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

doi:10.1016/j.ipl.2018.03.013

Information Processing Letters

Volume 136, August 2018, Pages 17-20

https://doi.org/10.1016/j.ipl.2018.03.013 Get rights and content

Highlights

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Given a parametric event-recording automaton and a parameter valuation, it is undecidable whether another valuation has the same untimed language.
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This undecidability result holds even for slightly different definitions of the untimed language.
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The problem asking for the existence of another parameter valuation with the same set of traces is left as future work.

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).

Introduction

Timed automata (TAs) [1] are a useful formalism to model and formally verify systems involving timing hard constraints and concurrency. TAs benefit from numerous decidability results, including the emptiness of the accepted language. However, the universality and the language inclusion are undecidable for timed automata [1]. Therefore, subclasses have been proposed. The language inclusion becomes decidable for event-recording automata (ERAs) [2].

Parametric timed automata (PTAs) [3] extend TAs with timing parameters: this very expressive formalism can model systems where timing constants are uncertain or unknown, at the cost of most interesting problems to be undecidable [4]. The mere emptiness of the valuation set for which a given location is reachable (“reachability-emptiness”) is undecidable [3].

Restricting the syntax of a formalism may bring decidability: the language inclusion undecidable for TAs [1] becomes decidable for ERAs [2]. In contrast, the reachability emptiness problem for PTAs remains undecidable for a subclass of PTAs with only open inequalities [5].

In [6], we proposed parametric event-recording automata (PERAs), and showed that the reachability-emptiness problem remains undecidable for PERAs. Although it seems that our proof idea can be extended to most problems where the language (i.e., the transition labels) does not play a role (which would include unavoidability-emptiness [7]), it remains open whether language problems undecidable for PTAs become or not decidable for PERAs. In [8], we showed that the following language preservation problem is undecidable for PTAs: “given a PTA and a reference parameter valuation, does there exist another valuation with the same untimed language?”. This problem has connections with the robustness of timed systems, as it asks whether other valuations of the timing constants may lead to the same discrete behavior. The set of valuations with the same untimed language can also be used to perform optimization of some constants without impacting the system's (untimed) behavior.

We show here that the language preservation problem is undecidable for PERAs, and remains undecidable for different definitions of the language. This quite surprising result comes in contrast with the difference of decidability between TAs and ERAs in the non-parametric setting.

Section snippets

Preliminaries

Throughout this paper, we assume a set X of clocks, i.e., real-valued variables that evolve at the same rate. A clock valuation is a function $μ : X \to R_{+}$ . We write $\vec{0}$ for the clock valuation that assigns 0 to all clocks. Given $d \in R_{+}$ , $μ + d$ denotes the valuation such that $(μ + d) (x) = μ (x) + d$ , for all $x \in X$ . Given $x \in X$ , we define the reset of a valuation μ, denoted by ${[μ]}_{x}$ , as follows: ${[μ]}_{x} (x^{'}) = 0$ if $x^{'} = x$ , and ${[μ]}_{x} (x^{'}) = μ (x^{'})$ otherwise.

We assume a set P of parameters, i.e., unknown rational-valued constants. A

Encoding a 2-counter machine into a PERA

We propose in this section an encoding of a 2-counter machine (2CM) into a PERA. This encoding is adapted from the PTA encoding of [8] to our setting of PERAs, and is therefore not a main contribution of this work.

Fix a deterministic 2CM M. Recall that such a machine has two non-negative counters $C_{1}$ and $C_{2}$ (the value of which is initially 0), and a finite number of states and of transitions of the form:

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when in state $s_{i}$ , increment $C_{k}$ and go to $s_{j}$ ;
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when in state $s_{i}$ , if $C_{k} = 0$ then go to $s_{j}$ else

Main result

Theorem 1

The language preservation problem for PERAs is undecidable.

The proof of undecidability of [8] for PTAs strongly relies on the fact that all transitions were labeled with the same action a. This reasoning cannot be kept here, as the transitions of our modified encoding in Section 3 are labeled with different actions so as to reset different clocks. Therefore, it is not possible to know in advance the language of the accepting run of the 2CM (if any).

For example, assume a run of the 2CM made of

Conclusion

We proved that the language preservation remains undecidable for a subclass of PTAs, namely parametric event-recording automata. We believe that the L/U-automata restrictions considered in the additional undecidability results of [8] could apply to our setting, and undecidability would still hold for “L/U-PERAs”. A more challenging future work is to study the trace preservation problem of [8] that considers not only the actions but also the locations.

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Cited by (2)

Language preservation problems in parametric timed automata
2020, Logical Methods in Computer Science
Language Preservation Problems in Parametric Timed Automata
2018, arXiv

¹: This work is partially supported by the ANR national research program PACS (ANR-14-CE28-0002).

²: This research is mainly supported by the startup grant M4081588.020.500000 of School of Computer Science and Engineering in Nanyang Technological University.

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