Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1909.04118

Structural Commutation Relations for Stochastic Labelled Graph Grammar Rule Operators

Eric Mjolsness
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show how to calculate the operator algebra and the operator Lie algebra of a stochastic labelled-graph grammar. More specifically, we carry out a generic calculation of the product (and therefore the commutator) of time-evolution operators for any two labelled-graph grammar rewrite rules, where the operator corresponding to each rule is defined in terms of elementary two-state creation/annihilation operators. The resulting graph grammar algebra has the following properties: (1) The product and commutator of two such operators is a sum of such operators with integer coefficients. Thus, the algebra and the Lie algebra occurs entirely at the structural (or graph-combinatorial) level of graph grammar rules, lifted from the level of elementary creation/annihilation operators (an improvement over [1], Propositions 1 and 2). (2) The product of the off-diagonal (state-changing) parts of two such graph rule operators is a sum of off-diagonal graph rule operators with non-negative integer coefficients. (3) These results apply whether the semantics of a graph grammar rule leaves behind hanging edges (Theorem 1), or removes hanging edges (Theorem 2). (4) The algebra is constructive in terms of elementary two-state creation/annihilation operators (Corollaries 3 and 8). These results are useful because dynamical transformations of labelled graphs comprise a general modeling framework, and algebraic commutators of time-evolution operators have many analytic uses including designing simulation algorithms and estimating their errors.

Explore related subjects

cs.FLmath-phmath.MPphysics.bio-phq-bio.QM

Keep this discovery

BibTeXRIS

Eric Mjolsness. 2019-09-09. Structural Commutation Relations for Stochastic Labelled Graph Grammar Rule Operators. https://arxiv.org/abs/1909.04118

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↗