Muroya (U. B’ham. & RIMS, Kyoto U.)

A Graph-Rewriting Perspective of the Beta-Law

Dan R. GhicaTodd Waugh Ambridge

(University of Birmingham)

LOLA 2018 (Oxford), 7 July 2018

Koko Muroya(University of Birmingham

& RIMS, Kyoto University)

work in progress

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation

“A function can be applied to a value before evaluation without changing the outcome”

2

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

3

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

4

basic operations(nat, int, float, ...)

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

5

algebraicdata structures

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

6

recursion

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

7

conditionalstatement

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

8

algebraic effects& handlers

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

9

control operators

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

justification by (operational) semantics, but how?

10

Muroya (U. B’ham. & RIMS, Kyoto U.)

Call-by-value beta-law

golden standard of (functional) program equivalence and compiler optimisation… respected by most intrinsic/extrinsic language extensions

justification by (operational) semantics, but how?

11

Muroya (U. B’ham. & RIMS, Kyoto U.)

Question

Given an extension of untyped λ-calculus,

what semantic property of the extension

validates the call-by-value beta-law?

12

Muroya (U. B’ham. & RIMS, Kyoto U.)

Question

Given an extension of untyped λ-calculus,

what operational-semantic property of the extension

validates the call-by-value beta-law?

13

Muroya (U. B’ham. & RIMS, Kyoto U.)

Question

Given an extension of untyped λ-calculus,

what operational-semantic property of the extension

validates the call-by-value beta-law?

14

Answer?

Muroya (U. B’ham. & RIMS, Kyoto U.)

Question

Given an extension of untyped λ-calculus,

what operational-semantic property of the extension

validates the call-by-value beta-law?

15

Answer?

A formal answer is yet to be stated…

But a graph-rewriting perspective provides:

● a useful & robust method

● key observations

Muroya (U. B’ham. & RIMS, Kyoto U.)

Methodology

Given an operational semantics of an extended λ-calculus:

define the contextual equivalence by:

prove the beta-law:

and observe some sufficient condition.

16

closed term

basic constant

Muroya (U. B’ham. & RIMS, Kyoto U.)

Methodology

Given an operational semantics of an extended λ-calculus:

define the contextual equivalence by:

prove the beta-law:

and observe some sufficient condition.

17

closed term

basic constant

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

18

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step reduction

19

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step reduction

… obscures a sub-term of interest :-(

20

redex searching(i.e. decomposition into evaluation context & redex)

obscures `t`

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

21

closed term

basic constant

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

22

graph representationwith unique open edge

closed term

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

23

distinguished edge/node as

“token”

closed term

graph representationwith unique open edge

Muroya (U. B’ham. & RIMS, Kyoto U.)

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

● redex searching (moving the token)

● rewriting (replacing a sub-graph)

Which operational semantics?

24

Muroya (U. B’ham. & RIMS, Kyoto U.)

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

● redex searching (moving the token)

● rewriting (replacing a sub-graph)

Which operational semantics?

25

garbage

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

… keeps a sub-term of interest traceable :-)

26

Muroya (U. B’ham. & RIMS, Kyoto U.)

Which operational semantics?

● easy to extend (esp. by nondeterminism, observables)

● easy to prove a contextual equivalence

small-step “token-guided” graph-rewriting

● visible interaction between the token and a sub-graph

○ redex searching○ rewriting

step-wise reasoning to prove a contextual equivalence27

vs

27

Muroya (U. B’ham. & RIMS, Kyoto U.)

Methodology

Given operational semantics of an extended λ-calculus:

define the contextual equivalence by:

prove the beta-law:

and observe some sufficient condition.

28

closed term

same basic constants

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

Given operational semantics:

define the contextual equivalence by:

prove the beta-law:

and observe some sufficient condition.

29

closed term

same basic constants

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

Given operational semantics:

30

closed term

linear variable

“linear” recursion

basic constant

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

Given operational semantics:

define the contextual equivalence by:

prove the beta-law:

and observe some sufficient condition.

31

closed term

same basic constants

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

… prove the beta-law:

32

same “graph-context”with matching

interface

no garbage created

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

… prove the beta-law by step-wise reasoning:

33

no garbage created

same “graph-context”with matching

interface

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

… prove the beta-law by step-wise reasoning:

1. redex searching “within”

graph-context

2. rewriting “in” graph-context

3. visiting the hole

34

same “graph-context”with matching

interface

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (1/6)

Case study: linear λ-calculus + “linear” recursion

35

searching stopped

at a value

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (2/6)

Case study: linear λ-calculus + “linear” recursion

36

searching stopped

at a value

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (3/6)

Case study: linear λ-calculus + “linear” recursion

37

a redex found

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (4/6)

Case study: linear λ-calculus + “linear” recursion

38

right-to-left call-by-value

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (5/6)

Case study: linear λ-calculus + “linear” recursion

39

right-to-left call-by-value

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching “within” graph-context (6/6)

Case study: linear λ-calculus + “linear” recursion

40

a redex found

Muroya (U. B’ham. & RIMS, Kyoto U.)

1. redex searching within graph-context (6 cases)

observation: only one node is inspected at each step

Case study: linear λ-calculus + “linear” recursion

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context (1/3)

Case study: linear λ-calculus + “linear” recursion

42

call-by-value beta-reduction

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context (2/3)

Case study: linear λ-calculus + “linear” recursion

43

call-by-value beta-reduction

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context

observation: the hole is not involved

Case study: linear λ-calculus + “linear” recursion

44

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context (3/3)

Case study: linear λ-calculus + “linear” recursion

45

reduction for recursion

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context (3/3)

Case study: linear λ-calculus + “linear” recursion

46

reduction for recursion

`G` contains:

● all reachable nodes

from `μ`

● hence,

○ none of the hole

○ or, all of the hole

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context (3/3)

Case study: linear λ-calculus + “linear” recursion

47

reduction for recursion

the hole is

● not involved

● or, duplicated as a

part of `G`

Muroya (U. B’ham. & RIMS, Kyoto U.)

2. rewriting “in” graph-context

observation: the hole is not involved, or is duplicated as a

whole

observation 2: each rewriting step is “history-free”

Case study: linear λ-calculus + “linear” recursion

48

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

49

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

50

1

right-to-left call-by-value

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

51

2

searching stopped

at a value

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

52

3

right-to-left call-by-value

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

53

4

searching stopped

at a value

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

54

5a redex found

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

55

6beta-reduction

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1/1)

Case study: linear λ-calculus + “linear” recursion

56

6 0

Muroya (U. B’ham. & RIMS, Kyoto U.)

3. visiting the hole (1 case)

observation: the hole is reduced

Case study: linear λ-calculus + “linear” recursion

57

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

Given operational semantics:

define the contextual equivalence by:

prove the beta-law by step-wise reasoning:

and observe some sufficient condition.

58

closed term

same basic constants

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case study: linear λ-calculus + “linear” recursion

… prove the beta-law by step-wise reasoning,

and observe that:

1. redex searching only inspects one node at each step2. rewriting preserves, duplicates or simply reduces a

beta-redex.3. rewriting is “history-free”.

59

Muroya (U. B’ham. & RIMS, Kyoto U.)

Case studies so far

… prove the beta-law by step-wise reasoning,

and observe that:

1. redex searching only inspects one node at each step2. rewriting preserves, duplicates or simply reduces a

beta-redex.3. rewriting is “history-free”.

60

✓ untyped pure λ-calculus

✓ basic operations, recursion, if-statement

✓ control operators: call/cc, shift/reset

● algebraic effects & handlersmethod needs to be slightly

adjusted

Muroya (U. B’ham. & RIMS, Kyoto U.)

Question

Given an extension of untyped λ-calculus,

what operational-semantic property of the extension

validates the call-by-value beta-law?

61

Answer?

A formal answer is yet to be stated…

But a graph-rewriting perspective provides:

● a useful & robust method

● key observations