- Linear logic
In

mathematical logic ,**linear logic**is a type ofsubstructural logic that denies thestructural rule s of "weakening" and "contraction". The interpretation is of "hypotheses as resources": every hypothesis must be consumed "exactly once" in a proof. This differs from usual logics such as classical orintuitionistic logic where the governing judgement is of "truth", which may be freely used as many times as necessary. To give an example, fromproposition s "A" and "A" ⇒ "B" one may conclude "A" ∧ "B" as follows:#

Modus ponens (or implication elimination) on the assumptions "A" and "A" ⇒ "B" to conclude "B".

# Conjunction of the assumption "A" and (1) to conclude "A" ∧ "B".This is often symbolically represented as a

sequent : "A", "A" ⇒ "B" Unicode|⊢ "A" ∧ "B". Both lines in the above proof "consume" the fact that "A" is true; this "freeness" of truth is usually what is desired in formal mathematics.However, truth is often too abstract or unwieldy when applied to statements about the world. For example, suppose one has a quart of

cream from which one can make a pound ofbutter . If the cream is used to make butter, then it cannot be concluded that one has both cream and butter. Yet, the logical schema outlined above leads one to conclude that`cream`,`cream`⇒`butter`Unicode|⊢`cream`∧`butter`(here,`cream`stands for the proposition "I have a quart of cream", and`butter`stands for "I have a pound of butter").The failure of ordinary logic to accurately model this activity is due to the nature of cream, butter, and "resources" in general: the quantity of resources is not a free fact to be used or disposed at will, like truth, but rather must be carefully accounted in every "state change". The accurate statement about making butter is:

: From a quart of cream and a process to convert a quart of cream into a pound of butter, one may obtain a pound of butter.

In linear logic this is written:

`cream`,`cream`Unicode|⊸`butter`Unicode|⊩`butter`, using different connectives (Unicode|⊸ instead of ⇒) and a different notion of logical entailment (Unicode|⊩ instead of Unicode|⊢).Linear logic was proposed by the French

logician Jean-Yves Girard in 1987.**Linear connectives**The

logical connective s are re-examined in this resource-interpretation; each connective splits into "multiplicative" and "additive" versions, which correspond to "simultaneous" and "alternative" presence, respectively. To motivate the connectives, let us use the example of avending machine .**Multiplicative conjunction**, also called "tensor" or "times" (written Unicode|⊗), denotes simultaneous occurrence of resources, to be used as you (the consumer) direct. For example, if you buy a stick of gum and a bottle of soft drink, then you are requesting`gum`Unicode|⊗`drink`. It is up to you in which order they appear; in fact, in principle you even have the option to mix them up. That doesn't really amount to anything when the conjuncts are atomic; but in, for example, (`gum`Unicode|⊗`drink`) Unicode|⊗`candy`, you have the power to choose the gum first, then the candy bar, and finally the drink. Indeed, Unicode|⊗ is anassociative andcommutative operation, so this is equivalent to`gum`Unicode|⊗ (`candy`Unicode|⊗`drink`). The constant 1 is used to denote the absence of any resource; it functions as a unit of tensor: "A" Unicode|⊗ 1 ≡ 1 Unicode|⊗ "A" ≡ "A".**Additive conjunction**, also called "with" (written &) represents alternative occurrence of resources, the choice of which you control. If in the vending machine there is a packet of chips, a candy bar, and a can of soft drink, all worth the same price, then for that price you can get exactly one of these products. After the purchase, you will have`candy`&`chips`&`drink`, "i.e.", exactly one of the conjuncts. You cannot use Unicode|⊗ for this outcome, because you cannot get all of these items "simultaneously" for the price of one. This operation is also both associative and commutative.Additive conjunction has a unit "top" (written Unicode|⊤, with "A" & Unicode|⊤ ≡ Unicode|⊤ & "A" ≡ "A"); it represents a lack of alternative or an inability to choose. It is often used when the exact accounting of resources is burdensome or impossible. For example, if you don't actually care what you get from the machine, or indeed whether you get anything at all, then the outcome may be expressed as Unicode|⊤. This unit can be used together with Unicode|⊗ to define a "minimal" composition of resources: if you want a candy bar at least, but possibly something else also, then the desired outcome is

`candy`Unicode|⊗ Unicode|⊤.**Multiplicative disjunction**, also called "par" or "parallelisation" (written Unicode|⅋, an upside-downampersand ) represents simultaneous occurrence of resources, to be used as the machine (the producer) directs. If there is a single button which, when pushed, will dispense both a stick of gum and a bottle of soft drink, then this may be represented as`gum`Unicode|⅋`drink`. The difference between this and the multiplicative conjunction`gum`Unicode|⊗`drink`is that now the machine chooses in what order they will be dispensed. For gum and a drink, this probably does not matter. However, for a coffee vending machine that dispenses both the coffee and the coffee cup,`cup`Unicode|⅋`coffee`and`cup`Unicode|⊗`coffee`are very different machines. As with the conjunctions, par is associative and commutative. Its unit is "bottom" (written ⊥), which stands for the empty goal: imagine pushing the "coin return" lever without inserting any money.**Additive disjunction**, also called "plus" (written ⊕) represents alternative occurrence of resources, the choice of which the machine controls. For example, suppose the vending machine permits gambling ("i.e.", "insert a dollar and win a candy bar, a soft drink, or an all-expenses-paid vacation"). Then the outcome of the purchase is`candy`⊕`drink`⊕`vacation`. You know that one of the choices will be produced, but you have no control over the result; indeed, it's consistent with this description that the machine might "never" produce a vacation. Note the difference from additive conjunction: if you have`candy`&`drink`&`vacation`, then you have the power to choose the vacation if you wish. Once again, this operation is associative and commutative. Its unit is the constant 0, which represents a lack of outcome, catastrophic failure, or inability of the machine to comply with its programming.**Linear implication**. The conjunctions and disjunctions above define the state of the world, but the description is static. For state change, linear logic defines the connective of "linear implication" (written Unicode|⊸), sometimes also known as "entails", "multimap", or "lolli" because of its lollipop-like shape. As a resource, "A" Unicode|⊸ "B" means a method to consume resource "A" to achieve resource "B". If the machine is apenny smasher , then its ability to smash a coin can be described as`penny`Unicode|⊸`smashed penny`. Note that the implication itself is a resource that must obey the principle of single consumption.**Exponential connectives**. The collection of connectives so far are excellent for describing states and transitions, but they are too weak if one needs the usual notion of "truth". This is obviously very desirable because a discussion about the actual world should not preclude standard mathematical reasoning. Linear logic uses an idea frommodal logic to embed the usual logic by means of a pair of exponential operators.

* Re-use or copying is allowed for propositions using the "of course" exponential operator (written !). Logically, two occurrences of !"A" as hypotheses may be contracted into a single occurrence. This is related to the conjunctions in that the consumer or user has the power to decide how often "A" will appear.

* The collection of goals is allowed to be extended with propositions using the "why not" operator (written ?). Logically, any fact can be weakened by including an additional conclusion ?"A". This is related to the disjunctions in that the producer or machine has the power to decide how often "A" will appear.Under the resource interpretation, ! encodes "arbitrary production" and ? encodes "arbitrary consumption".An actual vending machine could be specified as a complicated combination of the above connectives, describing all the allowed behaviours of the machine.

**Flavours of linear logic**Linear logic has many restrictions and variants. The primary axis of variation is along the classical/intuitionistic divide. "Classical linear logic" (CLL) is the original linear logic as proposed by Girard. In CLL every connective has a

dual . The following is a two-sided presentation of CLL as asequent calculus :Any proof can be transformed into one that doesn't use the cut rule.

"Linear implication" is not included in this table, but it is definable in CLL using linear negation and multiplicative disjunction: "A" Unicode|⊸ "B" ≡ "A"

^{⊥}Unicode|⅋ "B". This is familiar from other classical logics: for example, the usual implication ⇒ is similarly definable: "A" ⇒ "B" ≡ ?"A"^{⊥}Unicode|⅋ "B".Such definitions of course require a notion of "linear negation", but in classical logic one can use duals: the dual of "A", written "A"

^{⊥}is defined as follows.The logical units have similar duals; for example: Unicode|⊤^{⊥}= 0. Similarly, ! is dual to ?.The rule for duals is that an item may be moved from one side to the other by transforming it into its dual.Intuitionistic linear logic (ILL) allows only a single conclusion. Unlike CLL, connectives in ILL do not have perfect duals. Indeed, the connectives "par" and "why not" (?), and the propositional constant "bottom" (⊥), are absent in ILL because their introduction requires multiple conclusions. As a result, linear implication is a basic connective in ILL.

Other variants of linear logic variously allow or disallow certain connectives, giving rise to logics with varying complexity. The following are the most common variants.

* Multiplicative linear logic or MLL. This variant allows only the multiplicative connectives "tensor" and "par" (and their units). It is decidable, but the decision problem is

NP-complete .

* Multiplicative additive linear logic or MALL, which adds the additive connectives to MLL. This variant is also decidable with aPSPACE-complete decision problem.

* Multiplicative exponential linear logic or MELL, which is MLL plus the exponential operators. The decision problem for MELL is currently open.

* Multiplicative additive exponential linear logic or MAELL, which has all the above connectives. This variant isundecidable .

* Full Intuitionistic Linear Logic or FILL. This variant allows independent multiplicative connectives "tensor", "par" and "linear implication". This is just likeIntuitionistic logic , where one has independent connectives of conjunction, disjunction and implication.There are also first- and higher-order extensions of linear logic, but their development is standard (See

first-order logic andhigher-order logic .)The closest sub-structural cousins of linear logic are:

*

Affine logic , which extends linear logic with thestructural rule of weakening. The connectives "one" and "top" are indistinguishable in affine logic.

*Strict logic orrelevant logic , which extends linear logic with the structural rule of contraction.

*Non-commutative logic orordered logic which**removes**the structural rule of exchange from linear logic. Multiplicative conjunction divides further into a pair of "fuse"s ("left fuse" and "right fuse").**ee also***

Linear types

*Logic of unity (LU)

*Proof net s

*Geometry of interaction

*Game semantics

*Intuitionistic logic

*Computability logic

*Ludics

*Chu space s

*Uniqueness type **References*** Girard, Jean-Yves. " [

*http://iml.univ-mrs.fr/~girard/linear.pdf Linear logic*] ", Theoretical Computer Science, London Mathematical 50:1, pp. 1-102, 1987.

* Girard, Jean-Yves, Lafont, Yves, and Taylor, Paul. "Proofs and Types". Cambridge Press, 1989. (An electronic version is online at [*http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html*] .)

* Troelstra, A.S. "Lectures on Linear Logic". CSLI (Center for the Study of Language and Information) Lecture Notes No. 29. Stanford, 1992.**External links*** [

*http://www.csl.sri.com/users/lincoln/ Patrick Lincoln*] 's excellent [*http://www.csl.sri.com/~lincoln/papers/sigact92.ps Introduction to Linear Logic*] (Postscript)

* Introduction to Linear Logic by Torben Brauner [*http://www.brics.dk/LS/96/6/BRICS-LS-96-6/BRICS-LS-96-6.html*]

* A taste of linear logic by Philip Wadler [*http://homepages.inf.ed.ac.uk/wadler/topics/linear-logic.html*]

* [*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 Note, Volume 316, 2004.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Logic programming**— is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy s [1958] advice taker proposal, logic is used as a purely declarative… … Wikipedia**Logic**— For other uses, see Logic (disambiguation). Philosophy … Wikipedia**Linear temporal logic**— (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths such as that a condition will eventually be true, that a condition will be true until another fact becomes true,… … Wikipedia**Logic and the philosophy of mathematics in the nineteenth century**— John Stillwell INTRODUCTION In its history of over two thousand years, mathematics has seldom been disturbed by philosophical disputes. Ever since Plato, who is said to have put the slogan ‘Let no one who is not a geometer enter here’ over the… … History of philosophy**Linear partial information**— (LPI) is a method of making decisions based on insufficient or fuzzy information. LPI was introduced in 1970 by Polish Swiss mathematician Edward Kofler (1911 2007) to simplify decision processes. Comparing to other methods the LPI fuzziness is… … Wikipedia**Logic Pro**— Logic 8 Developer(s) Apple Inc. Stable release 9.1.5 / 2011 08 08 Operating system … Wikipedia**logic, history of**— Introduction the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic There was a medieval tradition according to which the Greek philosopher … Universalium**Linear temporal logic**— Lineare temporale Logik (LTL oder Linear temporal logic) ist ein Modell temporaler Logik mit zeitlichen Modalitäten. In LTL, können Formeln über die Zukunft von Pfaden aufgestellt werden, wie dass eine Bedingung irgendwann wahr wird, eine… … Deutsch Wikipedia**Linear extension**— In order theory, a branch of mathematics, a linear extension of a partial order is a linear order (or total order) that is compatible with the partial order. Contents 1 Definitions 2 Order extension principle 3 Related results … Wikipedia**Linear Time Temporal Logic**— Die Computation Tree Logic (kurz CTL) ist eine Temporale Logik, die speziell zur Spezifikation und Verifikation von Computersystemen dient. Meist wird sie auch mit CTL* bezeichnet. CTL bezeichnet dann eine spezielle Teilmenge der CTL* Formeln.… … Deutsch Wikipedia