Decidability Results Research Articles

Decidability Results for Metric and Layered Temporal Logics

We study the decidability problem for metric and layered temporal logics. The logics we consider are suitable to model time granularity in various contexts, and they allow one to build granular temporal models by referring to the "natural scale" in any component of the model and by properly constraining the interactions between differently-grained components. A monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered, and the theory of finitely-layered metric temporal structures is shown to be decidable.

Shuffle products and other operations in the class of primitive spaces

A Boolean space is a compact Hausdorff zero-dimensional space. Boolean spaces arise as Stone duals of Boolean algebras and the duals of the countable Boolean algebras are the countably baaed (or equivalently metrizable) Boolean spaces. Among the latter spaces, primitive spaces and more particularly finitary spaces form two very peculiar subclasses. A countably baaed Boolean space is primitive if it admits a basis of pi clopen sets, where X is pi if for each clopen U C X, either X z U or X Z X \ U. It is finitary if the set of homeomorphism types of its pi clopen subsets is finite. These spaces have been introduced through utterly different means by Hanf [2], Pierce [21] and also by Paljutin 1201. They were also studied by Dobbertin [I], Heindorf [12], Myers [18], the author [6] and others ([17], [23] . . . ). Primitive spaces can be handled by easy-to-use combinatorial tools : their diagrams ([2] or [7]) and the structure of their homeomorphism types is at present fairly well understood ([22], [12], [7]). Their pl ace in Ketonen’s hierarchy ([13]) is well-known, at least in the finitary case ([12]). In the class of primitive spaces, Tarski’s cube problem (X+X+X E X implies X+X E X, see [2l] or [S]) has a positive answer, but otherwise the diagram technique enables to produce easy examples of pathological behaviour of product of spaces ([22], [S]). It should be noted that the primitive spaces bear some surprising relationships with the variable-free fragment of the modal system K4 ([9]) and that Paljutin has used finitary spaces to obtain some decidability results ([20]). Finally, it turns out that many Lindenbaum-Tarski algebras of important first-order theories (such as abelian groups, well-orders and linear orders, Boolean algebras, . . ) are primitive ([lS]). It is well-known that the class of primitive spaces is closed under the formation of finite topological sums and finite Cartesian products ([22], [24]). In this paper, we introduce other operations on Boolean spaces under which the class of primitive

Some Decidability Results for ℤ[G]‐Modules when G is Cyclic of Squarefree Order

AbstractWe extend the analysis of the decision problem for modules over a group ring ℤ[G] to the case when G is a cyclic group of squarefree order. We show that separated ℤ[G]‐modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ℤ[G]‐modules. The paper includes a short analysis of the decision problem for the theories of (finitely generated) modules over ℤ[ζm], where m is a positive integer and ζm is a primitive mth root of 1.Mathematics Subject Classification: 03C60, 03B25.

Behavioural Equivalence for Infinite Systems—Partially Decidable!

For finite-state systems non-interleaving equivalences are computationally<br />at least as hard as interleaving equivalences. In this paper we show<br />that when moving to infinite-state systems, this situation may change<br />dramatically.<br />We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.<br />We first study equivalences on Basic Parallel Processes, BPP, a process<br />calculus equivalent to communication free Petri nets. For this simple<br />process language our two notions of non-interleaving equivalences agree.<br />More interestingly, we show that they are decidable, contrasting a result of<br />Hirshfeld that standard interleaving language equivalence is undecidable.<br />Our result is inspired by a recent result of Esparza and Kiehn, showing<br />the same phenomenon in the setting of model checking.<br />We follow up investigating to which extent the result extends to larger<br />subsets of CCS and TCSP. We discover a significant difference between<br />our non-interleaving equivalences. We show that for a certain non-trivial<br />subclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets.

Undecidability of bisimilarity for Petri nets and some related problems

The main result shows the undecidability of (strong) bisimilarity for labelled (place/transition) Petri nets. The technique of the proof applies to the language (or trace) equivalence and the reachability set equality as well, which yields stronger versions with simpler proofs of already known results. The paper also contains two decidability results. One concerns the Petri nets which are deterministic up to bisimilarity, the other concerns semilinear bisimulations and extends the result of Christensen et al. (1993) for basic parallel processes.

Generalized OM-Bisimulation

History preserving bisimilarity and maximality preserving bisimilarity take full account of the interplay between causality and branching, and they are congruences w.r.t. action refinement for suitable classes of Petri nets. But even for finite safe Petri nets they are based on possibly infinite transition systems, and it is not obvious how to decide them. OM-bisimulation gives a characterization of history preserving bisimulation for safe nets without internal moves, and it allows a decidability result. In this paper we generalize OM-bisimulation and this decidability result to safe nets with internal moves, and also treat the case of maximality preserving bisimulation.

New Decidability Results Concerning Two-Way Counter Machines

The authors study some decision questions concerning two-way counter machines and obtain the strongest decidable results to date concerning these machines. In particular, it is shown that the emptiness, containment, and equivalence (ECE, for short) problems are decidable for two-way counter machines whose counter is reversal-bounded (i.e., the counter alternates between increasing and decreasing modes at most a fixed number of times). This result is used to give a simpler proof of a recent result in which the ECE problems for two-way reversal-bounded pushdown automata accepting bounded languages (i.e., subsets of $w_1^*\ldots w_k^*$ for some nonnull words $w_1,\ldots,w_k$) are decidable. Other applications concern decision questions about simple programs. Finally, it is shown that nondeterministic two-way reversal-bounded multicounter machines are effectively equivalent to finite automata on unary languages, and hence their ECE problems are decidable also.

On the solvability of domino snake problems

In this paper we present an extensive treatment of tile connectability problems, sometimes called domino snake problems. The interest in such problems stems from their relationship to classical tiling problems, which have been established as an important, simple and useful tool for obtaining basic lower bound results in complexity and computability theory. We concentrate on the following two contrasting results: The general snake problem is undecidable in a half-plane (due to Ebbinghaus), but is decidable in the whole plane. This surprising decidability result was announced without proof by Myers in 1979. We provide here the first full proof, and show that the problem is actually PSPACE-complete. We also prove many results concerning the difficulty of variants of these general snake problems and their extension to infinite snakes. In addition, we establish a resemblance between snake problems and classical tiling problems, considering the corresponding bounded, unbounded and recurring cases.

Generalizations of Unification

We define a general notion of (syntactic) unification, the ( π → τ → ) -unification, whose special cases are matching, unification, semi-unification, and weak-unification. We settle all the implications between the various cases of ( π → τ → ) -unification leading to a classification and a decidability result. We show that some distinctions holding for finite terms collapse for infinite terms. We give some positive results on the existence of most general weak-unifiers.

Testing for unboundedness of fifo channels

Undecidability of the unboundedness problem for specification models allowing fifo channels was proved a few years ago by Brand and Zafiropulo. The paper investigates a testing approach of that problem. Dealing with the general framework of systems communicating through fifo channels, we find a sufficient condition for unboundedness based on a relation between the nodes of the reachability tree. The construction of the resulting reduced tree can then be applied as well to communicating finite-state machines as to fifo nets. Moreover, the test extends existing decidability results. As a matter of fact, it becomes a decision procedure for a class of systems strictly including linear and monogeneous systems, which are the two essential classes in which decidability is already known. In order to conclude our study on a practical view, we show that a few modifications of the relation make the test available for Estelle specifications.

On the Σ2-theory of the upper semilattice of Turing degrees

A first-order sentence Φ is Σ2 if there is a quantifier-free formula Θ such that Φ has the form . The Σ2-theory of a structure for a language ℒ is the set of Σ2-sentences of ℒ true in . It was shown independently by Lerman and Shore (see [Le, Theorem VII.4.4]) that the Σ2-theory of the structure = 〈D, ≤ 〉 is decidable, where D is the set of degrees of unsolvability and ≤ is the standard ordering of D. This result is optimal in the sense that the Σ3-theory of is undecidable, a result due to J. Schmerl. (For a proof, see [Le, Theorem VII.4.5]. As Lerman has pointed out, this proof should be corrected by defining θσ to be ∀xσ1(x) rather than ∀x(ψ(x)→ σ1(x)).) Nonetheless, in this paper we extend the decidability result of Lerman and Shore by showing that the Σ2-theory of is decidable, where ⋃ is the least upper bound operator and 0 is the least degree. Of course ⋃ is definable in , but many interesting degree-theoretic results are expressible as Σ2-sentences in the language of ∪ but not as Σ2-sentences in the language of . For instance, Simpson observed that the Posner-Robinson cupping theorem could be used to show that for any nonzero degrees a, b, there is a degree g such that b ≤ a ⋃ g, and b ⋠ g (see [PR, Corollary 6]). However, the Posner-Robinson technique does not seem to suffice to decide the Σ2-theory of ∪. We introduce instead a new method for coding a set into the join of two other sets and use it to decide this theory.

On decidable varieties of Heyting algebras

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

Parallel generation of infinite images

We introduce a syntactic model for generating sets of finite and infinite images where a finite image can be viewed as an array over finite alphabet and an infinite image is an array with finite number of columns and infinite number of rows. This model, called image grammar, can be considered as a generalization of classical Chromsky grammar. We study various types of infinite image grammars using Nivat's derivation which is an infinite unrestrictive derivation, Bvichi's (respectively Muller's) which is defined as infinite derivations selected by some repetitive set (respectively sets of rewriting rules). Then we study the combinatorial and language theoretical properties such as complexity measure, closure properties and decidability results. In terms of complexity measure we give a strict infinite hierarchy. We have extended Nivat's and Eilenberg's theorems to infinite images. Unfortunately we prove that Biichi's and McNaughton's theorems cannot be extended to infinite images. We also characterize these families in terms of deterministic image co-substitution and ω-languages

Cardinality problems of compositions of morphisms and inverse morphisms

We show that we cannot effectively determine whether, for morphismsα i ,β i ,card (uα 0 −1 α 1) =card(uβ 0 −1 β 1) for all wordsu over the domain alphabets of the two given compositions. In contrast it is decidable for morphismsα i ,β i and a regular setR whethercard(uα 0 α 1 −1 ) =card(uβ 0 β 1 −1 ) for all wordsu inR. In order to prove the latter result we give a characterization of the multiplicity functions of simple finite automata by using cardinalities of compositions of the above form. Finally, we show that the above decidability result also holds when we consider rational functions rather than morphisms.

FINITE IMAGES GENERATED BY GL-SYSTEMS

In this paper, we introduce a new device called GL-systems (i.e. Grammar-Lindenmayer systems) for generating finite images. GL-systems are sequential/parallel systems in which the horizontal rules form a Chomskian grammar and the vertical rules form a DTOL system. We obtain hierarchical results of various types of GL-systems. We study some properties like, closure properties, combinatorial results, pumping lemma and decidability results. An algebraic characterization for GL-systems (with variation) is presented.

A decidability result for homomorphic representation of automata by 205-1205-1205-1

A decidability result for homomorphic representation of automata by 205-1205-1205-1

On a representation of tree automata

Sequential compositions of tree automata, called selective products, are introduced as a connection of processors computing functions. Decidability results are established for the unary case. These results are related to classical automata theory.

On infinite words obtained by selective substitution grammars

We present a method of generating infinite words from selective substitution grammars introduced by Rozenberg (1977). Comparison is made with some of the well-known limiting processes and limit language families and certain closure properties are examined. A technique is given for obtaining infinite non-repetitive words. Several decidability results are established.

Recurrent words and simultaneous growth in T0L systems

We prove decidability results for recurrent words in T0L schemes and systems, thereby settling some recently posed open problems. However, the problems are shown to be P space-hard. These investigations are motivated by some questions from Markov DT0L systems. As the main tool for proving these results we introduce the notion of simultaneous growth, which is interesting in its own right. We prove that, for an ET0L language L over an alphabet Σ and a given subalphabet Δ, it is decidable whether for every positive integer n there is a word w in L such that in w the number of occurrences of every letter in Δ is at least n.

Filtered and stable boolean powers are relativized full boolean powers

The question whether the extended theorem of Feferman and Vaught (cf. Comer [7], Volger [13]) can be applied to filtered boolean powers with arbitrary base was raised by J. H. Schmerl [11]. It can be answered with the help of a relativized version of the theorem (cf. Comer [7], Volger [12], [14]) which seems to have escaped notice. Therefore we shall present in this note a precise formulation of the relativized version and apply it to filtered boolean powers with arbitrary base and to stable boolean powers with finite base and finite group. The former application was mentioned in Comer [7]. The latter application yields a decidability result which seems to be new. The question whether the second application can be extended to arbitrary bases remains open.