Linear_logic

From Wikipedia, the free encyclopedia Linear logic Linear logic Linear logic is a substructural logic invented by Jean-Yves Girard as a refinement of both classical and intuitionistic logic, combining the symmetries of the former with many of the constructive properties of the latter. Proof-theoretically, it derives from an analysis of classical sequent calculus in the absence of the structural rules of weakening and contraction. This has the effect that certain propositions (e.g., (A ∧ (A ⇒ B)) ⇒ (A ∧ B)) which are classically valid are not provable in linear logic, although both classical and intuitionistic logic can be encoded in linear logic by means of additional connectives (the socalled exponentials). Intuitively, rejecting the weakening and contraction rules transforms logic from being about persistent truths to being about ephemeral resources. This makes linear logic useful for reasoning about the world in many simple situations where ordinary logic is not. One example: churning cream into butter. Suppose we represent a quart of cream by the proposition A, and a pound of butter by B. To state the fact that a quart of cream can be churned into a pound of butter, we might write the implication A ⇒ B. But in ordinary (classical or intuitionistic) logic, from A and A ⇒ B one can conclude A ∧ B. So, ordinary logic leads us to believe that we can churn our cream and have it too! The rejection of weakening and contraction allows linear logic to avoid this kind of spurious reasoning, as well as to express many different situations by employing additional logical connectives that refine the ordinary ones under a resource interpretation. you (the consumer) direct. For example, if you buy a stick of gum and a bottle of soft drink, then you are requesting gum ⊗ 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 ⊗ drink) ⊗ candy, you have the power to choose the gum first, then the candy bar, and finally the drink. Indeed, ⊗ is an associative and commutative operation, so this is equivalent to gum ⊗ (candy ⊗ drink). The constant 1 is used to denote the absence of any resource; it functions as a unit of tensor: A ⊗ 1 ≡ 1 ⊗ 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 ⊗ 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 ⊤, with A & ⊤ ≡ ⊤ & 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 ⊤. This unit can be used together with ⊗ 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 ⊗ ⊤. Multiplicative disjunction, also called par or parallelisation (written ⅋, an upsidedown ampersand) 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 Linear connectives The logical connectives 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 a vending machine. Multiplicative conjunction, also called tensor or times (written ⊗), denotes simultaneous occurrence of resources, to be used as 1 From Wikipedia, the free encyclopedia drink, then this may be represented as gum ⅋ drink. The difference between this and the multiplicative conjunction gum ⊗ 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 ⅋ coffee and cup ⊗ 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 ?), sometimes also known as entails, multimap, or lolli because of its lollipop-like shape. As a resource, A ? B means a method to consume resource A to achieve resource B. If the machine is a penny smasher, then its ability to smash a coin can be described as penny ? 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 Linear logic should not preclude standard mathematical reasoning. Linear logic uses an idea from modal 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 a sequent 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 ? B ≡ A⊥ ⅋ B. This is familiar from other 2 From Wikipedia, the free encyclopedia classical logics: for example, the usual implication ⇒ is similarly definable: A ⇒ B ≡ ?A⊥ ⅋ 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. (A ⊗ B)⊥ = A⊥ ⅋ B⊥ (A & B)⊥ = A⊥ ⊕ B⊥ (A ⊕ B)⊥ = A⊥ & B⊥ (A ⅋ B)⊥ = A⊥ ⊗ B⊥ The logical units have similar duals; for example: ⊤⊥ = 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 a PSPACE-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 is undecidable. • Full Intuitionistic Linear Logic or FILL. This variant allows independent multiplicative connectives tensor, par and linear implication. This is just like Intuitionistic 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 Linear logic standard (See first-order logic and higher-order logic.) The closest sub-structural cousins of linear logic are: • Affine logic, which extends linear logic with the structural rule of weakening. The connectives one and top are indistinguishable in affine logic. • Strict logic or relevant logic, which extends linear logic with the structural rule of contraction. • Non-commutative logic or ordered logic which removes the structural rule of exchange from linear logic. Multiplicative conjunction divides further into a pair of fuses (left fuse and right fuse). See also • • • • • • • • • • Linear types Logic of unity (LU) Proof nets Geometry of interaction Game semantics Intuitionistic logic Computability logic Ludics Chu spaces Uniqueness type References • Girard, Jean-Yves. 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 [1].) • Troelstra, A.S. Lectures on Linear Logic. CSLI (Center for the Study of Language and Information) Lecture Notes No. 29. Stanford, 1992. External links • Patrick Lincoln’s excellent Introduction to Linear Logic (Postscript) • Introduction to Linear Logic by Torben Brauner [2] • A taste of linear logic by Philip Wadler [3] • Linear Logic by Roberto Di Cosmo and Dale Miller. The Stanford Encyclopedia of Philosophy (Fall 2006 Edition), Edward N. Zalta (ed.). 3 From Wikipedia, the free encyclopedia • Overview of linear logic programming by Dale Miller. In Linear Logic in Computer Science, edited by Ehrhard, Girard, Ruet, Linear logic and Scott. Cambridge University Press. London Mathematical Society Lecture Note, Volume 316, 2004. 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