Jürgen Avenhaus

On Conditional Rewrite Systems with Extra Variables and Deterministic Logic Programs

We study deterministic conditional rewrite systems, i.e. conditional rewrite systems where the extra variables are not totally free but ’input bounded’. If such a system R is quasi-reductive then → r is decidable and terminating. We develop a critical pair criterion to prove confluence if R is quasi-reductive and strongly deterministic. We apply our results to prove Horn clause programs to be uniquely terminating.

The Nielsen reduction and P-complete problems in free groups

Abstract The complexity of some classical algorithmic problems in free groups is studied. Problems like the generalized word problem, to determine shortest coset representatives, to decide whether two subgroups are equal or isomorphic, to decide whether a set of generators is independent or to decide whether a subgroup has finite index, are known to be decidable for free groups. We show that these problems are P-complete under log-space reducibility. This is proved by encoding the computations of a deterministic polynomially time bounded TM into a subgroup of a free group and implementing the Nielsen reduction, one of the main tools for solving algorithmic problems in free groups, in polynomial time.

Higher Order Conditional Rewriting and Narrowing

First order conditional rewrite systems R have been extensively studied. If R is confluent and terminating, then narrowing is a sound and complete procedure to compute all solutions of a goal s = t modulo R. Recently there has been developed a satisfactory way to combine higher order terms and unconditional rewriting. In this paper we first show that this approach can be carried over to conditional higher order rewrite systems. Then we study narrowing using higher order rewrite systems. A naive translation of first order narrowing may lead to unsolvable unification problems. So we restrict to ”quasi first order” goals and ”simple” rewrite systems.

Logicality of conditional rewrite systems

A conditional term rewriting system is called logical if it has the same logical strength as the underlying conditional equational system. In this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term rewriting systems.

On expressing commutativity by finite Church-Rosser presentations : a note on commutative monoids

Soit M un monoide commutatif infini. Supposons que M possede une presentation finie ayant la propriete de «Church-Rosser». Si M est simplifiable ou si la presentation est speciale, alors M est soit le groupe cyclique libre soit le monoide cyclique libre

On the complexity of intersection and conjugacy problems in free groups

Abstract Nielsen type arguments have been used to prove some problems in free group (e.g., the generalized word problem) [2] to be P-complete. In this paper we extend this approach. Having a Nielsen reduced set of generators for subgroups H and K one can solve a lot of intersection and conjugacy problems in polynomial time in a uniform way. We study the solvability of (i) ∃ h ∈ H , k ∈ K : hx = yk in F , and (ii) ∃ w ∈ F: w l Hw = K characterize the set of solutions. This leads for (i) to an algorithm for computing a set of generators for H ∩ K (and a new proof that free groups have the Howson property). For (ii) this gives a fast solution of Moldavanskiis conjugacy problem; an algorithm for computing the normal hull of H then gives a representation of all solutions. All the algorithms run in polynomial time and the decision problems are proved to be P-complete under log-space reducibility.

P-Complete Problems in Free Groups

A powerfull tool for solving algorithmic problems in free groups is the concept of Nielsen reduction. Since the Nielsen reduction can be done in polynomial time, many problems such as the generalized word problem, the equality and the isomorphism problem for subgroups are solvable in polynomial time. Even more, we show that these problems are P — complete under log-space reducibility.

Logicality of Conditional Rewrite Systems

A conditional term rewriting system is called logical if it has the same logical strength as the underlying conditional equational system. In this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term rewriting systems.

Theorembeweisen in hierarchischen bedingten Spezifikationen

Zusammenfassung. In der Arbeit werden klausale Spezifikationen zur Beschreibung von Programmen auf der Ebene des funktionalen Entwurfs betrachtet. Die Axiome solch einer Spezifikation

Groups presented by finite two-monadic Church-Rosser Thue systems

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