soft linear logic

Soft linear logic


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:

!ΓA!Γ!A(prom)Γ,ACΓ,!AC(der)Γ,!A,!ACΓ,!AC(ctr)ΓCΓ,!AC(wk)\frac{!\Gamma\vdash A}{!\Gamma\vdash !A}(prom) \qquad \frac{\Gamma,A \vdash C}{\Gamma,!A\vdash C}(der) \qquad \frac{\Gamma,!A,!A\vdash C}{\Gamma,!A\vdash C}(ctr) \qquad \frac{\Gamma\vdash C}{\Gamma,!A\vdash C}(wk)

Soft linear logic is obtained by observing that these rules are equivalent to soft promotion, multiplexing, and digging:

ΓA!Γ!A(sp)Γ,A,,A nCΓ,!AC(mpx)Γ,!!ACΓ,!AC(dig)\frac{\Gamma\vdash A}{!\Gamma\vdash !A}(sp) \qquad \frac{\Gamma,\overbrace{A,\dots,A}^n\vdash C}{\Gamma,!A\vdash C}(mpx) \qquad \frac{\Gamma,!!A\vdash C}{\Gamma,!A\vdash C}(dig)

and then removing digging. Thus, the only rules governing !! in soft linear logic are soft promotion and multiplexing.

Categorical semantics

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 !AA n!A \to A^{\otimes n} for all nn. 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 AA in a symmetric monoidal category with countable products:

!A= nA n, !A = \prod_{n\in\mathbb{N}} A^{\otimes n},

though it does not in general define a comonoid at all, is a lax monoidal functor with transformations !AA n!A \to A^{\otimes n}. The latter are just the product projections, while the lax monoidal structure !A!B!(AB)= n(AB) n!A \otimes !B \to !(A\otimes B) = \prod_n (A\otimes B)^{\otimes n} has components

!A!B=( nA n)( nB n)A nB n(AB) n. !A \otimes !B = \left(\prod_n A^{\otimes n}\right)\otimes \left(\prod_n B^{\otimes n}\right) \to A^{\otimes n} \otimes B^{\otimes n} \xrightarrow{\cong} (A\otimes B)^{\otimes n}.

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 nA^{\otimes n} we use an appropriate equalizer, we could force permutation-invariance of the transformations !AA n!A \to A^{\otimes n}.

  • computational complexity?
    • implicit computational complexity?
    • polynomial time?
  • light linear logic?
  • naive set theory?
  • ultrafinitism


  • 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 13:45:42. See the history of this page for a list of all contributions to it.