Algorithm to decide causality between arbitrary events along a path in the labeled λ-calculus

Develop an algorithm that, given a reduction path p in the labeled λ-calculus (T(Λ_E, Λ_V)) under the relation →β and two events occurring on p, decides whether one event causally precedes the other according to the path-based causal relation defined for that system.

Background

To extend causality beyond the "good" λ-expressions, the paper introduces labeled terms T(Λ_E, Λ_V) and a β-reduction that substitutes one occurrence at a time while updating event and variable histories. A notion of causality for successive events is specified, and an induced causal relation for ordinary λ-calculus is described by comparing paths in the richer labeled system and the ordinary system.

Despite these constructions, the paper notes that there is no known general algorithm to determine whether two arbitrary events on a given reduction path are causally related in the labeled system. This open question targets the existence of an effective decision procedure in that setting.