Local Dependence and Guarding

TL;DR

This paper explores the logical properties of Local Dependence Logic (LFD), establishing its connections with the guarded fragment of first-order logic (GF), and transferring complexity and model-theoretic results between them.

Contribution

It provides a new translation between GF and LFD, enabling the transfer of known results and deepening understanding of their relationship and computational properties.

Findings

Established a translation from GF to LFD and vice versa.

Transferred complexity bounds and model properties between GF and LFD.

Demonstrated the entanglement of local dependence and guarding in logical frameworks.

Abstract

We study LFD, a base logic of functional dependence introduced by Baltag and van Benthem (2021) and its connections with the guarded fragment GF of first-order logic. Like other logics of dependence, the semantics of LFD uses teams: sets of permissible variable assignments. What sets LFD apart is its ability to express local dependence between variables and local dependence of statements on variables. Known features of LFD include decidability, explicit axiomatization, finite model property, and a bisimulation characterization. Others, including the complexity of satisfiability, remained open so far. More generally, what has been lacking is a good understanding of what makes the LFD approach to dependence computationally well-behaved, and how it relates to other decidable logics. In particular, how do allowing variable dependencies and guarding quantifiers compare as logical devices? We provide a new compositional translation from GF into LFD, and conversely, we translate LFD into GF in an `almost compositional' manner. Using these two translations, we transfer known results about GF to LFD in a uniform manner, yielding, e.g., tight complexity bounds for LFD satisfiability, as well as Craig interpolation. Conversely, e.g., the finite model property of LFD transfers to GF. Thus, local dependence and guarding turn out to be intricately entangled notions.