finite type set and introduce the notion of a finitely typed (infinite) monoid. Following this approach we investigate type-respecting mappings and construct a new type of block product that more adequately deals with infinite monoids. We exhibit two classes of finitely typed groups which exactly characterize TC0 and L(Maj[<]) via inverse morphisms.