Matthias Jantzen

Monadic Thue systems

Certain infinite Thue systems over a finite alphabet are studied, in particular, systems S⊆∑∗×(∑∪{e}) such that for each aϵ∑∪{e}, the set {u| (u,a)ϵS} is a context-freelanguage. The syntactic structure of sets of ancestors and sets of descendants is considered, as well as that of unions of congruence classes, taken over (infinite) context-free languages or regular sets. The common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable (even for finite systems). The word problem for confluent systems of this type is shown to be tractable. The question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation.

The residue of vector sets with applications to decidability problems in Petri nets

A set K of integer vectors is called right-closed, if for any elementmeK all vectors m′≧m are also contained in K. In such a case K is a semilinear set of vectors having a minimal generating set res(K), called the residue of K. A general method is given for computing the residue set of a right-closed set, provided it satisfies a certain decidability criterion.

Petri net algorithms in the theory of matrix grammars

This paper shows that the languages over a one-letter alphabet generated by context-free matrix grammars are always regular. Moreover we give a decision procedure for the question of whether a context-free matrix language is finite. Hereby we strengthen a result of [Mk 92] and settle a number of open questions in [DP 89]. Both results are obtained by a reduction to Petri net problems.

Language Theory of Petri Nets

Petri nets where multiple arcs are allows and the capacity of the places need not be bounded are here called Place/Transition systems. The restrictions of the possible finite or infinite occurence sequences of a P/T-system to the transitions are called transition sequences and give the basis to define families of formal languages related to classes of P/T-systems.

The power of synchronizing operations on strings

Abstract Operations on strings and languages, such as shuffle, iterated shuffle, inverse shuffle and cancellation, have been used to describe sequentialized execution histories of concurrent processes. The power of these operations and their relation to the usual AFL-operations is studied and it is shown that flow expressions [11, 12], event expressions [8–10] and even very restricted variants of them define all the recursively enumerable sets. The family of recursively enumerable languages is equal to the least full trio which is in addition closed under iterated shuffle, and it is also equals the the smallest family of languages containing the finite sets and closed under (a) shuffle, iterated shuffle, and inverse shuffle; (b) shuffle, iterated shuffle, and cancellation; (c) product, iterated shuffle, and cancellation with finite sets; (d) product, iterated shuffle, and inverse shuffle with regular sets; (e) product, iterated shuffle, homomorphisms, and inverse homomorphisms. The family of languages definable by shuffle expressions [6, 12] is incomparable with the family of computation sequence sets [2–5].

Extending regular expressions with iterated shuffle

It is shown that every finite expression which uses the operations union, product, Kleene star, and iterated shuffle in any order, starting with finite sets, defines a language which can be recognized non-deterministically by some multicounter machine in quasirealtime. It is known that this family is in general not closed with respect to iterated shuffle. As a consequence of the characterization each such language is in NSPACE(log n) and thus in P. However, if P ≠ NP, then also neither P nor NSPACE(log n) are closed under iterated shuffle. The proof uses the new concept of so-called shuffle schemes and a number of results on algebraic language theory.

On a special monoid with a single defining relation

Abstract We show that no finite union of congruence classes [w], w being an arbitrary element of the free monoid {a, b}∗ with unit 1, is a context-free language if the congruence is defined by the single pair (abbaab, 1). This congruence is neither confluent nor even preperfect. The monoid formed by its congruence classes is a group which has infinitely many isomorphic proper subgroups.

A note on a special one-rule semi-Thue system

Abstract We show that the special semi-Thue system S1 = {(abba, λ)} has no equivalent finite semi-Thue system which is uniquely terminating, i.e. canonical. This gives another example of a Thue system with a decidable word problem, but solving it using a canonical string rewriting system is possible only by introducing new additional symbols. In contrast to the example obtained recently by Kapur and Narendran (1984) this system presents a monoid which is in fact a group.

Refining the hierarchy of blind multicounter languages and twist-closed trios

We introduce the new families (k, r)-RBC of languages accepted in quasi-realtime by one-way counter automata having k blind counters, of which at least r are reversal-bounded. It is proved, that these families form a strict and linear hierarchy of semi-AFLs within the the family BLIND = M∩(C1) of blind multicounter languages with generator C1 = {w ∈ {a1, b1}* | |w|a1= |w|b1}. This thereby combines the families BLIND and RBC from [13] to one strict hierarchy and generalizes and sharpens Greibachs results. The strict inclusions between the k-counter families (k, r)-RBC are proved using linear algebra techniques. We also study the language theoretic monadic operation twist [18,20], in connection with the semi-AFLs of languages accepted by multicounter and multipushdown acceptors, all restricted to reversal-bounded behavior. It is verified, that the family (k, r)-RBC is twist-closed if and only if r = 0, in which case (k, 0)-RBC = M(Ck), Ck being the k-fold shuffle of disjoint copies of C1. We characterize the family M∩(PAL) of languages accepted in quasi-realtime by nondeterministic one-way reversal-bounded multipushdown acceptors as the least twist-closed trio Mtwist(PAL) generated by the set of palindromes PAL = {w ∈ {a, b}* | w = wrev}.

Cancellation in context-free languages: enrichment by reduction

The following problem is shown to be decidable: Given a context-free grammar G and a string wϵX∗, does there exist a string uϵL(G) such that w is obtained from u by deleting all substrings ui that are elements of the symmetric Dyck set D∗1? The intersection of any two context-free languages can be obtained from only one context-free language by cancellation either with the smaller semi-Dyck set D′∗1 ⊂ D∗1 or with D∗1 itself. Also, the following is shown here for the first time: if the set EQ: = {xnx-n ¦ n ϵ N} ⊂ D′∗1 is used for this cancellation, then each recursively enumerable set can be obtained from linear context-free languages. Previous work has shown that cancellation of substrings from the semi-Dyck language D′∗1 or from any of the former languages D∗1, or EQ, allows one to obtain the following from the context-free languages: any terminal Petri net language [14, 16], the intersections of any context-free with a terminal Petri net language, and the nested iterated substitution thereof [15]. We generalize this characterization by showing that linear context-free languages suffice for generating terminal Petri net languages. In proving this we obtain a new closure property of the family of Petri net languages which is not shared by the context-free sets.