Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1511.09011

Quantifier Alternation for Infinite Words

Théo Pierron·Thomas Place·Marc Zeitoun
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the expressive power of quantifier alternation hierarchy of first-order logic over words. This hierarchy includes the classes $Σ_i$ (sentences having at most $i$ blocks of quantifiers starting with an $\exists$) and $\mathcal{B}Σ_i$ (Boolean combinations of $Σ_i$ sentences). So far, this expressive power has been effectively characterized for the lower levels only. Recently, a breakthrough was made over finite words, and decidable characterizations were obtained for $\mathcal{B}Σ_2$ and $Σ_3$, by relying on a decision problem called separation, and solving it for $Σ_2$. The contribution of this paper is a generalization of these results to the setting of infinite words: we solve separation for $Σ_2$ and $Σ_3$, and obtain decidable characterizations of $\mathcal{B}Σ_2$ and $Σ_3$ as consequences.

Explore related subjects

cs.FL

Keep this discovery

BibTeXRIS

Théo Pierron, Thomas Place, Marc Zeitoun. 2015-11-29. Quantifier Alternation for Infinite Words. https://arxiv.org/abs/1511.09011

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond the Turing threshold: Productive grammars generate essentially undecidable languages

Emil Post's productive sets are not even semi-computable, let alone computable, being thus essentially incomputable. Accordingly, formal languages whose set of words is a (completely) productive set are essentially undecidable. In this article, I elaborate on Post productivity from the viewpoint of formal language theory: I design formal grammars that emulate the construction of productive sets of natural numbers and are thus beyond Turing-decidability.

cs.FL↗
arXiv2026

RAGTIMER 1.0: Rapid Rare-Event Partial State Space Construction for Stochastic VAS (extended version)

Transient reachability analysis of rare events in Continuous-Time Stochastic Vector Addition Systems (CTSVAS) such as Chemical Reaction Networks (CRNs) has proven a formidable challenge to cutting-edge tools. Underlying a CTSVAS is a continuous-time Markov chain (CTMC), and CTMC transient reachability analysis calls for Probabilistic Model Checking (PMC). This analysis requires the explicit representation of a model's entire state space. Rare events occur with extremely low probability, compounding the challenge of probabilistic analysis. In CRNs, it is imperative to verify the probability of rare events; even a low concentration of a species can have pathological consequences. This paper presents the RAGTIMER 1.0 tool, which efficiently builds a partial state space for a CTSVAS by enumerating traces to a rare event of interest and expanding them to exploit concurrency and cycles, providing a guaranteed lower bound on the probability of a rare event. Guaranteed lower bounds are particularly useful in synthetic biological applications because they indicate how and when a rare event can be experimentally observed. RAGTIMER is an attractive alternative to existing rare event analysis methods for CTSVAS models. It outperforms existing PMC tools and refutes multiple probability estimates from rare-event stochastic simulation on multiple challenging CRN models. RAGTIMER uses optimized data structures, a simple input format, and memory-safe Rust code to improve the scalability and accessibility of PMC for industry professionals.

cs.FL↗
arXiv2026

Execution-Time Opacity Logic: A Logic for Ensuring ET-Opacity in Timed Systems

Ensuring confidentiality in Cyber-Physical Systems is critical, especially when attackers exploit execution times to infer sensitiveinformation. Traditional opacity models are inadequate for timed systems, as verifying opacity in Timed Automata is undecidable. To address this challenge, we propose Execution-Time Opacity Logic (ETOL), a new formalism that specifies opacity by requiring that for every execution satisfying a secret formula, there exists another execution of the same duration that does not satisfy it. ETOL guarantees that timing observations cannot reveal confidential agent activities. We present a decidable and efficient verification framework based on zone-based model checking, supported by a dedicated algorithm that systematically identifies duration-equivalent executions. Our approach is validated through an ATM case study, showing that ETOL enables efficient verification of execution-time confidentiality under timing attacks. We also developed a prototype tool for the ETOL logic that supports symbolic model checking over timed systems. It allows users to verify ETOL formulas based on clock-constrained execution paths.

cs.FL↗