The document introduces linear logic and provides examples of proofs in linear logic using natural deduction. It discusses key concepts in linear logic including linear implication (-○), contexts, the restriction that each hypothesis can only be used once, and the introduction and extraction rules for linear implication. It also covers other linear logic connectives like falsehood, AND, OR, storage (!A), and provides examples of proofs using these connectives. Finally, it discusses computations in linear logic through typed lambda calculus and a linear Lisp machine that avoids garbage collection by not allowing data sharing.

1. Introduction of Linear Logic
Sosuke MORIGUCHI

2. Agenda
 Linear Logic
 Viewpoint of Linear Logic
 Definition and Examples
 Computations
 Typed Lambda Calculus
 Linear Lisp Machine

3. Book
 LECTURES ON LINEAR LOGIC
 A. S. Troelstra
 CSLI Lectures Notes Number 29
 You can download pdf file of this book from
http://standish.stanford.edu/bin/object?00000065
 Chapter 6 is mainly used.

4. Viewpoint of Linear Logic
 In linear logic, we can use each hypothesis
just once.
 Strictly speaking, it’s not correct...
 This formula is not valid in linear logic.
(A -> B -> C) -> (A -> B) -> A -> C
 Because we need to use predicate “A” twice.
 This formula is valid.
(A -> B) -> (B -> C) -> A -> C
 Because each subformula is used only once.

5. Natural Deduction
 In this explanation, we use natural
deduction to construct proof trees.
 With contexts (i.e. lists of hypotheses).
 Syntax
 Γ├ A
 “Formula A is derived from context Γ.”
 Axioms
 A├ A
 Not Γ, A├ A

6. Linear Logic with
Natural Deduction
 Many handbooks for linear logic use
sequent calculus.
 But some of you may not know about sequent
calculus, so I won’t use.
 The book I read also explains about natural
deduction.
 In linear sequent calculus, there are neither
contraction nor weakening rules.

7. Linear Implication
 We use -○ for implication in linear logic.
 A -○ B means that “B is proved with just one
formula A”.
 We use following formulae as examples.
 (A -○ B) -○ (B -○ C) -○ A -○ C
 (A -○ B -○ C) -○ (A -○ B) -○ A -○ C

8. Linear Implication in
Natural Deduction
 Introduction rule
Γ, A├ B
Γ├ A -○ B
 Extract rule
Γ├ A -○ B Δ├ A
Γ, Δ├ B

9. Example
(A -○ B) -○ (B -○ C) -○ A -○ C
A –○ B├ A -○ B A├ A
B -○ C├ B -○ C A -○ B, A├ B
A -○ B, B -○ C, A├ C
A -○ B, B -○ C├ A -○ C
A -○ B├ (B -○ C) -○ A -○ C
├ (A -○ B) -○ (B -○ C) -○ A -○ C

10. Failure about proof 1
A -○ B -○ C├ A -○ B -○ C A├ A A-○ B├ A -○ B ├ A ??
A -○ B -○ C, A├ B -○ C A -○ B├ B
A -○ B -○ C, A -○ B, A├ C
A -○ B -○ C, A -○ B├ A -○ C
A -○ B -○ C├ (A -○ B) -○ A -○ C
├ (A -○ B -○ C) -○ (A -○ B) -○ A -○ C

11. Failure about proof 2
A -○ B -○ C├ A -○ B -○ C ├ A ?? A-○ B├ A -○ B A├ A
A -○ B -○ C├ B -○ C A -○ B, A├ B
A -○ B -○ C, A -○ B, A├ C
A -○ B -○ C, A -○ B├ A -○ C
A -○ B -○ C├ (A -○ B) -○ A -○ C
├ (A -○ B -○ C) -○ (A -○ B) -○ A -○ C

12. Other structures
(logical operators)
 NOT
 Or falsefood (with implication).
 AND
 OR
 There are two types for each operator.
 Contextual
 Context-free

13. Contextual and Context-free
 These kinds are based on operators’ rules.
 If a rule has a restriction for contexts of
hypotheses in the rule, then the operator is
contextual, otherwise context-free.
 Linear implication is context-free one.
 Contextual implication is also definable, but its
rules are hard to describe in natural deduction.

14. Falsehood
 Context-free
 0
Γ, A -○ 0├ 0
Γ├ A
 NOT A = A -○ 0.
 Above rule can be
viewed as “double
negation elimination”.
 Contextual
 ⏊
Γ, ⏊├ A
 I can’t explain why this
is “contextual” and left
one is “context-free”...

15. Example
((A -○ 0) -○ 0) -○ A
 ...very simple...
(A -○ 0) -○ 0├ (A -○ 0) -○ 0 A -○ 0├ A -○ 0
(A -○ 0) -○ 0, A -○ 0├ 0
(A -○ 0) -○ 0├ A
├ ((A -○ 0) -○ 0) -○ A

16. AND
 Context-free
 A★B
Γ├ A Δ├ B
Γ, Δ├ A★B
Γ├ A★B Δ, A, B├ C
Γ, Δ├ C
 Contextual
 A∩B
Γ├ A Γ├ B
Γ ├ A∩B
Γ├ A∩B Γ├ A∩B
Γ├ A Γ├ B

17. Example 1
(A -○ B)★C -○ A -○ B★C
A -○ B├ A -○ B A├ A
A, A -○ B├ B C├ C
(A -○ B)★B├ (A -○ B)★B A, A -○ B, C├ B★C
(A -○ B)★C, A├ B★C
(A -○ B)★C├ A -○ B★C
├ (A -○ B)★C -○ A -○ B★C

18. Example 2
(A -○ B)∩(A -○ C) -○ A -○ B∩C
(A -○ B)∩(A -○ C)├ (A -○ B)∩(A -○ C) (A -○ B)∩(A -○ C)├ (A -○ B)∩(A -○ C)
(A -○ B)∩(A -○ C)├ A -○ B A├ A (A -○ B)∩(A -○ C)├ A -○ C A├ A
(A -○ B)∩(A -○ C), A├ B (A -○ B)∩(A -○ C), A├ C
(A -○ B)∩(A -○ C), A├ B∩C
(A -○ B)∩(A -○ C)├ A -○ B∩C
├ (A -○ B)∩(A -○ C) -○ A -○ B∩C

19. OR
 Context-free
 A+B
 OR is hard to describe
in natural deduction,
like contextual
implication.
 Dual operator of ★.
 ~(A★B) = ~A+~B
 ~A = A -○ 0
 Contextual
 A∪B
 Dual operator of ∩.
 ~(A∩B) = ~A∪~B
Γ├ A
Γ├ A∪B
Γ├ A∪B
A, Δ├ C B, Δ├ C
Γ, Δ├ C

20. Example
(C -○ A)∪(C -○ B) -○ C -○ A∪B
C -○ A├ C -○ A C├ C C -○ B├ C -○ B C├ C
C -○ A, C├ A C -○ B, C├ B
(C -○ A)∪(C -○ B)├ (C -○ A)∪(C -○ B) C -○ A, C├ A∪B C -○ B, C├ A∪B
(C -○ A)∪(C -○ B), C├ A∪B
(C -○ A)∪(C -○ B)├ C -○ A∪B
├ (C -○ A)∪(C -○ B) -○ C -○ A∪B

21. Special Operator : Storage !A
 A storage formula !A denotes any number
of formula A.
Γ├ !B Δ, !B, !B├ A Γ├ !B Δ├ A
Γ, Δ├ A Γ, Δ├ A
Γ├ !B Δ, B├ A !Γ├ B
Γ, Δ├ A !Γ├ !B
 !Γ means that all hypotheses in Γ are storage
formulae.

22. Example
(A -○ B -○ C) -○ (A -○ B) -○ !A -○ C
A -○ B -○ C├ A -○ B -○ C A├ A A -○ B├ A -○ B A├ A
!A├ !A A -○ B -○ C, A├ B -○ C !A├ !A A -○ B, A├ B
A -○ B -○ C, !A├ B -○ C A -○ B, !A├ B
!A├ !A A -○ B -○ C, A -○ B, !A, !A├ C
A -○ B -○ C, A -○ B, !A├ C
A -○ B -○ C, A -○ B├ !A -○ C
A -○ B -○ C├ (A -○ B) -○ !A -○ C
├ (A -○ B -○ C) -○ (A -○ B) -○ !A -○ C

23. Computations
 Typed Lambda Calculus
 Linear Lisp Machine

24. Curry-Howard Isomorphism
 Relationship between typed lambda terms
and logical formulae.
 typing rule  derivation rule (axiom)
 typed lambda term  proof
 Here, we use intuitional linear logic, which
does not have double negation elimination
rule.

25. Linear Lambda Terms
Γ, x : A├ t : B
Γ├ λxA. t : A -○ B
Γ├ s : A -○ B Δ├ t : A
Γ, Δ├ (s t) : B
...Other rules are omitted.
(They are same forms as formulae.)

26. Example 1
 λfA-○B. λgB-○C. λaA. (g (f a))
: (A -○ B) -○ (B -○ C) -○ A -○ C
 λs(A-○B)★C.λaA.let (xA-○B★yC)=s in (x a)★y
: (A -○ B)★C -○ A -○ B★C
 λs(A-○B)∩(A-○C).λaA.((π0 s) a)∩((π1 s) a)
: (A -○ B)∩(A -○ C) -○ A -○ B∩C
 λs(C-○A)∪(C-○B).λcC. match s with
| fC-○A=>(κ0 (f c)) | gC-○B=>(κ1 (g c)).
: (C -○ A)∪(C -○ B) -○ C -○ A∪B

27. Example 2
 λfA-○B-○C.λgA-○B.λx!A.let (s!A,t!A)=copy(x) in
(let aA = load(s) in (f a)
let bA = load(t) in (g b))
: (A -○ B -○ C) -○ (A -○ B) -○ !A -○ C
 λfA-○B-○C.λgA-○B.λx!A.let (s!A,t!A)=copy(x) in
let aA = load(s) in let bA = load(t) in
((f a) (g b))
: (A -○ B -○ C) -○ (A -○ B) -○ !A -○ C
 ...etc

28. Linear Lisp
 Lively Linear Lisp – ‘Look Ma, No Garbage!’
 Henry G. Baker
 Nimble Computer Corporation
 ACM SIGPLAN Notices 27, 8(Aug. 1992),
89-98.

29. Linear Lisp (Machine)
 Linear Lisp has
 nil, cons cells and symbols.
 11 (atomic) operations.
 And some logical function.
 Linear Lisp does not require garbage
collection.
 Linear Lisp is not based on linear logic...
 Only “NO DATA SHARING IN PROGRAMS”.

30. Operations (1/3)
 rn : n-th register (variable)
 (<-> r1 r2)
 swap r1, r2
 (<-> r1 (CAR r2))
 (<-> r1 (CDR r2))
 r1 and r2 must be distinct, and r2 must not be
ATOM (neither nil nor symbol).

31. Operations (2/3)
 (NULL r1)
 Predicate for (= r1 NIL)
 (ATOM r1)
 Predicate for (or (= r1 NIL) (symbol? r1))
 (EQ r1 r2)
 Precondition : (and (ATOM r1) (ATOM r2))
 (:= r1 ‘foo)
 Precondition : (ATOM r1)

32. Operations (3/3)
 (:= r1 r2)
 Precondition : (and (ATOM r1) (ATOM r2))
 (CONS r1 r2)
 r2 becomes (cons r1 r2), and r1 becomes nil.
 (PUSH r1 r2)
 Same as CONS
 (POP r1 r2)
 Precondition :(and (NULL r1) (not (ATOM r2)))
 r2 becomes (cdr r2), and r1 becomes (car r2).

33. Proposition
 Reference counts of all cons cell in linear
lisp programs are 1.
 Cons cells are created when CONS (or PUSH)
operations are executed, and destroyed when
POP operations are executed.
 So if one cons cell is not accessible, then it is
destroyed soon.

34. CAUTION!
 Previous proposition is not one on the
paper.
 In the paper, auther uses “free register”,
which is infinite long list and elements of the
list are all nil.
 Reference counts of all lists are always identically 1.
 All cons cells are always accessible – i.e. live.
 No garbage is created.
 The name of the paper would be based on these
propositions.

35. Other operations
 We can construct some operations with
above operations.
 (FREE r1)
 r1 becomes nil.
 (COPY r1 r2)
 If r2 is nil, r2 becomes the same as r1.
 Not sharing cell.
 (EQUAL r1 r2)
 Recursive list equality.

36. FREE
(FREE r1) :
(if (not (NULL r1)
(if (ATOM r1) (:= r1 ‘nil)
(progn (PUSH r2 sp) (POP r2 r1)
(FREE r1)
(<-> r2 r1)
(FREE r1)
(POP r2 sp))))

37. COPY
(COPY r1 r2) : ; r2 = ‘nil
(if (not (NULL r1))
(if (ATOM r1) (:= r2 r1)
(progn
(PUSH t1 sp) (PUSH t2 sp)
(POP t1 r1) (COPY r1 r2)
(<-> t1 r1) (<-> t2 r2) (COPY r1 r2)
(<-> t1 r1) (<-> t2 r2)
(PUSH t1 r1) (PUSH t2 r2)
(POP t2 sp) (POP t1 sp))))

38. EQUAL
(EQUAL r1 r2) :
(or (and (ATOM r1) (ATOM r2) (EQ r1 r2))
(and (not (ATOM r1)) (not (ATOM r2))
(progn
(PUSH t1 sp) (PUSH t2 sp)
(POP t1 r1) (POP t2 r2)
(prog1
(and (EQUAL r1 r2)
(progn (<-> t1 r1) (<-> t2 r2)
(prog1 (EQUAL r1 r2)
(<-> t1 r1) (<-> t2 r2))))
(PUSH t1 r1) (PUSH t2 r2)
(POP t2 sp) (POP t1 sp)))))

39. Traditional Lisp Interpreter
 Auther says that we can program a
traditional lisp interpreter, but there is no
such a program in the paper.
 Instead of it, there is a metacircular linear lisp
interpreter.
 It needs quote, back-quote, and macros...
 The interpreter is inefficient, due to the
extra expense of copying.

40. And ...
 The rest of paper is about how to use
linear lisp in multiprocessing...
 But I don’t want to talk about that.
 So, that’s all for my talk.