Regular Languages Research Articles

Binary Coded Unary Regular Languages

[Formula: see text] is a binary coded unary regular language, if there exists a unary regular language [Formula: see text] such that [Formula: see text] is in [Formula: see text] if and only if the binary representation of [Formula: see text] is in [Formula: see text]. If a unary language [Formula: see text] is accepted by a minimal deterministic finite automaton (DFA) [Formula: see text] with [Formula: see text] states, then its binary coded version [Formula: see text] is regular and can be accepted by a DFA [Formula: see text] using at most [Formula: see text] states, but at least [Formula: see text] states. There are witness languages matching these upper and lower bounds exactly, for each [Formula: see text]. More precisely, if [Formula: see text] uses [Formula: see text] states in the initial segment and [Formula: see text] states in the loop, where [Formula: see text] is odd and [Formula: see text], then the minimal [Formula: see text] for [Formula: see text] consists of a preamble with at most [Formula: see text] but at least [Formula: see text] states, except for [Formula: see text] with no preamble, and a kernel with at most [Formula: see text] but at least [Formula: see text] states. Also these lower bounds are matched exactly by witness languages, for each [Formula: see text]. If the length of the loop is fixed, the size of the preamble is bounded by [Formula: see text]. The conversion in the opposite way is not always granted: there are binary regular languages the unary versions of which are not even context free. The conversion of a unary nondeterministic finite automaton (NFA) to a binary NFA uses [Formula: see text] states and introduces a binary version of Chrobak normal form.

Constructing Concise Characteristic Samples for Acceptors of Omega Regular Languages

A characteristic sample for a language $L$ and a learning algorithm $\textbf{L}$ is a finite sample of words $T_L$ labeled by their membership in $L$ such that for any sample $T \supseteq T_L$ consistent with $L$, on input $T$ the learning algorithm $\textbf{L}$ returns a hypothesis equivalent to $L$. Which omega automata have characteristic sets of polynomial size, and can these sets be constructed in polynomial time? We address these questions here. In brief, non-deterministic omega automata of any of the common types, in particular B\"uchi, do not have characteristic samples of polynomial size. For deterministic omega automata that are isomorphic to their right congruence automata, the fully informative languages, polynomial time algorithms for constructing characteristic samples and learning from them are given. The algorithms for constructing characteristic sets in polynomial time for the different omega automata (of types B\"uchi, coB\"uchi, parity, Rabin, Street, or Muller), require deterministic polynomial time algorithms for (1) equivalence of the respective omega automata, and (2) testing membership of the language of the automaton in the informative classes, which we provide.

Once-Marking and Always-Marking 1-Limited Automata

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as [Formula: see text]-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case, the size gap from [Formula: see text]-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of [Formula: see text]-limited automata, once-marking[Formula: see text]-limited automata and always-marking[Formula: see text]-limited automata, and study their descriptional complexity. We prove that once-marking [Formula: see text]-limited automata still exhibit a double exponential size gap to one-way deterministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic [Formula: see text]-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always-marking [Formula: see text]-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation.

Latvian Quantum Finite State Automata for Unary Languages

We design Latvian quantum finite state automata (LQFAs) recognizing unary regular languages with isolated cut point [Formula: see text]. From an architectural viewpoint, we suitably combine two LQFAs recognizing with isolated cut point, respectively, the finite part and the ultimately periodic part any given unary regular language [Formula: see text] consists of. In particular, both these LQFAs incorporate a sub-module discriminating strings on the basis of their length. Both the number of basis states and the isolation around the cut point of the resulting LQFAs for [Formula: see text] exponentially depend on the size of the minimal deterministic finite state automaton for [Formula: see text]. Moreover, the recognition of [Formula: see text] tends to becoming deterministic as the number of the basis states employed in the length-discriminating sub-module grows.

On decision problems concerning contextual insertions and deletions

The notions of stability, anti-stability, and error-correctability of a language that is modified by making contextual insertions in the words of the language were introduced in a previous paper by Bottoni et al. in 2011, where it was shown that these properties are decidable for regular languages. The authors proposed investigating the decidability of these properties for other classes of languages. Here, we derive necessary and sufficient conditions for a class of languages to have decidable stable, anti-stable, and error-correctable properties, and use these conditions to exhibit general classes of languages (strictly greater than the regular languages) for which the properties are decidable, and also simple classes (the first such classes) for which the properties are undecidable. We obtain identical results for the case when contextual deletions (instead of insertions) are made in the words of the language, and also with mixes of insertions and deletions. Our constructions also demonstrate that certain general problems involving nondeterministic generalized sequential machines (GSMs) applied to languages accepted by deterministic machine models are decidable, which is surprising as the deterministic language families do not need to be closed under GSM mappings.

Language Styles, Recovery Strategies and Users’ Willingness to Forgive in General Artificial Intelligence Service Recovery: A Mixed Study

As the prevalence of generative artificial intelligence (GenAI) in the service sector continues to grow, the impact of the language style and recovery strategies utilized during service failures remains insufficiently explored. This study, grounded in the theory of social presence and dual-process theory, employed a mixed-method approach combining questionnaire surveys and event-related potential (ERP) experiments to investigate the effect of different language styles (rational vs. humorous) and recovery strategies (gratitude vs. apology) on users’ willingness to forgive during the GenAI service recovery process. It further delves into the chained mediating role of perceived sincerity and social presence in this process. The findings revealed that a humorous language style was more effective in enhancing users’ willingness to forgive compared to a rational style, primarily through the enhancement of users’ perceived sincerity and sense of social presence; recovery strategies played a moderating role in this process, with the positive impact of perceived sincerity on social presence being significantly amplified when the GenAI service adopted an apology strategy. ERP results indicated that a rational language style significantly induced a larger N2 component (cognitive conflict) in apology scenarios, while a humorous style exhibited higher amplitude in the LPP component (positive emotional evaluation). This research unveils the intricate relationships between language style, recovery strategies, and users’ willingness to forgive in the GenAI service recovery process, providing important theoretical foundations and practical guidance for designing more effective GenAI service recovery strategies, and offering new insights into developing more efficacious GenAI service recovery tactics.

String editing under pattern constraints

We introduce the novel Nearest Pattern Constrained String (NPCS) problem of finding a minimum set Q of character mutation, insertion, and deletion edit operations sufficient to modify a string x to contain all contiguous substrings in a pattern set P and no contiguous substrings in a forbidden pattern set F. Letting Σ be the alphabet of allowed characters, and letting η and ϒ be the longest string length and sum of all string lengths in P∪F, respectively, we show that NPCS is fixed-parameter tractable in |P| with time complexity O(2|P|⋅ϒ⋅|Σ|⋅(|P|+η)(|x|+1)). Additionally, we consider a generalization of the NPCS problem in which we allow for constraints based on the membership of substrings in regular languages. In particular, we introduce a problem we denote String Editing under Substring in Language Constraints (StrEdit-SILC), where provided a wildcard-free string x∈Σ⁎, a finite set of regular languages R={L1,L2,…}, and a regular language LF, the objective is to find a minimum cost set of mutation, insertion, and deletion edit operations Q that suffice to convert the input string x into a string x′∈Σ⁎, where no substring has membership in LF, and ∀Li∈R, there exists a substring in Li. Here, letting Ψ and ϖ be the sum of all regular expression lengths and longest regular expression length for languages in R∪{LF}, respectively, and letting Cmid∈N be the maximum cost of an edit operation, we show that StrEdit-SILC is fixed-parameter tractable with respect to Ψ, having time complexity O(2Ψ⋅|x|⋅(ϖ⋅|Σ|+Cmid)). However, we also show that StrEdit-SILC is MAX-SNP-hard and otherwise difficult to approximate under stringent constraints.

Explaining results of path queries on graphs: Single-path results for context-free path queries

Many graph query languages use, at their core, path queries that yield node pairs (m,n) that are connected by a path of interest. For the end-user, such node pairs only give limited insight as to why this result is obtained, as the pair does not directly identify the underlying path of interest.In this paper, we propose the single-path semantics to address this limitation of path queries. Under single-path semantics, path queries evaluate to a single path connecting nodes m and n and that satisfies the conditions of the query. To put our proposal in practice, we provide an efficient algorithm for evaluating context-free path queries using the single-path semantics. Additionally, we perform a short evaluation of our techniques that shows that the single-path semantics is practically feasible, even when query results grow large.In addition, we explore the formal relationship between the single-path semantics we propose the problem of finding the shortest string in the intersection of a regular language (representing a graph) and a context-free language (representing a path query). As our formal results show, there is a distinction between the complexity of the single-path semantics for queries that use a single edge label and queries that use multiple edge labels: for queries that use a single edge label, the length of the shortest path is linearly upper bounded by the number of nodes in the graph; whereas for queries that use multiple edge labels, the length of the shortest path has a worst-case quadratic lower bound.

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

Myhill-Nerode Theorem for Higher-Dimensional Automata

We establish a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular if and only if it has finite prefix quotient. HDAs extend standard automata with additional structure, making it possible to distinguish between interleavings and concurrency. We also introduce deterministic HDAs and show that not all HDAs are determinizable, that is, there exist regular languages that cannot be recognised by a deterministic HDA. Using our theorem, we develop an internal characterisation of deterministic languages. Lastly, we develop analogues of the Myhill-Nerode construction and of determinacy for HDAs with interfaces.

Wadge degrees of Δ20$\mathbf{\Delta }^0_2$ omega‐powers

AbstractWe provide, for each natural number and each class among , , , a regular language whose associated omega‐power is complete for this class.

Hierarchical Sentiment Analysis Framework for Hate Speech Detection: Implementing Binary and Multiclass Classification Strategy

A significant challenge in automating hate speech detection on social media is distinguishing hate speech from regular and offensive language. These identify an essential category of content that web filters seek to remove. Only automated methods can manage this volume of daily data. To solve this problem, the community of Natural Language Processing is currently investigating different ways of hate speech detection. In addition to those, previous approaches (e.g., Convolutional Neural Networks, multi-channel BERT models, and lexical detection) have always achieved low precision without carefully treating other related tasks like sentiment analysis and emotion classification. They still like to group all messages with specific words in them as hate speech simply because those terms often appear alongside hateful rhetoric. In this research, our paper presented the hate speech text classification system model drawn upon deep learning and machine learning. In this paper, we propose a new multitask model integrated with shared emotional representations to detect hate speech across the English language. The Transformer-based model we used from Hugging Face and sentiment analysis helped us prevent false positives. Conclusion. We conclude that utilizing sentiment analysis and a Transformer-based trained model considerably improves hate speech detection across multiple datasets.

Binary distinguishability operation

This paper analyzes a new distinguishability operation for finite deterministic machines and languages. The research was inspired by the “Gedanken experiments” on sequential machines performed by Moore in 1956, and extends the study of the unary distinguishability operation to a binary one. Besides studying the new operation's properties, we give a tight bound of its state complexity on regular languages, including the case for finite languages.

State complexity bounds for projection, shuffle, up- and downward closure and interior on commutative regular languages

We consider the state complexity of projection, shuffle, up- and downward closure and interior on commutative regular languages. We deduce the state complexity bound n|Σ| for upward closure and downward interior, and (2nm)|Σ|, (nm)|Σ|, (n+m−1)|Σ| and (n+m−2)|Σ| for the shuffle on commutative regular, group, aperiodic and finite languages, respectively, with state complexities n and m over the alphabet Σ. We do not know whether these bounds are sharp. For projection, downward closure and upward interior, we give the sharp bound n. Our results are obtained by using the index and period vectors of a regular language, which we introduce in the present work and investigate w.r.t. the above operations and also union and intersection. Furthermore, we characterize the commutative aperiodic and commutative group languages in terms of these parameters and prove that a commutative regular language equals a finite union of shuffle products of commutative finite and commutative group languages.

On the power of local graph expansion grammars with and without additional restrictions

We study graph expansion grammars, a type of graph grammar that has recently been introduced with motivations in natural language processing. Graph expansion generalizes the well-known hyperedge replacement. In contrast to the latter, the former is able to generate graph languages of unbounded treewidth, like the set of all graphs. In an earlier paper, the complexity of the membership problem of the generated languages was studied, the main result being a polynomial parsing algorithm for local DAG expansion grammars (there called local DAG expansion grammars), a subclass of graph expansion grammars that generates directed acyclic graphs. Here, we study the generative power of local graph expansion grammars. While they, unrestricted, are able to simulate Turing machines, we identify natural restrictions that give rise to a pumping lemma and ensure that the generated languages have regular path languages as well as a semi-linear Parikh image.

Regular Expressions Avoiding Absorbing Patterns and the Significance of Uniform Distribution

Although regular expressions do not correspond univocally to regular languages, it is still worthwhile to study their properties and algorithms. For the average case analysis one often relies on the uniform random generation using a specific grammar for regular expressions, that can represent regular languages with more or less redundancy. Generators that are uniform on the set of expressions are not necessarily uniform on the set of regular languages. Nevertheless, it is not straightforward that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained using a more refined set that avoids some large class of equivalent expressions. In this paper we study a set of expressions that avoid a given absorbing pattern. It is shown that, although this set is significantly smaller than the standard one, the asymptotic average estimates for the size of the Glushkov automaton for these expressions does not differ from the standard case. Furthermore, for this set the asymptotic density of [Formula: see text]-accepting expressions is also the same as for the set of all standard regular expressions.

Constructing Deterministic Parity Automata from Positive and Negative Examples

We present a polynomial time algorithm that constructs a deterministic parity automaton (DPA) from a given set of positive and negative ultimately periodic example words. We show that this algorithm is complete for the class of $\omega$-regular languages, that is, it can learn a DPA for each regular $\omega$-language. For use in the algorithm, we give a definition of a DPA, that we call the precise DPA of a language, and show that it can be constructed from the syntactic family of right congruences for that language (introduced by Maler and Staiger in 1997). Depending on the structure of the language, the precise DPA can be of exponential size compared to a minimal DPA, but it can also be a minimal DPA. The upper bound that we obtain on the number of examples required for our algorithm to find a DPA for $L$ is therefore exponential in the size of a minimal DPA, in general. However we identify two parameters of regular $\omega$-languages such that fixing these parameters makes the bound polynomial.

Descriptional Complexity of Finite Automata – Selected Highlights

The state complexity, respectively, nondeterministic state complexity of a regular language L is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for L. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.

Decision Problems on Copying and Shuffling

We study decision problems of the form: given a regular or linear context-free language L, is there a word of a given fixed form in L, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations.

One-directional mapping for cryptographic security of the quasi-regular language sentences to interact light aviation systems subjects

One from the problems of the interaction processes (communications) protection between subjects of ultralight aviation systems that functioning (perform targeting) in an autonomous mode is considered – the problem of communications cryptographic protection in conditions when the interacting components information and communication capabilities are low An asymmetric cryptographic protection system based on the use of a modified regular language of communications is proposed, relevant to the low availability of information and computing resources.