Algorithmic causality decision in the full λ-calculus (or prove undecidability)

Develop an algorithm that, given two β-reduction events in the full λ-calculus, determines whether one event causally precedes the other along a reduction path under the notion of causality defined for λ-calculus reductions, or prove that this decision problem is undecidable.

Background

The work develops a formal notion of events and causality for reductions in λ-calculus. For a restricted class of terms ("good" λ-expressions), the reduction relation is terminating and satisfies the diamond property, enabling a unique, homotopy-invariant causal structure on paths and a clear criterion for when one event causes another.

For general (full) λ-expressions, the paper introduces a richer labeling scheme to handle duplication during substitution and defines a reduction relation that performs one-at-a-time substitutions. While a definition of causality for successive events is provided and an induced structure for ordinary λ-calculus is discussed, an effective decision procedure for arbitrary terms is not established. The stated open problem asks for an algorithm to decide causality in the full λ-calculus, or alternatively a proof that the problem is undecidable.