Chain-eye

This page defines an alternative terminology for a concept used in Benson's Definition of Unconditional Life. For further details, see that article.

Definition

A “chain-eye within a group” is a term proposed^[1] for the sort of “eye” every chain needs two of in Benson's Definition of Unconditional Life:

Given a chain ``C`` (i.e. a maximal set of connected stones of the same colour) which is one of the chains constituting a group ``G`` (i.e. a collection of chains of the same colour),

a chain-eye ``E`` of ``C`` within the group ``G`` is a chain (i.e. maximal connected set) of vacant points and possibly opposing stones, all of which are adjacent to ``C`` and none of which are adjacent to any point not in ``G`` or ``E``.

Significance

Given this definition, Benson's Theorem states that:

a collection of stones of one colour is pass-alive (cannot be captured however often the opponent moves)

if and only if

it is part of a group ``G`` every one of whose chains has at least two chain-eyes within ``G``.

Benson calls this latter condition “Unconditional Life” and calls a chain-eye of ``C`` in ``G`` a “small Black-” (or “White-”) “enclosed region of ``G`` vital to ``C``”.

Rationale

[1] I (Patrick Traill) propose this terminology because it seems to me:

• much closer to the normal term “eye”,
• considerably shorter than Benson’s,
• easier to link natural-sounding wiki text to a page called Chain-eye than to one called Small vital enclosed region.