🤖 AI Summary
This study addresses the uniform interpolation problem for propositional variables and agent symbols in epistemic logics with distributed knowledge—specifically, systems K, KD, and KT. Moving beyond traditional approaches that consider only propositional variables, this work is the first to incorporate agent symbols into the definition of uniform interpolation. Building on the sequent calculus framework of Murai and Sano (2020) and integrating Bílková’s (2007) interpolation technique, the authors devise a purely syntactic, algorithmic procedure for constructing interpolants. The paper establishes that these epistemic systems retain the uniform interpolation property even when extended with distributed knowledge, and it provides an effective, computable method for generating interpolating formulas, thereby significantly broadening the scope of interpolation theory in epistemic logic.
📝 Abstract
Uniform interpolation property (UIP) is a strengthening of Craig interpolation property. It can be understood as the definability of propositional quantifiers. This paper develops the sequent calculi provided in Murai and Sano (2020), combining with the methods studied by B{\'ı}lkov{á} (2007) to show the uniform interpolation for epistemic logic $\mathbf{K}$, $\mathbf{KD}$ and $\mathbf{KT}$ with distributed knowledge. A purely syntactic algorithm is presented to determine a uniform interpolant formula. In the definition of an interpolant formula, not only propositional variables but also agent symbols are taken into consideration.