Abstract

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(Fp 0 \Delta \Delta \Delta Fp n\Gamma1 ) is of length at least \Gamma n dn=2e \Delta , which is \Omega\Gamma/ n = p n). 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 2 n 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. 1 In...

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