nLab
!-modality

Idea

In full linear logic/linear type theory there is assumed a (comonadic) modality denoted “!” and called the exponential modality, whose role is, roughly, to give linear types also a non-linear interpretation. This is also called the “of course”-modality or the storage modality, and sometimes the “bang”-operation.

In classical linear logic (meaning with involutive de Morgan duality), the de Morgan dual of “!” is denoted “?” and called the “why not”-modality.

In categorical semantics of linear type theory the !-modality typically appears as a kind of Fock space construction. If one views linear logic as quantum logic (as discussed there), then this means that the !-modality produces free second quantization.

Categorical semantics

Everyone agrees that ! should be a comonad (and ? should be a monad), but there are different ways to proceed from there. The goal is to capture the syntactic rules allowing assumptions of the form $!A$ to be duplicated and discarded. The original definition from Seely was:

Definition

Let $C$ be an $\ast$-autonomous category with cartesian products. A Seely !-modality on $C$ is a comonad that is a strong monoidal functor from the cartesian monoidal structure to the $\ast$-autonomous monoidal structure, i.e. we have $!(A\times B)\cong !A \otimes !B$ coherently. (There is also a coherence axiom that should be imposed; see Mellies, section 7.3.)

(Note that in linear logic, the cartesian monoidal structure $\times$ is sometimes denoted by $\&$.) This implies that the Kleisli category of ! is a cartesian closed category, which is a categorical version of the translation of intuitionistic logic into linear logic.

Of course, the above definition depends on the existence of the cartesian product, and relies on the self-duality of an $\ast$-autonomous category to derive the rules for ? from the rules for !. A different definition that doesn’t require the existence of $\times$ was given by Benton, Bierman, de Paiva, and Hyland:

Definition

Let $C$ be a closed symmetric monoidal category; a !-modality on $C$ is a lax monoidal comonad such that every !-coalgebra naturally carries the structure of a comonoid object in the category of coalgebras, such that coalgebra maps are comonoid maps.

This definition implies that the category of all !-coalgebras (not just the free ones, i.e. its Kleisli category) is cartesian closed. Note that for a comonad on a poset, every coalgebra is free; thus the world of pure propositional “logic” doesn’t tell us whether to consider the Kleisli category or the Eilenberg-Moore category for the translation. A more even-handed approach is the following (see Mellies):

Definition

A linear-nonlinear adjunction is a monoidal adjunction $F : M \rightleftarrows L : G$ in which $L$ is symmetric monoidal and $M$ is cartesian monoidal. The induced !-modality is the comonad $F G$ on $L$.

This includes both of the previous definitions where $M$ is taken respectively to be the Kleisli category or the Eilenberg-Moore category of !.

On the other hand, the last two definitions are given only for “intuitionistic” linear logic, though in the $\ast$-autonomous case one could derive a ? from the ! by de Morgan duality. A definition not requiring the de Morgan duality and describing ! and ? together was given by Blute, Cockett, and Seely:

Definition

Let $C$ be a linearly distributive category with tensor product $\otimes$ and cotensor product $\parr$. A (!,?)-modality on $C$ consists of:

1. a $\otimes$-monoidal comonad ! and a $\parr$-comonoidal monad ?
2. ? is a !-strong monad, and ! is a ∞-strong comonad
3. all free !-coalgebras are naturally commutative $\otimes$-comonoids, and all free ∞-algebras are naturally commutative $\parr$-monoids.

Here a functor $F$ is strong with respect to a lax monoidal functor $G$ if there is a natural transformation $F A \otimes G B \to F(A\otimes G B)$ satisfying some natural axioms, and we similarly require compatibility of the monad and comonad structure transformations.

Modal term calculi

Girard’s original presentation of linear logic involved rules that explicitly assumed the presence of $!$ on hypotheses or on entire contexts, such as weakening and contraction:

$$\frac{\Gamma \vdash B}{\Gamma, !A\vdash B} \qquad \frac{\Gamma,!A,!A \vdash B}{\Gamma, !A\vdash B}$$

and “promotion”:

$$\frac{!\Gamma \vdash A}{!\Gamma\vdash !A}$$

If this is translated into a natural deduction style term calculus, the resulting rules are more complicated than those of most type formers. This can be avoided using adjoint type theory with two context zones, one “nonlinear” one where contraction and weakening are permitted (and admissible) and one “linear” one where they are not, with $!$ as a modality relating the two zones.

Such a “modal” presentation of linear logic was first introduced by Girard in his work on LU? and then developed by a number of other people such as Plotkin, Wadler, Benton, and Barber. See the references for details.

This presentation also generalizes naturally to dependent linear type theory, with the nonlinear type theory being dependent, and the linear types depending on the nonlinear ones but nothing depending on linear types. In this context, the $!$-modality decomposes into “context extension” and a “dependent sum”.

Related concepts

• exponential map

• multiplicative conjunction

• linear implication

References

The semantics of ! as a comonad is discussed in:

• R. A. G. Seely, Linear logic, $\ast$-autonomous categories and cofree coalgebras, Contemporary Mathematics 92, 1989. (pdf, ps.gz)

• Nick Benton, Gavin Bierman, Valeria de Paiva, Martin Hyland, Linear lambda-Calculus and Categorical Models Revisited (1992), citeseer

• R. F. Blute , J. R. B. Cockett and R. A. G. Seely. ! and ? – Storage as tensorial strength doi, pdf

• Paul-André Melliès, Categorical semantics of linear logic, 2009. pdf

• Martin Hyland and Andreas Schalk?, Glueing and orthogonality for models of linear logic, pdf

The relation to Fock space is discussed in:

• Richard Blute, Prakash Panangaden, R. A. G. Seely, Fock Space: A Model of Linear Exponential Types (1994) (web, pdf)

• Marcelo Fiore, Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 (pdf)

• Jamie Vicary, A categorical framework for the quantum harmonic oscillator, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 (arXiv:0706.0711)

  (in the context of finite quantum mechanics in terms of dagger-compact categories)

The interpretation of $\Omega^\infty \Sigma^\infty_+$ as an exponential in the context of Goodwillie calculus is due to

• Gregory Arone, Marja Kankaanrinta, The Goodwillie tower of the identity is a logarithm, 1995 (web)

based on

• Gregory Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, 1998 (pdf)

The modal approach to a term calculus for the $!$-modality can be found in:

• Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201-217, 1993.

• G. Plotkin. Type theory and recursion. In Proceedings of the Eigth Symposium of Logic in Computer Science, Montreal , page 374. IEEE Computer Society Press, 1993.

• N. Benton. A mixed linear and non-linear logic; proofs, terms and models. In Proceedings of Computer Science Logic ‘94, number 933 in LNCS. Verlag, June 1995.

• Philip Wadler. A syntax for linear logic. In Ninth International Coference on the Mathematical Foundations of Programming Semantics , volume 802 of LNCS . Springer Verlag, April 1993

• Andrew Barber, Dual Intuitionistic Linear Logic, Technical Report ECS-LFCS-96-347, University of Edinburgh, Edinburgh (1996), web