basic constructions:
strong axioms
further
(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Linear logic is sometimes thought of as being a logic for arguing about resource sensitive issues, but it can also be thought of categorically, or interpreted using Game Semantics, or as being related to Petri nets, or as a particular form of quantum logic. A bit more formally:
Linear logic is a substructural logic in which the contraction rule and the weakening rule are omitted, or at least have their applicability restricted.
In the original definition of (Girard 87) linear logic is the internal logic of/has categorical semantics in star-autonomous categories (Seely 89, prop. 1.5). But more generally linear logic came to refer to the internal logic of any possibly-non-cartesian symmetric closed monoidal category (then usually called multiplicative intuitionistic linear logic – MILL) or even polycategory (Szabo 78 see the history section and see also de Paiva 89, Blute 91, Benton-Bierman-de Paiva-Hyland 92, Hyland-de Paiva 93, Bierman 95, Barber 97, Schalk 04, Melliès 09). Under this interpretation, proof nets (or the associated Kelly-Mac Lane graphs) of linear logic are similar to string diagrams for monoidal categories.
Indeed, these more general senses of linear logic still faithfully follow the original motivation for the term “linear” as connoting “resource availability” explained below (and in Girard 87), since the non-cartesianness of the tensor product means the absence of a diagonal map and hence the impossibility of functions to depend on more than a single (linear) copy of their variables.
In addition to these usual “denotational” categorical semantics, linear logic also has an “operational” categorical semantics, called the Geometry of Interaction, in traced monoidal categories.
Although linear logic is traditionally presented in terms of inference rules, it was apparently discovered by Girard while studying coherent spaces.
Linear logic and linear type theory can be argued to be the proper incarnation of quantum logic (see there for references). In this context the linearity of the logic, hence the absence of diagonal maps in its categorical semantics (in non-cartesian symmetric monoidal categories) reflects the no-cloning theorem of quantum physics.
Unlike traditional logics, which deal with the truth of propositions, linear logic is often described as dealing with the availability of resources. A proposition, if it is true, remains true no matter how we use that fact in proving other propositions. By contrast, in using a resource $A$ to make available a resource $B$, $A$ itself may be consumed or otherwise modified. Linear logic deals with this by restricting our ability to duplicate or discard resources freely. For example, we have
from which we can prove
which by left contraction (duplication of inputs) in classical logic yields
Linear logic would disallow the contraction step and treat $\text{have cake},\; \text{have cake} \vdash A$ as explicitly meaning that two slices of cake yield $A$. Disallowing contraction then corresponds to the fact that we can’t turn one slice of cake into two (more’s the pity), so you can't have your cake and eat it too.
Linear logic can also be interpreted using a semantics of “games” or “interactions”. Under this interpretation, each proposition in a sequent represents a game being played or a transaction protocol being executed. An assertion of, for instance,
means roughly that if I am playing three simultaneous games of $P$, $Q$, and $R$, in which I am the left player in $P$ and $Q$ and the right player in $R$, then I have a strategy which will enable me to win at least one of them. Now the above statements about “resources” translate into saying that I have to play in all the games I am given and can’t invent new ones on the fly.
Linear logic is closely related to notions of relevant logic, which have been studied for much longer. The goal of relevant logic is to disallow statements like “if pigs can fly, then grass is green” which are true, under the usual logical interpretation of implication, but in which the hypothesis has nothing to do with the conclusion. Clearly there is a relationship with the “resource semantics”: if we want to require that all hypotheses are “used” in a proof then we need to disallow weakening.
Linear logic is usually given in terms of sequent calculus. There is a set of propositions (although as remarked above, to be thought of more as resources to be acquired than as statements to be proved) which we construct through recursion. Each pair of lists of propositions is a sequent (written as usual with ‘$\vdash$’ between the lists), some of which are valid; we determine which are valid also through recursion. Technically, the propositional calculus of linear logic also requires a set of propositional variables from which to start; this is usually identified with the set of natural numbers (so the variables are $p_0$, $p_1$, etc), although one can also consider the linear logic $LL(V)$ where $V$ is any initial set of propositional variables.
Here we define the set of propositions:
The terms “exponential”, “multiplicative”, and “additive” come from the fact that “exponentiation converts addition to multiplication”: we have $!{(A \& B)}\equiv !{A} \otimes !{B}$ and so on (see below).
However, the connectives and constants can also be grouped in different ways. For instance, the multiplicative conjunction $\otimes$ and additive disjunction $\oplus$ are both positive types, while the additive conjunction $\&$ and multiplicative disjunction $\parr$ are negative types. Similarly, the multiplicative truth $\mathbf{1}$ and the additive falsity $\mathbf{0}$ are positive, while the additive truth $\top$ and multiplicative falsity $\bot$ are negative. This grouping has the advantage that the similarity of symbols matches the adjective used.
But conversely, the natural grouping by multiplicative/additive, or equivalently by de Morgan dual pairs, has led many authors to alter Girard’s notation, in particular reverting to the category-theoretic $\times$ and $+$ for the additives $\&$ and $\oplus$, and introducing a different symbol such as $\odot$, $\bullet$ or (confusingly) $\oplus$ for Girard’s $\parr$. But on this page we will stick to Girard’s conventions for consistency.
In relevant logic, the terms “conjunction” and “disjunction” are often reserved for the additive versions $\&$ and $\oplus$, which are written with the traditional notations $\wedge$ and $\vee$. In this case, the multiplicative conjunction $A\otimes B$ is called fusion and denoted $A\circ B$, while the multiplicative disjunction $A\parr B$ is called fission and denoted $A+B$ (or sometimes, confusingly, $A\oplus B$). In relevant logic the symbol $\bot$ may also be used for the additive falsity, here denoted $\mathbf{0}$. Also, sometimes the additive connectives are called extensional and the multiplicatives intensional.
Sometimes one does not define the operation of negation, defining only $p^\perp$ for a propositional variable $p$. It is a theorem that every proposition above is equivalent (in the sense defined below) to a proposition in which negation is applied only to propositional variables.
We now define the valid sequents, where we write $A, B, C \vdash D, E, F$ to state the validity of the sequent consisting of the list $(A,B,C)$ and the list $(D,E,F)$ and use Greek letters for sublists:
The main point of linear logic is the restricted use of the weakening and contraction rules; if these were universally valid (applying to any $A$ rather than only to $!{A}$ or $?{A}$), then the additive and multiplicative operations would be equivalent (in the sense defined below) and similarly $!{A}$ and $?{A}$ would be equivalent to $A$, which would give us classical logic. On the other hand, one can also remove the exchange rule to get a variety of noncommutative logic?; one must then be careful about how to write the other rules (which we have been above).
As usual, there is a theorem of cut elimination showing that the cut rule and identity rule follow from all other rules and the special cases of the identity rule of the form $p \vdash p$ for a propositional variable $p$.
The propositions $A$ and $B$ are (propositionally) equivalent if $A \vdash B$ and $B \vdash A$ are both valid, which we express by writing $A \equiv B$. It is then a theorem that either may be swapped for the other anywhere in a sequent without affecting its validity. Up to equivalence, negation is an involution, and the operations $\&$, $\oplus$, $\otimes$, and $\parr$ are all associative, with respective identity elements $\top$, $\mathbf{0}$, $\mathbf{1}$, and $\bot$. These operations are also commutative (although this fails for the multiplicative connectives if we drop the exchange rule). The additive connectives are also idempotent (but the multiplicative ones are not).
There is a more refined notion of equivalence, where we pay attention to specific derivations $\pi: A \vdash B$ of sequents, and deem two derivations $\pi, \pi'$ of $A \vdash B$ Lambek-equivalent if they map to the same morphism under any categorical semantics $S$; see below. Given a pair of derivations $\pi: A \vdash B$ and $\rho: B \vdash A$, it then makes sense to ask whether they are Lambek-inverse to one another (i.e., whether $S(\rho) = S(\pi)^{-1}$ under any semantics), so that the derivations exhibit an isomorphism $S(A) \cong S(B)$ under any semantics $S$. This equivalence relation $A \equiv_{Lambek} B$ is strictly stronger than propositional equivalence. It should be observed that all equivalences $A \equiv B$ listed below are in fact Lambek equivalences.
We also have distributive laws that explain the adjectives ‘additive’, ‘multiplicative’, and ‘exponential’:
It is also a theorem that negation (except for the negations of propositional variables) can be defined (up to equivalence) recursively as follows:
The logical rules for negation can then be proved.
In this way, linear logic in the original sense (interpreted in star-autonomous categories) has a perfect de Morgan duality. But observe that more general variants (interpreted in more general symmetric monoidal categories) need not, see for instance (Hyland-de Paiva 93).
We can also restrict attention to sequents with one term on either side as follows: $\Gamma \vdash \Delta$ is valid if and only if $\bigotimes \Gamma \vdash \parr \Delta$ is valid, where $\bigotimes(A, B, C) \coloneqq A \otimes B \otimes C$, etc, and similarly for $\parr$ (using implicitly that these are associative, with identity elements to handle the empty sequence).
We can even restrict attention to sequents with no term on the left side and one term on the right; $A \vdash B$ is valid if and only if $\vdash A \multimap B$ is valid, where $A \multimap B \coloneqq A^\perp \parr B$. In this way, it's possible to ignore sequents entirely and speak only of propositions and valid propositions, eliminating half of the logical rules in the process. However, this approach is not as beautifully symmetric as the full sequent calculus.
The logic described above is full classical linear logic. There are many important fragments and variants of linear logic, such as:
multiplicative linear logic (MLL), which contains only $\otimes,\parr$ and their units $\mathbf{1},\bot$ as well as the negation $(-)^\perp$.
multiplicative-exponential linear logic (MELL), which contains only $\otimes,\parr,\mathbf{1},\bot,(-)^\perp$ and the exponential modalities $!,?$.
multiplicative-additive linear logic (MALL), which contains everything except the exponential modalities $!,?$.
multiplicative intuitionistic linear logic (MILL), which contains only $\otimes,\mathbf{1},\multimap$ (the latter now as a primitive operation); in particular there is no longer the involutive negation $(-)^\perp$. The sequents are also restricted to have only one formula on the right.
intuitionistic linear logic (ILL), which contains all the additive connectives $\&,\oplus,\mathbf{0},\top$ as well as the intutionistic multiplicatives $\otimes,\mathbf{1},\multimap$ and the exponential $!$, with one formula on the right as above. In this case all connectives are all independent of each other.
full intuitionistic linear logic (FILL), which in addition to ILL contains the multiplicative disjunction $\parr$, and perhaps the exponential $?$. (Sometimes ILL without $\parr,?$ is also called “full” intuitionistic linear logic.)
non-commutative linear logics (braided or not)
“light” and “soft” linear logics, which limit the use of ! to constrain the computational complexity of proofs
first-order linear logic, which adds quantifiers $\exists$ and $\forall$ (sometimes denoted $\bigvee$ and $\bigwedge$), either over a fixed domain or over varying types. These quantifiers are usually considered “additive”; for a theory that has a certain kind of “multiplicative quantifier” see bunched implication.
One can also consider adding additional rules to linear logic. For instance, by adding the weakening rule (but not the contraction rule) one obtains affine logic, whereas by adding contraction but not weakening one obtains relevance logic. Another rule that is sometimes considered is the mix rule.
We discuss the categorical semantics of linear logic. See also at relation between type theory and category theory.
One way to explain linear logic to a category theorist is to say that its models are *-autonomous categories with extra structure (Seely, 1989, prop. 1.5). (If the underlying category is a suplattice then these are commutative quantales, (Yetter 90))
Firstly, there is a monoidal ‘tensor’ connective $A \otimes B$. Negation $A^\bot$ is modelled by the dual object involution $(-)^*$, while linear implication $A\multimap B$ corresponds to the internal hom, which can be defined as $(A\otimes B^\bot)^\bot$. There is a de Morgan dual of the tensor called ‘par’, with $A \parr B = (A^\bot\otimes B^\bot)^\bot$. Tensor and par are the ‘multiplicative’ connectives, which roughly speaking represent the parallel availability of resources.
The ‘additive’ connectives $\&$ and $\oplus$, which correspond in another way to traditional conjunction and disjunction, are modelled as usual by products and coproducts. Seely (1989) notes that products are sufficient, as $*$-autonomy then guarantees the existence of coproducts; that is, they are also linked by de Morgan duality.
Recall also that linear logic recaptures the notion of a resource that can be discarded or copied arbitrarily by the use of the modal operator $!$ the !-modality: $!A$ denotes an ‘$A$-factory’, a resource that can produce zero or more $A$s on demand. It is modelled using a suitably monoidal comonad $!$ on the underlying $*$-autonomous category. There are various inequivalent ways to make this precise, however; see !-modality for discussion.
An LL sequent
is interpreted as a morphism
The comonoid structure on $!A$ then yields the weakening
and contraction
maps. The corresponding rules are interpreted by precomposing the interpretation of a sequent with one of these maps.
The (co)-Kleisli category of $!$ is cartesian closed, and the product there coincides with the product in the base category. The exponential (unsurprisingly for a Kleisli category) is $B^A \cong !A\multimap B$.
Particular monoidal and $*$-autonomous posets for modeling linear logic can be obtained by Day convolution from ternary frames. This includes Girard’s phase spaces as a particular example.
First-order linear logic is correspondingly modeled in a linear hyperdoctrine.
A different way to explain linear logic categorically (though equivalent, in the end) is to start with a categorical structure which lacks any of the connectives, but has sufficient structure to enable us to characterize them with universal properties. If we ignore the exponentials for now, such a structure is given by a polycategory. The polymorphisms
in a polycategory correspond to sequents $A,B,C \vdash D,E,F$ in linear logic. The multiplicative connectives are then characterized by representability and corepresentability properties:
and
(Actually, we should allow arbitrarily many unrelated objects to carry through in both cases.) The additives are similarly characterized as categorical products and coproducts, in a polycategorically suitable sense.
Finally, dual objects can be recovered as a sort of “adjoint”:
If all these representing objects exist, then we recover a $*$-autonomous category.
One merit of the polycategory approach is that it makes the role of the structural rules clearer, and also helps explain why $\parr$ sometimes seems like a disjunction and sometimes like a conjunction. Allowing contraction and weakening on the left corresponds to our polycategory being “left cartesian”; that is, we have “diagonal” and “projection” operations such as $Hom(A,A; B) \to Hom(A;B)$ and $Hom(;B) \to Hom(A,B)$. In the presence of these operations, a representing object is automatically a cartesian product; thus $\otimes$ coincides with $\&$. Similarly, allowing contraction and weakening on the right makes the polycategory “right cocartesian”, which causes corepresenting objects to be coproducts and thus $\parr$ to coincide with $\oplus$.
On the other hand, if we allow “multi-composition” in our polycategory, i.e. we can compose a morphism $A \to (B,C)$ with one $(B,C)\to D$ to obtain a morphism $A\to D$, then our polycategory becomes a PROP, and representing and corepresenting objects must coincide; thus $\otimes$ and $\parr$ become the same. This explains why $\parr$ has both a disjunctive and a conjunctive aspect. Of course, if in addition to multi-composition we have the left and right cartesian properties, then all four connectives coincide (including the categorical product and coproduct) and we have an additive category.
We can interpret any proposition in linear logic as a game between two players: we and they. The overall rules are perfectly symmetric between us and them, although no individual game is. At any given moment in a game, exactly one of these four situations obtains: it is our turn, it is their turn, we have won, or they have won; the last two states continue forever afterwards (and the game is over). If it is our turn (or if they have won), then they are winning; if it is their turn (or if we have won), then we are winning. So there are two ways to win: because the game is over (and a winner has been decided), or because it is forever the other players’ turn (either because the other players have no move or because every move results in the other players’ turn again).
This is a little complicated, but it's important in order to be able to distinguish the four constants:
The binary operators show how to combine two games into a larger game:
So we can classify things as follows:
To further clarify the difference between $\top$ and $\mathbf{1}$ (the additive and multiplicative versions of truth, both of which we win); consider $A \parr \top$ and $A \parr \mathbf{1}$. In $A \parr \top$, it is always their move (since it is their move in $\top$, hence their move in at least one game), so we win just as we win $\top$. (In fact, $A \parr \top \equiv \top$.) However, in $A \parr \mathbf{1}$, the game $\mathbf{1}$ ends immediately, so play continues as in $A$. We have won $\mathbf{1}$, so we only have to end the game to win overall, but there is no guarantee that this will happen. Indeed, in $\mathbf{0} \parr \mathbf{1}$, the game never ends and it is always our turn, so they win. (In $\bot \parr \mathbf{1}$, both games end immediately, and we win. In $A \otimes \mathbf{1}$, we must win both games to win overall, so this reduces to $A$; indeed, $A \otimes \mathbf{1} \equiv A$.)
Negation is easy:
There are several ways to think of the exponentials. As before, they have control in a conjunction, while we have control in a disjunction. Whoever has control of $!{A}$ or $?{A}$ chooses how many copies of $A$ to play and must win them all to win overall. There are many variations on whether the player in control can spawn new copies of $A$ or close old copies of $A$ prematurely, and whether the other player can play different moves in different copies (whenever the player in control plays the same moves).
Other than the decisions made by the player in control of a game, all moves are made by transmitting resources. Ultimately, these come down to the propositional variables; in the game $p$, we must transmit a $p$ to them, while they must transmit a $p$ to us in $p^\perp$.
A game is valid if we have a strategy to win (whether by putting the game in a state where we have won or by guaranteeing that it is forever their turn). The soundness and completeness of this interpretation is the theorem that $A$ is a valid game if and only if $\vdash A$ is a valid sequent. (Recall that all questions of validity of sequents can be reduced to the validity of single propositions.)
Game semantics for linear logic was first proposed by Andreas Blass, I believe in Blass (1992). The semantics here may or may not be the same as proposed by Blass.
Much as there are many exponential functions (say from $\mathbb{R}$ to $\mathbb{R}$), even though there is only one addition operation and one multiplication operation, so there can be many versions of the exponential operators $!$ and $?$. (However, there doesn't seem to be any analogue of the logarithm to convert between them.)
More precisely, if we add to the language of linear logic two more operators, $!'$ and $?'$, and postulate of them the same rules as for $!$ and $?$, we cannot prove that $!{A} \equiv !'{A}$ and $?{A} \equiv ?'{A}$. In contrast, if we introduce $\&'$, $\bot'$, etc, we can prove that the new operators are equivalent to the old ones.
In terms of the categorial interpretation above, there may be many comonads $!$; it is not determined by the underlying $*$-autonomous category. In terms of game/resource semantics, there are several slightly different interpretations of the exponentials.
One sometimes thinks of the exponentials as coming from infinitary applications of the other operations. For example:
All of these justify the rules for the exponentials, so again we see that there may be many ways to satisfy these rules.
The original article is
Reviews include
Jean-Yves Girard, Linear logic, its syntax and semantics (pdf)
Jean-Yves Girard, part III of Lectures on Logic, European Mathematical Society 2011
Daniel Murfet, Logic and linear algebra: an introduction (arXiv:1407.2650)
The categorical semantics of linear logic in star-autonomous categories originally appeared in
and for the special case of quantales in
The more general case of of multiplicative intuitionistic linear logic interpreted more generally in symmetric monoidal categories was systematically developed in
(that was before Girard introduced the “linear” terminology).
More recent articles exploring this include
This is reviewed/further discussed in
Richard Blute, Philip Scott, Category theory for linear logicians, 2004 (citeseer)
Paul-André Melliès , Categorial Semantics of Linear Logic, in Interactive models of computation and program behaviour, Panoramas et synthèses 27, 2009 (pdf)
Noncommutative linear logic is discussed for instance in
Further discussion of linear type theory is for instance in
Capter 7, Linear type theory pdf
Anders Schack-Nielsen, Carsten Schürmann, Linear contextual modal type theory pdf
See also
Discussion of application of linear logic to quantum logic, quantum computing and generally to quantum physics includes
Vaughan Pratt, Linear logic for generalized quantum mechanics, in Proc. of Workshop on Physics and Computation (PhysComp’92) (pdf, web)
Ross Duncan, Types for quantum mechanics, 2006 (pdf)
Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, section 9 of Quantum Logic and Nonclassical Logics, p. 127 in Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, 2009 North Holland
Ugo Dal Lago, Claudia Faggian, On Multiplicative Linear Logic, Modality and Quantum Circuits (arXiv:1210.0613)
Discussion for matrix factorization in Landau-Ginzburg models is in
Discussion of linear type theory without units is in
Discussion of inductive types in the context of linear type theory is in