$\Sigma^1_1$-Completeness of a Fragment of the Theory of Trees with Subtree Relation

Abstract

We consider the structure $IT_S$ of all labeled trees, called also infinite terms, in the first order language ${\cal L}$ with function symbols in a recursive signature $S$ of cardinality at least two and at least a symbol of arity two, with equality and a binary relation symbol $\sqsubseteq$ which is interpreted to be the subtree relation. The existential theory over ${\cal L}$ of this structure is decidable (see Tulipani [9]), but more complex fragments of the theory are undecidable. We prove that the $ \exists\Delta$ theory of the structure is in $\Sigma^1_1$, where $ \exists\Delta$ formulas are those in prenex form consisting of a string of unbounded existential quantifiers followed by a string of arbitrary quantifiers all bounded with respect to $\sqsubseteq$. Since the fragment of the theory was already known to be $\Sigma^1_1$-hard (see Marongiu and Tulipani [5]), it is now established to be $\Sigma^1_1$-complete.