Thue Systems Research Articles

McNaughton families of languages

In 1988 the Church–Rosser languages were introduced by McNaughton et al. as those languages that are recognized by finite, length-reducing and confluent string-rewriting systems using extra non-terminal symbols. Here we generalize this concept by considering classes of languages that are obtained by other types of string-rewriting systems. To honour Robert McNaughton's original contribution we call the resulting families of languages McNaughton families. Here it is shown that the concept of McNaughton families is as powerful as the notion of Turing machine or the notion of phrase-structure grammar. We investigate the relationships between the various McNaughton families, obtaining an extensive hierarchy of classes that includes many well-known language and complexity classes as well as some new classes. Further, we consider some closure and non-closure properties for those McNaughton families that are contained in the class of context-free languages, and we address the complexity of the fixed and the general membership problems for these families.

Decidability of Termination of Grid String Rewriting Rules

Termination of string rewriting is known undecidable. Termination of string rewriting with only one rule is neither known decidable nor known undecidable. This paper presents a decision procedure for rules $u\rightarrow v$ such that some letter b from u occurs as often or less often in v. We call such rules "grid" rules. By far most rules are grid rules. Grid rules cover all rules which terminate by a total division order. Thus total division orders are shown to be irrelevant for the termination problem of one-rule string rewriting.

A Completion Procedure for Finitely Presented Groups That Is Based on Word Cycles

A Knuth–Bendix-style completion procedure for groups is presented that, instead of working with sets of string-rewriting rules, manipulates finite sets of i>word cycles. A characterization is given for the resulting sets of persistent word cycles, from which it follows that the completion procedure terminates successfully if and only if the i>reduced word problem of the finite group presentation considered is a finite set. In this case the resulting set of persistent word cycles yields a finite canonical string-rewriting system for every linear reduction ordering.

Computing over K-modules

Kan extensions over the category of Sets provide a unifying framework for computation of group, monoid and category actions allowing a number of diverse problems to be solved with a generalised form of string rewriting. This paper extends these techniques to K-algebras and K-categories by using Gröbner basis techniques to compute Kan extensions over the category of K-modules.

Confluence Problems for Trace Rewriting Systems

Rewriting systems over trace monoids, briefly trace rewriting systems, generalize both semi-Thue systems and vector replacement systems. In [21], a particular trace monoid M is presented such that confluence is undecidable for the class of length–reducing trace rewriting systems over M. In this paper, we show that this result holds for every trace monoid, which is neither free nor free commutative. Furthermore we show that confluence for special trace rewriting systems over a fixed trace monoid is decidable in polynomial time.

Termination and derivational complexity of confluent one-rule string-rewriting systems

It is not known whether the termination problem is decidable for one-rule string-rewriting systems, though the confluence of such systems is decidable by Wrathall (in: Word Equations and Related Topics, Lecture Notes in Computer Science, vol. 572, 1992, pp. 237–246). In this paper we develop techniques to attack the termination and complexity problems of confluent one-rule string-rewriting systems. With given such a system we associate another rewriting system over another alphabet. The behaviour of the two systems is closely related and the termination problem for the new system is sometimes easier than for the original system. We apply our method to systems of the special type {apbq→t}, where t is an arbitrary word over {a,b}, and give a complete characterization for termination. We also give a complete analysis of the derivational complexity for the system {apbq→bnam}.

Proving convergence of self-stabilizing systems using first-order rewriting and regular languages

In the framework of self-stabilizing systems, the convergence proof is generally done by exhibiting a measure that strictly decreases until a legitimate configuration is reached. The discovery of such a measure is very specific and requires a deep understanding of the studied transition system. In contrast we propose here a simple method for proving convergence, which regards self-stabilizing systems as string rewrite systems, and adapts a procedure initially designed by Dershowitz for proving termination of string rewrite systems. In order to make the method terminate more often, we also propose an adapted procedure that manipulates "schemes", i.e. regular sets of words, and incorporates a process of scheme generalization. The interest of the method is illustrated on several nontrivial examples.

Semi-Thue Systems with an Inhibitor

A i>semi-Thue system with an inhibitor is one having a special symbol, called an i>inhibitor, that appears on the right side of every rule but does not appear on the left side of any rule. The main result of this paper is that the uniform halting problem is decidable for the class of such systems. The concept of i>inhibitor is related to the concept of i>well-behaved derivation in systems without an inhibitor. The latter concept has received some attention from those interested in the open question as to whether the uniform termination problem for one-rule semi-Thue systems is decidable.

Coset Enumeration Using Prefix Gröbner Bases: An Experimental Approach

AbstractThe authors study a new method for coset enumeration in finitely presented groups. Their method uses prefix Gröbner basis computation in the monoid ring ${\mathbb{K}}[{\cal M}]$, where ${\mathbb{K}}$ is a computable field and ${\cal M}$ a monoid presented by a convergent string-rewriting system. The method is compared to well-known methods for Todd-Coxeter enumeration, using examples from the literature where studies of these methods are reported. New insights into coset enumeration were gained using three different kinds of orderings, combined with new frameworks and strategies implemented in MRC 1.2.

An efficient automata approach to some problems on context-free grammars

Book and Otto (1993) solve a number of word problems for monadic string-rewriting systems using an elegant automata-based technique. In this note we observe that the technique is also very interesting from a pedagogical point of view, since it provides a uniform solution to several elementary problems on context-free languages.

Non-Looping String Rewriting

String rewriting reductions of the form $t\to_R^+ utv$ , called loops , are the most frequent cause of infinite reductions (non- termination). Regarded as a model of computation, infinite reductions are unwanted whence their static detection is important. There are string rewriting systems which admit infinite reductions although they admit no loops. Their non-termination is particularly difficult to uncover. We present a few necessary conditions for the existence of loops, and thus establish a means to recognize the difficult case. We show in detail four relevant criteria: (i) the existence of loops is characterized by the existence of looping forward closures ; (ii) dummy elimination , a non-termination preserving transformation method, also preserves the existence of loops; (iii) dummy introduction , a transformation method that supports subsequent dummy elimination, likewise preserves loops; (iv) bordered systems can be reduced to smaller systems on a larger alphabet, preserving the existence and the non-existence of loops. We illustrate the power of the four methods by giving a two-rule string rewriting system over a two-letter alphabet which admits an infinite reduction but no loop. So far, the least known such system had three rules.

Infinite Convergent String-rewriting Systems and Cross-sections for Finitely Presented Monoids

A finitely presented monoid has a decidable word problem if and only if it can be presented by some left-recursive convergent string-rewriting system if and only if it has a recursive cross-section. However, regular cross-sections or even context-free cross-sections do not suffice. This is shown by presenting examples of finitely presented monoids with decidable word problems that do not admit regular cross-sections, and that, hence, cannot be presented by left-regular convergent string-rewriting systems. Also examples of finitely presented monoids with decidable word problems are presented that do not even admit context-free cross-sections. On the other hand, it is shown that each finitely presented monoid with a decidable word problem has a finite presentation that admits a cross-section which is a Church–Rosser language. Finally we address the notion of left-regular convergent string-rewriting systems that are tractable.

Some undecidability results concerning the property of preserving regularity

A finite string-rewriting system R preserves regularity if and only if it preserves Σ-regularity, where Σ is the alphabet containing exactly those letters that have occurrences in the rules of R. This proves a conjecture of Gyenizse and Vágvölgyi (1997). In addition, some undecidability results are presented that generalize results of Gilleron and Tison (1995) from term-rewriting systems to string-rewriting systems. It follows that the property of being regularity preserving is undecidable for term-rewriting systems, thus answering another question of Gyenizse and Vágvölgyi (1997). Finally, it is shown that it is undecidable in general whether a finite, lengthreducing, and confluent string-rewriting system yields a regular set of normal forms for each regular language.

Contributions of Ronald V. Book to the theory of string-rewriting systems

Contributions of Ronald V. Book to the theory of string-rewriting systems

Infinite String Rewrite Systems and Complexity

We study the relation between time complexity and derivation work for the word problem of infinitely presented semigroups and groups. We introduce the notion of theworkof a derivation (defined as the sum of the lengths of all the rules used in the derivation, with multiplicity). The following results are proved:

Equational unification, word unification, and 2nd-order equational unification

For finite convergent term-rewriting systems it is shown that the equational unification problem is recursively independent of the equational matching problem, the word matching problem, and the 2nd- order equational matching problem. Apart from the latter these results are derived by considering term-rewriting systems on signatures that contain unary function symbols only (i.e., string-rewriting systems). Also for this special case 2nd-order equational matching is shown to be reducible to 1st-order equational matching. In addition, we present some new decidability results for simultaneous equational matching and unification. Finally, we compare the word unification problem to the 2nd- order equational unification problem.

Linear generalized semi-monadic rewrite systems effectively preserve recognizability

We introduce the notion of the generalized semi-monadic rewrite system, which is a generalization of well-known rewrite systems: the ground rewrite system, the monadic rewrite system, and the semi-monadic rewrite system. We show that linear generalized semi-monadic rewrite systems effectively preserve recognizability. We show that a tree language L is recognizable if and only if there exists a rewrite system R such that R∪ R −1 is a linear generalized semi-monadic rewrite system and that L is the union of finitely many ↔ R ★-classes. We show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi-monadic rewrite systems. For example, we show that for a rewrite system R effectively preserving recognizability, it is decidable if R is locally confluent. Moreover, we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse-free rewrite systems. Finally, as a consequence of our results on trees we get that restricted right-left overlapping string rewrite systems effectively preserve recognizability.

A polynomial algorithm testing partial confluence of basic semi-Thue systems

We give a polynomial algorithm solving the problem “is S partially confluent on the rational set R?” for finite, basic, noetherian semi-Thue systems. The algorithm is obtained by a polynomial reduction of this problem to the equivalence problem for deterministic 2-tape finite automata, which has been shown to be polynomially decidable in Friedman and Greibach (1982).

String rewriting and homology of monoids

Results of Anick (1986), Squier (1987), Kobayashi (1990), Brown (1992b), and others, show that a monoid with a finite convergent rewriting system satisfies a homological condition known as FP∞.In this paper we give a simplified version of Brown's proof, which is conceptual, in contrast with the other proofs, which are computational.We also collect together a large number of results and examples of monoids and groups that satisfy FP∞ and others that do not. These may provide techniques for showing that various monoids do not have finite convergent rewriting systems, as well as explicit examples with which methods can be tested.

On termination of confluent one-rule string-rewriting systems

The termination of a confluent one-rule string-rewriting system R = { → t} is reduced to that of another one-rule system R′ = {′ → t′} such that s′ is self-overlap-free (sof). A necessary and sufficient condition is given for termination of a one-rule system R = { → t} such that s is sof and s overlaps t once on both sides.