A Simple Logic of Functional Dependence

Abstract

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.

Finite Model Property and Bisimulation for LFD

“…where X ⊆ V is a finite set of variables, y ∈ V an individual variable, P ∈ τ a predicate symbol and x = (x 1 , ..., x n ) ∈ V ar(P) a finite string of variables. 2 Fixing notation, for any Y ⊆ V , we write s |= D X Y if s |= D X y holds for all y ∈ Y . We also skip the set brackets for singletons, writing D x Y for D {x} Y , and D x y for D {x} {y}.…”

“…The question whether LFD has the finite model property (FMP) w.r.t. dependence models remained an open problem [2]. Our main result is that LFD has the FMP, by a new application of Herwig's theorem on extending partial isomorphisms.…”

“…Many interesting notions of dependence (such as lineair dependence in vector spaces, temporal dependence in dynamical systems and strategic interaction in a multi-player game) can be formalized in LFD [2]. Moreover, LFD invites a natural epistemic interpretation where (sets of) variables may represent (groups of) agents, (joint) questions or objects.…”

“…Moreover, LFD invites a natural epistemic interpretation where (sets of) variables may represent (groups of) agents, (joint) questions or objects. [2] The dependence modalities then capture distributed knowledge or the interrogative modality, while the dependence atoms capture epistemic superiority or inquisitive implication (or other 'mixed' notions). More spectacularly, [3] introduces a complete and decidable dynamic-epistemic logic based on LFD with so-called 'reading events' as well as a notion of 'common distributed knowledge' that combines features of common knowledge and distributed knowledge.…”

Intelligence in Strategic Games

“…Nodes of the tree are labeled with maximal consistent sets and edges are labeled with coalitions. This construction has been used in logics of distributed knowledge before (Naumov & Tao, 2017a, 2017b, 2018c, 2018b, 2018aWáng &Ågotnes, 2020;Baltag & van Benthem, 2021).…”