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. 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.

Unsafe Order-2 Tree Languages Are Context-Sensitive

“…Although some of the analogous results have recently been obtained for unsafe grammars (those without the safety restriction) [16,7,14], many problems still remain open, such as the context-sensitiveness of higher-order languages. This is a pity, as many of the recent applications of higher-order grammars make use of unsafe ones.…”

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.…”

“…This allows us to define the semantics [[M, χ, ν]] of a term given a valuation χ for nonterminals and ν for variables 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.…”