basic constructions:
strong axioms
further
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
In formal logic, affine logic refers to substructural logics which omit the contraction rule but retain the weakening rule. Conversely, affine logic is linear logic adjoined with the weakening rule.
In terms of categorical semantics, affine logic is modeled by (symmetric) monoidal categories whose tensor unit $I$ is terminal, also known as semicartesian monoidal categories, an example of which is the category of affine spaces, whence the name.
(In contrast, the category of linear spaces, ie. vector spaces, serves as categorical semantics for the multiplicative fragment of linear logic [eg. Murfet 2014], where also the weakening rule is dropped.)
One might imagine that a more general notion of categorical semantics would be given by monoidal categories equipped with a natural (in $A$) family $A\to I$ of morphisms implementing weakening for each object. However, such an interpretation is in general badly behaved, unless one additionally requires these natural transformations to be monoidal, but it can be shown that this additional requirement already forces the tensor unit to be terminal (specifically, this follows from the component at $I$ being the identity).
An example of the badly behaved case – where the transformation is not monoidal, and the tensorial unit is not terminal – is given by the category Rel of relations, with cartesian product as tensor product (i.e., with $Rel$ as cartesian bicategory). Here a natural family of relations $A\to I$ is given by picking empty relations everywhere. In the corresponding interpretation of affine logic, any weakening yields an empty relation, which contradicts intuitive principles like for example that “adding a dummy variable to a proof and then substituting a closed term” should not change the semantics.
The substructural part of many forms of bunched logic are affine instead of linear, sometimes inadvertently, ultimately due to former technical problems with formulating dependent linear types (see review in Riley 2022 §1.7.2).
Affine BCK combinatory logic
Vaughan Pratt, Re: Affine, mailing list discussion (1997) [web]
Alexei P. Kopylov, Decidability of Linear Affine Logic, Information and Computation 164 1 (2001) 173-198 [doi:10.1006/inco.1999.2834]
Gianluigi Bellin , Two Paradigms of Logical Computation in Affine Logic?, in: Logic for Concurrency and Synchronisation, Trends in Logic 15 (2003) [doi:10.1007/0-306-48088-3_3]
Michael Shulman, Affine logic for constructive mathematics, Bulletin of Symbolic Logic 28 3 (2022) 327-386 [arXiv:1805.07518, doi:10.1017/bsl.2022.28]
See also:
Last revised on September 6, 2024 at 12:24:45. See the history of this page for a list of all contributions to it.