Priority Downward Closures

TL;DR

This paper introduces the concept of priority downward closures for message sequences in lossy channels with congestion control, providing algorithms to compute these closures for various language classes, extending the regular language framework.

Contribution

It is the first to study priority downward closures, offering algorithms for regular, one-counter, and context-free languages, advancing formal language analysis under priority constraints.

Findings

Priority downward closures are always regular.

Algorithms for computing priority downward closures are developed for multiple language classes.

This work extends the understanding of language abstractions in lossy, priority-based communication systems.

Abstract

When a system sends messages through a lossy channel, then the language encoding all sequences of messages can be abstracted by its downward closure, i.e. the set of all (not necessarily contiguous) subwords. This is useful because even if the system has infinitely many states, its downward closure is a regular language. However, if the channel has congestion control based on priorities assigned to the messages, then we need a finer abstraction: The downward closure with respect to the priority embedding. As for subword-based downward closures, one can also show that these priority downward closures are always regular. While computing finite automata for the subword-based downward closure is well understood, nothing is known in the case of priorities. We initiate the study of this problem and provide algorithms to compute priority downward closures for regular languages, one-counter languages, and context-free languages.