[Paper Review] On Forgetting in Tractable Propositional Fragments

Yisong Wang|arXiv (Cornell University)|Feb 10, 2015

Logic, Reasoning, and Knowledge24 references3 citations

TL;DR

This paper presents a resolution-based algorithm for forgetting in tractable propositional logic fragments, proving that forgetting is categorical in Horn, renamable Horn, q-Horn, double Horn, and Krom (2-CNF) fragments. It establishes complexity results showing that reasoning problems like var-equivalence and var-independence are tractable for Krom and co-NP-complete for DNF, while Horn variants are one level below CNF in the complexity hierarchy.

ABSTRACT

Distilling from a knowledge base only the part that is relevant to a subset of alphabet, which is recognized as forgetting, has attracted extensive interests in AI community. In standard propositional logic, a general algorithm of forgetting and its computation-oriented investigation in various fragments whose satisfiability are tractable are still lacking. The paper aims at filling the gap. After exploring some basic properties of forgetting in propositional logic, we present a resolution-based algorithm of forgetting for CNF fragment, and some complexity results about forgetting in Horn, renamable Horn, q-Horn, Krom, DNF and CNF fragments of propositional logic.

Motivation & Objective

• To address the lack of a general forgetting algorithm and computation-oriented study in tractable propositional logic fragments.
• To investigate the categoricity of forgetting—whether forgetting preserves fragment membership—in key tractable fragments like Horn, Krom, and q-Horn.
• To analyze the computational complexity of key reasoning problems involving forgetting, such as var-independence, var-equivalence, and var-match.
• To provide a comprehensive complexity classification of forgetting-related problems across CNF, DNF, Horn, ren-Horn, q-Horn, and Krom fragments.

Proposed method

• Proposes a resolution-based algorithm for computing forgetting results in CNF fragments, using clause resolution to eliminate specified atoms.
• Applies syntactic transformations via replacement of forgotten atoms with ⊤ and ⊥, followed by resolution to derive the forgetting result.
• Employs model-theoretic and proof-theoretic techniques to analyze the closure properties of fragments under forgetting.
• Uses complexity-theoretic analysis to classify reasoning problems involving forgetting into P, co-NP-complete, or Π₂^P-complete classes.
• Introduces and formalizes new reasoning problems: var-weak, var-strong, var-match, and var-entailment, to study knowledge base evolution.
• Leverages known results on strongest necessary and weakest sufficient conditions to relate forgetting to definability and logical interpolation.

Experimental results

Research questions

• RQ1 Is forgetting intractable for CNF fragments, and can a resolution-based algorithm mitigate this?
• RQ2 Does forgetting preserve membership in tractable fragments such as Horn, Krom, and q-Horn?
• RQ3 What is the computational complexity of var-equivalence, var-independence, and var-match in CNF, DNF, and tractable fragments?
• RQ4 How do forgetting-related reasoning problems like var-weak and var-entailment behave in these fragments?
• RQ5 Can the forgetting process be characterized via definability concepts such as strongest necessary and weakest sufficient conditions?

Key findings

• Forgetting is categorical in Horn, renamable Horn, q-Horn, double Horn, and Krom fragments, meaning the result remains within the same fragment.
• The var-equivalence and var-independence problems are co-NP-complete for DNF and CNF fragments, while they are tractable (P) for Krom fragments.
• For Horn, ren-Horn, and q-Horn fragments, all forgetting-related reasoning problems (var-independence, var-weak, var-strong, var-match, var-entailment) are co-NP-complete, one level below CNF’s Π₂^P-completeness.
• The var-match problem is Π₂^P-complete for CNF and co-NP-complete for Horn variants, indicating that checking if a formula is a forgetting result is hard but tractable in Krom.
• The strongest and weakest definition problems are Π₂^P-complete in general, but become co-NP-complete for Horn and q-Horn formulas, and tractable for Krom formulas.
• The resolution-based algorithm, while theoretically expensive, opens avenues for heuristic optimization through atom ordering and clause selection strategies.