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'''Linear logic''' is a [[substructural logic]] proposed by [[Jean-Yves Girard]] as a refinement of [[classical logic|classical]] and [[intuitionistic logic]], joining the [[Duality (mathematics)|dualities]] of the former with many of the [[Constructivism (mathematics)|constructive]] properties of the latter.<ref>{{cite journal|first1=Jean-Yves|last1=Girard|author1-link=Jean-Yves Girard|year=1987|title=Linear logic|journal=Theoretical Computer Science|volume=50|issue=1|pages=1–102|url=http://iml.univ-mrs.fr/~girard/linear.pdf|doi=10.1016/0304-3975(87)90045-4}}</ref>  Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as [[programming languages]], [[game semantics]], and [[quantum physics]],<ref>{{cite journal|first1=John|last1=Baez|author1-link=John Baez|first2=Mike|last2=Stay|year=2008|title=Physics, Topology, Logic and Computation: A Rosetta Stone|editor=[[Bob Coecke]]|journal=New Structures of Physics|url=http://math.ucr.edu/home/baez/rosetta.pdf}}</ref> as well as [[linguistics]],<ref>{{cite book|title=Dagstuhl Seminar 99341 on Linear Logic and Applications|first1=V.|last1=de Paiva|first2=J.|last2=van Genabith|first3=E.|last3=Ritter|year=1999|url=http://www.dagstuhl.de/Reports/99/99341.pdf}}</ref> particularly because of its emphasis on resource-boundedness, duality, and interaction.
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Linear logic lends itself to many different presentations, explanations and intuitions.
[[Proof theory|Proof-theoretically]], it derives from an analysis of classical [[sequent calculus]] in which uses of (the [[structural rule]]s) [[Idempotency of entailment|contraction]] and [[Monotonicity of entailment|weakening]] are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "truths", but also a way of manipulating ''resources'' that cannot always be duplicated or thrown away at will.  In terms of simple denotational models, linear logic may be seen as refining the interpretation of intuitionistic logic by replacing [[cartesian closed categories]] by [[symmetric monoidal categories]], or the interpretation of classical logic by replacing [[boolean algebras]] by [[C*-algebras]].
 
==Connectives, duality, and polarity==
 
===Syntax===
The language of ''classical linear logic'' (CLL) is defined inductively by the [[Backus-Naur Form|BNF notation]]
 
{| style="margin:auto"
|-
| {{math|<VAR>A</VAR>}}
| ::=
| {{math|<VAR>p</VAR> ∣ <VAR>p</VAR><sup>⊥</sup>}}
|-
|
| {{math|∣}}
| {{math| <VAR>A</VAR> {{Unicode|⊗}} <VAR>A</VAR> ∣ <VAR>A</VAR> {{Unicode|⊕}} <VAR>A</VAR>}}
|-
|
| {{math|∣}}
| {{math| <VAR>A</VAR> & <VAR>A</VAR> ∣ <VAR>A</VAR> {{Unicode|⅋}} <VAR>A</VAR>}}
|-
|
| {{math|∣}}
| {{math| 1 ∣ 0 ∣ {{Unicode|⊤}} ∣ ⊥}}
|-
|
| {{math|∣}}
| {{math| !<VAR>A</VAR> ∣ ?<VAR>A</VAR>}}
|}
 
Here {{math|<VAR>p</VAR>}} and {{math|<VAR>p</VAR><sup>⊥</sup>}} range
over [[Atomic formula|logical atoms]].  For reasons to be explained
below, the [[Logical connective|connectives]] {{Unicode|⊗}}, {{Unicode|⅋}}, 1, and
{{Unicode|⊥}} are called ''multiplicatives'', the connectives &,
{{Unicode|⊕}}, {{Unicode|⊤}}, and 0 are called ''additives'', and the
connectives ! and ? are called ''exponentials''.  We can further
employ the following terminology:
 
* {{Unicode|⊗}} is called "multiplicative conjunction" or "times" (or sometimes "tensor")
* {{Unicode|⊕}} is called "additive disjunction" or "plus"
* & is called "additive conjunction" or "with"
* {{Unicode|⅋}} is called "multiplicative disjunction" or "par"
* ! is pronounced "of course" (or sometimes "bang")
* ? is pronounced "why not"
 
Every proposition {{math|<VAR>A</VAR>}} in CLL has a '''dual''' {{math|<VAR>A</VAR><sup>⊥</sup>}}, defined as follows:
{| border="1" cellpadding="5" cellspacing="0" style="margin:auto"
|-
| colspan=3 align="center"| {{math|(<VAR>p</VAR>)<sup>⊥</sup>  {{=}}  <VAR>p</VAR><sup>⊥</sup>}}
| colspan=3 align="center"| {{math|(<VAR>p</VAR><sup>⊥</sup>)<sup>⊥</sup>  {{=}} <VAR>p</VAR>}}
|-
| colspan=3 align="center"| {{math|(<VAR>A</VAR> {{Unicode|⊗}} <VAR>B</VAR>)<sup>⊥</sup> {{=}} <VAR>A</VAR><sup>⊥</sup> {{Unicode|⅋}} <VAR>B</VAR><sup>⊥</sup>}}
| colspan=3 align="center"| {{math|(<VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>)<sup>⊥</sup> {{=}} <VAR>A</VAR><sup>⊥</sup> {{Unicode|⊗}} <VAR>B</VAR><sup>⊥</sup>}}
|-
| colspan=3 align="center"| {{math|(<VAR>A</VAR> {{Unicode|⊕}} <VAR>B</VAR>)<sup>⊥</sup> {{=}} <VAR>A</VAR><sup>⊥</sup> & <VAR>B</VAR><sup>⊥</sup>}}
| colspan=3 align="center"| {{math|(<VAR>A</VAR> & <VAR>B</VAR>)<sup>⊥</sup> {{=}} <VAR>A</VAR><sup>⊥</sup> {{Unicode|⊕}} <VAR>B</VAR><sup>⊥</sup>}}
|-
| colspan=3 align="center"| {{math|(1)<sup>⊥</sup> {{=}} ⊥}}
| colspan=3 align="center"| {{math|(⊥)<sup>⊥</sup> {{=}} 1}}
|-
| colspan=3 align="center"| {{math|(0)<sup>⊥</sup> {{=}} {{Unicode|⊤}}}}
| colspan=3 align="center"| {{math|({{Unicode|⊤}})<sup>⊥</sup> {{=}} 0}}
|-
| colspan=3 align="center"| {{math|(!<VAR>A</VAR>)<sup>⊥</sup> {{=}} ?(<VAR>A</VAR><sup>⊥</sup>)}}
| colspan=3 align="center"| {{math|(?<VAR>A</VAR>)<sup>⊥</sup> {{=}} !(<VAR>A</VAR><sup>⊥</sup>)}}
|-
|}
 
{| class="wikitable" style="float:right"
|+ Classification of connectives
|-
!    !! add  !! mul   !! exp
|-
! pos
| ⊕ 0 || ⊗ 1 || !
|-
! neg
| & ⊤ || ⅋ ⊥ || ?
|}
Observe that {{math|(-)<sup>⊥</sup>}} is an [[Involution (mathematics)|involution]], i.e., {{math|<VAR>A</VAR><sup>⊥⊥</sup> {{=}} <VAR>A</VAR>}} for all propositions. {{math|<VAR>A</VAR><sup>⊥</sup>}} is also called the ''linear negation'' of {{math|<VAR>A</VAR>}}.
 
The columns of the table suggest another way of classifying the connectives of linear logic, termed '''polarity''': the connectives negated in the left column ({{Unicode|⊗}}, {{Unicode|⊕}}, 1, 0, !) are called ''positive'', while their duals on the right ({{Unicode|⅋}}, &, ⊥, {{Unicode|⊤}}, ?) are called ''negative''; cf. table on the right.
 
''Linear implication'' is not included in the grammar of connectives, but is definable in CLL using linear negation and multiplicative disjunction, by {{math|<VAR>A</VAR> {{Unicode|⊸}} <VAR>B</VAR> :{{=}} <VAR>A</VAR><sup>⊥</sup> {{Unicode|⅋}} <VAR>B</VAR>}}. The connective {{Unicode|⊸}} is sometimes pronounced "[[lollipop]]", owing to its shape.
 
==Sequent calculus presentation==
 
One way of defining linear logic is as a [[sequent calculus]].  We use the letters {{math| &Gamma;}} and {{math| &Delta;}} to range over list of propositions {{math|<VAR>A</VAR><sub>1</sub>, ..., <VAR>A</VAR><sub>n</sub>}}, also called ''contexts''. A ''sequent'' places a context to the left and the right of the [[turnstile (symbol)|turnstile]], written {{math|&Gamma; {{tee}} &Delta;}}. Intuitively, the sequent asserts that the conjunction of {{math| &Gamma;}} [[Logical consequence|entails]] the disjunction of {{math| &Delta;}} (though we mean the "multiplicative" conjunction and disjunction, as explained below).  Girard describes classical linear logic using only ''one-sided'' sequents (where the left-hand context is empty), and we follow here that more economical presentation. This is possible because any premises to the left of a turnstile can always be moved to the other side and dualised.
 
We now give [[Sequent calculus#Inference rules|inference rules]] describing how to build proofs of sequents.<ref>Girard (1987), p.22, Def.1.15</ref>
 
First, to formalize the fact that we do not care about the order of propositions inside a context, we add the structural rule of
[[exchange rule|exchange]]:
 
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;'}}
|}
| {{math|(&Gamma;' a permutation of &Gamma;)}}
|}
 
(Alternatively we could accomplish the same thing by defining contexts to be [[multisets]] rather than lists.)
Note that we do '''not''' add the structural rules of weakening and contraction, because we do care about the
absence of propositions in a sequent, and the number of copies present.
 
Next we add ''initial sequents'' and ''cuts'':
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| &nbsp;
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} <VAR>A</VAR>, <VAR>A</VAR><sup>⊥</sup>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>}} || &nbsp;&nbsp; || &nbsp;&nbsp; || {{math|{{tee}} <VAR>A</VAR><sup>⊥</sup>, &Delta;}}
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} &Gamma;, &Delta;}}
|}
|}
 
The cut rule can be seen as a way of composing proofs, and initial sequents serve as the [[identity element|units]]
for composition.  In a certain sense these rules are redundant: as we introduce additional rules for building proofs below, we will maintain the property that arbitrary initial sequents can be derived from atomic initial sequents, and that whenever a sequent is provable it can be given a cut-free proof.  Ultimately, this [[canonical form]] property (which can be divided into the [[completeness of atomic initial sequents]] and the [[cut-elimination theorem]], inducing a notion of [[analytic proof]]) lies behind the applications of linear logic in computer science, since it allows the logic to be used in proof search and as a resource-aware lambda-calculus.
 
Now, we explain the connectives by giving ''logical rules''.  Typically in sequent calculus
one gives both "right-rules" and "left-rules" for each connective, essentially describing two modes of reasoning
about propositions involving that connective (e.g., verification and falsification).  In a one-sided presentation, one instead makes use of negation: the right-rules for a connective
(say {{Unicode|⅋}}) effectively play the role of left-rules for its dual ({{Unicode|⊗}}).  So, we should expect a certain [[Logical harmony|"harmony"]]
between the rule(s) for a connective and the rule(s) for its dual.
 
===Multiplicatives===
 
The rules for multiplicative conjunction ({{Unicode|⊗}}) and disjunction ({{Unicode|⅋}}):
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>}} || &nbsp;&nbsp; || &nbsp;&nbsp; || {{math|{{tee}} &Delta;, <VAR>B</VAR>}}
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} &Gamma;, &Delta;, <VAR>A</VAR> {{Unicode|⊗}} <VAR>B</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>, <VAR>B</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>}}
|}
|}
 
and for their units:
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| &nbsp;
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} 1}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, ⊥}}
|}
|}
 
Observe that the rules for multiplicative conjunction and disjunction are [[Admissible rule|admissible]] for
plain ''conjunction'' and ''disjunction'' under a classical interpretation
(i.e., they are admissible rules in [[Sequent calculus#The system LK|LK]]).
 
===Additives===
 
The rules for additive conjunction (&) and disjunction ({{Unicode|⊕}}) :
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>}} || &nbsp;&nbsp; || &nbsp;&nbsp; || {{math|{{tee}} &Gamma;, <VAR>B</VAR>}}
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} &Gamma;, <VAR>A</VAR> & <VAR>B</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR> {{Unicode|⊕}} <VAR>B</VAR>}}
|}
| width="25" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>B</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR> {{Unicode|⊕}} <VAR>B</VAR>}}
|}
|}
and for their units:
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| &nbsp;
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} &Gamma;, {{Unicode|⊤}}}}
|}
| width="50" |
| style="text-align: center;" | (no rule for {{math|0}})
|}
 
Observe that the rules for additive conjunction and disjunction are again admissible
under a classical interpretation.  But now we can explain the basis for the multiplicative/additive distinction
in the rules for the two different versions of conjunction: for the multiplicative connective ({{Unicode|⊗}}),
the context of the conclusion ({{math|&Gamma;, &Delta;}}) is split up between the premises, whereas
for the additive case connective (&) the context of the conclusion ({{math|&Gamma;}}) is carried whole into both
premises.
 
===Exponentials===
 
The exponentials are used to give controlled access to weakening and contraction.  Specifically, we add
structural rules of weakening and contraction for  ?'d propositions:<ref>Girard (1987), p.25-26, Def.1.21</ref>
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, ?<VAR>A</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, ?<VAR>A</VAR>, ?<VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, ?<VAR>A</VAR>}}
|}
|}
and use the following logical rules:
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} ?&Gamma;, <VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} ?&Gamma;, !<VAR>A</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} &Gamma;, <VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} &Gamma;, ?<VAR>A</VAR>}}
|}
|}
One might observe that the rules for the exponentials follow a different pattern from the rules for the other connectives,
and that there is no longer such a clear symmetry between the duals ! and ?.  This situation is remedied in alternative
presentations of CLL (e.g., the [[Logic of unity|LU]] presentation).
 
==Remarkable formulae==
 
In addition to the [[De Morgan's laws|De Morgan dualities]] described above, some important equivalences in linear logic include:
 
; Distributivity : <math>A\otimes(B\oplus C)\equiv(A\otimes B)\oplus(A\otimes C)</math>
; Exponential isomorphism : <math>\,!(A \& B)\equiv \,!A \otimes \,!B</math>
 
(Here <math>A\equiv B\quad=\quad(A\multimap B)\&(B\multimap A)</math>.)
 
The following is not in general an equivalence, only an implication:{{clarify|reason=What does the curly 'p' mean here? Moreover, in Girard (1987), p.50, semi-distributivity is mentioned, but the expressions look different from here.|date=September 2013}}
 
; Semi-distributivity : <math>(A\otimes(B\,\wp\,C))\multimap((A\otimes B)\,\wp\,C)</math>
 
==Encoding classical/intuitionistic logic in linear logic==
 
Both intuitionistic and classical implication can be recovered from linear implication by inserting exponentials: intuitionistic implication is encoded as {{math|!<VAR>A</VAR> {{Unicode|⊸}} <VAR>B</VAR>}}, and classical implication as {{math|!<VAR>A</VAR> {{Unicode|⊸}} ?<VAR>B</VAR>}}.<ref>[http://www.dicosmo.org/CourseNotes/LinLog/], chapter 2</ref> The idea is that exponentials allow us to use a formula as many times as we need, which is always possible in classical and intuitionistic logic.
 
Formally, there exists a translation of formulae of intuitionistic logic to formulae of linear logic in a way which guarantees that the original formula is provable in intuitionistic logic if and only if the translated formula is provable in linear logic. Using the [[Gödel–Gentzen negative translation]], we can thus embed classical first-order logic into linear first-order logic.
 
==The resource interpretation==
 
Lafont (1993) first showed how intuitionistic linear logic can be explained as a logic of resources, so providing the logical language with access to formalisms that can be used for reasoning about resources within the logic itself, rather than, as in classical logic, by means of non-logical predicates and relations.  [[Antony Hoare]] (1985)'s classical example of the vending machine can be used to illustrate this idea.
 
Suppose we represent a candy bar by the atomic proposition {{math|<VAR>candy</VAR>}}, and a dollar by {{math|<VAR>$1</VAR>}}.  To state the fact that a dollar will buy you one candy bar, we might write the implication {{math|<VAR>$1</VAR> &rArr; <VAR>candy</VAR>}}.  But in ordinary (classical or intuitionistic) logic, from {{math|<VAR>A</VAR>}} and {{math|<VAR>A</VAR> &rArr; <VAR>B</VAR>}} one can conclude {{math|<VAR>A</VAR> &and; <VAR>B</VAR>}}.  So, ordinary logic leads us to believe that we can buy the candy bar and keep our dollar!  Of course,
we can avoid this problem by using more sophisticated encodings, although typically such encodings suffer from the [[frame problem]]. However, the rejection of weakening and contraction allows linear logic to avoid this kind of spurious reasoning even with the "naive" rule. Rather than {{math|<VAR>$1</VAR> &rArr; <VAR>candy</VAR>}}, we express the property of the vending machine as a ''linear'' implication {{math|<VAR>$1</VAR>  {{Unicode|⊸}} <VAR>candy</VAR>}}.  From {{math|<VAR>$1</VAR>}} and this fact, we can conclude {{math|<VAR>candy</VAR>}}, but ''not'' {{math|<VAR>$1</VAR> {{Unicode|⊗}} <VAR>candy</VAR>}}.  In general, we can use the linear logic proposition {{math|<VAR>A</VAR> {{Unicode|⊸}} <VAR>B</VAR>}} to express the validity of transforming resource {{math|<VAR>A</VAR>}} into resource {{math|<VAR>B</VAR>}}.
 
Running with the example of the vending machine, let us consider the "resource interpretations" of the other multiplicative and additive connectives.  (The exponentials provide the means to combine this resource interpretation with the usual notion of persistent [[logical truth]].)
 
Multiplicative conjunction {{math|(<VAR>A</VAR> {{Unicode|⊗}} <VAR>B</VAR>)}} denotes simultaneous occurrence of resources, to be used as the consumer directs. For example, if you buy a stick of gum and a bottle of soft drink, then you are requesting {{math|<VAR>gum</VAR> {{Unicode|⊗}} <VAR>drink</VAR>}}. The constant 1 denotes the absence of any resource, and so functions as the unit of {{Unicode|⊗}}.
 
Additive conjunction {{math|(<VAR>A</VAR> & <VAR>B</VAR>)}} represents alternative occurrence of resources, the choice of which the consumer controls. If in the vending machine there is a packet of chips, a candy bar, and a can of soft drink, each costing one dollar, then for that price you can buy exactly one of these products.  Thus we write {{math|<VAR>$1</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> & <VAR>chips</VAR> & <VAR>drink</VAR>)}}.  We do ''not'' write {{math|<VAR>$1</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> {{Unicode|⊗}} <VAR>chips</VAR> {{Unicode|⊗}} <VAR>drink</VAR>)}}, which would imply that one dollar suffices for buying all three products together.  However, from  {{math|<VAR>$1</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> & <VAR>chips</VAR> & <VAR>drink</VAR>)}}, we can correctly deduce {{math|<VAR>$3</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> {{Unicode|⊗}} <VAR>chips</VAR> {{Unicode|⊗}} <VAR>drink</VAR>)}}, where {{math|<VAR>$3</VAR> :{{=}} <VAR>$1</VAR> {{Unicode|⊗}} <VAR>$1</VAR> {{Unicode|⊗}} <VAR>$1</VAR>}}.  The unit {{Unicode|⊤}} of additive conjunction can be seen as a wastebasket <!--or [[Garbage collection (computer science)|garbage collector]]--> for irrelevant alternatives. For example, we can write {{math|<VAR>$3</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> {{Unicode|⊗}} {{Unicode|⊤}})}} to express that three dollars will buy you a candy bar and something else (we don't care what).
 
Additive disjunction {{math|(<VAR>A</VAR> {{Unicode|⊕}} <VAR>B</VAR>)}} represents alternative occurrence of resources, the choice of which the machine controls.  For example, suppose the vending machine permits gambling: insert a dollar and the machine may dispense a candy bar, a packet of chips, or a soft drink.  We can express this situation as {{math|<VAR>$1</VAR> {{Unicode|⊸}} (<VAR>candy</VAR> {{Unicode|⊕}} <VAR>chips</VAR> {{Unicode|⊕}} <VAR>drink</VAR>)}}.  The constant 0 represents a product that cannot be made, and thus serves as the unit of {{Unicode|⊕}} (a machine that might produce {{math|<VAR>A</VAR>}} or {{math|0}} is as good as a machine that always produces {{math|<VAR>A</VAR>}} because it will never succeed in producing a 0).
 
Multiplicative disjunction {{math|(<VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>)}} is more difficult to gloss in terms of the resource interpretation, although we can encode back into linear implication, either as {{math|<VAR>A</VAR><sup>⊥</sup> {{Unicode|⊸}} <VAR>B</VAR>}} or {{math|<VAR>B</VAR><sup>⊥</sup> {{Unicode|⊸}} <VAR>A</VAR>}}.
 
== Other proof systems ==
 
=== Proof nets ===
{{main|Proof net}}
Introduced by [[Jean-Yves Girard]], proof nets have been created to avoid the ''bureaucracy'', that is all the things that make two derivations different in the logical point of view, but not in a "moral" point of view.
 
For instance, these two proofs are "morally" identical:
{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} <VAR>A</VAR>, <VAR>B</VAR>, <VAR>C</VAR>, <VAR>D</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} <VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>, <VAR>C</VAR>, <VAR>D</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} <VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>, <VAR>C</VAR> {{Unicode|⅋}} <VAR>D</VAR>}}
|}
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} <VAR>A</VAR>, <VAR>B</VAR>, <VAR>C</VAR>, <VAR>D</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} <VAR>A</VAR>, <VAR>B</VAR>, <VAR>C</VAR> {{Unicode|⅋}} <VAR>D</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} <VAR>A</VAR> {{Unicode|⅋}} <VAR>B</VAR>, <VAR>C</VAR> {{Unicode|⅋}} <VAR>D</VAR>}}
|}
|}
 
The goal of proof nets is to make them identical by creating a graphical representation of them.
 
== Semantics ==
 
=== Algebraic semantics ===
{{see also|Quantale}}
{{Empty section|date=April 2011}}
 
==Decidability/complexity of entailment==
 
The [[entailment]] relation in full CLL is [[Undecidable problem|undecidable]].<ref>For this and the below complexity results, see: {{cite journal|first1=Patrick|last1=Lincoln|first2=John|last2=Mitchell|first3=Andre|last3=Scedrov|first4=Natarajan|last4=Shankar|year=1992|title=Decision Problems for Propositional Linear Logic|journal=Annals of Pure and Applied Logic|volume=56|pages=239–311|doi=10.1016/0168-0072(92)90075-B}}</ref> Fragments of
CLL are often considered, for which the decision problem is more subtle:
 
* Multiplicative linear logic (MLL): only the multiplicative connectives.  MLL entailment is [[NP-complete]].
* Multiplicative-additive linear logic (MALL): only multiplicatives and additives (i.e., exponential-free).  MALL entailment is [[PSPACE-complete]].
* Multiplicative-exponential linear logic (MELL): only multiplicatives and exponentials.  The decidability of MELL entailment is currently open.
 
==Variants of linear logic==
 
Many variations of linear logic arise by further tinkering with the structural rules:
 
* [[Affine logic]], which forbids contraction but allows global weakening.
* [[Strict logic]] or [[relevant logic]], which forbids weakening but allows global contraction.
* [[Noncommutative logic|Non-commutative logic]] or ordered logic, which removes the rule of exchange, in addition to barring weakening and contraction.  In ordered logic, linear implication divides further into left-implication and right-implication.
 
Different intuitionistic variants of linear logic have been considered. When based on a single-conclusion sequent calculus presentation, like in ILL (Intuitionistic Linear Logic), the connectives {{Unicode|⅋}}, ⊥, and ? are absent, and linear implication is treated as a primitive connective. In FILL (Full Intuitionistic Linear Logic) the connectives {{Unicode|⅋}}, ⊥, and ? are present, linear implication is a primitive connective and, similarly to what happens in intuitionistic logic, all connectives (except linear negation) are independent.
There are also first- and higher-order extensions of linear logic, whose formal development is somewhat standard (see [[first-order logic]] and [[higher-order logic]]).
 
==See also==
{{Portal|Logic}}
 
* [[Linear type system]], a [[substructural type system]]
* [[Logic of unity]] (LU)
* [[Proof net]]s
* [[Geometry of interaction]]
* [[Game semantics]]
* [[Intuitionistic logic]]
* [[Computability logic]]
* [[Ludics]]
* [[Chu space]]s
* [[Uniqueness type]]
 
==References==
{{Reflist}}
 
==Further reading==
* Girard, Jean-Yves. ''[http://iml.univ-mrs.fr/~girard/linear.pdf Linear logic]'', Theoretical Computer Science, Vol 50, no 1, pp.&nbsp;1–102, 1987.
* Girard, Jean-Yves, Lafont, Yves, and Taylor, Paul.  ''[http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html Proofs and Types]''.  Cambridge Press, 1989.
* Hoare, C. A. R., 1985.  ''Communicating Sequential Processes''.  Prentice-Hall International. ISBN 0-13-153271-5
* Lafont, Yves, 1993.  ''Introduction to Linear Logic''.  Lecture notes from TEMPUS Summer School on ''Algebraic and Categorical Methods in Computer Science'', Brno, Czech Republic.
* [[A. S. Troelstra|Troelstra, A.S.]] ''Lectures on Linear Logic''.  CSLI (Center for the Study of Language and Information) Lecture Notes No. 29. Stanford, 1992.
* [[A. S. Troelstra]], H. Schwichtenberg (1996). ''Basic Proof Theory''.  In series ''Cambridge Tracts in Theoretical Computer Science'', Cambridge University Press, ISBN 0-521-77911-1.
* Di Cosmo, Roberto, and Danos, Vincent. ''[http://www.dicosmo.org/CourseNotes/LinLog/ The linear logic primer]''.
* [http://www.csl.sri.com/~lincoln/papers/sigact92.ps Introduction to Linear Logic] (Postscript) by [http://www.csl.sri.com/users/lincoln/ Patrick Lincoln]
* [http://www.brics.dk/LS/96/6/BRICS-LS-96-6/BRICS-LS-96-6.html Introduction to Linear Logic] by Torben Brauner
* [http://homepages.inf.ed.ac.uk/wadler/topics/linear-logic.html A taste of linear logic] by Philip Wadler
* [http://plato.stanford.edu/archives/fall2006/entries/logic-linear/ Linear Logic] by [http://www.pps.jussieu.fr/~dicosmo/index.html.en Roberto Di Cosmo] and [http://www.lix.polytechnique.fr/Labo/Dale.Miller/ Dale Miller]. The Stanford Encyclopedia of Philosophy  (Fall 2006 Edition), Edward N. Zalta (ed.).
* [http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/llp.pdf Overview of linear logic programming] by [http://www.lix.polytechnique.fr/Labo/Dale.Miller/ Dale Miller]. In ''Linear Logic in Computer Science'', edited by Ehrhard, Girard, Ruet, and Scott. Cambridge University Press. London Mathematical Society Lecture Notes, Volume 316, 2004.
 
{{Logic}}
 
{{DEFAULTSORT:Linear Logic}}
[[Category:Substructural logic]]
[[Category:Non-classical logic]]
[[Category:Linear logic| ]]

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