By Dilsun Kaynar, Nancy Lynch, Roberto Segala, Frits Vaandrager

This monograph offers the Timed Input/Output Automaton (TIOA) modeling framework, a simple mathematical framework to help description and research of timed (computing) platforms. Timed platforms are structures within which fascinating correctness or functionality houses of the method rely on the timing of occasions, not only at the order in their incidence. Timed platforms are hired in a variety of domain names together with communications, embedded structures, real-time working structures, and automatic keep an eye on. Many purposes concerning timed structures have robust security, reliability, and predictability standards, which make it vital to have equipment for systematic layout of structures and rigorous research of timing-dependent habit. The TIOA framework additionally helps description and research of timed disbursed algorithms -- allotted algorithms whose correctness and function rely on the relative speeds of processors, accuracy of neighborhood clocks, or communique hold up bounds. Such algorithms come up, for instance, in conventional and instant communications, networks of cellular units, and shared-memory multiprocessors. the necessity to turn out rigorous theoretical effects approximately timed allotted algorithms makes it very important to have an appropriate mathematical starting place. a major function of the TIOA framework is its aid for decomposing timed procedure descriptions. specifically, the framework features a thought of exterior habit for a timed I/O automaton, which captures its discrete interactions with its setting. The framework additionally defines what it capacity for one TIOA to enforce one other, in line with an inclusion courting among their exterior habit units, and defines notions of simulations, which supply adequate stipulations for demonstrating implementation relationships. The framework encompasses a composition operation for TIOAs, which respects exterior habit, and a inspiration of receptiveness, which suggests TIOA doesn't block the passage of time. The TIOA framework additionally defines the inspiration of a estate and what it ability for a estate to be a security or a liveness estate. It comprises effects that seize universal facts tools for exhibiting that automata fulfill houses. desk of Contents: advent / Mathematical Preliminaries / Describing Timed approach habit / Timed Automata / Operations on Timed Automata / houses for Timed Automata / Timed I/O Automata / Operations on Timed I/O Automata / Conclusions and destiny paintings

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**Additional info for The Theory of Timed I O Automata, Second Edition (Synthesis Lectures on Distributed Computing Theory) **

**Sample text**

Now suppose that e is integrable when viewed as a function. Then we say that τ satisfies e ≤ d(v) if, for every t1 , t2 ∈ dom(τ ) such that t1 ≤ t2 , v(t1 ) + d(v) ≤ t2 t1 e(t )dt ≤ v(t2 ), and τ satisfies e if, for every t1 , t2 ∈ dom(τ ) such that t1 ≤ t2 , v(t2 ) ≤ v(t1 ) + t2 t1 e(t )dt . Conventions for automata specifications: In all the examples of this monograph we assume the time axis T to be R and specify timed automata by using a variant of the TIOA language presented in [93, 56, 32, 33].

8. 5 IMPLEMENTATION RELATIONSHIPS Timed automata A1 and A2 are comparable if they have the same external interface, that is, if E1 = E2 . 1 1 In [86, 36, 80, 81], definitions of the set of traces of an automaton and of one automaton implementing another are based on closed and admissible executions only. The results we obtain in this monograph using the newer, more inclusive definition imply corresponding results for the earlier definition. 6. SIMULATION RELATIONS 41 Other preorders between timed automata could also be used as implementation relationships, for example, if A1 and A2 are comparable timed automata, we could consider: • Every closed trace of A1 is a trace of A2 .

Define trajectories τ0 , τ1 , τ2 , . . inductively by τ0 τi+1 = τ0 , = τi τi+1 for nonfinal i. 5 implies that for each nonfinal i, τi ≤ τi+1 . We define the concatenation τ0 τ1 to be the limit of the chain τ0 , τ1 , τ2 , . 4. 4 τ2 · · · HYBRID SEQUENCES In this section, we introduce the notion of a hybrid sequence, which is used to model a combination of changes that occur instantaneously and changes that occur over intervals of time. Our definition is parameterized by a set A of actions, which are used to model instantaneous changes and instantaneous synchronizations with the environment, and a set V of variables, which are used to model changes over intervals of time.