Undecidable Membership Problem Research Articles

Variations of checking stack automata: Obtaining unexpected decidability properties

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties:1.The deterministic version has a decidable membership problem but has an undecidable emptiness problem.2.The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.

Rational subsets and submonoids of wreath products

It is shown that membership in rational subsets of wreath products H≀V with H a finite group and V a virtually free group is decidable. On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath product Z≀Z with an undecidable membership problem.

On the extension problem for partial permutations

A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl*V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

Automatic verification of recursive procedures with one integer parameter

Context-free processes (BPA) have been used for dataflow analysis in recursive procedures with applications in optimizing compilers (Proceedings of FOSSaCS’99, Lecture Notes in Computer Science, Vol. 1578, Springer, Berlin, 1999, pp. 14–30). We introduce a more refined model called BPA(Z) that can model not only recursive dependencies, but also the passing of an integer parameter to a subroutine. Moreover, this parameter can be tested against conditions expressible in Presburger arithmetic. This new and more expressive model can still be analyzed automatically. We define Z-input 1-CM, a new class of 1-counter machines (cm) that take integer numbers as input, to describe sets of configurations of BPA(Z). We show that the Post∗ (the set of successors) of a set of BPA(Z)-configurations described by a Z-input 1-CM can be effectively constructed. The Pre∗ (set of predecessors) of a regular set can be effectively constructed as well. However, the Pre∗ of a set described by a Z-input 1-CM cannot be represented by a Z-input 1-CM, in general, and has an undecidable membership problem. Then we develop a new temporal logic based on reversal-bounded counter machines (i.e. machines which use counters such that the change between increasing and decreasing mode of each counter is bounded (J. Assoc. Comput. Mach. 25 (1978) 116) that can be used to describe properties of BPA(Z) and show that the model-checking problem is decidable.

UNDECIDABILITY, AUTOMATA, AND PSEUDOVARITIES OF FINITE SEMIGROUPS

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

A decidable equational theory with undecidable membership problem for finite algebras

A decidable equational theory with undecidable membership problem for finite algebras

Subsemigroups and complexity via the Presentation Lemma

The Krohn-Rhodes Theorem shows that any finite semigroup S can be built by cascading (via wreath product) the simple groups which divide S with trivial combinatorial “flip-flops”. The complexity of a semigroup is essentially the length of a shortest such decomposition (counting alternations of groups). It is an important open question whether complexity of finite semigroups is decidable. In this paper, after reviewing some local and semi-local structure theory of finite semigroups which distills the insights of the Rees-Sushkevych Theorem, we prove the Presentation Lemma. The Presentation Lemma gives a characterization of complexity n in terms of semi-local mapping properties and relational morphisms to transformation semigroups of lower complexity. Thus the Presentation Lemma is a bridge linking the local Green-Rees-Sushkevych coordinate picture of a single J-class to the global properties of the semigroup. As an application, we derive sufficient conditions for complexity of a finite semigroup S to be effectively computable by examining its local subsemigroups (those of the form eSe where e 2 = e GS). The Almost-Disjoint Semigroup Theorem, proved via the Presentation Lemma, and its corollaries enable one to determine the complexity of a large class of examples. A simple application of the Presentation Lemma also allows us to recover Tilson's theorem that the complexity of semigroups with two or fewer non-zero J-classes is effectively computable. Conditions, in terms of subsemigroups, guaranteeing parallelizability of computation are also obtained. A counterexample constructed using deep insights obtained from the Presentation Lemma shows that complexity need not be locally determined. Also a reformulation of the Presentation Lemma in the language of categories and functors is given. The Presentation Lemma will be used in future papers as a tool to attack the decidability of complexity and for constructing pseudo-varieties of semigroups with undecidable membership problem from those with decidable membership problem using the pseudo-variety operations, ∗, ∗ r , and m.