A Purely Model-Theoretic Proof of the Succinctness Gap Between CTL and CTL

1 The Branching Time Temporal Logics CTL and CTL + A transition system is a tuple T = (S, − →, L) where S is a set of states, − → ⊆ S × S a transition relation and L : P → 2 S a function that assigns to each q in some set P of atomic propositions the set of states L(q) in which q holds. Here we assume the transition relation to be total in the sense that for all s ∈ S there is a t ∈ S s.t. s − → t. A path is an infinite sequence π = s 0 , s Formulas of the branching time temporal logic CTL + are given by the following grammar. The formulas derived from ϕ are sometimes called state formulas whereas those derived from ψ are called path formulas. We consider path formulas only as genuine subformulas of state formulas. We use the usual abbreviations for ∧, →, ↔, etc. from propositional logic, and from temporal logic: Aψ := ¬E¬ψ, Fϕ := ttUϕ where tt := q ∨ ¬q for some q ∈ P, and Gϕ := ¬F¬ϕ.