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Inference of Abstraction for Grounded Predicate Logic

Hiroyuki Kido

TL;DR

The paper tackles how to ground predicate logic in data to achieve meaningful abstraction by introducing a data-to-model-to-formula joint distribution that treats formulas as abstractions of models and models as abstractions of data. It formalizes data support models, models support formulas, and a predicate-reasoning mechanism that yields empirical and probabilistic generalizations of logical consequence, including reasoning from possible and impossible information with a tunable parameter $\mu\in[0.5,1]$. The key contributions include a concrete probabilistic framework for predicative abstraction, a derivation of conditional-independence properties, and demonstrations of empirical-consequence generalization and two-mode applicability to simple tasks, along with proofs. The work offers a novel lens on symbol grounding and tractable predicate reasoning, and outlines directions for integrating learning of the mapping from data to models and vocabulary design for creative abstraction.

Abstract

An important open question in AI is what simple and natural principle enables a machine to reason logically for meaningful abstraction with grounded symbols. This paper explores a conceptually new approach to combining probabilistic reasoning and predicative symbolic reasoning over data. We return to the era of reasoning with a full joint distribution before the advent of Bayesian networks. We then discuss that a full joint distribution over models of exponential size in propositional logic and of infinite size in predicate logic should be simply derived from a full joint distribution over data of linear size. We show that the same process is not only enough to generalise the logical consequence relation of predicate logic but also to provide a new perspective to rethink well-known limitations such as the undecidability of predicate logic, the symbol grounding problem and the principle of explosion. The reproducibility of this theoretical work is fully demonstrated by the included proofs.

Inference of Abstraction for Grounded Predicate Logic

TL;DR

The paper tackles how to ground predicate logic in data to achieve meaningful abstraction by introducing a data-to-model-to-formula joint distribution that treats formulas as abstractions of models and models as abstractions of data. It formalizes data support models, models support formulas, and a predicate-reasoning mechanism that yields empirical and probabilistic generalizations of logical consequence, including reasoning from possible and impossible information with a tunable parameter . The key contributions include a concrete probabilistic framework for predicative abstraction, a derivation of conditional-independence properties, and demonstrations of empirical-consequence generalization and two-mode applicability to simple tasks, along with proofs. The work offers a novel lens on symbol grounding and tractable predicate reasoning, and outlines directions for integrating learning of the mapping from data to models and vocabulary design for creative abstraction.

Abstract

An important open question in AI is what simple and natural principle enables a machine to reason logically for meaningful abstraction with grounded symbols. This paper explores a conceptually new approach to combining probabilistic reasoning and predicative symbolic reasoning over data. We return to the era of reasoning with a full joint distribution before the advent of Bayesian networks. We then discuss that a full joint distribution over models of exponential size in propositional logic and of infinite size in predicate logic should be simply derived from a full joint distribution over data of linear size. We show that the same process is not only enough to generalise the logical consequence relation of predicate logic but also to provide a new perspective to rethink well-known limitations such as the undecidability of predicate logic, the symbol grounding problem and the principle of explosion. The reproducibility of this theoretical work is fully demonstrated by the included proofs.

Paper Structure

This paper contains 12 sections, 11 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proposals
  3. Data support models
  4. Models support formulas
  5. Predicate reasoning
  6. Evaluations
  7. Reasoning as learning
  8. Reasoning from possible information
  9. Reasoning from impossible information
  10. Applicability
  11. Conclusions and Future Work
  12. Proofs

Key Result

Proposition 9

Let $\alpha_{1},\alpha_{2}\in L$. $\alpha_{1}$ is conditionally independent of $\alpha_{2}$ and $D$ given $M$, i.e., $p(\alpha_{1}|\alpha_{2},M,D)=p(\alpha_{1}|M)$.

Figures (1)

  • Figure 1: The hierarchy shown on the left is our illustration of the existing work on the inference of propositional abstraction kido:24-1kido:24-2. The one shown on the right is an illustration of our work on the inference of predicative abstraction. The top layers are both distributions of data. The middle layer on the left is a distribution of models in propositional logic, i.e., valuations. The one on the right is a distribution of models in predicate logic, i.e., pairs of domains of discourse and valuation functions. The bottom layer on the left is a distribution of the truth values of the propositional formula, whereas the one on the right is the same type of distribution for the predicate formula.

Theorems & Definitions (28)

  • Example 1
  • Example 2
  • Definition 3
  • Example 4
  • Definition 5
  • Example 6
  • Example 7: Continued
  • Definition 8
  • Proposition 9
  • Example 10: Continued.
  • ...and 18 more