[Monday][yesterday], I said that I needed to introduce the sequent calculus, because it would be useful for describing things like linear logic. Today we’re going to take a quick look at linear logic – in particular, at *propositional* linear logic; you can expand to predicate linear logic in a way very similar to the way we went from propositional logic to first order predicate logic.
So first: what the heck *is* linear logic?
The answer is, basically, logic where statements are treated as *resources*. So using a statement in an inference step in linear logic *consumes* the resource. This is a strange notion if you’re coming from a regular predicate logic. For example, in regular predicate logic, if we have the statements: “A”, “A ⇒ B”, and “A ⇒ C”, we know that we can conclude “B ∧ C”. In linear logic, that’s not true: using either implication statement would *consume* the “A”. So we could infer “B”, or we could infer “C”, but we could *not* infer both.
When people talk about linear logic, and why it makes sense, they almost always use a vending machine analogy. Suppose I walk up to a vending machine, and I want to buy a soda and a candy bar. I’ve got 8 quarters in my pocket; the soda costs $1.50; the candy bar costs $.75.
In linear logic, I’d say something like the following (the syntax is wrong, but we’ll get to syntax later): (Q,Q,Q,Q,Q,Q,Q,Q), (Q,Q,Q,Q,Q,Q) ⇒ Soda, (Q,Q,Q) ⇒ Candy.
Using the rules, I can by a soda by “spending” 6 of my Qs. I wind up with “(Q,Q) ∧ Soda”, and “(Q,Q,Q) ⇒ Candy”. I’ve consumed 6 Qs, and I’ve consumed the “(Q,Q,Q,Q,Q,Q) ⇒ Soda” implication. I can’t do anything else; I don’t have enough Qs.
The basic statements in linear logic, with intuitive meanings are:
1. A ⊗ B. This is called *multiplicative conjunction*, also known as *simultaneous occurrence*. This means that I definitely have both A and B. This has an *identity unit* called “1”, such that A ⊗ 1 ≡ 1 ⊗ A ≡ A. 1 represents the idea of the absence of any resource.
2. A & B : *additive conjunction*, aka *internal choice*. I can have either A *or* B, and I get to pick which one. The unit is ⊤, pronounced “top”, and represents a sort of “I don’t care” value.
3. A ⊕ B. This is called *additive disjunction*, also known as *external choice*. It means that I get either A or B, but I don’t get to pick which one. The unit here is 0, and represents the lack of an outcome.
4. A ⅋ B : *multiplicative disjunction*, aka *parallel occurence*; I *need to have* both A and B at the same time. The unit for this is ⊥, pronounced “bottom”, and represents the absence of a goal. In the vending machine metaphor, think of it as the “cancel” button on the vending machine: I decided I don’t want any, so I’m not going to spend my resources.
5. A -o B : Linear implication. Consume resource A to *produce* resource B. The normal symbol for this looks like an arrow with a circle instead of an arrowhead; this operator is often called “lolly” because of what the normal symbol looks like. I’m stuck writing it as “-o”, because there’s no HTML entity for the symbol.
6. !A : Positive exponentiation, pronounced “Of course A”. This *produces* an arbitrary number of As. Equivalent to A ⊗ !A.
7. ?A : Negative exponentiation, pronounced “Why not A?”. This *consumes* As.
Ok. So, suppose I want to talk about buying lunch. I’ve got 10 dollars to buy lunch. I’ll be full if I have a salad, a coke, and a tuna sandwich. If I wanted to write “I’ve got a dollar” as “D”, “I have a salad” as “S”, “I have a coke” as “C”, “I have a tuna sandwich” as “T”, and finally, “I’m full” as “F”
* I can write “I have 10 dollars” in LL as: “(D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D⊗ D ⊗ D)”.
* I can write “Tuna sandwich and salad and coke” as a group of things that I want to have all of as: “T ⅋ S ⅋ C”.
* I can say that I’ll be full if I have lunch as “T ⅋ S ⅋ C -o F”
If I want to talk about buying lunch, I can describe the prices of the things I want using implication:
* A coke costs one dollar: “D -o C”; I can spend one dollar, and in return I get one coke.
* A salad costs 3 dollars: “(D ⊗ D ⊗ D) -o S”
* A tuna sandwich also costs three dollars: “(D ⊗ D ⊗ D) -o S”
Now, I can do some reasoning with these.
* By taking 1 of the dollars, I can get one C. That leaves me with “D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D ⊗ D⊗ D ⊗ C”
* By taking 3 D, I can get one S. “D ⊗ D ⊗ D ⊗ D ⊗ D⊗ D ⊗ C ⊗ S”.
* By taking 3 D, I can get one T. “D ⊗ D⊗ D ⊗ C ⊗ S ⊗ T”.
* Now I’ve got my lunch. I can eat it and be full, with three dollars left: “D ⊗ D⊗ D ⊗ F”.
Just from this trivial example, you should be able to see why linear logic is cool: the idea of being able to talk about *how* resources are used in an inference process or a computation is really valuable, and linear logic gives you the ability to really work with the concept of resource in a solid, formal way. If you think of it in terms of the Curry-Howard isomorphism [types-as-proofs concept from the simply typed lambda calculus][types-as-proofs], you can imagine using linear logic for types of values that are *consumed* by a computation – i.e., they’re no longer available once they’ve been used.
I’m going to adopt a slightly different format for the sequents for working in linear logic. The way that I produced the center bars in yesterdays post was really painful to write, and didn’t even end up looking particularly good. So, the way that I’m going to right the sequents in this post is to wrap the “top” and “bottom” of the sequent in curly braces, and separate them by a “⇒”, as in:

{GivenContext :- GivenEntailment } ⇒ { InferredContext :- InferredEntailment}

Now, let’s take a look at the sequent rules for propositional linear logic. I’m using the version of these rules from [Patrick Lincoln’s SIGACT ’92 paper][sigact92]. Yeah, I know that’s a bit of a big load there. Don’t worry about it too much; the important part is the concept described up above; the sequents are useful to look at when you have a hard time figuring out what some operator means in inference. For example, you can see the difference between & and ⊕ (which I found confusing at first) by looking at their sequents, to see what they do.
1. **Identity**: { } ⇒ { A :- A }
2. **Cut**: { Γ₁ :- A, Σ₁   Γ₂, A :- Σ₂ } ⇒ { Γ₁, Γ₂ :- Σ₁,Σ₂}
3. **Exchange Left**: { Γ₁, A, B, Γ₂ :- Σ } ⇒ { Γ₁, B, A, Γ₂ :- Σ }
4. **Exchange Right**: { Γ :- Σ₁, A, B, Σ₂ } ⇒ { Γ :- Σ₁, B, A, Σ₂}
5. **⊗ Left**: {Γ, A, B :- Σ} ⇒ { Γ, A ⊗ B :- Σ }
6. **⊗ Right**: { Γ₁ :- A, Σ₁   Γ₂ :- B, Σ₂} ⇒ { Γ₁, Γ₂ :- (A ⊗ B), Σ₁,Σ₂}
7. **-o Left**: { Γ₁ :- A, Σ₁  Γ₂, B :- Σ₂ } ⇒ { Γ₁,Γ₂, (A -o B) :- Σ₁,Σ₂}
8. **-o Right**: { Γ, A :- B, Σ} ⇒ { Γ :- A -o B, Σ}
9. **⅋ Left**: { Γ₁,A :- Σ₁  Γ₂ :- B, Σ₂ } ⇒ { Γ₁,Γ₂, (A ⅋ B) :- Σ₁,Σ₂}
10. **⅋ Right**: { Γ :- A, B, Σ} ⇒ { Γ :- A ⅋ B, Σ}
11. **& Left**: { Γ, A :- Σ } ⇒ { Γ,A & B :- Σ}/{ Γ, B :- Σ } ⇒ { Γ,A & B :- Σ}
12. **& Right**: { Γ :- A,Σ   Γ :- B,Σ} ⇒ { Γ :- (A & B), Σ }
13. **⊕ Left**: Γ,A :- Σ   Γ,B :- Σ} ⇒ { Γ,A ⊕ B :- Σ}
14. **⊕ Right**: {Γ :- A,Σ} ⇒ {Γ :- A ⊕ B, Σ}/{Γ :- B,Σ} ⇒ {Γ :- A ⊕ B, Σ}
15. **!W**: {Γ :- Σ} ⇒ {Γ,!A :- Σ}
16. **!C**: {Γ,!A,!A :- Σ} ⇒ { Γ,!A :- Σ }
17. **!D**: { Γ, A :- Σ} ⇒ { Γ,!A :- Σ}
18. **!S**: { !Γ :- A, ?Σ} ⇒ { !Γ :- !A, ?Σ}
19. **?W**: {Γ :- Σ} ⇒ {Γ:- ?A, Σ}
20. **?C**: {Γ :- ?A,?A, Σ} ⇒ { Γ :- ?A, Σ }
21. **?D**: { Γ :- A, Σ} ⇒ { Γ :- ?A, Σ}
22. **?S**: { !Γ, A :- ?Σ} ⇒ { !Γ,?A :- ?Σ}
23. **⊥ Exp Left**: { Γ :- A, Σ} ⇒ { Γ, A^⊥ :- Σ}
24. **⊥ Exp Right**: { Γ, A :- Σ} ⇒ { Γ :- A^⊥, Σ}
25. **0 Left**: { Γ,0 :- Σ } *(Nothing can be inferred)*
26. **⊤ Right**: { Γ :- ⊤,Σ}
27. **⊥ Left**: { ⊥ :- }
28. **⊥ Right**: {Γ :- Σ} ⇒ {Γ :- ⊥,Σ}
29. **1 Left**: { Γ :- Σ} ⇒ { Γ,1 :- Σ}
30. **1 Right**: { :- 1}
This is long enough that I’m not going to get into how this hooks into category theory today, except to point out that if you look carefully at the multiplication and exponentiations, they might seem a tad familiar.
[types-as-proofs]: http://goodmath.blogspot.com/2006/06/finally-modeling-lambda-calculus.html
[sigact92]: http://www.csl.sri.com/~lincoln/papers/sigact92.ps
[yesterday]: http://scienceblogs.com/goodmath/2006/07/a_brief_diversion_sequent_calc.php