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Allan van Hulst

Publications and source records attributed to Allan van Hulst.

6 recordsLinked to original sources
arXiv2022

A Result on the Small Quasi-Kernel Conjecture

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite and without self-loops. A source in a directed graph is a vertex having at least one ingoing arc. A quasi-kernel $Q\subseteq V(D)$ is an independent set in $D$ such that every vertex in $V(D)$ can be reached in at most two steps from a vertex in $Q$. It is an open problem whether every source-free directed graph has a quasi-kernel of size at most $|V(D)|/2$, a problem known as the small quasi-kernel conjecture (SQKC). The aim of this paper is to prove the SQKC under the assumption of a structural property of directed graphs. This relates the SQKC to the existence of a vertex $u\in V(D)$ and a bound on the number of new sources emerging when $u$ and its out-neighborhood are removed from $D$. The results in this work are of technical nature and therefore additionally verified by means of the Coq proof-assistant.

math.CO↗
arXiv2021

A Complete Axiom System for 1-Free Kleene Star Expressions under Bisimilarity: An Elementary Proof

Grabmayer and Fokkink recently presented a finite and complete axiomatization for 1-free process terms over the binary Kleene star under bismilarity equivalence (proceedings of LICS 2020, preprint available). A different and considerably simpler proof is detailed in this paper. This result, albeit still somewhat technical, only relies on induction and normal forms and is therefore also much closer to a potential rewriting algorithm. In addition, a complete verification in the Coq proof assistant of all results in this work is provided, but correctness does not depend upon any computer-assisted methodology.

cs.LO↗
arXiv2021

Kernels and Small Quasi-Kernels in Digraphs

A directed graph $D=(V(D),A(D))$ has a kernel if there exists an independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)-K$ has an ingoing arc $u\mathbin{\longrightarrow}v$ for some $u\in K$. There are directed graphs that do not have a kernel (e.g. a 3-cycle). A quasi-kernel is an independent set $Q$ such that every vertex can be reached in at most two steps from $Q$. Every directed graph has a quasi-kernel. A conjecture by P.L. Erdős and L.A. Székely (cf. A. Kostochka, R. Luo, and, S. Shan, arxiv:2001.04003v1, 2020) postulates that every source-free directed graph has a quasi-kernel of size at most $|V(D)|/2$, where source-free refers to every vertex having in-degree at least one. In this note it is shown that every source-free directed graph that has a kernel also has a quasi-kernel of size at most $|V(D)|/2$, by means of an induction proof. In addition, all definitions and proofs in this note are formally verified by means of the Coq proof assistant.

math.CO↗
arXiv2014

Maximally Permissive Controlled System Synthesis for Modal Logic

We propose a new method for controlled system synthesis on non-deterministic automata, which includes the synthesis for deadlock-freeness, as well as invariant and reachability expressions. Our technique restricts the behavior of a Kripke-structure with labeled transitions, representing the uncontrolled system, such that it adheres to a given requirement specification in an expressive modal logic. while all non-invalidating behavior is retained. This induces maximal permissiveness in the context of supervisory control. Research presented in this paper allows a system model to be constrained according to a broad set of liveness, safety and fairness specifications of desired behavior, and embraces most concepts from Ramadge-Wonham supervisory control, including controllability and marker-state reachability. Synthesis is defined in this paper as a formal construction, which allowed a careful validation of its correctness using the Coq proof assistant.

cs.FL↗
arXiv2011

A Process Algebra for Supervisory Coordination

A supervisory controller controls and coordinates the behavior of different components of a complex machine by observing their discrete behaviour. Supervisory control theory studies automated synthesis of controller models, known as supervisors, based on formal models of the machine components and a formalization of the requirements. Subsequently, code generation can be used to implement this supervisor in software, on a PLC, or embedded microprocessor. In this article, we take a closer look at the control loop that couples the supervisory controller and the machine. We model both event-based and state-based observations using process algebra and bisimulation-based semantics. The main application area of supervisory control that we consider is coordination, referred to as supervisory coordination, and we give an academic and an industrial example, discussing the process-theoretic concepts employed.

cs.LO↗