@MISC{Bojańczyk_, author = {Mikołaj Bojańczyk}, title = {}, year = {} }

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Abstract

The common fragment of CTL and LTL needs existential modalities The language “all paths belong to (ab) ∗ a(ab) ∗ c ω ” is definable in CTL, but not in ACTL, which is the fragment of CTL that only uses universal modalities. CTL [1] is the temporal branching time logic that uses the modalities: AϕUψ (on every path, there is a node with ψ, and all preceding nodes satisfy ϕ), AXϕ (every successor position satisfies ϕ) and AGϕ (on every path, every position satisfies ϕ). ACTL is the fragment of CTL, which uses the above modalities and does not allow negation. (Here we use mutually exclusive atomic propositions; if they are are not exclusive then negation is allowed next to atomic propositions.) Clearly, ACTL is a proper fragment of CTL; for instance, the CTL property “some node has label a ” is not definable in ACTL. A universal path property is one of the form “all paths belong to L”, for some infinite word language L. In [2], Maidl investigated the universal path properties expressible in CTL. However, the decidability results from [2] actually concerned not all of CTL, but only ACTL, and the following question was raised: is a CTL-definable universal path language necessarily ACTL-definable? A positive answer to this question would be consistent with the intuition that only universal path modalities are needed to express universal path properties. In this note, we show that this is not the case: Theorem 0.1 The language L = “all paths belong to (ab) ∗ a(ab) ∗ c ω ” is definable in CTL, but not ACTL. This example shows that ideas significantly different from those in [2] are needed to understand the universal path properties definable in CTL. Here, we view CTL as a logic that describes properties of infinite trees, where both infinite paths, and infinite node outdegree are allowed. The same results would apply for transition systems, and also finite trees. Before proceeding with the proof, we would like to remark the similarity of this “paradox ” with a result for firstorder logic over finite binary trees. In [4], Potthof showed