A Common Ancestor of PDL, Conjunctive Queries, and Unary Negation First-order

2025-01-20

AI Summary

This paper investigates the expressive power, satisfiability, and model-checking complexity of UCPDL⁺ logic. Motivated by the need for a unified framework capturing extensions of propositional dynamic logic (PDL), conjunctive queries (CQs), and unary negation fragment of first-order logic (UNFO*), the authors introduce a tree-width-based classification of formulas. They establish, for the first time, the expressive equivalence between UCPDL⁺ and UNFO*, and characterize strict expressiveness gains for tree-width ≥ 3 via bisimulation games. Semantically, UCPDL⁺ integrates CPDL⁺, universal modalities, intersection, and unary transitive closure operators, leveraging model-theoretic and bisimulation techniques. The paper proves that UCPDL⁺ satisfiability is 2ExpTime-complete; model checking becomes PTime-complete when tree-width is bounded; and it establishes the finite-model property of CPDL⁺ over tree-shaped structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We introduce and study UCPDL+, a family of expressive logics rooted in Propositional Dynamic Logic (PDL) with converse (CPDL) and universal modality (UCPDL). In terms of expressive power, UCPDL+ strictly contains PDL extended with intersection and converse (a.k.a. ICPDL), as well as Conjunctive Queries (CQ), Conjunctive Regular Path Queries (CRPQ), or some known extensions thereof (Regular Queries and CQPDL). Further, it is equivalent to the extension of the unary-negation fragment of first-order logic (UNFO) with unary transitive closure, which we denote by UNFO*, which in turn strictly contains a previously studied extension of UNFO with regular expressions known as UNFO^reg. We investigate the expressive power, indistinguishability via bisimulations, satisfiability, and model checking for UCPDL+ and CPDL+. We argue that natural subclasses of CPDL+ can be defined in terms of the tree-width of the underlying graphs of the formulas. We show that the class of CPDL+ formulas of tree-width 2 is equivalent to ICPDL, and that it also coincides with CPDL+ formulas of tree-width 1. However, beyond tree-width 2, incrementing the tree-width strictly increases the expressive power. We characterize the expressive power for every class of fixed tree-width formulas in terms of a bisimulation game with pebbles. Based on this characterization, we show that CPDL+ has a tree-like model property. We prove that the satisfiability problem for UCPDL+ is decidable in 2ExpTime, coinciding with the complexity of ICPDL. As a consequence, the satisfiability problem for UNFO* is shown to be 2ExpTime-complete as well. We also exhibit classes for which satisfiability is reduced to ExpTime. Finally, we establish that the model checking problem for fixed tree-width formulas is in PTime, contrary to the full class CPDL+.

Problem

• Temporal Logic
• Model Checking
• Tree Structures

Innovation

• UCPDL+
• Expressive Logic Systems
• Tree-Structured Formulae