Bad news on decision problems for patterns

We study the inclusion problem for pattern languages, which—due to Jiang et al. [T. Jiang, A. Salomaa, K. Salomaa, S. Yu, Decision problems for patterns, Journal of Computer and System Sciences 50 (1995) 53–63]—is known to be undecidable. More precisely, Jiang et al. demonstrate that there is no effective procedure deciding the inclusion for the class of all pattern languages over all alphabets. Most applications of pattern languages, however, consider classes over fixed alphabets, and therefore it is practically more relevant to ask for the existence of alphabet-specific decision procedures. Our first main result states that, for all but very particular cases, this version of the inclusion problem is also undecidable. The second main part of our paper disproves the prevalent conjecture on the inclusion of so-called similar E-pattern languages, and it explains the devastating consequences of this result for the intensive previous research on the most prominent open decision problem for pattern languages, namely the equivalence problem for general E-pattern languages.     

Monadic recursion schemes: The effect of constants

When the class of monadic recursion schemes is augmented by individual constants, some of the properties change. It becomes undecidable whether “S diverges” or “S is strongly equivalent to T” for S, T schemes with individual constants. The family of value languages generated by this class of schemes is the family of recursively enumerable languages. The subclass of free schemes with constants is also investigated. It remains decidable whether “S halts” or “S diverges” for S a free scheme with individual constants, but it becomes undecidable whether “T has a strongly equivalent free scheme” for T an arbitrary scheme with individual constants.     

Insertion languages

The operations of insertion ( ← ) and iterated insertion ( ←₁ ) are simple variants of Kleene's operations · and 1 [15] in a manner similar to the operations shuffle and iterated shuffle (see e.g. [13, 23, 20, 14]). Using the operation of iterated insertion, we can generate both the semi-Dyck and the two-sided Dyck languages from certain finite subsets of these languages. Thus the class of languages of the form $\text{S}{}^{\mathrm{\leftarrow }}{}^1$ for finite S forms a natural class of generalized Dyck languages. This class is equivalent to the class of pure unitary languages discussed in [6]. We investigate this class further, by examining for it the problems of equivalence, ambiguity, and determinism, all of which are easily decidable. On the other hand, we show that the problem “$\text{S}{}^{\mathrm{\leftarrow }}\text{1 ∩ T}{}^{\mathrm{\leftarrow }}\text{1 = {λ}?}$” is undecidable for finite, unambiguous S and T. Furthermore, by extending the regular expressions to include the operations ← and $\text{←}{}_1$, we obtain the class of insertion languages which generalizes both the regular languages and the Dyck languages, but is properly contained within the class of context-free languages. Our main result here is that the problem “$\text{L = ∑}{}^1\text{?}$” is undecidable for the class of insertion languages. From this result, it follows that the equivalence problem and the problem “IsL regular?” are also undecidable for this class.     

On the complexity of nonuniform wavelength-based machine

The wavelength-based machine, or simply w-machine, is an optical computational model, which is designed based on simultaneous movement of several wavelengths in a single light ray, and simultaneous effect of simple optical devices on them. In this paper, we investigate nonuniform complexity classes of w-machine, based on three complexity measures, namely, size, time, and word length. We show that the class of languages which can be generated by constant size nonuniform w-machines contain infinitely many Turing undecidable languages. Also, we show that polynomial size nonuniform w-machines generate all NP languages, and every NP-hard language requires at least polynomial time and polynomial size nonuniform w-machines to be generated. We prove that the class of languages which can be generated by polynomial size nonuniform w-machines is equal to NP/poly, and almost all languages require exponential size and polynomial time nonuniform w-machines to be generated.     

A simple characterization of uniform boundedness for a class of recursions

Detecting bounded recursions is a powerful optimization technique for recursive database query languages, as bounded recursions can be replaced by equivalent nonrecursive definitions. The problem is also of theoretical interest in that varying the class of recursions considered generates problem instances that vary from linearly decidable to NP-hard to undecidable. In this paper we review and clarify the existing definitions of boundedness. We then specify a class of recursions C such that membership in C guarantees that a certain simple condition is necessary and sufficient for boundedness. We use the notion of membership in C to unify and extend previous work on determining decidable classes of bounded recursions.     

Learning regular languages using RFSAs

Residual languages are important and natural components of regular languages and several grammatical inference algorithms naturally rely on this notion. In order to identify a given target language L, classical inference algorithms try to identify words which define identical residual languages of L. Here, we study whether it could be interesting to perform a tighter analysis by identifying inclusion relations between the residual languages of L. We consider the class of Residual Finite State Automata (RFSAs). An RFSA A is a NonDeterministic Automaton whose states corresponds to residual languages of the language L_A it recognizes. The inclusion relations between residual languages of L_A can be naturally materialized on A. We prove that the class of RFSAs is not polynomially characterizable. We lead some experiments which show that when a regular language is randomly drawn by using a nondeterministic representation, the number of inclusion relations between its residual languages is very important. Moreover, its minimal RFSA representation is much smaller than its minimal DFA representation. Finally, we design a new learning algorithm, DeLeTe2, based on the search for the inclusion relations between the residual languages of the target language. We give sufficient conditions for the identifiability of the target language. We experimentally compare the performance of DeLeTe2 to those of classical inference algorithms.     

On the language equivalence of NE star-patterns

A pattern is a finite string of constant and variable symbols. The language generated by a pattern is the set of all strings of constant symbols which can be obtained from the pattern by substituting (non-empty) strings for variables. The pattern languages are one of language family which is orthogonal to the Chomsky-type languages hierarchy. They have many applications, such as the extended regular expressions, for instance, in languages Perl, awk, etc. They are well applicable in machine learning as well. There are erasing and non-erasing patterns are used. For non-erasing pattern languages the equivalence of languages is decidable but the inclusion problem for them is undecidable. In extended regular expressions one can use union, concatenation and Kleene star to make more complex patterns. It is also known, that the equivalence problem of extended regular expressions is undecidable. However, the problem, whether the equivalence is decidable for patterns using only concatenation and star still open. In this paper there are some interesting results about inclusion properties and equivalences of some kinds of erasing and non-erasing pattern languages. We show that the equivalence problem of non-erasing patterns in some cases can be reduced to the decidability problem of some very special inclusion properties. These results may be useful to decide whether the language equivalence of non-erasing star-patterns is decidable or not.     

Monadic Thue systems

Certain infinite Thue systems over a finite alphabet are studied, in particular, systems $\text{S?∑}{}^1\text{×(∑∪{e})}$ such that for each a?∑∪{e}, the set {u| (u,a)?S} 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.     

Murg term rewrite systems

The union of a monadic and a right-ground term rewrite system is called a murg term rewrite system. We show that for murg TRSs the ground common ancestor problem is undecidable. We show that for a murg term rewrite system it is undecidable whether the set of descendants of a ground tree is a recognizable tree language. We show that it is undecidable whether a murg term rewrite system over Σ preserves Σ-recognizability.     

A characterization of (regular) circular languages generated by monotone complete splicing systems

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for monotone complete systems, which are finite circular splicing systems with rules of a simpler form. We show that a circular language L is generated by a monotone complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by monotone complete systems. We can also decide whether the language generated by a monotone complete system is regular. Finally, we point out that monotone complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule of a specific type is allowed. From our results on monotone complete systems, it follows that finite simple systems generate a class of languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.     

Some decision problems related to the reachability problem for Petri nets

In this paper, we show that (1) the question to decide whether a given Petri net is consistent, M_o-reversible or live is reduced to the reachability problem in a unified manner, (2) the reachability problem for Petri nets is equivalent to the equality problem and the inclusion problem for the sets of all firing sequences of two Petri nets, (3) the equality problem for the sets of firing sequences of two Petri nets with only two unbounded places under homomorphism is undecidable, (4) the coverability and reachability problems are undecidable for generalized Petri nets in which a distinguished transition has priority over the other transitions, and (5) the reachability problem is undecidable for generalized Petri nets in which some transitions can reset a certain place to zero marking.     

Permutation rewriting and algorithmic verification

We propose a natural subclass of regular languages (Alphabetic Pattern Constraints, APC) which is effectively closed under permutation rewriting, i.e., under iterative application of rules of the form ab → ba. It is well-known that regular languages do not have this closure property, in general. Our result can be applied for example to regular model checking, for verifying properties of parametrized linear networks of regular processes, and for modeling and verifying properties of asynchronous distributed systems. We also consider the complexity of testing membership in APC and show that the question is complete for PSPACE when the input is an NFA, and complete for NLOGSPACE when it is a DFA. Moreover, we show that both the inclusion problem and the question of closure under permutation rewriting are PSPACE-complete when we restrict to the class APC.     

Distributed synthesis is simply undecidable

The distributed synthesis problem of safety and reachability languages is known to be undecidable. In this article, we establish that this is the case for very simple languages, namely for safety and reachability specifications in the intersection of LTL and ACTL.     

Highly Constrained Unification Grammars

Unification grammars are widely accepted as an expressive means for describing the structure of natural languages. In general, the recognition problem is undecidable for unification grammars. Even with restricted variants of the formalism, off-line parsable grammars, the problem is computationally hard. We present two natural constraints on unification grammars which limit their expressivity and allow for efficient processing. We first show that non-reentrant unification grammars generate exactly the class of context-free languages. We then relax the constraint and show that one-reentrant unification grammars generate exactly the class of mildly context-sensitive languages. We thus relate the commonly used and linguistically motivated formalism of unification grammars to more restricted, computationally tractable classes of languages.     

Safety in grammatical protection systems

In the context of operating system protection mechanisms,safety refers to the ability to decide who can obtained certain rights to resources by some future sequence of command invocations. Harrison, Ruzzo and Ullman have shown that in general safety is undecidable. On the other hand Jones, Lipton and Snyder have analyzed a simple system is which safety is decidable in time linear in the size of the system. This paper presents a large class of operating system protection mechanisms for which a polynomial time decision procedure for the safety question can be given. Extensions are then exhibited that are P-space complete andNP-complete.     

Petri nets and regular languages

It is shown that the regularity problem for firing sequence sets of Petri nets is decidable. For the proof, new techniques to characterize unbounded places are introduced. In the class $\text{L}$₀ of terminal languages of labelled Petri nets the regularity problem in undecidable. In addition some lower bounds for the undecidability of the equality problems in $\text{L}$₀ and $\text{L}$ are given. L₀^λ is shown to be not closed under complementation without reference to the reachability problem.     

Future scientific programming on parallel machines

A language for large scientific applications should facilitate encoding and debugging of programs at the highest level possible. At the same time it should facilitate generation of efficient code for parallel machines. Often these two requirements are conflicting, and trade-offs must be made. Functional and other declarative languages offer relief on both counts. The use of higher-order functions, especially in carried forms, can raise the level of programming dramatically. In addition, such languages often have straightforward operational semantics, thereby providing tremendous opportunities for parallel execution. Programs written in declarative languages thus eliminate the problem of “detecting parallelism.” This paper illustrates programming in one such language, Id Nouveau, and contrasts it with programming in Fortran. Using an excerpt from an application known as Simple, it is shown how a program can be composed in Id Nouveau from small functions that directly relate to the mathematical and physical concepts of the problem. The difficulty of expressing these concepts in Fortran is discussed. Finally, it is shown that by performing simple transformations, such as in-line substitution of functions, the resulting Id Nouveau code becomes as efficient as an equivalent Fortran program written to run efficiently on a parallel machine.     

On the monotonicity of (LDL) logic programs with set

LDL is one of the recently proposed logical query languages, which incorporate set, for data and knowledge base systems. Since LDL programs can simulate negation, they are not monotonic in general. On the other hand, there are monotonic LDL programs. This paper addresses the natural question of “When are the generally nonmonotonic LDL programs monotonic?” and investigates related topics such as useful applications for monotonicity. We discuss four kinds of monotonicity, and examine two of them in depth. The first of the two, called “ω-monotonicity”, is shown to be undecidable even when limited to single-stratum programs. The second, called “uniform monotonicity”, is shown to implyω-monotonicity. We characterize the uniform monotonicity of a program (i) by a relationship between its Bancilhon-Khoshafian semantics and its LDL semantics, and (ii) with a useful property called subset completion independence. Characterization (ii) implies that uniformly monotonie programs can be evaluated more efficiently by discarding dominated facts. Finally, we provide some necessary and/or sufficient, syntactic conditions for uniform monotonicity. The conditions pinpoint (a) enumerated set terms, (b) negations of membership and inclusion, and (c) sharing of set terms as the main source for nonuniform monotonicity.