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Then, satisfying a formula of CT L on this model is de ned by: S1 KS, s0 |= P if and only if P (s0 ) S2 KS, s0 |= F AND g if and only if KS, s0 |= f and KS, s0 |= g KS, s0 |= NOT f if and only if there are not KS, s0 |= f S3 KS, s0 |= Ef if and only if resulting from s0 such as KS, |= f KS, s0 |= Af if and only if resulting from s0 , KS, |= f P1 If f is a state formula, KS, |= f if and only if KS, s0 |= f P2 KS, |= F AND g if and only if KS, |= f and KS, |= g KS, |= NOT f if and only if NOT (KS, |= f ).



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keys would have to be stored in each access point, which would increase the administrative overhead WPA also replaces the insecure WEP encryption with the new temporal key integrity protocol (TKIP) ciphering algorithm TKIP effectively prevents attackers from generating the ciphering key by launching a brute-force attack on the network The encryption is not as strong as required in the 80211i standard, but can be implemented on current WLAN chipsets designed for WEP In theory, access points can support WPA and WEP devices at the same time In practice, only a few access points available on the market support this As the security of the network is only as good as the weakest link, the use of WPA only makes a lot of sense for company networks.





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P3 KS, |= F Ug if and only if i such as KS, i |= g and jn such as pr PROM(em ) / In practice, CT L is enriched by abbreviations to simplify the expression of properties: (OR) f OR g NOT (NOT f AND NOT g) (true) true NOT P OR P (false) f alse NOT true (F) F f true U f (G) G f NOT F NOT f (W) f W g f U g OR G f F f means that f is true for a suf x of the considered sequence G f means that f is true for all suf xes of the considered sequence Contrary to f Ug, note that f Wg (W for weak until ) does not imply that g will be true for a suf x In this case, f is true for all suf xes.

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[BRO 88] also introduced stuttering equivalence, which gives a behavioral characterization of the temporal logic CT L _X, namely CT L without using its next time operator 125 Decidability of bisimulation and of evaluation of formulas Let us now consider the fundamental decidability results concerning the bisimulation of Petri nets and the evaluation of formulas of temporal logic for Petri nets 4 shows that all generic properties are decidable (ie boundedness, accessibility, etc) Moreover, the complexity of the veri cation of a few properties is completely characterized (eg the boundness property is EX Pspace-complete), while for others the problem is still open (accessibility is EX Pspace-hard but the algorithm of decision is not primitive recursive) As has been seen in the previous sections, using temporal logic and bisimulation to characterize the behavior of a labeled Kripke structure is a better method than using generic properties.

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For private networks, the concurrent use of WPA an WEP could be desirable if the access point and all PCs of the network are already WPA compatible, while other devices like PDAs only support WEP PDAs usually transmit only small amounts of data and thus do not pose a great security risk After final publication of the 80211i standard, the WiFi Alliance, which promotes the IEEE WLAN standards (http://wwwwifiorg), adapted its WPA certification program accordingly WPA2 is an implementation of the 80211i standard and is backward compatible to WPA This means that a WPA2 certified access point is also able to support WPA devices For further backward compatibility, a WPA2 access point can also support legacy WEP clients if WPA/WPA2 is turned off In addition to WPA s TKIP encryption algorithm, WPA2 now supports the stronger AES (advanced encryption standard) encryption.

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Also, it can be expected that the decision procedures are more dif cult, but the problem of accessibility is easily expressed in LT L and CT L EXAMPLE 1217 in LT L, m not accessible from (R, m0 ) (R, m0 ) |= G OR p P p = m(p) in CT L, m non-accessible since (R, m0 ) (R, m0 ) |=AG OR p P p = m(p) 1251 Undecidability results The usual technique for showing that a problem is undecidable consists of reducing it to another problem known to be undecidable The more general problem we will study is the termination of a program THEOREM 123 (Termination of a program with parameters) The problem of the termination of a program prog, parameterized by an integer x is undecidable Proof We show this result by contradiction.

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