scc-aware automata complementation

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.

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 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.

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.

Determining dynamical properties of automata networks (AN) and succinct graph representations (SGR) suffers from high computational complexity, yet tight lower bounds—especially for succinct systems—remain largely unestablished. Method: We introduce the first general circuit gadget constructible in deterministic logarithmic space (O(log n)), integrating finite model theory with dynamical systems theory to build a logspace meta-reduction framework. Contribution/Results: We establish the first universal Rice-type undecidability lower bound for succinct dynamical systems. Specifically, we prove that deciding any nontrivial dynamical property over AN or SGR is NL-hard (and thus at least as hard as nondeterministic logspace), with reductions provably constrained to deterministic logspace. This yields the first tight, unified lower-bound tool for classifying the complexity of succinct dynamical systems, resolving a long-standing gap in succinct computation theory.

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.

Traditional cascade products suffer from exponential blowup in both states and alphabet size, rendering them inefficient for automaton decomposition. This work proposes the $\Sigma$-chain product model, which constructs a hierarchical compositional structure by restricting each component to depend only on the input alphabet and the preceding component. The model achieves, for the first time, linear-size automaton compositions, significantly enhancing succinctness while preserving expressive power; it enables exponential compression of representation size and is equivalent to the cascade product for certain classes of automata. Furthermore, the paper establishes theoretical connections between $\Sigma$-chain products and permutation-reset automata as well as subclasses of star-free regular languages, precisely characterizing their recognition capabilities.

This work addresses the ambiguity problem in finite-state automata by proposing a general unambiguation algorithmic framework. The approach generalizes the classical subset construction method, generating successor states on demand while preserving the original automaton’s structure. It supports partial unambiguation for full, finite, and polynomial degrees of ambiguity and naturally extends to weighted automata. The proposed algorithm computes states in polynomial time, thereby achieving—for the first time—an efficient construction of unambiguous automata applicable across multiple ambiguity levels and automaton models.

This study investigates the succinctness of chained co-Büchi automata (COCOA) for representing ω-regular languages and analyzes their state complexity under Boolean operations. By constructing specific families of languages, the work compares the expressive power of COCOA with deterministic parity automata and examines the state blowup induced by conjunction, disjunction, and complementation. It is shown for the first time that COCOA can be exponentially more succinct than deterministic parity automata—even when all component automata within the chain are deterministic—yet they inevitably suffer exponential state explosion under Boolean operations and complementation. These results highlight the inherent trade-off between the compact representational capacity of COCOA and their fundamental limitations in compositional system design.

Existing approaches to handling universal quantification based on nondeterministic Büchi automata rely on complementation, which incurs high computational costs. This work proposes a knowledge compilation method tailored to alternating safety automata, introducing for the first time a normal form that supports efficient existential and universal quantifier projection. By structurally transforming the transition function using binary decision diagrams (BDDs), the method eliminates the need for complementation and enables stepwise elimination of quantifier sequences. A prototype implementation demonstrates substantial performance gains over state-of-the-art techniques on QPTL satisfiability benchmarks, confirming both the feasibility and superiority of the proposed approach.