“An n! lower bound on formula size”

We introduce a new Ehrenfeucht-Fraisse game for proving lower bounds on the size of first-order formulas. Up until now such games have only been used to prove bounds on the operator depth of formulas, not their size. We use this game to prove that the CTL⁺ formula Occur_n≡E[Fp₁∧F p₂∧···∧F_n ] which says that there is a path along which the predicates p₁ through p_n occur in some order; requires size n! to express in CTL. Our lower bound is optimal. It follows that the succinctness of CTL+ with respect to CTL is exactly Θ(n). Wilke (1999) had shown that the succinctness was at least exponential. We also use our games to prove all optimal Θ(n) lower bound on the number of boolean variables needed for a weak reachability logic (ℛL ^w) to polynomially embed the language LTL. The number of booleans needed for full reachability logic RC and the transitive closure logic FO²(TC) remain open (Immerman and Vardi, 1997; Alechina and Immerman, 2000)