Relations and signalling strengths

The contexts included in the supremum are part of the declaration. An influence relation for one fixed context and the union of influence witnesses over several contexts are different objects. A diamond norm is used only after a complete channel reduction has been defined. It is not assigned without justification to an arbitrary multi-time probability functional Watrous (Chapters 2–3).

Direct influence is not assumed to be transitive or acyclic. A future order extraction must explicitly:

1. prove the needed order properties; or

2. specify an order-forming operation and prove what it represents.

For an acyclic directed relation, reflexive transitive closure gives a partial order. For a relation with cycles, quotienting strongly connected components gives an order of components, not an injective representation of the original events. Such a quotient must not be left implicit.

In complete definite-order channel networks, influence is bounded by scheduled ancestry. Equality requires more than the existence of a wire: a replacement channel may ignore that wire entirely CheungVIIJ,CheungXXIIA.