[Paper Review] Common equivalence and size after forgetting

Paolo Liberatore|arXiv (Cornell University)|Jun 19, 2020

Logic, Reasoning, and Knowledge61 references4 citations

TL;DR

This paper investigates the trade-off between formula size and logical equivalence after forgetting variables in propositional logic, showing that introducing new variables can drastically reduce size while preserving common equivalence. It proves that minimizing after forgetting is polynomial-time for single-head, acyclic formulas but becomes NP-hard when new variables are allowed, with a tight linear relationship between minimal formula size and vertex cover size in a reduction to graph theory.

ABSTRACT

Forgetting variables from a propositional formula may increase its size. Introducing new variables is a way to shorten it. Both operations can be expressed in terms of common equivalence, a weakened version of equivalence. In turn, common equivalence can be expressed in terms of forgetting. An algorithm for forgetting and checking common equivalence in polynomial space is given for the Horn case; it is polynomial-time for the subclass of single-head formulae. Minimizing after forgetting is polynomial-time if the formula is also acyclic and variables cannot be introduced, NP-hard when they can.

Motivation & Objective

• To analyze how forgetting variables in propositional formulas can unexpectedly increase size, undermining its purpose of simplification.
• To investigate whether introducing new variables can reduce the size of a formula after forgetting, restoring compactness.
• To formalize the relationship between formula minimization after forgetting and computational complexity, particularly in Horn and single-head subclasses.
• To establish a tight connection between minimal common-equivalent formulas and minimum vertex covers in graphs, demonstrating complexity bounds.

Proposed method

• Uses common equivalence—a weakened form of logical equivalence that preserves consequences on shared variables—as the formalism to compare formulas after forgetting.
• Reduces the problem of minimizing a formula after forgetting to a graph-theoretic problem by constructing a formula whose minimal common-equivalent forms correspond to vertex covers in a derived graph.
• Constructs a transformation from a propositional formula to a new formula with auxiliary variables, ensuring that resolving out the new variables recovers the original formula.
• Proves that for single-head, acyclic formulas, minimizing after forgetting is solvable in polynomial time by exploiting structural constraints.
• Demonstrates that when new variables are allowed, the minimization problem becomes NP-hard by reduction from the minimum vertex cover problem.
• Establishes a linear-size correspondence between minimal common-equivalent formulas and minimum vertex covers, showing that formula size grows linearly with cover size.

Experimental results

Research questions

• RQ1 Can introducing new variables reduce the size of a formula after forgetting, despite the initial increase due to forgetting?
• RQ2 What is the computational complexity of minimizing a formula after forgetting, both when new variables are forbidden and when they are allowed?
• RQ3 Is there a structural characterization of formulas for which minimization after forgetting remains tractable?
• RQ4 How does the size of a minimal common-equivalent formula relate to combinatorial optimization problems like vertex cover?

Key findings

• Minimizing a formula after forgetting is polynomial-time solvable for single-head, acyclic formulas, due to their structural constraints.
• When new variables are not allowed, minimization after forgetting remains polynomial-time for single-head, acyclic formulas.
• When new variables are allowed, the problem becomes NP-hard, even for Horn formulas, due to a reduction to the minimum vertex cover problem.
• There exists a tight linear relationship between the size of a minimal common-equivalent formula and the size of a minimum vertex cover in a derived graph: formula size = 6|E| + 8|V| + k, where k is the cover size.
• Every minimal common-equivalent formula corresponds to a vertex cover, and every vertex cover of size k yields a common-equivalent formula of size 6|E| + 8|V| + k.
• The construction ensures that resolving out the auxiliary variables in the new formula recovers the original formula, proving common equivalence.