Soft linear logic is a version of linear logic in which the exponential modality is weakened. This has as a particular consequence that the only “definable functions” (in various senses) are computable in polynomial time?.
The usual rules of the exponential modality $!$ in (intuitionistic) linear logic are promotion, dereliction, contraction, and weakening:
Soft linear logic is obtained by observing that these rules are equivalent to soft promotion, multiplexing, and digging:
and then removing digging. Thus, the only rules governing $!$ in soft linear logic are soft promotion and multiplexing.
Inspecting the rules for the soft $!$, we see that soft promotion says that $!$ is a lax monoidal functor (or equivalently a functor between multicategories), while multiplexing says that we have a (presumably natural) transformation $!A \to A^{\otimes n}$ for all $n$. On a non-thin category one would presumably ask for some coherence axioms on these transformations.
Intriguingly, however, the naive formula for a “cofree comonoid” on an object $A$ in a symmetric monoidal category with countable products:
though it does not in general define a comonoid at all, is a lax monoidal functor with transformations $!A \to A^{\otimes n}$. The latter are just the product projections, while the lax monoidal structure $!A \otimes !B \to !(A\otimes B) = \prod_n (A\otimes B)^{\otimes n}$ has components
In particular, therefore, any symmetric monoidal poset with countable meets admits a modality of this form satisfying the rules of the soft exponential. For a non-thin category, one could take a further limit in an attempt to impose whatever axioms are desired of the structure on $!$. For instance, if instead of $A^{\otimes n}$ we use an appropriate equalizer, we could force permutation-invariance of the transformations $!A \to A^{\otimes n}$.
Yves Lafont, Soft linear logic and polynomial time, Theoretical Computer Science, Volume 318, Issues 1–2, 2004, Pages 163-180, doi, ps
Marco Gaboardi, Jean-Yves Marion, Simona Ronchi Della Rocca, Soft Linear Logic and Polynomial Complexity Classes, Electronic Notes in Theoretical Computer Science, Volume 205, 2008, Pages 67-87, doi
Richard McKinley, Soft Linear Set Theory, The Journal of Logic and Algebraic Programming, Volume 76, Issue 2, 2008, doi
Last revised on July 3, 2018 at 17:45:42. See the history of this page for a list of all contributions to it.