The last installment of this series of three posts on characteristic formulae deals with characteristic formulae for behavioural equivalences over timed automata.
The first construction of a characteristic formula for timed bisimilarity over timed automata I am aware of was presented in the paper
François Laroussinie, Kim G. Larsen, and Carsten Weise. From Timed Automata to Logic - and Back. Jirí Wiedermann, Petr Hájek (Eds.): Mathematical Foundations of Computer Science 1995, 20th International Symposium, MFCS'95, Prague, Czech Republic, August 28 - September 1, 1995, Proceedings. Lecture Notes in Computer Science 969 Springer 1995, ISBN 3-540-60246-1.
In that paper, the authors propose a timed version of Hennessy-Milner logic Lnu, and show thateach timed automaton can be uniquely characterized by a single formula in that logic up to timed bisimilarity.
The construction in that paper relied on the region graph of the timed automaton, and thus produces "large" formulae. (See yesterday's post on the number of regions in a timed automaton.) In the setting of timed automata without invariants, the construction by Laroussinie, Larsen and Weise was improved upon in the paper
Luca Aceto, Anna Ingólfsdóttir, Mikkel Lykke Pedersen, and Jan Poulsen. Characteristic Formulae for Timed Automata. ITA 34(6): 565-584 (2000).
That paper offers characteristic formula constructions in the real-time logic Lnu for several behavioural relations between (states of) timed automata. The behavioural relations studied in op. cit. are timed (bi)similarity, timed ready simulation, faster-than bisimilarity and timed trace inclusion. The characteristic formulae delivered by the constructions offered in the aforementioned paper have size which is linear in that of the timed automaton they logically describe.
Finally, the paper
Luca Aceto, Patricia Bouyer, Augusto Burgueño, Kim G. Larsen. The power of reachability testing for timed automata. Theor. Comput. Sci. 300(1-3): 411-475 (2003)
uses a property language characterizing the power of reachability testing over timed automata in the context of "test automata" to provide a definition of characteristic properties with respect to a timed version of the ready simulation preorder, for nodes of τ-free, deterministic timed automata.
I am not aware of a timed counterpart of the results by Browne, Clarke and Grümberg on the characterization of bisimulation-like equivalences over Kripke structures using characteristic formulae in fragments of CTL. Is Timed CTL expressive enough to describe characteristic formulae for timed bisimilarity? This has been an item on my "to do" list for quite some time, but I have never had the guts to start working on the problem yet.