complementation constructions

Complementation of Emerson–Lei automata has long lacked efficient algorithms, both theoretically and practically, particularly when dealing with complex acceptance conditions. This work introduces the Emerson–Lei elevator automaton model and presents the first efficient complementation algorithm tailored to this structure. By integrating structured automaton analysis, classification of strongly connected components, and targeted optimization strategies, the proposed method achieves asymptotically superior complexity compared to the best existing approaches. Experimental evaluation demonstrates that its implementation in the Spot tool significantly outperforms current state-of-the-art complementation techniques for general Emerson–Lei automata, offering both theoretical improvements and practical performance gains.

Higher-dimensional automata (HDA) are often too rigid in their formalism, limiting their direct applicability in modeling and reasoning. This work systematically integrates several weakened variants—such as HDA with interfaces, partial HDA, ST-automata, and relational HDA—and demonstrates, through formal language theory, automata transformations, and algebraic analysis, that these variants fall into only two distinct classes at the language level: those closed under inclusion and those that are not. The paper’s core contributions include the first proof that partial HDA satisfy a Kleene theorem and admit determinization, alongside the establishment of a unified framework that clarifies the expressive power boundaries among all considered variants. These results lay the foundational groundwork for regular expression characterizations and determinization procedures for partial HDA.

To address the state explosion problem in NFA complementation caused by traditional determinization, this paper proposes an efficient complement construction that avoids full determinization. Our method comprises two key innovations: (1) an inverse powerset construction that reverse-simulates the powerset process underlying DFA complementation; and (2) two structure-aware symbolic transition techniques that exploit common NFA features—including ε-transitions, state sharing, and local determinism—to minimize intermediate automaton size. Semantic correctness is preserved throughout. Experimental evaluation on large-scale benchmarks demonstrates that our approach reduces the number of states in the complement automaton by 62%–89% on average compared to the classical subset-construction-then-complement method, while accelerating runtime by one to two orders of magnitude. These gains significantly enhance the practicality and scalability of NFA complementation in applications such as formal verification, regular expression negation, and program analysis.

This paper investigates the computational complexity of model checking monadic second-order (MSO) properties on Boolean and finite-alphabet automata networks. Determining the hardness of nontrivial MSO queries over dynamical graphs—specifically, whether such properties are decidable or tractable under structural constraints. Method: The analysis integrates MSO logic, computational complexity theory, treewidth-based decomposition, and finite dynamical systems modeling, applied to bounded-alphabet (including Boolean) and nondeterministic networks of bounded treewidth. Contribution/Results: First Rice-type complexity lower bounds for nontrivial MSO problems over bounded-alphabet and bounded-treewidth networks are established; it is proven that every nontrivial MSO property is either NP-hard or coNP-hard on Boolean automata networks—revealing a sharp complexity dichotomy between trivial and nontrivial properties; and classical Rice’s undecidability theorem is tightened to the finite-domain setting. These results establish fundamental intractability barriers for algorithmic analysis of constrained-state systems, thereby extending theoretical boundaries of formal verification and decidability in complex dynamical systems.

This paper resolves the long-standing open problem posed by Grigorchuk et al. concerning the undecidability of freeness for automaton semigroups and automaton monoids. The authors devise a novel encoding technique based on the Post Correspondence Problem (PCP), embedding this canonical undecidable problem into automaton algebraic structures over a fixed finite alphabet while precisely controlling the generated relations. This construction establishes the undecidability of freeness testing and, in a unified manner, derives the undecidability of several fundamental decision problems: left-cancellativity, separability, homomorphic extendability, and freeness of presentations (in the monoid case). By integrating automata theory, semigroup algebra, and computability theory, the work provides the first systematic characterization of intrinsic computational limits governing core semantic properties of automaton algebras.

This study investigates the complexity and decidability of the number of “turns” in accepting computations of pushdown automata and one-counter automata when recognizing context-free languages. Employing tools from formal language and automata theory, along with constructive proofs and undecidability arguments, the work establishes that boundedness of turn complexity is undecidable and reveals a non-recursive trade-off in expressive power between the two automaton models. The main contributions include the construction of an infinite strict hierarchy of turn complexity classes bounded by distinct sublinear functions, and the demonstration that there exist languages requiring sublinear—but not constant—turn complexity, with growth rates slower than any given sublinear function, thereby refining the theoretical framework of finite-turn automata.

This work addresses two core challenges in model checking—complementation and inclusion checking for Büchi automata—by proposing a modular solution framework. The approach decomposes automata into strongly connected components and applies structure-aware strategies to tailor complementation algorithms for each component. It introduces a novel on-the-fly emptiness-checking technique targeting simple generalized Rabin pairs, enabling immediate termination upon satisfaction of stopping conditions. Additionally, efficient heuristic strategies are devised to accelerate modular inclusion verification. Experimental results demonstrate that the method significantly outperforms existing tools in both robustness and efficiency, often achieving speedups of several orders of magnitude on standard benchmarks, thereby establishing itself as one of the most robust solutions currently available.

This work addresses the limited expressiveness of traditional visibly pushdown automata by proposing a novel model called Visibly Recursive Automata (VRA), which extends procedural automata systems through mutually recursive classical automata. Introducing the new notion of “co-determinism,” the paper replaces conventional determinism constraints while preserving expressive power, thereby enabling efficient algorithms for crucial operations such as complementation. By integrating formal language theory, automata composition, and complexity analysis, the study rigorously demonstrates that VRA strictly subsumes existing models in expressiveness. Furthermore, it establishes that co-determinism enjoys favorable algorithmic properties, providing a solid foundation for efficient practical implementations.

This work extends classical finite automaton theory—traditionally confined to free strings over unstructured alphabets—to settings where the alphabet itself is a directed graph (finite or infinite), and string concatenation is constrained by vertex adjacency in the graph. By integrating algebraic language theory, graph theory, and automata models, the paper establishes foundational results for this framework: it proves adapted versions of Kleene’s theorem and the Myhill–Nerode theorem, demonstrates that the class of regular languages is not closed under complementation, and develops determinization and minimization algorithms while analyzing prefix and suffix quotient structures. The study not only constructs a comprehensive theoretical foundation for automata over graph alphabets but also outlines a pathway for generalization to more abstract presimplicial alphabets.

This work challenges the traditional Chomsky hierarchy by highlighting its neglect of how an observer’s perception of input influences language recognizability. Introducing the notion of an “observer,” the paper proposes an orthogonal observational hierarchy that classifies formal languages based on the structure of observer functions rather than computational power. Integrating formal language theory, automata models, and poset analysis, the authors define observational complexity and parameterized complexity classes, proving that multiset observers correspond precisely to permutation-closed language classes. They construct an observational hierarchy exhibiting a diamond-shaped sublattice structure and demonstrate that under full-profile observation, P_O = NP_O, thereby revealing that computational hardness and structural blindness are independent phenomena.