Доклад для RailsCLub 2014.

1. Abstract Machines for
Great Good
A Gentle Introduction

2. _epicmonkey
epicmonkey
mail@epicmonkey.org

7. Grimm Brothers
• (Grand) Mother tells a story to
her (grand) children
• Very beginnings of culture
• Find and collect stories that
passed through the sieve of
history

8. Germany before unification
• A collection of different
• People mostly spoke German,
English, France
• Collect stories to bind the
German speaking peoples
together

9. The Story of a Boy Who Went Forth to Learn Fear
David Hockey

10. JFK Moon Speech
We choose to go to the moon. We choose to go to
the moon in this decade and do the other things,
not because they are easy, but because they are
hard, because that goal will serve to organize and
measure the best of our energies and skills,
because that challenge is one that we are willing
to accept, one we are unwilling to postpone, and
one which we intend to win, and the others, too.

11. • 1920s: Birth of Combinatory Logic
• Schonfinkel to eliminate bound variables
• Briefly covered high order functions
• Von Neumann's doctoral thesis on set theory
• Curry's combinators
History of Lambda Calculus

12. • 1930s: Birth of Lambda calculus
• Alonzo Church invented Lambda calculus
(explicit formal convertion rules)
• Representation of positive integers — Church
numerals
• Turing published first fixed-point combinator
History of Lambda Calculus

13. History of Lambda Calculus
• 1940s and 1950s:
• Research sort of is frozen, though - - -
• Simple type theory
• Weak reduction

14. History of Lambda Calculus
• 1960+:
• John McCarthy Computer language LISP
• Peter Landin — abstract machine
• Bohm's CuCh language
• Types

15. Lambda Calculus
λx.x
argument body
}
lambda term
applicationabstraction

16. λx.λy.x
λx.λy.y
λf.λx.x
λf.λx.f x
λf.λx.f (f x)
λm.λn.λf.λx.((m f) ((n f) x))
λp.λq.((p (λx.λy.x)) q)
Basic Lambda
True
False
Zero
One
Two
Add
Or

17. ((λ (l) (((λ (f) ((λ (g) (f (λ (x) ((g g) x)))) (λ (g) (f (λ (x) ((g g) x)))))) (λ (r) (λ (l) ((((λ (p) (λ (x) (λ (y) ((p x)
y)))) ((λ (p) (p (λ (x) (λ (y) (λ (x) (λ (y) y)))))) l)) (λ (x) (λ (x) (λ (y) x)))) ((λ (p) ((λ (s) (((λ (x) (λ (y) ((((λ
(f) ((λ (g) (f (λ (x) ((g g) x)))) (λ (g) (f (λ (x) ((g g) x)))))) (λ (r) (λ (x) (λ (y) ((((λ (p) (λ (x) (λ (y) ((p x)
y)))) ((λ (p) (p (λ (x) (λ (y) (λ (x) (λ (y) y)))))) x)) y) (((λ (p) (λ (q) (λ (h) ((h p) q)))) ((λ (p) (p (λ (x) (λ (y)
x)))) x)) ((r ((λ (p) (p (λ (x) (λ (y) y)))) x)) y))))))) x) y))) (((λ (x) (λ (y) ((((λ (f) ((λ (g) (f (λ (x) ((g g) x))))
(λ (g) (f (λ (x) ((g g) x)))))) (λ (r) (λ (x) (λ (y) ((((λ (p) (λ (x) (λ (y) ((p x) y)))) ((λ (p) (p (λ (x) (λ (y) (λ (x)
(λ (y) y)))))) x)) y) (((λ (p) (λ (q) (λ (h) ((h p) q)))) ((λ (p) (p (λ (x) (λ (y) x)))) x)) ((r ((λ (p) (p (λ (x) (λ
(y) y)))) x)) y))))))) x) y))) (r (((λ (p) (λ (l) ((((λ (f) ((λ (g) (f (λ (x) ((g g) x)))) (λ (g) (f (λ (x) ((g g) x))))))
(λ (r) (λ (p) (λ (l) ((((λ (p) (λ (x) (λ (y) ((p x) y)))) ((λ (p) (p (λ (x) (λ (y) (λ (x) (λ (y) y)))))) l)) (λ (x) (λ (x)
(λ (y) x)))) ((((λ (p) (λ (x) (λ (y) ((p x) y)))) (p ((λ (p) (p (λ (x) (λ (y) x)))) l))) (((λ (p) (λ (q) (λ (h) ((h p)
q)))) ((λ (p) (p (λ (x) (λ (y) x)))) l)) ((r p) ((λ (p) (p (λ (x) (λ (y) y)))) l)))) ((r p) ((λ (p) (p (λ (x) (λ (y) y))))
l)))))))) p) l))) (λ (x) (((λ (x) (λ (y) ((λ (n) ((n (λ (x) (λ (x) (λ (y) y)))) (λ (x) (λ (y) x)))) (((λ (m) (λ (n) ((n
(λ (n) (λ (f) (λ (x) (((n (λ (g) (λ (h) (h (g f))))) (λ (u) x)) (λ (u) u)))))) m))) x) y)))) x) p))) s))) (((λ (p) (λ (q)
(λ (h) ((h p) q)))) p) (λ (x) (λ (x) (λ (y) x)))))) (r (((λ (p) (λ (l) ((((λ (f) ((λ (g) (f (λ (x) ((g g) x)))) (λ (g) (f
(λ (x) ((g g) x)))))) (λ (r) (λ (p) (λ (l) ((((λ (p) (λ (x) (λ (y) ((p x) y)))) ((λ (p) (p (λ (x) (λ (y) (λ (x) (λ (y)
y)))))) l)) (λ (x) (λ (x) (λ (y) x)))) ((((λ (p) (λ (x) (λ (y) ((p x) y)))) (p ((λ (p) (p (λ (x) (λ (y) x)))) l))) (((λ
(p) (λ (q) (λ (h) ((h p) q)))) ((λ (p) (p (λ (x) (λ (y) x)))) l)) ((r p) ((λ (p) (p (λ (x) (λ (y) y)))) l)))) ((r p) ((λ
(p) (p (λ (x) (λ (y) y)))) l)))))))) p) l))) (λ (x) (((λ (x) (λ (y) ((λ (p) ((p (λ (x) (λ (y) y))) (λ (x) (λ (y) x))))
(((λ (x) (λ (y) ((λ (n) ((n (λ (x) (λ (x) (λ (y) y)))) (λ (x) (λ (y) x)))) (((λ (m) (λ (n) ((n (λ (n) (λ (f) (λ (x) (((n
(λ (g) (λ (h) (h (g f))))) (λ (u) x)) (λ (u) u)))))) m))) x) y)))) x) y)))) x) p))) s)))) ((λ (p) (p (λ (x) (λ (y) y))))
l))) ((λ (p) (p (λ (x) (λ (y) x)))) l)))))) l))
(((λ (p) (λ (q) (λ (h) ((h p) q)))) (λ (f) (λ (x) (f (f (f x)))))) (((λ (p) (λ (q) (λ (h) ((h p) q)))) (λ (f) (λ (x) (f
x)))) (((λ (p) (λ (q) (λ (h) ((h p) q)))) (λ (f) (λ (x) (f (f x))))) (λ (x) (λ (x) (λ (y) x)))))))
Not so Basic Lambda /
Quicksort

22. Abstract Machine
• How does one design an abstract machine?
• Why do some abstract machines operate on λ-
terms directly whereas others operate on
compiled λ-terms?
• How does one prove the correctness of an
abstract machine?

24. Abstract Machine
• Does not need tree-shaped data structure to interpreter
code — much better performance
• Does not compile code to a native machine instructions
— freedom and architecture-independent
• AM compiles code to a sequence of (abstract)
instructions, defines reduction strategy, evaluation order
and data representations
• Call-by-name (lazy) and call-by-value (eager) defined by
an evaluation strategy that results into different machines

25. Abstract Machine
• Landin invented SECD machine
• Plotkin proved its correctness
• Felleisen invented CEK machine
• Many others proposed modifications and other
discoverables

27. Abstract Machine for
Arithmetic Expressions
• Arithmetic expression:
• a ::= N | a1 + a2 | a1 − a2 | . . .
• Instruction set:
• C(N) — push integer N on stack
• ADD — pop two integers, push their sum
• SUB — pop two integers, push their difference

28. Evaluation scheme
• E(N) = C(N)
• E(a1 + a2) = E(a1); E(a2); ADD
• E(a1 - a2) = E(a1); E(a2); SUB
• Example: E(3+(2-1)) = C(3); C(2); C(1); SUB; ADD

29. Transitions
Machine state afterMachine state before

30. Evaluation
Example: E(3 + (2 - 1))

31. SECD Machine
S — Stack (holding intermediate results and return
addresses)
E — Environment (giving values to symbols)
C — Code (instructions yet to be executed)
D — Dump (S.E.C)

32. SECD Machine
• Instruction set (forget for a moment about Wikipedia):
• ACCESS(n) — push n-th field of the environment
• CLOSURE(c) — push closure of code c with current environment
• APPLY — pop function closure and argument, perform
application
• RETURN — terminate current function, jump back to caller
• LET* — pop value and add it to environment
• ENDLET* — discard first entry of environment

33. De Bruijn Indices
λx.λy.x
λx.λy.y
λf.λx.x
λf.λx.f x
λf.λx.f (f x)
True
False
Zero
One
Two
λx1.λx0.x0
λx1.λx0.x1
λx1.λx0.x0
λx1.λx0.x1 x0
λx1.λx0.x1 (x1 x0)

34. Evaluation scheme
• E(n) = ACCESS(n)
• E(λa) = CLOSURE(E(a); RETURN)
• E(let a in b) = E(a); LET; E(b); ENDLET
• E(a b) = E(a); E(b); APPLY

35. Transitions
Machine state afterMachine state before

36. Evaluation
Example: (λx. x+1) 2 = CLOSURE(c); C(2); APPLY
where c = ACCESS(1); C(1) ; ADD; RETURN

37. Tail call optimization
• tail call optimization drops extra RETURN
instruction enabling function to return result
directly to function and saving stack space
• E(λa) = CLOSURE(E(a); RETURN)

38. Evaluation scheme
• T(let a in b) = E(a); LET; T(b)
• T(a b) = E(a); E(b); TAILAPPLY
• T(a) = E(a); RETURN
• E(n) = ACCESS(n)
• E(λa) = CLOSURE(T(a))
• E(let a in b) = E(a); LET; E(b); ENDLET
• E(a b) = E(a); E(b); APPLY

39. Evaluation
Machine state afterMachine state before

40. K Machine
S — Stack (holding intermediate results and return
addresses)
E — Environment (giving thunks to symbols)
C — Code (instructions yet to be executed)

41. Evaluation scheme
• E(n) = ACCESS(n)
• E(λa) = GRAB; E(a)
• E(a b) = PUSH(E(b)); E(a)
• ACCESS — start evaluating the N-th thunk in the environment
• PUSH(c) — push a thunk for code c
• GRAB — pop one argument and add it to environment
Example: (λx. x+1) 2 = PUSH(C(2)); GRAB; ACCESS(1); C(1); ADD

42. Is it any good?

43. Yes

44. Wanna know more?
• ZAM-, B-, G- machines
• BLC
• Cata-, Hylo-, Ana- morphisms as a reduction
• P-numerals vs Church version
• Ultimately, Read T.S.Eliot

46. Thank you!
_epicmonkey
epicmonkey
mail@epicmonkey.org