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1233 Temporal logic and Petri nets The speci cation of propositional temporal logic formulas for Petri nets implies the de nition of atomic properties Since formulas of temporal logic are evaluated on the reachability graph, a state of the model is a reachable marking Then, any Boolean expression whose scope is the set of reachable markings is appropriate In practice, the corresponding expressions are easily evaluated Let p be the marking of p in the current state Here are some examples of formulas of interest: Two places p1 , p2 are mutually exclusive: AG (p1 p2 = 0) For any reachable marking, place p will inevitably be marked: AG AF (p > 0) For any reachable marking, place p can always be marked: AG EFF (p > 0).

For any execution sequence, place p is in nitely marked and in nitely unmarked: AG F (p > 0) AND F (p = 0) A transition t is live (always rable in the future of any state): AG EFF ANDp P p Pr (p, y) e As will be illustrated in the nal example, it is possible to reason on the rability of a transition However, this is not actually the ring itself because it would require evaluation of the evolution of the marking between two successive states As a consequence, the language CT L is extended by considering the operator X{E} , whose semantics is de ned by: |= X{e} f if and only if KS, 1 |= f and the rst transition of is labeled by e In this section, let us assume that a transition of a Petri net is never labeled as an empty string.

5

The veri cation methods described above can be easily extended to this new logic, which is at the same time state- and event-based Let us assume that the labeling function of a net is the identity We can express that a transition t is red an in nite number of times in all sequences by: AG F X{t} true To study the decidability of the veri cation of temporal logic formulas of Petri nets, the following cases need to be considered: state-based propositional logic CT L (and its fragments) which prohibits these new operators;.

The input noise power pnoise,1 is, by de nition, the thermal (Johnson) noise power from the source at 290 K (about 17 C) into a matched load, the available noise power at that temperature. This theoretical noise level is pnoise,1 = kT0 B, where k is Boltzmann s constant, T0 is 290 K, and B is noise bandwidth. The value of NT = kT0 is approximately 4 10 21 W, or 174 dBm, per Hz bandwidth.1 [Resistors also have icker noise, which dominates at low frequencies (Egan, 2000, p. 119).] The input signal power psignal,1 is the available source power of the signal. The output powers are also de ned into a matched load. The ratio of output power to input power then meets the de nition of available gain (see Appendix G). Figure 3.1 shows a noise gure test setup where some of the variables have circum exes (hats) to identify them with this theoretical setup. Note that the impedance of the source and load must, in general, be changed for each device under test (DUT), the source impedance to correspond to the speci ed source and the load impedance to match the impedance at the DUT output. The noise factor is the factor by which the inherent random noise of the source resistance at 290 K would have to increase to account for the additional output noise that is actually produced by the DUT. An alternate representation of module noise is noise temperature, which is the increase in source temperature that could have accounted for the module noise contribution. We will include both representations in some of the development that follows.

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