[Paper Review] Finitely-valued propositional dynamic logic

Igor Sedlár|arXiv (Cornell University)|Dec 22, 2020

Logic, Reasoning, and Knowledge19 references4 citations

TL;DR

This paper introduces a finitely-valued extension of Propositional Dynamic Logic (PDL) where both formula evaluations in states and accessibility relations between states are many-valued, using finite FL-algebras. It establishes decidability and completeness for PDL over finite commutative integral FL-algebras with canonical constants, providing the first such results for non-crisp many-valued PDL with graded accessibility relations.

ABSTRACT

We study a many-valued generalization of Propositional Dynamic Logic where formulas in states and accessibility relations between states of a Kripke model are evaluated in a finite FL-algebra. One natural interpretation of this framework is related to reasoning about costs of performing structured actions. We prove that PDL over any finite FL-algebra is decidable. We also establish a general completeness result for a class of PDLs based on commutative integral FL-algebras with canonical constants.

Motivation & Objective

• To develop a many-valued generalization of PDL where both formula evaluations and accessibility between states are valued in a finite FL-algebra, enabling reasoning about costs or weights of actions.
• To overcome the limitations of classical PDL, which treats actions as crisp (yes/no) and cannot express efficiency or cost differences between actions.
• To establish general decidability and completeness results for PDL over finite FL-algebras, particularly focusing on commutative integral FL-algebras with canonical constants.
• To provide a formal framework for reasoning about weighted or costed actions in dynamic systems, such as in weighted transition systems or resource-aware computation.
• To lay the groundwork for future extensions, including the integration of test and Kleene star operators in the many-valued setting.

Proposed method

• The framework uses finite FL-algebras (residuated lattices with a distinguished 0) to interpret both state formulas and accessibility relations, allowing graded truth values and weighted transitions.
• The logic is defined using the Kleene plus operator as primitive instead of the Kleene star, due to technical issues in canonical model construction when using the star.
• Decidability is proven via a generalized filtration technique, extending the smallest filtration method to the many-valued setting with finite algebras.
• Completeness is established using a novel canonical model construction based on the greatest filtration, tailored for commutative integral FL-algebras with canonical constants.
• The approach generalizes two-valued PDL techniques but adapts them to handle non-Boolean algebras and graded accessibility relations.
• The proof relies on strong completeness and non-modal homomorphisms to show that invalid formulas can be falsified in canonical models.

Experimental results

Research questions

• RQ1 Can a many-valued PDL be developed where both formula evaluations and accessibility relations are graded using finite FL-algebras?
• RQ2 Is it possible to prove decidability for PDL over finite FL-algebras when accessibility relations are many-valued?
• RQ3 Can a general completeness result be established for PDL based on finite commutative integral FL-algebras with canonical constants?
• RQ4 How can the standard PDL operators, such as Kleene star and test, be adapted to the many-valued setting without breaking completeness or decidability?
• RQ5 What are the practical interpretations and applications of this framework in weighted systems or description logics?

Key findings

• The paper proves that PDL over any finite FL-algebra is decidable, establishing the first general decidability result for many-valued PDL with graded accessibility relations.
• A general completeness result is established for PDL based on finite commutative integral FL-algebras with canonical constants, using a novel canonical model construction.
• The use of Kleene plus instead of Kleene star is essential for the completeness proof, as the standard Kleene star leads to failures in key lemmas in canonical models.
• The test operator cannot be directly incorporated into the current framework without significant modifications to the completeness argument, as it breaks the required properties in canonical models.
• The framework provides a formal basis for reasoning about costs, efficiency, or resources in dynamic systems, such as weighted transition systems or cost-aware programs.
• The results generalize two-valued PDL techniques to the many-valued case, but require new constructions, especially for canonical models, to handle non-Boolean algebras.