1. Abstract machines & interpreters

2. (Term rewriting semantics) recap

3. (→*) (→) = (→β) ∪ (→α) ∪ (→η)
(→) = (→β) ∪ (→α) ∪ (→η)
(→*)
A non-deterministic pre-order (i.e., reflexive, transitive)

4. α α α α α e0 e1 e2 e3 e4 e5

5. α α α α α e0 e1 e2 e3 e4 e5 β η e6

6. α α α α α e0 e1 e2 e3 e4 e5 β η e6 β β β

7. α α α α α e0 e1 e2 e3 e4 e5 β β
e10 e7 β β
e11 e8 β e9

8. * * α α α α α e0 e1 e2 e3 e4 e5 β β e10 e7 β β e11 e8 β e9 e12
Confluence (i.e., Church-Rosser Theorem) α α α α α e0 e1 e2 e3 e4 e5 β β
e10 e7 β β
e11 e8 β * e9 *
e12

9. { {
((λ (x) E0) E1) →β
E0[x ← E1]
redex

10. (E0 E1)[x ← E] = (E0[x ← E] E1[x ← E])
x[x ← E] = E
y[x ← E] = y where y ≠ x
(E0 E1)[x ← E] = (E0[x ← E] E1[x ← E])
(λ (x) E0)[x ← E] = (λ (x) E0)
(λ (y) E0)[x ← E] = (λ (y) E0[x ← E])
where y ≠ x and y ∉ FV(E)
β-reduction cannot occur when y ∈ FV(E)

11. (...
(λ (x)
((λ (a) (λ (x) a))
(λ (b) x)))
...)
... (λ (x)
(λ (x) (λ (b) x)))

12. (λ (y) E0)[x ← E] = (λ (y) E0[x ← E])
(...
(λ (x)
((λ (a) (λ (x) a))
(λ (b) x)))
...)
(λ (y) E0)[x ← E] = (λ (y) E0[x ← E])
where y ≠ x and y ∉ FV(E)

13. Evaluation contexts ℰ ::= (ℰ e) | (v ℰ) | □ v ::= (λ (x) e)
e ::= (λ (x) e) | (e e)
| x
ℰ ::= (ℰ e) | (v ℰ) | □

14. (((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))
Context and redex { { r
ℰ[(v v)] =
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))
ℰ = (□ (λ (w) w))
r = ((λ (x) ((λ (y) y) x)) (λ (z) z))

15. Abstract Machines

16. (e0, env) ⇓ ((λ (x) e2), env’) (e1, env) ⇓ v1 (e2, env’[x ↦ v1]) ⇓ v2
((e0 e1), env) ⇓ v2
((λ (x) e), env) ⇓ ((λ (x) e), env)
(x, env) ⇓ env(x)

17. Big-step (Natural) Semantics

18. (e0 e1), env
e’, env’

19. e0 e1
(e0 e1), env
e’, env’

20. (define (interp e env)
(match e
[(? symbol? x)
(hash-ref env x)]
[`(λ (,x) ,e0)
`(clo (λ (,x) ,e0) ,env)]
[`(,e0 ,e1)
(define v0 (interp e0 env))
(define v1 (interp e1 env))
(match v0
[`(clo (λ (,x) ,e2) ,env)
(interp e2 (hash-set env x v1))])]))

21. (define (interp e env)
(match e
[(? symbol? x)
(hash-ref env x)]
[`(λ (,x) ,e0)
`(clo (λ (,x) ,e0) ,env)]
[`(,e0 ,e1)
(define v0 (interp e0 env))
(define v1 (interp e1 env))
(match v0
[`(clo (λ (,x) ,e2) ,env)
(interp e2 (hash-set env x v1))])]))

22. (e0, env) ⇓ ((λ (x) e2), env’) (e1, env) ⇓ v1 (e2, env’[x ↦ v1]) ⇓ v2
((e0 e1), env) ⇓ v2
((λ (x) e), env) ⇓ ((λ (x) e), env)
(x, env) ⇓ env(x)

23. (((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

24. ((λ (x) …), ∅) ⇓ ((λ (x) …), ∅)
(((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

25. ((λ (z) z), ∅) ⇓ ((λ (z) z), ∅)
… ⇓ ((λ (x) …), ∅)
((λ (z) z), ∅) ⇓ ((λ (z) z), ∅)
(((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

26. (e0, env) ⇓ ((λ (x) e2), env’) (e1, env) ⇓ v1 (e2, env’[x ↦ v1]) ⇓ v2
((e0 e1), env) ⇓ v2
((λ (x) e), env) ⇓ ((λ (x) e), env)
(x, env) ⇓ env(x)

27. (((λ (y) y) x)), [x ↦ ((λ (z) z), ∅)]) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

28. … ⇓ ((λ (y) y), [x ↦ ((λ (z) z), ∅)])
(((λ (y) y) x)), [x ↦ ((λ (z) z), ∅)]) ⇓ …
(((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

29. (x, [x ↦ ((λ (z) z), ∅)]) ⇓ ((λ (z) z), ∅)
… ⇓ ((λ (y) y), [x ↦ ((λ (z) z), ∅)])
… ⇓ ((λ (z) z), ∅)
(((λ (y) y) x)), [x ↦ ((λ (z) z), ∅)]) ⇓ …
(((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

30. (y, [y ↦ ((λ (z) z), ∅), x ↦ ((λ (z) z), ∅)]) ⇓ ((λ (z) z), ∅)
(x, [x ↦ ((λ (z) z), ∅)]) ⇓ ((λ (z) z), ∅)
… ⇓ ((λ (y) y), [x ↦ ((λ (z) z), ∅)])
… ⇓ ((λ (z) z), ∅)
(((λ (y) y) x)), [x ↦ ((λ (z) z), ∅)]) ⇓ …
(((λ (x) …) (λ (z) z)), ∅) ⇓ …
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

31. (((λ (x) …) (λ (z) z)), ∅) ⇓ ((λ (z) z), ∅)
((λ (w) w), ∅) ⇓ ((λ (w) w), ∅)
(((λ (x) …) (λ (z) z)), ∅) ⇓ ((λ (z) z), ∅)
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

32. (((λ (x) …) (λ (z) z)), ∅) ⇓ ((λ (z) z), ∅)
(z, [z ↦ ((λ (w) w), ∅)]) ⇓ ((λ (w) w), ∅)
((λ (w) w), ∅) ⇓ ((λ (w) w), ∅)
(((λ (x) …) (λ (z) z)), ∅) ⇓ ((λ (z) z), ∅)
( , ∅) ⇓ …
(((λ (x) ((λ (y) y) x)) (λ (z) z)) (λ (w) w))

33. Review Meta-circular interpreters (reuse all features from host)
Term-rewriting (non-deterministic but consistent)
outermost and innermost beta-reductions =>
Lazy (CBN) and Eager (CBV) when stop at λ
Using a grammar for evaluation contexts, we force CBV
Substitution environments and big-step reductions
Equivalent to a closure-creating interpreter

34. Try evaluating using context&redex and then big-step
Exercises
Try evaluating using context&redex and then big-step
(((λ (z) z) (λ (u) (u u))) (λ (x) x))
(((λ (f) (λ (g) ((f g) f))) (λ (z) z)) (λ (u) (u u)))