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.     

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.     

The undecidability of form equivalence for context-free and E0L forms

This paper studies the decidability status of various equivalence problems in form theory. Most of our discussions concern the notion of a language form. We compare the form equivalence problem between language forms with the ordinary equivalence problem between languages. However, the main results deal with L forms and grammar forms under strict interpretations. We prove that the form equivalence problem is undecidable for (a) context-free grammar forms, and (b) E0L forms. The proofs of these results are based on our investigations concerning language forms.     

On languages of automaton counter machines

A class of formal languages (ACML) acceptable by automaton counter machines is considered. This class is shown to be close with respect to the operations of union, regular intersection, concatenation, infinite iteration, homomorphism, and inverse homomorphism. It follows from here that this class is a full abstract family of languages [7] with all the properties following from this. Furthermore, the ACML is close with respect to intersection and substitution but is not closed with respect to complement and reverse. For the ACML class, the problems of emptiness and recognition of words of a language given by an automaton counter machine are decidable, but the problems of inclusion and equivalence of languages are undecidable. A comparison with other classes of languages (regular, context-free, context-sensitive, and Petri-net languages) is performed.     

The inclusion problem for simple languages

A deterministic pushdown acceptor is called a simple machine when it is restricted to have only one state, operate in real-time, and accept by empty store. While the existence of an effective procedure for deciding equivalence of languages accepted by these simple machines is well-known, it is shown that this family is powerful enough to have an undecidable inclusion problem. It follows that the inclusion problems for the LL(k) languages and the free monadic recursion schemes that do not use an identity function are also undecidable.     

On two-dimensional pattern-matching languages and their decision problems

We propose two new classes of two-dimensional array languages, called existential matching languages (EXMLs) and universal matching languages (UNMLs). These languages are closely related to two-dimensional pattern matching, and thus suited for studying them formally. In this paper, basic properties of these languages and decidability (or undecidability) of several problems about them are investigated. We show that, because of the two-dimensionality, several decision problems, such as the emptiness problem for UNMLs, the universe problem for EXMLs, and the equivalence problems for both languages, are undecidable. Thus we cannot decide, for example, whether two two-dimensional pattern-matching tasks are equivalent.     

Some decidability problems of sentential formst †

Various decidability problems of SF-languages, i.e. sets of sentential forms, of Chomsky grammars are studied. Among the problems are the equivalence, ambiguity, equality to a regular set, and regularity of SF-languages, and the SF-ness of regular languages. All of these problems are undecidable for type 0 grammars, and many of them are decidable for minimal linear grammars. Strongest possible undecidability and decidability results are looked for.     

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.     

On the Expressive Power of Temporal Concurrent Constraint Programming Languages: (invited talk)

The tcc paradigm is a formalism for timed concurrent constraint programming. Several tcc languages differing in their way of expressing infinite behavior have been proposed in the literature. In this work we study the expressive power of some of these languages. In particular, we show that: (1) recursive procedures with parameters can be encoded into parameterless recursive procedures with dynamic scoping, and viceversa. (2) replication can be encoded into parameterless recursive procedures with static scoping, and viceversa. (3) the languages from (1) are strictly more expressive than the languages from (2). Furthermore, we show that behavioral equivalence is undecidable for the languages from (1), but decidable for the languages from (2). The undecidability result holds even if the process variables take values from a fixed finite domain.(Joint work with Mogens Nielsen and Frank D. Valencia)     

The membership and equivalence problems for picture languages

The paper describes two principal results: (1) there is a nondeterministic polynomial-time algorithm to determine whether an arbitrary picture is a member of a context-free picture language and (2) the equivalence problem for regular picture languages is not decidable (not even partially decidable). This solves open questions described in earlier publications. Furthermore, these results are best possible. That is, the membership problem for context-free picture languages was earlier shown to be NP-hard and the complement of the equivalence problem for regular picture languages is easily seen to be partially decidable.     

Derivation languages of grammar forms†

The family of derivation languages or Szilard languages associated with a grammar form is studied. It is shown that Szilard equivalence, regular completeness and regular sufficiency are decidable properties. Although the decidability of left Szilard equivalence remains open, we reduce this problem to the equivalence problem for a special class of s-grammar forms     

Refining the Undecidability Border of Weak Bisimilarity

Weak bisimilarity is one of the most studied behavioural equivalences. This equivalence is undecidable for pushdown processes (PDA), process algebras (PA), and multiset automata (MSA, also known as parallel pushdown processes, PPDA). Its decidability is an open question for basic process algebras (BPA) and basic parallel processes (BPP). We move the undecidability border towards these classes by showing that the equivalence remains undecidable for weakly extended versions of BPA and BPP. In fact, we show that the weak bisimulation equivalence problem is undecidable even for normed subclasses of BPA and BPP extended with a finite constraint system.     

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.     

NTS languages are deterministic and congruential

A context-free grammar is said to be NTS if the set of sentential forms it generates is unchanged when the rules are used both ways. We prove here that such grammars generate deterministic languages which are finite unions of congruence classes. Moreover, we show that this family of languages is closed under reversal and intersection with regular sets. A forthcoming paper will prove that, for this class, the equivalence problem is decidable.     

On sentential forms of context-free grammars

Summary There is no algorithm for deciding whether two linear context-free grammars generate the same sentential forms. The equivalence problem for propagatingOL-systems is undecidable. The finiteness problem forOL-systems is decidable.SF-languages, i.e., languages which equal the set of sentential forms of a context-free grammar, possess some of the properties of context-free languages but their family is not closed under any of the ordinary operations.     

正则语言的一个特征

Myhill-Nerode定理利用等价关系描述了正则语言的一个重要特征,它是有限自动机理论中的一个经典、优美的结果。为了将Myhill-Nerode定理推广到更一般的情形,引入了有限自动机M上的状态转移半群和Σ*上的M-半群,讨论了其若干性质。在此基础上,将Myhill-Nerode定理中的等价关系一般化,给出了正则语言的一个新的特征定理,Myhill-Nerode定理成为该定理的一个推论。讨论了正则语言的最一般的特征,提出了有待进一步研究的问题。     

The complexity of some decision problems about two-dimensional array grammars

The complexity of several decision problems concerning two-dimensional isometric array grammars (IAG) is studied. Because of the two-dimensional and the “isometric” properties of IAG, many decision problems become very hard to solve even for regular array grammars (RAG), the lowest subclass of IAG in the Chomsky-like hierarchy. In this paper, it is shown that the membership problems for RAGs and for context-free array grammars (CFAG) are both NP-complete. The emptiness problem and the equivalence problem for RAGs are shown to be undecidable.     

Regular grammars with truth values in lattice-ordered monoid and their languages

In this study, we introduce the concept of lattice-valued regular grammars. Such grammars have become a necessary tool for the analysis of fuzzy finite automata. The relationship between lattice-valued finite automata (LA) and lattice-valued regular grammars (LRG) are discussed and we get the following results, for a given LRG, there exists an LA such that they accept the same languages, and vice versa. We also show the equivalence between deterministic lattice-valued regular grammars and deterministic lattice-valued finite automata.     

From Syntactic Theories to Interpreters: Automating the Proof of Unique Decomposition

Developing syntactic theories for reasoning about programming languages usually involves proving a unique-decomposition lemma. The proof of such a lemma is tedious, error-prone, and is usually attempted many times during the design of a theory. We therefore investigate the automation of such proofs.We map the unique-decomposition lemma to the problems of checking equivalence and ambiguity of syntactic definitions. Because checking these properties of context-free grammars is undecidable, we work with regular tree grammars and use algorithms on finite tree automata to perform the checking. To make up for the insufficient expressiveness of regular tree grammars, we extend the basic framework with built-in types and constants, contexts, and polymorphic types.Our implementation extends an earlier system by Xiao et al. [16] that translates semantic specifications expressed as syntactic theories to interpreters. We have successfully used the combined system to generate interpreters and verify the unique-decomposition lemma for a number of examples.     

Undecidability in decentralized supervision

It is well known that supervisor synthesis problems involving partial observations generally have high computational complexity, but was only recently shown (in the preliminary version of this article) that the solvability of decentralized supervisor synthesis problems may be undecidable, even in cases where all pertinent languages are represented as finite automata. The result is established by reduction from the halting problem for Turing machines. Alternative proofs, discovered independently by other authors, employ a reduction from Post's correspondence problem.