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Additional info for The Mathematics of Syntactic Structure: Trees and Their Logics
For any finite η it is simple to construct an L^ p formula that requires, for instance, every point with any children at all to have at least n + 1 children. While there are infinitely many trees satisfying this formula, there 32 James Rogers can only be one satisfying assignment in Nn—the trivial tree (containing just the root)—for any translation of it into the language of SnS. Lemma 16 Every set offinite trees definable in SnS for some η < ω is recognizable. This is a consequence of the fact that, if we restrict ourselves to sets of finite trees, Rabin's tree automata reduce to ordinary tree automata.
In the remainder of the paper we will mostly use LQ 0 and refer to it as L2. The change does not limit the power of the language since both the predicate and individual constants correspond to free (global) variables. In fact, if one uses a tool to compile the automata from the formulas, one must use free variables since the compilation process cannot handle constants. We deviate slightly from Rogers' definition of the language in that we include (Ξ explicitly and use X(x) and χ e X interchangeably.
X n ) is satisfied in a tree 7 = (T, Pj, D j , Lj, Ι χ , Ig>) relative to an assignment s mapping individual variables in χ to individuals in Τ and labels X to subsets of T. We extend this to a function f interpreting terms: s(t) $(t) = < iDc(t) k Mt) if t € χ or t € X ifteSC if t e IP. Then Τ satisfies φ(χ, X) with s in the standard way: 7>tT) T(=u(t)[s] A SitjeKU), Τ μ (3Χ)[φ(Χ)] [s] Ä 7 \ =