Reduction strategies for CBN and CBV Structural operational style - - PowerPoint PPT Presentation

▶
reduction strategies for cbn and cbv
SMART_READER_LITE
LIVE PREVIEW

Reduction strategies for CBN and CBV Structural operational style - - PowerPoint PPT Presentation

Jul 27, 2023 124 likes •1.28k views

Reduction strategies for CBN and CBV Structural operational style and Felleisen style Bas Broere April 24th 2018 Bas Broere Reduction strategies for CBN and CBV 1/22 Introduction Disclaimer I have (almost) no experience in functional

slide-1
SLIDE 1

Reduction strategies for CBN and CBV

Structural operational style and Felleisen style Bas Broere April 24th 2018

Bas Broere — Reduction strategies for CBN and CBV 1/22

slide-2
SLIDE 2

Introduction

Disclaimer I have (almost) no experience in functional programming.

Bas Broere — Reduction strategies for CBN and CBV 2/22

slide-3
SLIDE 3

Introduction

Disclaimer I have (almost) no experience in functional programming. Question: What is a functional programming language?

Bas Broere — Reduction strategies for CBN and CBV 2/22

slide-4
SLIDE 4

Introduction

Disclaimer I have (almost) no experience in functional programming. Question: What is a functional programming language? ”A language that takes functions seriously”

Bas Broere — Reduction strategies for CBN and CBV 2/22

slide-5
SLIDE 5

Introduction

Disclaimer I have (almost) no experience in functional programming. Question: What is a functional programming language? ”A language that takes functions seriously” A language that emphasises what needs to be calculated, instead of how this needs to be done, i.e. what does it mean to calculate something?

Bas Broere — Reduction strategies for CBN and CBV 2/22

slide-6
SLIDE 6

Introduction

Disclaimer I have (almost) no experience in functional programming. Question: What is a functional programming language? ”A language that takes functions seriously” A language that emphasises what needs to be calculated, instead of how this needs to be done, i.e. what does it mean to calculate something? Important: It supports passing functions as arguments, returning them as value etc. Basically, you can do everything with them.

Bas Broere — Reduction strategies for CBN and CBV 2/22

slide-7
SLIDE 7

Setting up an FP language

Bas Broere — Reduction strategies for CBN and CBV 3/22

slide-8
SLIDE 8

Setting up an FP language

1 Take a λ-calculus, so:

Terms: M, N ::= x | λx.M | M N; Define β-reduction as the smallest congruence relation s.t. (λx.M) N →β M[x := N];

Bas Broere — Reduction strategies for CBN and CBV 3/22

slide-9
SLIDE 9

Setting up an FP language

1 Take a λ-calculus, so:

Terms: M, N ::= x | λx.M | M N; Define β-reduction as the smallest congruence relation s.t. (λx.M) N →β M[x := N];

2 Choose your reduction strategy (topic of today);

Bas Broere — Reduction strategies for CBN and CBV 3/22

slide-10
SLIDE 10

Setting up an FP language

1 Take a λ-calculus, so:

Terms: M, N ::= x | λx.M | M N; Define β-reduction as the smallest congruence relation s.t. (λx.M) N →β M[x := N];

2 Choose your reduction strategy (topic of today); 3 Add some stuff like primitive data-types and operations;

Bas Broere — Reduction strategies for CBN and CBV 3/22

slide-11
SLIDE 11

Setting up an FP language

1 Take a λ-calculus, so:

Terms: M, N ::= x | λx.M | M N; Define β-reduction as the smallest congruence relation s.t. (λx.M) N →β M[x := N];

2 Choose your reduction strategy (topic of today); 3 Add some stuff like primitive data-types and operations; 4 Make an execution model.

Bas Broere — Reduction strategies for CBN and CBV 3/22

slide-12
SLIDE 12

Evaluation strategies

First: Choose evaluation strategy.

Bas Broere — Reduction strategies for CBN and CBV 4/22

slide-13
SLIDE 13

Evaluation strategies

First: Choose evaluation strategy. Difference evaluation and reduction strategies An evaluation strategy tells you when you need to evaluate an expression.

Bas Broere — Reduction strategies for CBN and CBV 4/22

slide-14
SLIDE 14

Evaluation strategies

First: Choose evaluation strategy. Difference evaluation and reduction strategies An evaluation strategy tells you when you need to evaluate an expression. A reduction strategy tells you how to reduce a complex expression.

Bas Broere — Reduction strategies for CBN and CBV 4/22

slide-15
SLIDE 15

Evaluation strategies

Idea:

Bas Broere — Reduction strategies for CBN and CBV 5/22

slide-16
SLIDE 16

Evaluation strategies

Idea: Add integer value n to terms: M, N ::= n | x | λx.M | M N Values (results of evaluations): v ::= n | λx.M

Bas Broere — Reduction strategies for CBN and CBV 5/22

slide-17
SLIDE 17

Evaluation strategies

Idea: Add integer value n to terms: M, N ::= n | x | λx.M | M N Values (results of evaluations): v ::= n | λx.M Call-by-value (CBV) The argument expression is evaluated, and the resulting value is bound to the corresponding variable in the function.

Bas Broere — Reduction strategies for CBN and CBV 5/22

slide-18
SLIDE 18

Evaluation strategies

Idea: Add integer value n to terms: M, N ::= n | x | λx.M | M N Values (results of evaluations): v ::= n | λx.M Call-by-value (CBV) The argument expression is evaluated, and the resulting value is bound to the corresponding variable in the function. Call-by-name (CBN) The arguments to a function are not evaluated before the function is called.

Bas Broere — Reduction strategies for CBN and CBV 5/22

slide-19
SLIDE 19

Reduction strategies for CBV

Structural operation style (SOS) One-step reductions: (λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

Bas Broere — Reduction strategies for CBN and CBV 6/22

slide-20
SLIDE 20

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-21
SLIDE 21

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-22
SLIDE 22

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-23
SLIDE 23

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-24
SLIDE 24

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-25
SLIDE 25

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-26
SLIDE 26

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.x) 1 → 1 (λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 7/22

slide-27
SLIDE 27

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) [ ((λx.x) 1) (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-28
SLIDE 28

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) [ ((λx.x) 1) (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-29
SLIDE 29

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) [ ((λx.x) 1) (λx.x) ] → (λx.λy.y x) [ 1 (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-30
SLIDE 30

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.λy.y x) [ ((λx.x) 1) (λx.x) ] → (λx.λy.y x) [ 1 (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-31
SLIDE 31

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

((λx.x) 1) (λx.x) → 1 (λx.x) (λx.λy.y x) [ ((λx.x) 1) (λx.x) ] → (λx.λy.y x) [ 1 (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-32
SLIDE 32

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

((λx.x) 1) (λx.x) → 1 (λx.x) (λx.λy.y x) [ ((λx.x) 1) (λx.x) ] → (λx.λy.y x) [ 1 (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-33
SLIDE 33

Examples

Recution steps

(λx.M) v → M[x := v] (βv) M → M′ M N → M′ N (app-l) N → N′ v N → v N′ (app-r)

(λx.x) 1 → 1 ((λx.x) 1) (λx.x) → 1 (λx.x) (λx.λy.y x) [ ((λx.x) 1) (λx.x) ] → (λx.λy.y x) [ 1 (λx.x) ]

Bas Broere — Reduction strategies for CBN and CBV 8/22

slide-34
SLIDE 34

Features/limitations of SOS in CBV

We cannot reduce under λ-abstractions (weak reduction). So this is NOT a thing:

M → M′ λx.M → λx.M′

Bas Broere — Reduction strategies for CBN and CBV 9/22

slide-35
SLIDE 35

Features/limitations of SOS in CBV

We cannot reduce under λ-abstractions (weak reduction). So this is NOT a thing:

M → M′ λx.M → λx.M′

We need to reduce N to a value if we want to evaluate (λx.M)N (call-by-value). So also this is NOT a thing:

(λx.M)(N P) → M[x := N P]

Bas Broere — Reduction strategies for CBN and CBV 9/22

slide-36
SLIDE 36

Features/limitations of SOS in CBV

We cannot reduce under λ-abstractions (weak reduction). So this is NOT a thing:

M → M′ λx.M → λx.M′

We need to reduce N to a value if we want to evaluate (λx.M)N (call-by-value). So also this is NOT a thing:

(λx.M)(N P) → M[x := N P]

In the application M N, we need to reduce M to a value first (left-to-right), because:

N → N′ v N → v N′ (app-r).

Bas Broere — Reduction strategies for CBN and CBV 9/22

slide-37
SLIDE 37

Features/limitations of SOS in CBV

We cannot reduce under λ-abstractions (weak reduction). So this is NOT a thing:

M → M′ λx.M → λx.M′

We need to reduce N to a value if we want to evaluate (λx.M)N (call-by-value). So also this is NOT a thing:

(λx.M)(N P) → M[x := N P]

In the application M N, we need to reduce M to a value first (left-to-right), because:

N → N′ v N → v N′ (app-r).

It is deterministic, i.e. for every M there is at most one M′ s.t. M → M′.

Bas Broere — Reduction strategies for CBN and CBV 9/22

slide-38
SLIDE 38

Multiple reductions

When making a reduction sequence we have 3 possibilities:

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-39
SLIDE 39

Multiple reductions

When making a reduction sequence we have 3 possibilities:

1 The reductions end in a value (termination):

M → M1 → M2 → . . . → v.

Example: (λx.λy.y x) ((λx.x)1) (λx.x) → (λx.λy.y x) 1 (λx.x) → (λy.y 1) (λx.x) → (λx.x) 1 → 1

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-40
SLIDE 40

Multiple reductions

When making a reduction sequence we have 3 possibilities:

1 The reductions end in a value (termination):

M → M1 → M2 → . . . → v.

Example: (λx.λy.y x) ((λx.x)1) (λx.x) → (λx.λy.y x) 1 (λx.x) → (λy.y 1) (λx.x) → (λx.x) 1 → 1

2 The reduction never ends (divergence):

M → M1 → M2 → . . . → Mn → . . ..

Example:

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-41
SLIDE 41

Multiple reductions

When making a reduction sequence we have 3 possibilities:

1 The reductions end in a value (termination):

M → M1 → M2 → . . . → v.

Example: (λx.λy.y x) ((λx.x)1) (λx.x) → (λx.λy.y x) 1 (λx.x) → (λy.y 1) (λx.x) → (λx.x) 1 → 1

2 The reduction never ends (divergence):

M → M1 → M2 → . . . → Mn → . . ..

Example: ω ω, where ω = λx.x x.

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-42
SLIDE 42

Multiple reductions

When making a reduction sequence we have 3 possibilities:

1 The reductions end in a value (termination):

M → M1 → M2 → . . . → v.

Example: (λx.λy.y x) ((λx.x)1) (λx.x) → (λx.λy.y x) 1 (λx.x) → (λy.y 1) (λx.x) → (λx.x) 1 → 1

2 The reduction never ends (divergence):

M → M1 → M2 → . . . → Mn → . . ..

Example: ω ω, where ω = λx.x x.

3 Error: M → M1 → M2 → . . . → Mn →, where Mn is not

a value, but also does not reduce.

Example:

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-43
SLIDE 43

Multiple reductions

When making a reduction sequence we have 3 possibilities:

1 The reductions end in a value (termination):

M → M1 → M2 → . . . → v.

Example: (λx.λy.y x) ((λx.x)1) (λx.x) → (λx.λy.y x) 1 (λx.x) → (λy.y 1) (λx.x) → (λx.x) 1 → 1

2 The reduction never ends (divergence):

M → M1 → M2 → . . . → Mn → . . ..

Example: ω ω, where ω = λx.x x.

3 Error: M → M1 → M2 → . . . → Mn →, where Mn is not

a value, but also does not reduce.

Example: ω 2 → 2 2 →.

Bas Broere — Reduction strategies for CBN and CBV 10/22

slide-44
SLIDE 44

Reduction strategies for CBV

Reduction contexts (felleisen style) Head reductions: (λx.M) v →ǫ M[x := v] (βv)

Bas Broere — Reduction strategies for CBN and CBV 11/22

slide-45
SLIDE 45

Reduction strategies for CBV

Reduction contexts (felleisen style) Head reductions: (λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), with E ::= [ ] | E b | v E terms with a hole [ ] in it.

Bas Broere — Reduction strategies for CBN and CBV 11/22

slide-46
SLIDE 46

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-47
SLIDE 47

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-48
SLIDE 48

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-49
SLIDE 49

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-50
SLIDE 50

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-51
SLIDE 51

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-52
SLIDE 52

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-53
SLIDE 53

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-54
SLIDE 54

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x) E = [ ]

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-55
SLIDE 55

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x) E = [ ] → (λx.x) 1

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-56
SLIDE 56

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x) E = [ ] → (λx.x) 1 E = [ ]

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-57
SLIDE 57

Example

Reduction contexts (Felleisen style)

(λx.M) v →ǫ M[x := v] (βv) M →ǫ M′ E[M] → E[M′] (context), E ::= [ ] | E b | v E

Same example as before: (λx.λy.y x) ((λx.x) 1) (λx.x) E = (λx.λy.x y) [ ] (λx.x) → (λx.λy.y x) 1 (λx.x) E = [ ] (λx.x) → (λy.y 1) (λx.x) E = [ ] → (λx.x) 1 E = [ ] → 1

Bas Broere — Reduction strategies for CBN and CBV 12/22

slide-58
SLIDE 58

Equivalence SOS and Felleisen

Remember the earlier example: (λx.x) 1 → 1 (λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x) This corresponds to the context:

Bas Broere — Reduction strategies for CBN and CBV 13/22

slide-59
SLIDE 59

Equivalence SOS and Felleisen

Remember the earlier example: (λx.x) 1 → 1 (λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x) This corresponds to the context: E =

Bas Broere — Reduction strategies for CBN and CBV 13/22

slide-60
SLIDE 60

Equivalence SOS and Felleisen

Remember the earlier example: (λx.x) 1 → 1 (λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x) This corresponds to the context: E = ((λx.λy.y x) [ ]) (λx.x) with head reduction (λx.x) 1 →ǫ 1.

Bas Broere — Reduction strategies for CBN and CBV 13/22

slide-61
SLIDE 61

Equivalence SOS and Felleisen

Remember the earlier example: (λx.x) 1 → 1 (λx.λy.y x) ((λx.x) 1) → (λx.λy.y x) 1 (λx.λy.y x) ((λx.x) 1) (λx.x) → (λx.λy.y x) 1 (λx.x) This corresponds to the context: E = ((λx.λy.y x) [ ]) (λx.x) with head reduction (λx.x) 1 →ǫ 1. Equivalence SOS and Felleisen approach are one-to-one!

Bas Broere — Reduction strategies for CBN and CBV 13/22

slide-62
SLIDE 62

Result

From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*.

Bas Broere — Reduction strategies for CBN and CBV 14/22

slide-63
SLIDE 63

Result

From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*. *: The determinism is also a consequence of the following theorem: Unique decomposition theorem For all terms M, there exists at most one reduction context E and one term N, s.t. M = E[N] and N can reduce by head-reduction.

Bas Broere — Reduction strategies for CBN and CBV 14/22

slide-64
SLIDE 64

Result

From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*. *: The determinism is also a consequence of the following theorem: Unique decomposition theorem For all terms M, there exists at most one reduction context E and one term N, s.t. M = E[N] and N can reduce by head-reduction. Proof: Induction on the terms.

Bas Broere — Reduction strategies for CBN and CBV 14/22

slide-65
SLIDE 65

Reduction strategies for CBN

Call-by-name The arguments to a function are not evaluated before the function is called (perform β-reduction asap).

Bas Broere — Reduction strategies for CBN and CBV 15/22

slide-66
SLIDE 66

Reduction strategies for CBN

Call-by-name The arguments to a function are not evaluated before the function is called (perform β-reduction asap). Structural operation style (SOS) (λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

Bas Broere — Reduction strategies for CBN and CBV 15/22

slide-67
SLIDE 67

Reduction strategies for CBN

Call-by-name The arguments to a function are not evaluated before the function is called (perform β-reduction asap). Structural operation style (SOS) (λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l) Felleisen style (λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), with E ::= [ ] | E M

Bas Broere — Reduction strategies for CBN and CBV 15/22

slide-68
SLIDE 68

Reduction strategies for CBN

Call-by-name The arguments to a function are not evaluated before the function is called (perform β-reduction asap). Structural operation style (SOS) (λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l) Felleisen style (λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), with E ::= [ ] | E M, so E ::= [ ] M1 . . . Mn.

Bas Broere — Reduction strategies for CBN and CBV 15/22

slide-69
SLIDE 69

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-70
SLIDE 70

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-71
SLIDE 71

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-72
SLIDE 72

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-73
SLIDE 73

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z) → (λz.z) ((λx.x) 1)

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-74
SLIDE 74

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z) → (λz.z) ((λx.x) 1) → (λx.x) 1

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-75
SLIDE 75

Examples

Structural operation style (SOS)

(λx.M)N → M[x := N] (βn) M → M′ M N → M′ N (app-l)

The same example once again, in SOS-style: (λx.λy.y x) ((λx.x) 1) (λz.z) → (λy.y ((λx.x) 1)) (λz.z) → (λz.z) ((λx.x) 1) → (λx.x) 1 → 1.

Bas Broere — Reduction strategies for CBN and CBV 16/22

slide-76
SLIDE 76

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-77
SLIDE 77

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λx.x)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-78
SLIDE 78

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-79
SLIDE 79

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-80
SLIDE 80

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-81
SLIDE 81

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ]

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-82
SLIDE 82

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ] → (λz.z) ((λx.x) 1)

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-83
SLIDE 83

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ] → (λz.z) ((λx.x) 1) E = [ ]

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-84
SLIDE 84

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ] → (λz.z) ((λx.x) 1) E = [ ] → (λx.x) 1

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-85
SLIDE 85

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ] → (λz.z) ((λx.x) 1) E = [ ] → (λx.x) 1 E = [ ]

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-86
SLIDE 86

Examples

Felleisen style

(λx.M)N → M[x := N] (βn) M → M′ E[M] → E[M′](context), E ::= [ ] | E M.

And in Felleisen style: (λx.λy.y x) ((λx.x) 1) (λz.z) E = [ ] (λz.z) → (λy.y ((λx.x) 1)) (λz.z) E = [ ] → (λz.z) ((λx.x) 1) E = [ ] → (λx.x) 1 E = [ ] → 1.

Bas Broere — Reduction strategies for CBN and CBV 17/22

slide-87
SLIDE 87

CBV vs CBN

Some examples: Termination:

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-88
SLIDE 88

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) →

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-89
SLIDE 89

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1;

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-90
SLIDE 90

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) →

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-91
SLIDE 91

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . .

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-92
SLIDE 92

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around.

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-93
SLIDE 93

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency:

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-94
SLIDE 94

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: (λx.x + x) M:

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-95
SLIDE 95

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: (λx.x + x) M: CBV evaluates M once, CBN twice;

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-96
SLIDE 96

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: (λx.x + x) M: CBV evaluates M once, CBN twice; (λx.1) M:

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-97
SLIDE 97

CBV vs CBN

Some examples: Termination: CBN: (λx.1) (ω ω) → 1; CBV: (λx.1) (ω ω) → (λx.1) (ω ω) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: (λx.x + x) M: CBV evaluates M once, CBN twice; (λx.1) M: CBV evaluates M once, CBN not at all.

Bas Broere — Reduction strategies for CBN and CBV 18/22

slide-98
SLIDE 98

CBN in CBV language

Bas Broere — Reduction strategies for CBN and CBV 19/22

slide-99
SLIDE 99

CBN in CBV language

Introduce functions λx.M to delay computation of M (thunks).

Bas Broere — Reduction strategies for CBN and CBV 19/22

slide-100
SLIDE 100

CBN in CBV language

Introduce functions λx.M to delay computation of M (thunks).

1 x = x () (application);

Bas Broere — Reduction strategies for CBN and CBV 19/22

slide-101
SLIDE 101

CBN in CBV language

Introduce functions λx.M to delay computation of M (thunks).

1 x = x () (application); 2 λx.M = λx.M;

Bas Broere — Reduction strategies for CBN and CBV 19/22

slide-102
SLIDE 102

CBN in CBV language

Introduce functions λx.M to delay computation of M (thunks).

1 x = x () (application); 2 λx.M = λx.M; 3 M N = M (λx.N),

where () is a constant of type unit and x ∈ FV (N).

Bas Broere — Reduction strategies for CBN and CBV 19/22

slide-103
SLIDE 103

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

=λx.x + x (λy.M)

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-104
SLIDE 104

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-105
SLIDE 105

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-106
SLIDE 106

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-107
SLIDE 107

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

1

=(λx.x () + x ()) (λy.M)

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-108
SLIDE 108

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

1

=(λx.x () + x ()) (λy.M) →β(λy.M) () + (λy.M )()

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-109
SLIDE 109

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

1

=(λx.x () + x ()) (λy.M) →β(λy.M) () + (λy.M )() →βM + M

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-110
SLIDE 110

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

1

=(λx.x () + x ()) (λy.M) →β(λy.M) () + (λy.M )() →βM + M

1

=M () + M () →β M + M

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-111
SLIDE 111

Example

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

(λx.x + x) M

3

= λx.x + x (λy.M)

2

=(λx.x + x) (λy.M)

1

=(λx.x () + x ()) (λy.M) →β(λy.M) () + (λy.M )() →βM + M

1

=M () + M () →β M + M So we need to evaluate M twice now, just as in CBN.

Bas Broere — Reduction strategies for CBN and CBV 20/22

slide-112
SLIDE 112

Types

Rules

x

1

= x (), λx.M

2

= λx.M M N

3

= M (λx.N)

Effect on types:

1 M : A, then M : A; 2 M : A → B, then M : (unit → A) → B; 3 basetype = basetype.

Bas Broere — Reduction strategies for CBN and CBV 21/22

slide-113
SLIDE 113

CBV in CBN language

This is hard!

Bas Broere — Reduction strategies for CBN and CBV 22/22

slide-114
SLIDE 114

CBV in CBN language

This is hard! We will read about it :)

Bas Broere — Reduction strategies for CBN and CBV 22/22