The topology of syntax relations of a formal language

TL;DR

This paper introduces a method to construct Grothendieck's topology on syntax diagrams of a formal language using neighborhood grammars, linking syntax structures to semantics via sheaves of sets.

Contribution

It presents a novel approach to formal language syntax and semantics using category theory, specifically Grothendieck's topology and sheaf theory, on syntax diagrams.

Findings

Defines a topology on syntax diagrams based on neighborhood grammars

Establishes a functorial relationship between syntax diagrams and meanings

Uses sheaves to model compositionality in semantic analysis

Abstract

The method of constructing of Grothendieck's topology basing on a neighbourhood grammar, defined on the category of syntax diagrams is described in the article. Syntax diagrams of a formal language are the multigraphs with nodes, signed by symbols of the language's alphabet. The neighbourhood grammar allows to select correct syntax diagrams from the set of all syntax diagrams on the given alphabet by mapping an each correct diagram to the cover consisted of the grammar's neighbourhoods. Such the cover gives rise to Grothendieck's topology on category of correct syntax diagrams extended by neighbourhoods' diagrams. An each object of the category may be mapped to the set of meanings (abstract senses) of this syntax construction. So, the contrvariant functor from category of correct syntax diagrams to category of sets is defined. The given category of contravariant functors likes to be seen as the convenient means to think about relations between syntax and semantic of a formal language. The sheaves of set defined on category of contravariant functors are the objects that satisfy of compositionality principle defined in the semantic analysis.