# On the Distance Between Timed Automata.

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

### Abstract

Some fundamental problems in the class of non-deterministic timed automata, like the problem of inclusion of the language accepted by timed automaton $A$ (e.g., the implementation) in the language accepted by $B$ (e.g., the specification) are, in general, undecidable.
In order to tackle this disturbing problem we show how to effectively construct deterministic timed automata $A_d$ and $B_d$ that are discretizations (digitizations) of the non-deterministic timed automata $A$ and $B$ and differ from the original automata by at most $\frac{1}{6}$ time units on each occurrence of an event.
Language inclusion in the discretized timed automata is decidable and it is also decidable when instead of $L(B)$ we consider $\overline{L(B)}$, the closure of $L(B)$ in the Euclidean topology:
if $L(A_d) \nsubseteq L(B_d)$ then $L(A) \nsubseteq L(B)$ and if
$L(A_d) \subseteq L(B_d)$ then $L(A) \subseteq \overline{L(B)}$.

Moreover, if $L(A_d) \nsubseteq L(B_d)$ we would like to know how far away is $L(A_d)$ from being included in $L(B_d)$. For that matter we define the distance between the languages of timed automata as the limit on how far away a timed trace of one timed automaton can be from the closest timed trace of the other timed automaton.
We then show how one can decide under some restriction whether the distance between two timed automata is finite or infinite.
Originalsprache englisch FORMATS 2019 É André, M. Stoelinga Springer, Cham 199-215 17 11750 978-3-030-29662-9 978-3-030-29661-2 https://doi.org/10.1007/978-3-030-29662-9_12 Veröffentlicht - 13 Aug 2019 17th International Conference on Formal Modeling and Analysis of Timed Systems - Amsterdam, NiederlandeDauer: 27 Aug 2019 → 29 Aug 2019

### Publikationsreihe

Name Lecture Notes in Computer Science 11750

### Konferenz

Konferenz 17th International Conference on Formal Modeling and Analysis of Timed Systems FORMATS 2019 Niederlande Amsterdam 27/08/19 → 29/08/19

Timed Automata
Inclusion
Trace
Digitization
Automata
Euclidean
Closure
Discretization
Specification
Restriction
Topology
Unit
Language

### Dies zitieren

Rosenmann, A. (2019). On the Distance Between Timed Automata. in É. André, & M. Stoelinga (Hrsg.), FORMATS 2019 (Band 11750, S. 199-215). (Lecture Notes in Computer Science; Band 11750). Springer, Cham. https://doi.org/10.1007/978-3-030-29662-9_12
FORMATS 2019. Hrsg. / É André; M. Stoelinga. Band 11750 Springer, Cham, 2019. S. 199-215 (Lecture Notes in Computer Science; Band 11750).

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

Rosenmann, A 2019, On the Distance Between Timed Automata. in É André & M Stoelinga (Hrsg.), FORMATS 2019. Bd. 11750, Lecture Notes in Computer Science, Bd. 11750, Springer, Cham, S. 199-215, Amsterdam, Niederlande, 27/08/19. https://doi.org/10.1007/978-3-030-29662-9_12
Rosenmann A. On the Distance Between Timed Automata. in André É, Stoelinga M, Hrsg., FORMATS 2019. Band 11750. Springer, Cham. 2019. S. 199-215. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-030-29662-9_12
Rosenmann, Amnon. / On the Distance Between Timed Automata. FORMATS 2019. Hrsg. / É André ; M. Stoelinga. Band 11750 Springer, Cham, 2019. S. 199-215 (Lecture Notes in Computer Science).
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abstract = "Some fundamental problems in the class of non-deterministic timed automata, like the problem of inclusion of the language accepted by timed automaton $A$ (e.g., the implementation) in the language accepted by $B$ (e.g., the specification) are, in general, undecidable.In order to tackle this disturbing problem we show how to effectively construct deterministic timed automata $A_d$ and $B_d$ that are discretizations (digitizations) of the non-deterministic timed automata $A$ and $B$ and differ from the original automata by at most $\frac{1}{6}$ time units on each occurrence of an event.Language inclusion in the discretized timed automata is decidable and it is also decidable when instead of $L(B)$ we consider $\overline{L(B)}$, the closure of $L(B)$ in the Euclidean topology:if $L(A_d) \nsubseteq L(B_d)$ then $L(A) \nsubseteq L(B)$ and if $L(A_d) \subseteq L(B_d)$ then $L(A) \subseteq \overline{L(B)}$. Moreover, if $L(A_d) \nsubseteq L(B_d)$ we would like to know how far away is $L(A_d)$ from being included in $L(B_d)$. For that matter we define the distance between the languages of timed automata as the limit on how far away a timed trace of one timed automaton can be from the closest timed trace of the other timed automaton.We then show how one can decide under some restriction whether the distance between two timed automata is finite or infinite.",
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N2 - Some fundamental problems in the class of non-deterministic timed automata, like the problem of inclusion of the language accepted by timed automaton $A$ (e.g., the implementation) in the language accepted by $B$ (e.g., the specification) are, in general, undecidable.In order to tackle this disturbing problem we show how to effectively construct deterministic timed automata $A_d$ and $B_d$ that are discretizations (digitizations) of the non-deterministic timed automata $A$ and $B$ and differ from the original automata by at most $\frac{1}{6}$ time units on each occurrence of an event.Language inclusion in the discretized timed automata is decidable and it is also decidable when instead of $L(B)$ we consider $\overline{L(B)}$, the closure of $L(B)$ in the Euclidean topology:if $L(A_d) \nsubseteq L(B_d)$ then $L(A) \nsubseteq L(B)$ and if $L(A_d) \subseteq L(B_d)$ then $L(A) \subseteq \overline{L(B)}$. Moreover, if $L(A_d) \nsubseteq L(B_d)$ we would like to know how far away is $L(A_d)$ from being included in $L(B_d)$. For that matter we define the distance between the languages of timed automata as the limit on how far away a timed trace of one timed automaton can be from the closest timed trace of the other timed automaton.We then show how one can decide under some restriction whether the distance between two timed automata is finite or infinite.

AB - Some fundamental problems in the class of non-deterministic timed automata, like the problem of inclusion of the language accepted by timed automaton $A$ (e.g., the implementation) in the language accepted by $B$ (e.g., the specification) are, in general, undecidable.In order to tackle this disturbing problem we show how to effectively construct deterministic timed automata $A_d$ and $B_d$ that are discretizations (digitizations) of the non-deterministic timed automata $A$ and $B$ and differ from the original automata by at most $\frac{1}{6}$ time units on each occurrence of an event.Language inclusion in the discretized timed automata is decidable and it is also decidable when instead of $L(B)$ we consider $\overline{L(B)}$, the closure of $L(B)$ in the Euclidean topology:if $L(A_d) \nsubseteq L(B_d)$ then $L(A) \nsubseteq L(B)$ and if $L(A_d) \subseteq L(B_d)$ then $L(A) \subseteq \overline{L(B)}$. Moreover, if $L(A_d) \nsubseteq L(B_d)$ we would like to know how far away is $L(A_d)$ from being included in $L(B_d)$. For that matter we define the distance between the languages of timed automata as the limit on how far away a timed trace of one timed automaton can be from the closest timed trace of the other timed automaton.We then show how one can decide under some restriction whether the distance between two timed automata is finite or infinite.

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