Introducing Second-Order Spider Diagrams for Defining Regular Languages

The spider diagram logic forms a fragment of the constraint diagram logic and was designed to be primarily used as a diagrammatic software specification tool. Our interest is in using the logical basis of spider diagrams and the existing known equivalences between certain logics, formal language theory classes and some automata to inform the development of diagrammatic logics. Such developments could have many advantages, one of which would be aiding software engineers who are familiar with formal languages and automata to more intuitively understand diagrammatic logics. In this paper we consider relationships between spider diagrams of order (an extension of spider diagrams) and the star-free subset of regular languages. We extend the concept of the language of a spider diagram to encompass languages over arbitrary alphabets. Furthermore, the product of spider diagrams is introduced. This operator is the diagrammatic analogue of language concatenation. We establish that star-free languages are definable by spider diagrams of order equipped with the product operator and, based on this relationship, spider diagrams of order are as expressive as first order monadic logic of order.

We prove that the class of all languages that are definable in /spl Sigma//sub 1//sup 1/(FO/sup 2/), that is, in (non-monadic) existential second-order logic with only two first-order variables, coincides with the regular languages. This provides an alternative logical description of regularity to both the traditional one in terms of monadic second-order logic, due to Buchi and Trakhtenbrot, and the more recent ones in terms of prefix fragments of /spl Sigma//sub 1//sup 1/, due to Eiter, Gottlob and Gurevich. Our result extends to more general settings than words. Indeed, definability in /spl Sigma//sub 1//sup 1/(FO/sup 2/) coincides with recognizability by appropriate notions of automata on a large class of objects, including /spl omega/-words, trees, pictures and, more generally, all weakly deterministic, triangle-free transition systems.

By a well-known result due to Büchi and Trakhtenbrot, all monadic second-order sentences over words describe regular languages. In this paper, we investigate prefix classes of general second-order logic. Such a prefix class is called regular, if each of its sentences describes a regular language, and nonregular otherwise. Recently, the regular and nonregular prefix classes of existential second order logic (Σ1 1) were exhaustively determined. We briefly recall these results and continue this line of research by systematically investigating the syntactically more complex prefix classes Σ1 k(Q) of second-order logic for each integer k > 1 and for each first-order quantifier prefix Q. We give an exhaustive classification of the regular and nonregular prefix classes of this form, and derive of complexity results for the corresponding model checking problems. We also give a brief survey of recent results on the complexity of evaluating existential second-order logic over graphs, and a list of interesting open problems.KeywordsModel CheckUndirected GraphRegular LanguagePredicate VariableWord StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we define three classes of languages of bi-infinite words, namely local bi-ω languages, recognizable bi-ω languages and Buchi local bi-ω languages as subclasses of the class of regular bi-ω languages and prove some basic results. We observe that the class of recognizable bi-ω languages coincides with the well-known class of rational bi-adherence languages and show that the class of regular bi-ω languages is the class of morphic images of Buchi local bi-ω languages. We provide learning algorithms for Buchi local bi-ω languages and regular bi-ω languages.

Learning approximately regular languages with reversible languages

Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free (d-union-free, for short) languages. In this paper [Formula: see text]-free nondeterministic variants of 1CFPAs are used to define n-union-free languages. The defined language class is shown to be properly between the classes of (nondeterministic) union-free and d-union-free languages (in case of at least binary alphabet). In case of unary alphabet the class of n-union-free languages coincides with the class of union-free languages. Some properties of the new subregular class of languages are discussed, e.g., closure properties. On the other hand, a regular expression is in union normal form if it is a finite union of union-free expressions. It is well known that every regular expression can be written in union normal form, i.e., all regular languages can be described as finite unions of (nondeterministic) union-free languages. It is also known that the same fact does not hold for deterministic union-free languages, that is, there are regular languages that cannot be written as finite unions of d-union-free languages. As an important result here we show that every regular language can be defined by a finite union of n-union-free languages. This fact also allows to define n-union-complexity of regular languages.

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of omega-tree languages derived from weak regular omega-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived omega-tree languages to learning the underlying class of omega-word languages, for any class of omega-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [Maler and Pneuli, Inform. Comput., 1995] for the class of weak regular omega-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi omega-word acceptors. A previous claim of an algorithm to learn regular omega-trees due to Jayasrirani, Begam and Thomas [Jayasrirani et al., ICGI, 2008] is unfortunately incorrect, as shown in [Angluin, YALEU/DCS/TR-1528, 2016].

Concatenation hierarchies are natural classifications of regular languages. All such hierarchies are built through the same construction process: one starts from an initial, specific class of languages (the basis) and builds new levels using two generic operations. Concatenation hierarchies have gathered a lot of interest since the early 70s, notably thanks to an alternate logical definition: each concatenation hierarchy can be defined as the quantification alternation hierarchy within a variant of first-order logic over words (while the hierarchies differ by their bases, the variants differ by their set of available predicates). Our goal is to understand these hierarchies. A typical approach is to look at two decision problems: membership and separation. In the paper we are interested in the latter, which is more general. For a class of languages $C$ , C-separation takes two regular languages as input and asks whether there exists a third one in $C$ including the first one and disjoint from the second one. Settling whether separation is decidable for the levels within a given concatenation hierarchy is among the most fundamental and challenging questions in formal language theory. In all prominent cases, it is open, or answered positively for low levels only. Recently, a breakthrough was made using a generic approach for a specific kind of hierarchies: those with a finite basis. In this case. separation is always decidable for levels 1/2. 1 and 3/2. Our main theorem is similar but independent: we consider hierarchies with possibly infinite bases, but that contain only group languages. An example is the group hierarchy introduced by Pin and Margolis: its basis consists of all group languages. Another example is the quantifier alternation hierarchy of first-order logic with modular predicates FO( $ , MOD): its basis consists of the languages that count the length of words modulo some number. Using a generic approach, we show that for any such hierarchy, if separation is decidable for the basis, then it is decidable as well for levels 1/2, 1 and 3/2 (we actually solve a more general problem called covering). This complements the aforementioned result nicely: all bases considered in the literature are either finite or made of group languages. Thus, one may handle the lower levels of any prominent hierarchy in a generic way.

We show how appropriately chosen functions f which we call distinguishing can be used to make deterministic finite automata backward deterministic. These ideas have been exploited to design regular language classes called f-distinguishable which are identifiable in the limit from positive samples. Special cases of this approach are the k-reversible and terminal distinguishable languages as discussed in [1,3,5,15,16]. Here, we give new characterizations of these language classes. Moreover, we show that all regular languages can be approximated in the setting introduced by Kobayashi and Yokomori [12,13]. Finally, we prove that the class of all function-distinguishable languages is equal to the class of regular languages.

Word classes (‘parts-of-speech’, ‘syntactic categories’, ‘lexical categories’) are the fundamental building blocks of linguistic expressions in all natural human languages. They have been investigated since antiquity and continue to play a central role in modern linguistics. Today an increasingly important role is assigned to the information that is specified in the lexical entry of a word in the lexicon (including, of course, information about its category membership), both in formal and in functional approaches to grammar. Furthermore, there is a growing awareness that the traditional set of word classes is biased towards the better studied European languages and needs to be revised to accommodate unfamiliar word classes in nonEuropean languages.

A Boolean closed full trio is a class of languages that is closed under the Boolean operations (union, intersection, and complementation) and rational transductions. It is well-known that the regular languages constitute such a Boolean closed full trio. It is shown here that every such language class that contains any non-regular language already includes the whole arithmetical hierarchy (and even the one relative to this language). A consequence of this result is that aside from the regular languages, no full trio generated by one language is closed under complementation. Our construction also shows that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.

Model checking is often cited as a success story for transitioning and engineering ideas rooted in logics and automata to practice. In this paper, we discuss how the efforts aimed at improving the scope and effectiveness of model checking tools have revived the study of logics and automata leading to unexpected theoretical advances whose impact is not limited to model checking. In particular, we describe how our efforts to add context-free specifications to software model checking led us to the model of nested words as a representation of data with both a linear ordering and a hierarchically nested matching of items. Such dual structure occurs in diverse corners of computer science ranging from executions of structured programs where there is a well-nested matching of entries to and exits from functions and procedures, to XML documents with the hierarchical structure specified by start-tags matched with end-tags. Finite-state acceptors of nested words define the class of regular languages of nested words that has all the appealing theoretical properties that the class of regular word languages enjoys. We review the emerging theory of nested words, its extension to nested trees, and its potential applications.

Closure properties of linear context-free tree languages with an application to optimality theory

This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential if there exists a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S . It was known that there are deterministic linear context-free languages which are not Church-Rosser congruential, but the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012.

We show that the class of string languages generated by linear context-free rewriting systems is equal to the class of output languages of deterministic tree-walking transducers. From equivalences that have previously been established we know that this class of languages is also equal to the string languages generated by context-free hypergraph grammars, multicomponent tree-adjoining grammars, and multiple context-free grammars and to the class of yields of images of the regular tree languages under finite-copying top-down tree transducers.