Linear logic

Template:Short description Template:Use dmy dates Template:Redirect Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter.Template:Sfn 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 (because linear logic can be seen as the logic of quantum information theory),Template:Sfn as well as linguistics,Template:Sfn particularly because of its emphasis on resource-boundedness, duality, and interaction.

Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and 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 (closed) categories, or the interpretation of classical logic by replacing Boolean algebras by C*-algebras.Template:Citation needed

Connectives, duality, and polarityEdit

SyntaxEdit

Template:Anchor The language of classical linear logic (CLL) is defined inductively by the BNF notation

Template:Math	::=	Template:Math
	Template:Math	Template:Math
	Template:Math	Template:Math
	Template:Math	Template:Math
	Template:Math	Template:Math

Here Template:Math and Template:Math range over logical atoms. For reasons to be explained below, the connectives ⊗, ⅋, 1, and ⊥ are called multiplicatives, the connectives &, ⊕, ⊤, and 0 are called additives, and the connectives ! and ? are called exponentials. We can further employ the following terminology:


Symbol	Name
⊗	multiplicative conjunction	times	tensor
⊕	additive disjunction	plus
&	additive conjunction	with
Template:Anchor⅋	multiplicative disjunction	par
!	of course	bang
?	why not	quest

Binary connectives ⊗, ⊕, & and ⅋ are associative and commutative; 1 is the unit for ⊗, 0 is the unit for ⊕, ⊥ is the unit for ⅋ and ⊤ is the unit for &.

Every proposition Template:Math in CLL has a dual Template:Math, defined as follows:

Template:Math			Template:Math
Template:Math			Template:Math
Template:Math			Template:Math
Template:Math			Template:Math
Template:Math			Template:Math
Template:Math			Template:Math

Classification of connectives
	add	mul	exp
pos	⊕ 0	⊗ 1	!
neg	& ⊤	⅋ ⊥	?

Observe that Template:Math is an involution, i.e., Template:Math for all propositions. Template:Math is also called the linear negation of Template:Math.

The columns of the table suggest another way of classifying the connectives of linear logic, termed Template:Em: the connectives negated in the left column (⊗, ⊕, 1, 0, !) are called positive, while their duals on the right (⅋, &, ⊥, ⊤, ?) are called negative; cf. table on the right.

Template:Em is not included in the grammar of connectives, but is definable in CLL using linear negation and multiplicative disjunction, by Template:Math. The connective ⊸ is sometimes pronounced "lollipop", owing to its shape.

Sequent calculus presentationEdit

One way of defining linear logic is as a sequent calculus. We use the letters Template:Math and Template:Math to range over lists of propositions Template:Math, also called contexts. A sequent places a context to the left and the right of the turnstile, written Template:Math. Intuitively, the sequent asserts that the conjunction of Template:Math entails the disjunction of Template:Math (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 inference rules describing how to build proofs of sequents.Template:Sfn

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:

Template:Math

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:

Template:Math

Template:Math			Template:Math

Template:Math

The cut rule can be seen as a way of composing proofs, and initial sequents serve as the 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 ⅋) effectively play the role of left-rules for its dual (⊗). So, we should expect a certain "harmony" between the rule(s) for a connective and the rule(s) for its dual.

MultiplicativesEdit

The rules for multiplicative conjunction (⊗) and disjunction (⅋):

Template:Math			Template:Math

Template:Math

Template:Math

and for their units:



Template:Math

Template:Math

Observe that the rules for multiplicative conjunction and disjunction are admissible for plain conjunction and disjunction under a classical interpretation (i.e., they are admissible rules in LK).

AdditivesEdit

The rules for additive conjunction (&) and disjunction (⊕) :

Template:Math			Template:Math

Template:Math

Template:Math

and for their units:



Template:Math

(no rule for Template:Math)

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 (⊗), the context of the conclusion (Template:Math) is split up between the premises, whereas for the additive case connective (&) the context of the conclusion (Template:Math) is carried whole into both premises.

ExponentialsEdit

The exponentials are used to give controlled access to weakening and contraction. Specifically, we add structural rules of weakening and contraction for Template:Math'd propositions:Template:Sfn

Template:Math

and use the following logical rules, in which Template:Math stands for a list of propositions each prefixed with Template:Math:

Template:Math

One might observe that the rules for the exponentials follow a different pattern from the rules for the other connectives, resembling the inference rules governing modalities in sequent calculus formalisations of the normal modal logic S4, and that there is no longer such a clear symmetry between the duals Template:Math and Template:Math. This situation is remedied in alternative presentations of CLL (e.g., the LU presentation).

Remarkable formulasEdit

In addition to the De Morgan dualities described above, some important equivalences in linear logic include:

Distributivity

Template:Math

By definition of Template:Math as Template:Math, the last two distributivity laws also give:

Template:Math

(Here Template:Math is Template:Math.)

Exponential isomorphism

Template:Math

Linear distributions

A map that is not an isomorphism yet plays a crucial role in linear logic is:

Template:Math

Linear distributions are fundamental in the proof theory of linear logic. The consequences of this map were first investigated in Cockett & Seely (1997) and called a "weak distribution".Template:Sfn In subsequent work it was renamed to "linear distribution" to reflect the fundamental connection to linear logic.

Other implications

The following distributivity formulas are not in general an equivalence, only an implication:

Template:Math

Encoding classical/intuitionistic logic in linear logicEdit

Both intuitionistic and classical implication can be recovered from linear implication by inserting exponentials: intuitionistic implication is encoded as Template:Math, while classical implication can be encoded as Template:Math or Template:Math (or a variety of alternative possible translations).Template:Sfn 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 formulas of intuitionistic logic to formulas of linear logic in a way that 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 interpretationEdit

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. Tony Hoare (1985)'s classic example of the vending machine can be used to illustrate this idea.

Suppose we represent having a candy bar by the atomic proposition Template:Math, and having a dollar by Template:Math. To state the fact that a dollar will buy you one candy bar, we might write the implication Template:Math. But in ordinary (classical or intuitionistic) logic, from Template:Math and Template:Math one can conclude Template:Math. 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,Template:Clarify 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 Template:Math, we express the property of the vending machine as a linear implication Template:Math. From Template:Math and this fact, we can conclude Template:Math, but not Template:Math. In general, we can use the linear logic proposition Template:Math to express the validity of transforming resource Template:Math into resource Template:Math.

Running with the example of the vending machine, 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 Template:Math 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 Template:Math. The constant 1 denotes the absence of any resource, and so functions as the unit of ⊗.

Additive conjunction Template:Math 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 Template:Math. We do not write Template:Math, which would imply that one dollar suffices for buying all three products together. However, from Template:Math, we can correctly deduce Template:Math, where Template:Math. The unit ⊤ of additive conjunction can be seen as a wastebasket for unneeded resources. For example, we can write Template:Math to express that with three dollars you can get a candy bar and some other stuff, without being more specific (for example, chips and a drink, or $2, or $1 and chips, etc.).

Additive disjunction Template:Math 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 Template:Math. The constant 0 represents a product that cannot be made, and thus serves as the unit of ⊕ (a machine that might produce Template:Math or Template:Math is as good as a machine that always produces Template:Math because it will never succeed in producing a 0). So unlike above, we cannot deduce Template:Math from this.

Other proof systemsEdit

Proof netsEdit

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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:

Template:Math

The goal of proof nets is to make them identical by creating a graphical representation of them.

SemanticsEdit

Template:Expand section

Algebraic semanticsEdit

Template:See also

Decidability/complexity of entailmentEdit

The entailment relation in full CLL is undecidable.Template:Refn When considering fragments of CLL, the decision problem has varying complexity:

Multiplicative linear logic (MLL): only the multiplicative connectives. MLL entailment is NP-complete, even restricting to Horn clauses in the purely implicative fragment,Template:Sfn or to atom-free formulas.Template:Sfn
Multiplicative-additive linear logic (MALL): only multiplicatives and additives (i.e., exponential-free). MALL entailment is PSPACE-complete.<ref name="Lincoln+92" />
Multiplicative-exponential linear logic (MELL): only multiplicatives and exponentials. By reduction from the reachability problem for Petri nets,Template:Sfn MELL entailment must be at least EXPSPACE-hard, although decidability itself has had the status of a longstanding open problem. In 2015, a proof of decidability was published in the journal Theoretical Computer Science,Template:Sfn but was later shown to be erroneous.Template:Sfn
Affine linear logic (that is linear logic with weakening, an extension rather than a fragment) was shown to be decidable, in 1995.Template:Sfn

VariantsEdit

Many variations of linear logic arise by further tinkering with the structural rules:

Affine logic, which forbids contraction but allows global weakening (a decidable extension).
Strict logic or relevant logic, which forbids weakening but allows global contraction.
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 ⅋, ⊥, and ? are absent, and linear implication is treated as a primitive connective. In FILL (Full Intuitionistic Linear Logic) the connectives ⅋, ⊥, 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).