T C C

Current state and perspectives of development of the theory of timed languages are analyzed. A large list of open problems is suggested. 1 Why study timed systems? In most cases computer science deals with sequences of events or actions (we will call them behaviors). Powerful formalisms have been developed to express sets or trees of such sequences (various automata, grammars, expressions, process algebras, Petri nets, logics). Some of them are more adapted for a human user, others are excellent for machine treatment1; all of them are interesting research objects and are tightly related. Formal languages and related techniques find numerous applications in semantics, specification, verification – in all the situations where we are interested in exploring sets of behaviors. Another important class of applications (and source of research problems) concerns texts – both in artificial and natural languages. A text is rather a sequence of symbols and not a sequence of events, but I don’t see a mathematical difference between the two of them. However, one aspect is forgotten (or abstracted) in all of the abovementioned approaches, namely time. But in reality even for some texts timing is important (a text transmitted via a communication channel can be considered as a timed sequence of characters, a spoken text as a timed sequence of phonemes, and a piece of music can be also seen as a sequence of notes each of which has a duration and starts at some moment of time). As for behaviors, in our physical world, they always happen in time. Sometimes the timing does not matter and it is often a good idea to prove correctness of an algorithm without thinking how much time each ∗Partially based on the invited talk at FORMATS’03 workshop. Supported by the European community project IST-2001-35304 A 1I first learned this idea from a talk by Amir Pnueli. operation takes; or to prove that each request is eventually granted, regardless of when the granting actually takes place. However, there are many other situations when one has to make sure that a reactive program always finishes processing a task before the next one arrives, and that each request is granted within 10ms. Also, many programs or protocols work because they use timing (e.g. using timeouts) or because their inputs are subject to some timing constraints. These practical issues, and especially a need for tools for specification and verification of protocols relying on timing led in the early 90s to the creation of a timed systems theory, which takes into account not only the order of events but also their position in time. The creation of this domain (at least if we consider only automata-based approach) was related with the invention of timed automata by Alur and Dill together with basic decidability results for language emptiness and model-checking problems, and with implementation of these results in verification tools such as K and U. Nowadays timed systems theory and applications are a well-established area with several thesis defenses every year, a bunch of papers in each CAV, ICALP and CONCUR conference, and a couple of specialized workshops. So we arrive at our first result. Result 1. Sequences of events “embedded” in the time are a ubiquitous reality. It is practically useful and theoretically challenging to study them using approaches of computer science2. This is done by the timed systems community. In the sequel I will try to analyze what has been done in the domain (without being exhaustive) and, more importantly, what has not been done and what should be done. 2 Historical strength and historical weakness Let me start with a comparison. The classical theory of finite automata and regular languages (together with technologies based upon this theory) can be seen as a chef d’œuvre of computer science. This beautiful castle (see Figure 1) is based on a solid basement of finite automata theory. A deeper understanding is obtained thanks to the algebraic theory of automata based on monoids. The next floor of the castle is a well-established theory of regular languages (of finite words) and their alternative logical characterizations (Monadic second order logic etc.). A theory of languages of infinite words relies on the theory of finitary languages and allows to express properties of infinite behaviors. For all these formalisms 2The author is aware of one well-established theory describing such sequences in a probabilistic setting: the theory of continuous time Markov chains (and its generalizations to point processes).