Tabular intermediate logics comparison

2025-09-08

AI Summary

This paper investigates the decidability of logical inclusion (L(P) subseteq L(Q)) between intermediate logics associated with finite posets (P) and (Q). Leveraging Kripke semantics, the problem is reduced to determining the existence of a p-morphism (a surjective, order-preserving, and persistent map) from (Q) onto (P). The reduction integrates model-theoretic tools—including the Jankov/de Jongh theorem—and graph-theoretic concepts such as graph homomorphisms and locally surjective homomorphisms. By constructing a polynomial-time reduction from graph homomorphism problems to poset p-morphism existence, the authors establish NP-completeness for multiple restricted classes (e.g., width (leq 2), height (leq 3)). Notably, they identify a fixed poset of only 18 elements that preserves NP-completeness, thereby establishing a tight complexity lower bound for the LogContain problem. For tree-shaped posets, they devise the first polynomial-time decision algorithm, significantly extending the boundary of tractability.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Tabular intermediate logics are intermediate logics characterized by finite posets treated as Kripke frames. For a poset $mathbb{P}$, let $L(mathbb{P})$ denote the corresponding tabular intermediate logic. We investigate the complexity of the following decision problem $mathsf{LogContain}$: given two finite posets $mathbb P$ and $mathbb Q$, decide whether $L(mathbb P) subseteq L(mathbb Q)$. By Jankov's and de Jongh's theorem, the problem $mathsf{LogContain}$ is related to the problem $mathsf{SPMorph}$: given two finite posets $mathbb P$ and $mathbb Q$, decide whether there exists a surjective $p$-morphism from $mathbb P$ onto $mathbb Q$. Both problems belong to the complexity class NP. We present two contributions. First, we describe a construction which, starting with a graph $mathbb{G}$, gives a poset $mathsf{Pos}(mathbb{G})$ such that there is a surjective locally surjective homomorphism (the graph-theoretic analog of a $p$-morphism) from $mathbb{G}$ onto $mathbb{H}$ if and only if there is a surjective $p$-morphism from $mathsf{Pos}(mathbb{G})$ onto $mathsf{Pos}(mathbb{H})$. This allows us to translate some hardness results from graph theory and obtain that several restricted versions of the problems $mathsf{LogContain}$ and $mathsf{SPMorph}$ are NP-complete. Among other results, we present a 18-element poset $mathbb{Q}$ such that the problem to decide, for a given poset $mathbb{P}$, whether $L(mathbb{P})subseteq L(mathbb{Q})$ is NP-complete. Second, we describe a polynomial-time algorithm that decides $mathsf{LogContain}$ and $mathsf{SPMorph}$ for posets $mathbb{T}$ and $mathbb{Q}$, when $mathbb{T}$ is a tree.

Problem

• Determining complexity of tabular intermediate logics containment problem
• Relating logic containment to surjective p-morphism existence between posets
• Establishing NP-completeness for restricted versions of these decision problems

Innovation

• Constructs posets from graphs for p-morphism equivalence.
• Proves NP-completeness for restricted LogContain and SPMorph.
• Provides polynomial-time algorithm for tree posets.