Universal Higher Order Grammar

TL;DR

This paper introduces a formal framework based on extended type theory for defining languages through provability, enabling precise representation of logical and semantic features without special syntactic types.

Contribution

It develops a novel formalism that characterizes a class of languages via provability in an extended type theory, including effective procedures for finite rule-based languages and extensions for semantics.

Findings

Class of languages matches logically closed languages definable in the formalism.

Effective methods for constructing compact type-theoretic representations for rule-based languages.

Extension with context type allows modeling of intensional and dynamic semantics.

Abstract

We examine the class of languages that can be defined entirely in terms of provability in an extension of the sorted type theory (Ty_n) by embedding the logic of phonologies, without introduction of special types for syntactic entities. This class is proven to precisely coincide with the class of logically closed languages that may be thought of as functions from expressions to sets of logically equivalent Ty_n terms. For a specific sub-class of logically closed languages that are described by finite sets of rules or rule schemata, we find effective procedures for building a compact Ty_n representation, involving a finite number of axioms or axiom schemata. The proposed formalism is characterized by some useful features unavailable in a two-component architecture of a language model. A further specialization and extension of the formalism with a context type enable effective account of intensional and dynamic semantics.