A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and we shall mostly follow the notation and definitions given in that reference. As usual, we shall assume a countably infinite set Delta of action names, ranged over by a and b, and a distinguished action 62 Delta. Let Act = Delta [ fg, be the set of actions, ranged over by and . Following [13], we define a monoid (X; 0) to be: ffl left cancellative iff (x y = x z) y = z) and ffl anti symmetric iff (x y = 0) x = y = 0) Examples of left cancellative anti symmetric monoids include: 2 ffl The singleton set (1; 0) ffl The natural numbers (N; 0) ffl The ....

....hold for many of the models of time that have been considered in the literature on timed process algebras, e.g. the natural numbers, the non negative rationals and the non negative reals. The definition of time domain that we have chosen in this paper is due to Jeffrey, Schneider and Vaandrager [13] and suits the purpose of this paper well. However, it is certainly not the only one possible, and several ones have been proposed in the literature (see [4] for a series of examples) Complete axiomatizations of behavioural equivalences for several timed process algebras have been presented in ....

A.S.A. Jeffrey, S. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, 1993. Also available as Computer Science Report 11/93, University of Sussex.

....namely that if time can pass in two steps, then it can also pass in a single step. Finally, axiom A3 says that if time can pass with an amount d, then it is possible to associate states with all times in the interval [0; d] in a consistent way. For a further discussion of this axiom we refer to [11, 6]. 4.3 Composition and Hiding Intuitively, the composition of two timed I O automata is their Cartesian product, with the added requirement that automata synchronize on shared actions and on passage of time. The synchronization of shared actions models communication between system components: if ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....(D; 0) induces a binary order of precedence defined by: 8t; u 2 D : t u ( 9v 2 D : t v = u A6 There exists an absorption element 1 is defined such that 8t 2 D : t 1 = 1. So 8t 2 D : t 1. The assumptions A1 A4 imply that the time domain is a left cancellative anti symmetric monoid [JSV93] Examples of time domains include: the natural numbers (nat [ f1g; 0) the positive rational numbers, etc. NOTE: For the moment, Traian considers that the time domain is nat. 4.4 Derived Types The Lotos NT type declaration is very general. It allows the definition of more familiar types ....

A. Jeffrey, S. Schneider, and F. Vaandrager. A Comparison of Additivity Axioms in Timed Transition Systems. Technical Report cs1193, COGS, University of Sussex, 1993.

.... that says that there must be a trajectory of consistent states through the interval [t; t 0 ] Since our timed transition system satisfies the image finiteness condition (that is, for any B and t 0 there are at most finitely many B 0 such that hB; ti ; hB 0 ; t 0 i) it follows from [17] that our model also satisfies this stronger trajectory condition. A consistent causality based view on a timed process algebra 15 consistency result, where TT (E [ B ] denotes the set of timed event traces of E [ B ] Theorem 11. For all B 2 TPA : TT (E [ B ] T [ B ] The proof of ....

A. Jeffrey, S. Schneider, and F. Vaandrager. A comparison of additivity axioms in timed transition systems. Technical Report CS-R9366, Centre for Mathematics and Computer Science (1993)

....that if time can pass in two steps, then it can also pass in a single step. The trajectory axiom S2 is a kind of converse to S1; it says that any time passage step can be filled in with states for each intervening time, in a consistent way. For a further discussion of this axiom we refer to [LV93, JSV93]. The difference is just the explicit indication of the amount of elapsed time in the time passage action instead of using a .now function that associates the current time to a state. 4.2. Timed Traces Executions of timed automata correspond to what are called sampling computations in [MP93] ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....that if time can pass in two steps, then it can also pass in a single step. The trajectory axiom S2 is a kind of converse to S1; it says that any time passage step can be filled in with states for each intervening time, in a consistent way. For a further discussion of this axiom we refer to [30, 21]. 4.2 Timed Traces Executions of timed automata correspond to what are called sampling computations in [31] they provide information about a run of a system at a countable number of points in time. In [30] a notion of timed execution is also defined for timed automata: these are alternating ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....to those settings. For convenience, we use R as our domain of times in this paper. The need for densetime models has been well discussed in [4] However, for the purpose of generality we could have parameterized our timed automata by an arbitrary (possibly discrete) time domain in the sense of [27, 53, 28]. We do not assume a general lower bound on the time between events, or an upper bound on the number of instantaneous actions; this choice is also made in e.g. 7, 2, 9, 25, 48, 53, 61] but still distinguishes our model from many others, e.g. 11, 17, 20, 50, 52, 55, 62] The cost of this ....

....for providing the environment for an active research interchange that led to many improvements in our work. A Other Axioms for Timed Automata We consider the relationship between axioms S2 and S2 , as defined in Section 2.1. The relationship between the two axioms is also investigated in [28]. Define a semi timed automaton to be a timed automaton, except that it does not have to satisfy S2, but only the weaker (and simpler) axiom S2 . It is immediate from the definition of a trajectory that each timed automaton is semi timed. In this appendix, we consider the reverse implication. ....

[Article contains additional citation context not shown here]

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....that if time can pass in two steps, then it can also pass in a single step. The trajectory axiom S2 is a kind of converse to S1; it says that any time passage step can be filled in with states for each intervening time, in a consistent way. For a further discussion of this axiom we refer to [30, 21]. 4.2 Timed Traces Executions of timed automata correspond to what are called sampling computations in [31] they provide information about a run of a system at a countable number of points in time. In [30] a notion of timed execution is also defined for timed automata: these are alternating ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....to those settings. For convenience, we use R as our domain of times in this paper. The need for densetime models has been well discussed in [4] However, for the purpose of generality we could have parameterized our timed automata by an arbitrary (possibly discrete) time domain in the sense of [27, 53, 28]. We do not assume a general lower bound on the time between events, or an upper bound on the number of instantaneous actions; this choice is also made in e.g. 7, 2, 9, 25, 48, 53, 61] but still distinguishes our model from many others, e.g. 11, 17, 20, 50, 52, 55, 62] The cost of this ....

....for providing the environment for an active research interchange that led to many improvements in our work. A Other Axioms for Timed Automata We consider the relationship between axioms S2 and S2 0 , as defined in Section 2.1. The relationship between the two axioms is also investigated in [28]. Define a semi timed automaton to be a timed automaton, except that it does not have to satisfy S2, but only the weaker (and simpler) axiom S2 0 . It is immediate from the definition of a trajectory that each timed automaton is semi timed. In this appendix, we consider the reverse implication. ....

[Article contains additional citation context not shown here]

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....s:now , then there is an s 00 with s 00 :now = t such that s 0 Gamma s 00 and s 00 Gamma s. It is immediate from the definition of a trajectory that S5 ) S5 0 . In this appendix we discuss the reverse implication. The relationship between the two axioms is also investigated in [10]. As the time domain for our timed automata we have chosen the set R 0 of nonnegative real numbers. We could have chosen a different time domain though, or parametrized our automata with an arbitrary time domain as in [9, 20] In order to state the axioms for timed automata, all we need is the ....

A. Jeffrey, S. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems, 1993. In preparation.

....that if time can pass in two steps, then it can also pass in a single step. The trajectory axiom S2 is a kind of converse to S1; it says that any time passage step can be filled in with states for each intervening time, in a consistent way. For a further discussion of this axiom we refer to [LV93, JSV93]. 6 The difference is just the explicit indication of the amount of elapsed time in the time passage action instead of using a .now function that associates the current time to a state. Action Transducers and Timed Automata 19 4.2. Timed Traces Executions of timed automata correspond to what ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

....if time can pass in two steps, then it can also pass in a single step. Finally, the trajectory axiom A3 says that if time can pass with an amount d, then it is possible to associate states with all times in the interval [0; d] in a consistent way. For a further discussion of this axiom we refer to [18, 11]. 2.3 Composition and Hiding Intuitively, the composition of two timed I O automata is their Cartesian product, with the added requirement that automata synchronize on shared actions and on passage of time. The synchronization of shared actions models communication between system components: if ....

A.S.A. Jeffrey, S.A. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, November 1993.

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