# The language preservation problem is undecidable for parametric event-recording automata

## TL;DR

This paper proves that the language preservation problem remains undecidable for parametric event-recording automata, even under various restrictions and different definitions of the language, highlighting fundamental computational limits.

## Contribution

It extends the undecidability results from parametric timed automata to the more restricted class of PERAs, showing the problem's robustness.

## Key findings

- Language preservation problem is undecidable for PERAs.
- Undecidability persists under various language definitions.
- Results highlight fundamental limits in parametric automata analysis.

## Abstract

Parametric timed automata (PTA) extend timed automata with unknown constants ("parameters"), at the price of undecidability of most interesting problems. The (untimed) language preservation problem ("given a parameter valuation, can we find at least one other valuation with the same untimed language?") is undecidable for PTAs. We prove that this problem remains undecidable for parametric event-recording automata (PERAs), a subclass of PTAs that considerably restrains the way the language can be used; we also show it remains undecidable even for slightly different definitions of the language, i.e., finite sequences of actions ending in or passing infinitely often through accepting locations, or just all finite untimed words (without accepting locations).