that satisfies one also satisfies the other. This is denoted It can be seen that A and E are duals, being universal and existential computation path quantifiers respectively: . Furthermore, so are G and F. Hence an instance of De Morgan's laws can be formulated in CTL: It can be shown using such identities that a subset of the CTL temporal connectives is adequate if it contains , at least one of and at least one of and the boolean connectives. The important equivalences below are called the expansion laws; they allow to unfold the verification of a CTL connective towards its successors in time. Examples Let "P" mean "I like chocolate" and Q mean "It's warm outside." AG.P "I will like chocolate from now on, no matter what happens." EF.P "It's possible I may like chocolate some day, at least for one day." AF.EG.P "It's always possible (AF) that I will suddenly start liking chocolate for the rest of time." (Note: not just the rest of my life, since my life is finite, while G is infinite). EG.AF.P "Depending on what happens in the future (E), it's possible that for the rest of time (G), I'll be guaranteed at least one (AF) chocolate-liking day still ahead of me. However, if something ever goes wrong, then all bets are off and there's no guarantee about whether I'll ever like chocolate." The two following examples show the difference between CTL and CTL*, as they allow for the until operator to not be qualified with any path operator (A or E): AG(PUQ) "From now until it's warm outside, I will like chocolate every single day. Once it's warm outside, all bets are off as to whether I'll like chocolate anymore. Oh, and it's guaranteed to be warm outside eventually, even if only for a single day." EF((EX.P)U(AG.Q)) "It's possible that: there will eventually come a time when it will be warm forever (AG.Q) and that before that time there will always be some way to get me to like chocolate the next day (EX.P)." Relations with other logics Computation tree logic (CTL) is a subset of CTL* as well as of the modal μ calculus. CTL is also a fragment of Alur, Henzinger and Kupferman's alternating-time temporal logic (ATL). Computation tree logic (CTL) and linear temporal logic (LTL) are both a subset of CTL*. CTL and LTL are not equivalent and they