CTL+ is Exponentially more Succinct than CTL

It is proved that CTL+ is exponentially more succinct than CTL. More precisely, it is shown that every CTL formula (and every modal -calculus formula) equivalent to the CTL+ formula E(Fp0 ^ ^ Fpn 1) is of length at least n dn=2e , which is (2n=pn). This matches almost the upper bound provided by Emerson and Halpern, which says that for every CTL+ formula of length n there exists an equivalent CTL formula of length at most 2n log n. It follows that the exponential blow-up as incurred in known conversions of nondeterministic Büchi word automata into alternation-free -calculus formulas is unavoidable. This answers a question posed by Kupferman and Vardi. The proof of the above lower bound exploits the fact that for every CTL ( calculus) formula there exists an equivalent alternating tree automaton of linear size. The core of the proof is an involved cut-and-paste argument for alternating tree automata.