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Sheaving-a universal construction for semantic compositionality.

Steven Phillips1

  • 1National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8566, Japan.

Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences
|December 17, 2019
PubMed
Summary
This summary is machine-generated.

Category theory offers a novel framework for understanding semantic compositionality, integrating algebraic and geometric perspectives. This approach uses sheaving to model how meaning arises from constituent parts and their relationships, grounding it in topological spaces.

Keywords:
category theorycompositionalitysheafsheavingtopology

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Area of Science:

  • Cognitive Science
  • Theoretical Linguistics
  • Category Theory
  • Mathematical Foundations of Semantics

Background:

  • Semantic compositionality, the principle that complex meanings derive from constituent parts and their structure, is central to cognitive capacities like systematicity.
  • Current cognitive science debates mechanistic explanations for compositionality, with key proposals including a 'language of thought' versus a 'geometry of thought'.
  • A formal divide exists between algebraic and geometric approaches to understanding meaning.

Purpose of the Study:

  • To propose a category theory perspective on the semantics of compositionality, bridging formal divides in cognitive science.
  • To explore sheaving as a universal construction for integrating local knowledge and grounding meaning in topological spaces.
  • To demonstrate how topological structures can convey semantic information.

Main Methods:

  • Utilized category theory, specifically sheaving, a construction merging algebra and geometry/topology.
  • Developed a framework where meaning is grounded in the topological space of inclusion relations between open sets.
  • Illustrated the approach with three examples demonstrating topology's role in conveying semantic information.

Main Results:

  • Sheaving provides a universal method for inference from local knowledge, with meaning intrinsically linked to an underlying topological space.
  • Topological inclusion relations between open sets serve as a model for conveying semantic information.
  • Category (sheaf) theory offers a general, integrative framework for semantic compositionality.

Conclusions:

  • Category theory, particularly through sheaving, presents a unified mathematical framework for semantic compositionality.
  • This approach integrates disparate perspectives (algebraic and geometric) in cognitive science.
  • The framework suggests that meaning composition can be understood through structures derived from topological spaces.