Micah Adler and N. Immerman, An n! Lower Bound On Formula Size, ACM Transactions on Computational Logic, 4(3) (2003), 296-314. Preliminary version appeared in LICS '01, 197-206. (LICS '01, 197-206.

Abstract: 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,

which says that there is a path along which 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 Theta(n). Wilke had shown that the succinctness was at least exponential [Wi99].

We also use our games to prove an optimal Omega(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 RL and the transitive closure logic FO²(TC) remain open [IV97,AI00].

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