[Paper Review] Timed Discrete-Event Systems are Synchronous Product Structures

Liyong Lin|arXiv (Cornell University)|Jul 18, 2018

Petri Nets in System Modeling24 references3 citations

TL;DR

This paper proves that timed discrete-event systems (TDES) are inherently synchronous product structures by constructing a generalized synchronous product from transformed activity and timer automata. The key contribution is a model transformation that enables TDES synthesis using standard synchronous product tools, avoiding explicit tick enumeration and reducing state explosion through modular composition of constituent automata.

ABSTRACT

Timed discrete-event systems (TDES), which is a modelling formalism proposed by Brandin and Wonham, can be used for modelling scheduling and production planning problems. This paper aims to show that TDES are essentially synchronous product structures. The proof is constructive in the sense that a generalized synchronous product rule is provided to generate a TDES from the activity automaton and the timer automata (that is, the syntactic description of the TDES) after some model transformation. We then also explain how the generalized synchronous product operation can be reduced into the standard synchronous product operation and how to reduce the number of (refined) events introduced in the model transformation. Thus, any software that can compute synchronous products can be used to compute a TDES from its activity automaton and its timer automata, after the model transformation.

Motivation & Objective

• To resolve an open problem in TDES formalism: whether a TDES can be systematically constructed from its activity and timer automata.
• To address the state explosion problem in TDES by avoiding explicit enumeration of tick events.
• To enable efficient supervisor synthesis by leveraging modular composition via synchronous product operations.
• To provide a constructive method that maps TDES into standard synchronous product frameworks for tool compatibility.

Proposed method

• Transform the activity automaton and each timer automaton by adding refined event labels to track timer state changes.
• Introduce placeholder events (e.g., Eσ, Dσ, σ!, ¬σ!) to encode event and timer interactions during synchronization.
• Define a generalized synchronous product operation that synchronizes events between the transformed activity automaton and each timer automaton based on shared time and event labels.
• Perform relabeling to convert the generalized product into a standard synchronous product, preserving semantic equivalence.
• Use event hiding to recover original TDES semantics from the refined product structure.
• Limit the number of refined events by using only transitions present in the original activity automaton, avoiding exponential blowup.

Experimental results

Research questions

• RQ1 Can a TDES be systematically constructed from its activity automaton and timer automata, resolving an open problem in Brandin (1994)?
• RQ2 How can the generalized synchronous product operation be defined to correctly model TDES behavior with time intervals?
• RQ3 Can the generalized synchronous product be reduced to a standard synchronous product for compatibility with existing tools?
• RQ4 What model transformations are required to avoid explicit tick enumeration and reduce state explosion?
• RQ5 Can modular composition of activity and timer automata improve efficiency in TDES supervisor synthesis?

Key findings

• TDES are formally shown to be synchronous product structures, resolving an open problem from Brandin (1994).
• A constructive model transformation enables the generation of a TDES from its activity and timer automata via generalized synchronous product operation.
• The generalized synchronous product can be reduced to a standard synchronous product through event relabeling, enabling tool compatibility.
• The number of refined events introduced during transformation is bounded by the number of transitions in the activity automaton, preventing exponential blowup.
• The approach supports modular supervisor synthesis by treating activity and timer automata as holons, improving efficiency and scalability.