A Theory of Formalisms for Representing Knowledge

TL;DR

This paper introduces a unifying theoretical framework for knowledge representation formalisms in AI, demonstrating that all universal and intertranslatable formalisms are essentially equivalent up to offline compilation.

Contribution

It develops a general framework capturing various knowledge formalisms and proves their recursive isomorphism, addressing longstanding debates in AI knowledge representation.

Findings

All universal formalisms are recursively isomorphic.

Pairwise intertranslatable formalisms with the padding property are recursively isomorphic.

Universal formalisms are equivalent up to offline compilation.

Abstract

There has been a longstanding dispute over which formalism is the best for representing knowledge in AI. The well-known "declarative vs. procedural controversy" is concerned with the choice of utilizing declarations or procedures as the primary mode of knowledge representation. The ongoing debate between symbolic AI and connectionist AI also revolves around the question of whether knowledge should be represented implicitly (e.g., as parametric knowledge in deep learning and large language models) or explicitly (e.g., as logical theories in traditional knowledge representation and reasoning). To address these issues, we propose a general framework to capture various knowledge representation formalisms in which we are interested. Within the framework, we find a family of universal knowledge representation formalisms, and prove that all universal formalisms are recursively isomorphic. Moreover, we show that all pairwise intertranslatable formalisms that admit the padding property are also recursively isomorphic. These imply that, up to an offline compilation, all universal (or natural and equally expressive) representation formalisms are in fact the same, which thus provides a partial answer to the aforementioned dispute.