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Generalizing Analogical Inference from Boolean to Continuous Domains

Francisco Cunha, Yves Lepage, Miguel Couceiro, Zied Bouraoui

TL;DR

The paper addresses the gap between Boolean analogical inference and continuous regression by first showing a counterexample that undermines a key generalization bound in the Boolean setting, then introducing a unified framework based on generalized means that extends analogical reasoning to real-valued domains. It provides a complete characterization of analogy-preserving functions AP_{(p;q)} with a closed-form form, shows how this framework yields both worst-case and average-case regression guarantees, and demonstrates how the Boolean case fits within the broader continuous theory. The results lay a theoretical foundation for analogical regression and offer implementable, distance-based guarantees that bridge discrete and continuous domains, with practical directions for empirical evaluation and integration with neural architectures. Overall, the work advances a general theory of analogical inference across domains and opens avenues for reliable, interpretable analogy-driven learning in regression tasks.

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.

Generalizing Analogical Inference from Boolean to Continuous Domains

TL;DR

The paper addresses the gap between Boolean analogical inference and continuous regression by first showing a counterexample that undermines a key generalization bound in the Boolean setting, then introducing a unified framework based on generalized means that extends analogical reasoning to real-valued domains. It provides a complete characterization of analogy-preserving functions AP_{(p;q)} with a closed-form form, shows how this framework yields both worst-case and average-case regression guarantees, and demonstrates how the Boolean case fits within the broader continuous theory. The results lay a theoretical foundation for analogical regression and offer implementable, distance-based guarantees that bridge discrete and continuous domains, with practical directions for empirical evaluation and integration with neural architectures. Overall, the work advances a general theory of analogical inference across domains and opens avenues for reliable, interpretable analogy-driven learning in regression tasks.

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.

Paper Structure

This paper contains 24 sections, 9 theorems, 60 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Analogical Inference from Boolean to General Cases
  4. Analogy-preserving functions and analogy based classification.
  5. Limitations and Drawbacks
  6. An Illustrative Counter-example
  7. Failure of analogical inference.
  8. Analogy-Preserving Functions
  9. Parameterized Analogical Proportions
  10. Analogical Roots and Extensions
  11. Characterizing Analogy-Preserving Functions
  12. Analogy-based Regression
  13. Explicit Description of AP functions
  14. Performance Guarantees
  15. The Probabilistic View
  16. ...and 9 more sections

Key Result

Theorem 1

The class of AP functions is exactly the class $\mathcal{L}$ of affine functions, i.e., functions $f:\mathbb{B}^n\rightarrow \mathbb{B}$ of the form $f=a_1x_1+\ldots+a_nx_n+b$, for $a_i\in \mathbb{B}$ and where $+$ is the addition over the 2-element field $\mathbb{B}$.

Theorems & Definitions (29)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • Remark 10
  • ...and 19 more