[Paper Review] Losing Connection:the Modal Logic of Definable Link Deletion

Dazhu Li|arXiv (Cornell University)|Nov 13, 2018

Logic, Reasoning, and Knowledge16 references4 citations

TL;DR

This paper introduces Definable Sabotage Modal Logic (SdML), a modal logic for modeling graph games where an adversarial player deletes links based on definable node properties. It presents a first-order translation, a novel bisimulation framework, and proves SdML's undecidability, lack of finite and tree model properties, and incompleteness relative to dynamic-epistemic logics.

ABSTRACT

In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics, and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed which leads to a characterization theorem for the logic as a fragment of first-order logic, and a further investigation is made of its expressive power against hybrid modal languages. Next, we discuss how to axiomatize this logic of link deletion, using dynamic-epistemic logics as a contrast. Finally, we show that our new modal logic lacks both the tree model property and the finite model property, and that its satisfiability problem is undecidable.

Motivation & Objective

• To formalize a realistic graph game where an adversary deletes links based on definable node properties, modeling real-world communication disruptions.
• To develop a new modal logic, SdML, that captures the strategic dynamics of such link deletion under definable conditions.
• To provide a first-order translation for SdML and prove its correctness, enabling model-theoretic analysis.
• To introduce a novel bisimulation notion that characterizes SdML as a fragment of first-order logic.
• To investigate the expressive power of SdML relative to hybrid and dynamic-epistemic logics, and to axiomatize it using recursion axioms in an extended hybrid framework.

Proposed method

• Proposes a new modal language with a dynamic modality $[-]$ that deletes all outgoing links from the current state to nodes satisfying a specified atomic property.
• Defines a Kripke-style semantics for SdML over directed graphs, where transitions are modified by definable link deletions.
• Constructs a non-trivial first-order translation for SdML that preserves truth across models, proving correctness via induction on formula structure.
• Introduces a new notion of bisimulation—definable sabotage bisimulation—that characterizes SdML as a fragment of first-order logic.
• Uses spy-point techniques from hybrid logic to prove undecidability, adapting methods from [13] and [2].
• Axiomatizes SdML using recursion axioms in an extended hybrid language, contrasting with standard dynamic-epistemic logic approaches.

Experimental results

Research questions

• RQ1 How can a modal logic be designed to capture local, definable link deletion in graph-based games where the adversary acts based on node properties?
• RQ2 What is the expressive power of SdML compared to standard modal logic, dynamic-epistemic logic, and hybrid logics?
• RQ3 Can a first-order translation be constructed for SdML that correctly captures its semantics?
• RQ4 Does SdML satisfy the finite model property or the tree model property, and what are the implications for decidability?
• RQ5 Is the satisfiability problem for SdML decidable, and what are the sources of its undecidability?

Key findings

• SdML is not expressible as a fragment of first-order logic invariant under standard bisimulation, but it is characterized by a novel definable sabotage bisimulation.
• The logic SdML lacks both the finite model property and the tree model property, as demonstrated via model constructions.
• The satisfiability problem for SdML is undecidable, proven using spy-point techniques adapted from hybrid logic literature.
• SdML is strictly more expressive than standard modal logic and dynamic-epistemic logics like public announcement logic, yet it is not axiomatizable via standard recursion axioms.
• The logic SdML is not equivalent to SML (sabotage modal logic), and the two differ in expressive power, with SdML capturing more complex link deletion patterns.
• A first-order translation for SdML is constructed and proven correct, providing a bridge between modal and first-order logic for this class of graph games.