An n ! lower bound on formula size

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Abstract
We introduce a new Ehrenfeucht--Fraïssé 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[F p 1 ∧ F p 2 ∧ … ∧ F p n ], which says that there is a path along which the predicates p 1 through p n all occur, 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 had shown that the succinctness was at least exponential [Wilke 1999].We also use our games to prove an optimal Ω( n ) lower bound on the number of boolean variables needed for forward reachability logic (RL f ) to polynomially embed the language CTL + . The number of booleans needed for full reachability logic RL and the transitive closure logic FO 2 (TC) remain open [Immerman and Vardi 1997; Alechina and Immerman 2000].