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Chapter 2 Linear Logic CHAPTER 2 LINEAR LOGIC INEAR logic was introduced by Girard (1987) as a resource conscious L logic, that is a logic where the number of occurrences of a formula mat- ters. This is not true in classical logic. The structural rule of contraction from classical logic Γ,A,A` ∆ Γ ` A, A, ∆ [LC] [RC] Γ,A` ∆ Γ ` A, ∆ tells us that whenever we can prove something using the formula A as a premiss twice, we can also prove it using A just once. When we interpret A as, for example, a lemma we use in a proof, it makes sense to say that reproving the lemma each time we use it is unnecessary. But when we want to interpret A as a resource, a physical as opposed to an ideal entity, then it makes sense to reject the rule of contraction, at least globally. The number of occurrences of our resources generally do matter to us. Similarly, the rule of weakening ` Γ ` A, ∆ Γ ∆ [LW ] [RW] Γ,A` ∆ Γ ` ∆ states that whenever we can prove ∆ with assumptions Γ we can add an extra assumption A and still have a valid proof. Again, if we interpret A as a resource we can’t just get it from thin air, we have to justify how we obtained it. So global availability of weakening is also undesirable from a resource conscious point of view. For our intended linguistic applications, the absence of contraction and weakening seems like a good starting point, since, in general, the number of occurrences of words in a sentence is quite relevant to the grammaticality of the sentence. 8 Linear Logic When we remove contraction and weakening, however, we are faced with some choices to make when we want to write down the rules for con- junction. Two possible pairs of sequent rules exist Γ,A,B ` ∆ Γ ` A, ∆Γ0 ` B,∆0 [L∧] [R∧] Γ,A∧ B ` ∆ Γ, Γ0 ` A ∧ B,∆, ∆0 Γ,A ` ∆ Γ ` A, ∆Γ` B,∆ i [L∧0] [R∧0] Γ,A0 ∧ A1 ` ∆ Γ ` A ∧ B,∆ depending on whether in the right rules we take the union of the contexts of the two premisses or demand the contexts are the same. These choices force the corresponding left rule upon us if we want our system to satisfy cut elimination. We will call the conjunction which takes the union of the contexts the mul- tiplicative conjunction in linear logic, and write it as ‘⊗’. We will call the con- junction which demands the context formulas of the premisses to be equal the additive conjunction, and write it as ‘&’ We face the symmetrical choice for the rules& for disjunction, where we will also have a multiplicative instantiation ‘ ’ and an additive instantiation ‘⊕’. 2.1 Sequent Calculus I will now present the sequent calculus for linear logic. The group of identity rules should contain no surprises. Note that the [Cut] rule is multiplicative. Identity Γ,A` ∆Γ0 ` A, ∆0 [Ax] [Cut] A ` A Γ, Γ0 ` ∆, ∆0 Linear negation (.)⊥ has the obvious rules. Negation Γ ` A, ∆ Γ,A` ∆ [L⊥] [R⊥] Γ,A⊥ ` ∆ Γ ` A⊥, ∆ 2.1.1 Multiplicatives For the multiplicative conjunction ‘⊗’ (tensor) and disjunction ‘ ’& (par) the context of the conclusion is the union of the contexts of the premisses of the A ⊗ B A B rule. A way to look at a formula is that it gives both an and& a resource. It is difficult to come up with a similar intuition for ‘ ’ except perhaps by noting it is the De Morgan dual of ‘⊗’. 2.1 Sequent Calculus 9 −◦ Linear implication ‘ ’ is not a primitive& connective of classical linear −◦ ⊥ logic, and is defined as A B =def A B. For completeness, I will present the rules for linear implication along with& the other multiplicatives. In an in- tuitionistic setting ‘−◦ ’ will replace ‘ ’, as par makes essential use of multiple formulas on the right hand side of the sequent. Multiplicatives Γ,A,B ` ∆ Γ ` A, ∆Γ0 ` B,∆0 [L⊗] [R⊗] Γ,A⊗ B ` ∆ Γ, Γ0 ` A ⊗ B,∆, ∆0 0 0 & Γ,A` ∆Γ,B ` ∆ Γ ` A, B, ∆ & & & [L ] [R ] Γ, Γ0,A B ` ∆, ∆0 Γ ` A B,∆ Γ ` A, ∆Γ0,B ` ∆0 Γ,A` B,∆ [L−◦ ] [R−◦ ] Γ, Γ0,A−◦ B ` ∆, ∆0 Γ ` A−◦ B,∆ 1 ⊥ ⊗ & The multiplicative units and are the identities for and respec-& tively. A⊗1 a` A, which should remind you of multiplication, and A ⊥a` A. Multiplicative Units ` Γ ∆ [L1] [R1] Γ, 1 ` ∆ ` 1 ` [L⊥] Γ ∆ [R⊥] ⊥` Γ `⊥, ∆ 2.1.2 Additives For the additives ‘&’ (with) and ‘⊕’ (plus) the contexts of the premisses of the rule must be the same. A formula A & B represents a choice between A or B (as opposed to A ⊗ B, where you will receive both A and B). Similarly A ⊕ B means it is unknown whether we have an A or a B resource. Additives Γ,A ` ∆ Γ ` A, ∆Γ` B,∆ i [L&] [R&] Γ,A0 & A1 ` ∆ Γ ` A & B,∆ Γ,A` ∆Γ,B ` ∆ Γ ` A , ∆ [L⊕] i [R⊕] Γ,A⊕ B ` ∆ Γ ` A0 ⊕ A1, ∆ The additive units 0 and > are the identities for ⊕ and & respectively. A ⊕ 0 is equivalent to A, again arithmetic serves as a mnemonic, and A&> is equivalent to A. Note that there is no [L>] or [R0] rule. Additive Units [L0] [R>] Γ, 0 ` ∆ Γ `>, ∆ 10 Linear Logic Another equivalence reminiscent of arithmetic, relating the multiplica- tives and the additives is A ⊗ (B ⊕ C) ` (A ⊗ B) ⊕ (A ⊗ C) (A ⊗ B) ⊕ (A ⊗ C) ` A ⊗ (B ⊕ C) whereas we have a similar relation between & and .& & & A & (B & C) ` (A B)&(A C) & & (A & B)&(A C) ` A (B & C) 2.1.3 Exponentials The connectives ‘?’ (why not) and ‘!’ (bang) are called the exponentials. The idea is that on the left hand side !A should be interpreted as an arbitrary number of occurrences of the formula A (we choose how many), whereas ?A specifies an unknown quantity of occurrences of the formula A. In the structural rules of the system this will be reflected by licensing the use of the structural rules of weakening and contraction for resources marked with an exponential, as shown below. Under this interpretation, the left rule for ‘!’ specifies that if something is derivable with a single formula A then it is also derivable if we are allowed to use an arbitrary number of occurrences of A. The left rule for ‘?’ would then indicate that we could accommodate for an unknown quantity of A formulas only if all context formulas can be used as many times as necessary. Readers familiar with modal logic will recognize these logical rules as those of the modal logic S4. We will discuss this connection in more detail in Section 3.4, where we will introduce more fine-grained structural modalities. Exponentials Γ,A` ∆ !Γ ` A, ?∆ [L!] [R!] Γ, !A ` ∆ !Γ `!A, ?∆ !Γ,A`?∆ Γ ` A, ∆ [L?] [R?] !Γ, ?A `?∆ Γ `?A, ∆ Structural Rules Γ,B,A,Γ0 ` ∆ Γ ` ∆,B,A,∆0 [LP ] [RP] Γ,A,B,Γ0 ` ∆ Γ ` ∆,A,B,∆0 Γ, !A, !A ` ∆ Γ `?A, ?A, ∆ [LC] [RC] Γ, !A ` ∆ Γ `?A, ∆ ` ` Γ ∆ [LW ] Γ ∆ [RW] Γ, !A ` ∆ Γ `?A, ∆ 2.1 Sequent Calculus 11 The general theme here is important: we note that certain global struc- tural rules are too strong for our purposes, and we reintroduce controlled versions of these structural rules. This theme will return in Chapter 3. We again have an equivalence relating the multiplicatives, additives and exponentials, reminding us of the arithmetical 2a+b =2a · 2b. !(A & B) ` !A⊗!B !A⊗!B ` !(A & B) ?(A ⊕ B) ` ?A ?& B ?A ?& B ` ?(A ⊕ B) 2.1.4 Quantifiers The first order quantifiers, which quantify over entities in our model, and second order quantifiers, which quantify over formulas in our logic can be added to propositional linear logic to produce first and second order linear logic. First Order Quantifiers Γ,A[x := v] ` ∆ Γ ` A, ∆ [L∀] [R∀] Γ, ∀x.A ` ∆ Γ `∀x.A, ∆ Γ,A` ∆ Γ ` A[x := v], ∆ [L∃] [R∃] Γ, ∃x.A ` ∆ Γ `∃x.A, ∆ Second Order Quantifiers Γ,A[X := B] ` ∆ Γ ` A, ∆ [L∀] [R∀] Γ, ∀X.A ` ∆ Γ `∀X.A, ∆ Γ,A` ∆ Γ ` A[X := B], ∆ [L∃] [R∃] Γ, ∃X.A ` ∆ Γ `∃X.A, ∆ Both the first and the second order rules for [R∀] and [L∃] have the con- dition that there is no free occurrence of x (resp. X)inΓ or ∆. 2.1.5 Cut Elimination One of the most important questions to ask of any logical system is the ques- tion whether it is consistent, or in other words if there are sequents which are unprovable in the logic. Suppose, for example, there exists a proof D of the sequent ` 0. We could extend this proof as follows. D [L0] . Γ, 0 ` ∆ ` 0 [Cut] Γ ` ∆ 12 Linear Logic Therefore, if there exists a proof of ` 0 every sequent would be derivable in the sequent calculus, and our logic would be inconsistent. But if we look at the hypothetical proof D, which rule could be the last rule in this proof? There is no [R0] rule and all other logical rules introduce logical connectives other than 0. None of the structural rules apply either, as ` 0 is not a valid conclusion for any of those rules. This means that the last rule must have been [Cut]. So if we are able to eliminate the [Cut] rule we thereby show the non-existence of a proof of ` 0. Another way to look at consistency is to say it is impossible to have both ⊥ a proof of a formula A and a proof of the negation A of that formula. If we ⊥ have a proof D1 of A and a proof D2 of A we can combine them to prove the empty sequent and finally to conclude `⊥.
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