# 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.

Chain-eye last edited by PJTraill on November 19, 2019 - 22:57