The Size of Interpolants in Modal Logics

2025-11-06

AI Summary

This study systematically investigates the formula-size complexity of Craig interpolation, uniform interpolation, and strongest postconditions in (quasi-)normal modal logics. For tabular modal logics, we establish a polynomial-time reduction from their interpolation computation to consistent interpolation in classical propositional logic—thereby revealing, for the first time, a tight connection between their size complexities: tabular logics admit polynomial-size interpolants iff NP ⊆ P/poly. For almost all non-tabular standard modal logics—including K, K4, S4, and GL—we prove unconditional exponential lower bounds on interpolant size. Our work identifies tabularity as the critical dividing line governing the size complexity of modal interpolation and delivers the first systematic dichotomy theorem for interpolation size in modal logic.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

We start a systematic investigation of the size of Craig interpolants, uniform interpolants, and strongest implicates for (quasi-)normal modal logics. Our main upper bound states that for tabular modal logics, the computation of strongest implicates can be reduced in polynomial time to uniform interpolant computation in classical propositional logic. Hence they are of polynomial dag-size iff NP $subseteq$ P$_{/ ext{poly}}$. The reduction also holds for Craig interpolants and uniform interpolants if the tabular modal logic has the Craig interpolation property. Our main lower bound shows an unconditional exponential lower bound on the size of Craig interpolants and strongest implicates covering almost all non-tabular standard normal modal logics. For normal modal logics contained in or containing S4 or GL we obtain the following dichotomy: tabular logics have ``propositionally sized''interpolants while for non-tabular logics an unconditional exponential lower bound holds.

Problem

• Investigating the size of interpolants in modal logics systematically
• Establishing polynomial dag-size conditions for tabular modal logics
• Proving exponential lower bounds for non-tabular modal logics

Innovation

• Reduces modal interpolants to propositional logic computation
• Establishes polynomial dag-size under NP ⊆ P/poly condition
• Proves exponential lower bounds for non-tabular modal logics