nLab affine logic

Idea

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.)

Categorical semantics

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.

Examples

Example

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).

Related concepts

• Affine BCK combinatory logic

References

• 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:

• Wikipedia, Affine logic