Generalizing Analogical Inference from Boolean to Continuous Domains

TL;DR

This paper extends analogical inference from Boolean to continuous domains using a unified framework based on generalized means, enabling analogical reasoning in regression and real-valued functions.

Contribution

It introduces a new unified framework for analogical inference in real-valued domains, generalizing previous Boolean-based models and characterizing analogy-preserving functions.

Findings

Counterexample shows existing bounds fail even in Boolean case

Framework supports analogical inference in continuous functions

Provides error bounds under smoothness assumptions

Abstract

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct for functions close to affine. These results have informed the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains. In this paper, we revisit analogical inference from a foundational perspective. We first present a counterexample showing that existing generalization bounds fail even in the Boolean setting. We then introduce a unified framework for analogical reasoning in real-valued domains based on parameterized analogies defined via generalized means. This model subsumes both Boolean classification and regression, and supports analogical inference over continuous functions. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. Our results offer a general theory of analogical inference across discrete and continuous domains.