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Re: Affine
From: Vladimir Alexiev <vladimir@cs.ualberta.ca>
Date: Thu, 26 Jun 1997 10:21:08 -0600
I think the original question is still not answered: why is
affine logic called affine? I would be very disappointed if the
answer is "because you obtain it by loosening up linear logic a
bit, just like you obtain affine geometry by loosening up
linear geometry a bit". I'd imagine that the *character* of the
two "loosening-ups" should be similar.
Affine logic is linear logic with weakening. Weakening may be
understood as a form of translation. The algebraic rationale for
rejecting weakening is that the signature contains a constant which is
preserved by homorphisms. Weakening is refuted in this way by many
closed categories: pointed sets, abelian groups, finite dimensional
vector spaces, etc. The binary operation of the latter two also
refutes contraction, which however is ok for pointed sets. Affine
algebra/geometry is the other way round, having a binary operation that
refutes contraction but no constant refuting weakening (affine
geometry's missing origin).
>From the perspective of Chu spaces over K, weakening asserts the
continuity of the unique function
A -> 1
which is equivalent to the requirement that A\perp contain all K
constants. For K >= 2 this implies that A contains no constants.
Vaughan Pratt
References:
- Re: Affine
- From: Vladimir Alexiev <vladimir@cs.ualberta.ca>