Structure-Preserving Uncertainty Propagation in First-Order Proof Search

2026-08-10

AI Summary

This work addresses the challenge of precisely propagating quantified uncertainty in first-order logic reasoning with non-ground clauses while preserving the original logical structure. Building upon the GK theorem prover, it introduces a structured uncertainty propagation mechanism that avoids global grounding by leveraging bounded first-order proof search and dependency graph traversal. The approach retains proof history to reconstruct premises and integrates positive and negative evidence at the level of intermediate atoms, explicitly distinguishing states of support, conflict, and ignorance. It further accommodates prioritized default rules with exception handling. To the best of our knowledge, this is the first method to enable structured uncertainty propagation within non-ground first-order proofs. Case studies and simulations reproduce benchmark results and demonstrate its comparative advantages and limitations relative to probabilistic logic programming and probabilistic answer set programming in terms of semantic expressiveness and computational capabilities.

Abstract

GK is a query-directed first-order prover that extends ordinary resolution-based proof search with explicit positive and negative claims, numerical confidence values, and prioritized default rules with exceptions. It works directly with non-ground clauses, including equality and function terms. Candidate proofs are found by bounded first-order proof search; exception conditions of defaults are checked by further bounded searches, recursively when exceptions themselves depend on defaults. This avoids requiring a finite global grounding, while allowing incomplete searches to be reported as such. This paper adds structure-preserving quantitative reporting to that framework. Retained proof histories are used in two calculations. The first reconstructs the uncertain ground premises used by each proof and computes the probability that at least one retained proof is available, without counting shared premises independently. The second resolves positive and negative support at intermediate atoms before that support is propagated through later rules; the same calculation evaluates uncertain exception conditions for individual rule applications. Reports separate positive support, negative support, conflict, and ignorance and identify detected incomplete calculations or fallbacks. The implementation performs bounded reconstruction and dependency traversal after proof search and still requires no global grounding. Analytic examples and independent simulators reproduce the reference calculations on their stated fragments. Comparisons with probabilistic logic, probabilistic ASP, default logic, and goal-directed ASP identify cases of agreement, semantic difference, unsupported translation, and incomplete computation.

Problem

• uncertainty propagation
• first-order proof search
• structure-preserving
• default rules
• incomplete reasoning

Innovation

• structure-preserving uncertainty propagation
• first-order proof search
• default reasoning with exceptions
• bounded reconstruction
• probabilistic support aggregation