The IO and OI Hierarchies Revisited

Abstract

We study languages of λ-terms generated by IO and OI unsafe grammars. These languages can be used to model meaning representations in the formal semantics of natural languages following the tradition of Montague [25]. Using techniques pertaining to the denotational semantics of the simply typed λ-calculus, we show that the emptiness and membership problems for both types of grammars are decidable. In the course of the proof of the decidability results for OI, we identify a decidable variant of the λ-definability problem, and prove a stronger form of Statman's finite completeness Theorem [35].

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

“…We now take the monotone applicative structure (see [21,24]) M = (Mα)α∈Sorts where Mo is the two element lattice, with maximal element ⊤ and minimal element ⊥. Intuitively, ⊤ means nonempty language and ⊥ means empty language.…”

“…Least fixed point models of schemes induce an interpretation on infinite trees by finite approximations. An infinite tree has value ⊤ iff it represents a non-empty language [21]. The important point is that the semantics of a term and that of the infinite tree generated from the term coincide.…”

“…In this paper, we consider non-deterministic higher-order recursion schemes as recognizers of languages of finite trees. In other words we consider higher-order OI grammars [12,21]. This is an expressive formalism covering many other models such as indexed grammars [2], ordered multi-pushdown automata [5], or the more general higher-order pushdown automata with collapse [17] (cf.…”