Full Reduction in Open Strict Calculus Alvaro Garc a IMDEA - - PowerPoint PPT Presentation

Full Reduction in Open Strict Calculus

´ Alvaro Garc´ ıa IMDEA Software 2011

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Motivation

◮ Typing rules for dependent type systems:

Γ ⊢ t : τ τ =β τ ′ Γ ⊢ t : τ ′

(cons)

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Motivation

◮ Typing rules for dependent type systems:

Γ ⊢ t : τ τ =β τ ′ Γ ⊢ t : τ ′

(cons)

◮ Usually by-value (strict) semantics.

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Target Strategies

Full-reducing strategies in open strict calculus:

◮ Full normal forms (NF).

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Target Strategies

Full-reducing strategies in open strict calculus:

◮ Full normal forms (NF). ◮ Free variables.

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Target Strategies

Full-reducing strategies in open strict calculus:

◮ Full normal forms (NF). ◮ Free variables. ◮ By-value semantics.

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Plotkin’s theory λN

β (λx.B) N ⊲ [N/x]B

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Plotkin’s theory λN

β (λx.B) N ⊲ [N/x]B µ

M ⊲ M′ M N ⊲ M′ N

ν

N ⊲ N′ M N ⊲ M N′

ξ

B ⊲ B′ λx.B ⊲ λx.B′

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Plotkin’s theory λN

β (λx.B) N ⊲ [N/x]B µ

M ⊲ M′ M N ⊲ M′ N

ν

N ⊲ N′ M N ⊲ M N′

ξ

B ⊲ B′ λx.B ⊲ λx.B′

ρ M ⊲ M τ M ⊲ N

N ⊲ P M ⊲ P

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Plotkin’s theory λN

β (λx.B) N = [N/x]B µ

M = M′ M N = M′ N

ν

N = N′ M N = M N′

ξ

B = B′ λx.B = λx.B′

ρ M = M τ M = N

N = P M = P

σ M = N

N = M

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Plotkin’s theory λV

βV (λx.B) N ⊲ [N/x]B (N ∈ Val)

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Plotkin’s theory λV

βV (λx.B) N ⊲ [N/x]B (N ∈ Val) µ

M ⊲ M′ M N ⊲ M′ N

ν

N ⊲ N′ M N ⊲ M N′

ξ

B ⊲ B′ λx.B ⊲ λx.B′

5 / 1

Plotkin’s theory λV

βV (λx.B) N ⊲ [N/x]B (N ∈ Val) µ

M ⊲ M′ M N ⊲ M′ N

ν

N ⊲ N′ M N ⊲ M N′

ξ

B ⊲ B′ λx.B ⊲ λx.B′

ρ M ⊲ M τ M ⊲ N

N ⊲ P M ⊲ P

5 / 1

Plotkin’s theory λV

βV (λx.B) N = [N/x]B (N ∈ Val) µ

M = M′ M N = M′ N

ν

N = N′ M N = M N′

ξ

B = B′ λx.B = λx.B′

ρ M = M τ M = N

N = P M = P

σ M = N

N = M

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Values and Irreducible Forms (I)

◮ Values:

Val ::= x | λx.B

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Values and Irreducible Forms (I)

◮ Values:

Val ::= x | λx.B

◮ Weak normal forms in λV :

WNF V ::= λx.B | StuckW StuckW ::= x WNF ∗ | ((λx.B) StuckW ) WNF ∗

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Values and Irreducible Forms (I)

◮ Values:

Val ::= x | λx.B

◮ Weak normal forms in λV :

WNF V ::= λx.B | StuckW StuckW ::= x WNF ∗ | ((λx.B) StuckW ) WNF ∗ Note that Val = (WNF V − StuckW ) ∪ {x}.

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Values and Irreducible Forms (II)

◮ Normal forms in λV :

NF V ::= λx.NF V | StuckV StuckV ::= x NF V

∗

| ((λx.NF V ) StuckV ) NF V

∗

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Values and Irreducible Forms (II)

◮ Normal forms in λV :

NF V ::= λx.NF V | StuckV StuckV ::= x NF V

∗

| ((λx.NF V ) StuckV ) NF V

∗

Note that Val = (NF V − StuckV ) ∪ {x}.

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Call-by-value Strategy (cbv)

Uniform standard strategy in λV :

var x → x

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Call-by-value Strategy (cbv)

Uniform standard strategy in λV :

var x → x abs λx.B → λx.B

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Call-by-value Strategy (cbv)

Uniform standard strategy in λV :

var x → x abs λx.B → λx.B red M → λx.B

N → N′ [N′/x]B → B′ M N → B′

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Call-by-value Strategy (cbv)

Uniform standard strategy in λV :

var x → x abs λx.B → λx.B red M → λx.B

N → N′ [N′/x]B → B′ M N → B′

app M → M′ ≡ λx.B

N → N′ M N → M′ N′

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Standardisation Theorem

Theorem

λV ⊢ M ⊲ N iff s.r. M ⊲1 . . . ⊲1 N

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Standardisation Theorem

Theorem

λV ⊢ M ⊲ N iff s.r. M ⊲1 . . . ⊲1 N

Corolary

λV ⊢ M ⊲ N, N ∈ Val iff M

cbv

→ N′, N′ ∈ Val

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Standardisation Theorem

Theorem

λV ⊢ M ⊲ N iff s.r. M ⊲1 . . . ⊲1 N

Corolary

λV ⊢ M ⊲ N, N ∈ Val iff M

cbv

→ N′, N′ ∈ Val Weak reduction is not enoguh for implementing joinability!

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Full Reduction and Confluence

◮ Restricting NF’s as arguments breaks Church-Rosser:

M ≡ (λx.(λy.z) (x (λx.x x))) (λx.x x)

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Hybrid Strategies

Example: call-by-name and normal order in λN.

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Hybrid Strategies

Example: call-by-name and normal order in λN. Call-by-name (cbn): x

cbn

→ x λx.B

cbn

→ λx.B

• M

cbn

→ M′ ≡ λx.B [N/x]B

cbn

→ S M N

cbn

→ S

• M

cbn

→ M′ ≡ λx.B M N

cbn

→ M′ N

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Hybrid Strategies

Example: call-by-name and normal order in λN. Call-by-name (cbn): Normal order (nor): x

cbn

→ x x

nor

→ x λx.B

cbn

→ λx.B B

nor

→ B′ λx.B

nor

→ λx.B′

• M

cbn

→ M′ ≡ λx.B [N/x]B

cbn

→ S M N

cbn

→ S

• M

cbn

→ M′ ≡ λx.B [N/x]B

nor

→ S M N

nor

→ S

• M

cbn

→ M′ ≡ λx.B M N

cbn

→ M′ N

• M

cbn

→ M′ ≡ λx.B M′ nor → M′′ N

nor

→ N′′ M N

nor

→ M′′ N′′ Subsidiary Hybrid

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Hybrid Strategies

Example: call-by-name and normal order in λN. Call-by-name (cbn): Normal order (nor): x

cbn

→ x x

nor

→ x λx.B

cbn

→ λx.B B

nor

→ B′ λx.B

nor

→ λx.B′

• M

cbn

→ M′ ≡ λx.B [N/x]B

cbn

→ S M N

cbn

→ S

• M

cbn

→ M′ ≡ λx.B [N/x]B

nor

→ S M N

nor

→ S

• M

cbn

→ M′ ≡ λx.B M N

cbn

→ M′ N

• M

cbn

→ M′ ≡ λx.B M′ nor → M′′ N

nor

→ N′′ M N

nor

→ M′′ N′′ Subsidiary Hybrid

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Rule Template

var x → x

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Rule Template

var x → x abs

B

la

→ B′ λx.B → λx.B′

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Rule Template

var x → x abs

B

la

→ B′ λx.B → λx.B′

red M → λx.B

N

ar1

→ N′ [N′/x]B → B′ M N → B′

12 / 1

Rule Template

var x → x abs

B

la

→ B′ λx.B → λx.B′

red M → λx.B

N

ar1

→ N′ [N′/x]B → B′ M N → B′

app M → M′ ≡ λx.B

N

ar2

→ N′ M N → M′ N′

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The Beta Cube

Booleans la, ar1 and ar2 encode the occurrence of the coloured premises in the rule template.

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The Beta Cube

Booleans la, ar1 and ar2 encode the occurrence of the coloured premises in the rule template. (⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• la
• ar1
• ar2
• 13 / 1

The Beta Cube

Booleans la, ar1 and ar2 encode the occurrence of the coloured premises in the rule template. (⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• la
• 0 weakness

1 fullness ar1

• 0 non-strictness

1 strictness ar2

• 0 headness

1 non-headness

13 / 1

The Beta Cube

Booleans la, ar1 and ar2 encode the occurrence of the coloured premises in the rule template. (⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• la
• 0 weakness

1 fullness ar1

• 0 non-strictness

1 strictness ar2

• 0 headness

1 non-headness

nf wnf

• hnf
• whnf
• 13 / 1

Hybridisation Operator

x

sub

→ x B

sub

→ B′ λx.B

sub

→ λx.B′

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

sub

→ S M N

sub

→ S

• M

sub

→ M′ ≡ λx.B N

sub

→ N′ M N

sub

→ M′ N′ Subsidiary

14 / 1

Hybridisation Operator

x

sub

→ x x

hyb

→ x B

sub

→ B′ λx.B

sub

→ λx.B′ B

hyb

→ B′ λx.B

hyb

→ λx.B′

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

sub

→ S M N

sub

→ S

• M

sub

→ λx.B N

hyb

→ N′ [N′/x]B

hyb

→ S M N

hyb

→ S

• M

sub

→ M′ ≡ λx.B N

sub

→ N′ M N

sub

→ M′ N′

• M

sub

→ M′ ≡ λx.B M′ hyb → M′′ N

hyb

→ N′ M N

hyb

→ M′′ N′ Subsidiary Hybrid

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Hybrids in the Non-strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• nor

= hyb(101, cbn)

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Hybrids in the Non-strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• nor

= hyb(101, cbn) hn = hyb(101, he)

15 / 1

Hybrids in the Non-strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• nor

= hyb(101, cbn) hn = hyb(101, he) hr = hyb(he, cbn)

15 / 1

Hybrids in the Non-strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• nor

= hyb(101, cbn) hn = hyb(101, he) hr = hyb(he, cbn)

Conjecture

nor = hn = hyb(101, 001)

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Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv)

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV !

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

ha

→∗ (λx.B) N

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

ha

→∗ (λx.B) N

ha

→∗ (λx.B) NV

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

ha

→∗ (λx.B) N

ha

→∗ (λx.B) NV

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

ha

→∗ (λx.B) N

ha

→∗ (λx.B) NV

ha

→∗ (λx.B) NNF

16 / 1

Hybrids in the Strict Face of the Cube

(⊑)

AOR CBV

• 101
• 001
• 110
• 010
• HE
• CBN
• ha

= hyb(aor, cbv) ha is not standard in λV ! M N

ha

→∗ (λx.B) N

ha

→∗ (λx.B) NV

ha

→∗ (λx.B) NNF

ha

→1 [NNF/x]B

16 / 1

Standard Hybridisation Operator

x

sub

→ x B

sub

→ B′ λx.B

sub

→ λx.B′

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

sub

→ S M N

sub

→ S

• M

sub

→ M′ ≡ λx.B N

sub

→ N′ M N

sub

→ M′ N′ Subsidiary

17 / 1

Standard Hybridisation Operator

x

sub

→ x x

hyb

→ x B

sub

→ B′ λx.B

sub

→ λx.B′ B

hyb

→ B′ λx.B

hyb

→ λx.B′

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

sub

→ S M N

sub

→ S

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

hyb

→ S M N

hyb

→ S

• M

sub

→ M′ ≡ λx.B N

sub

→ N′ M N

sub

→ M′ N′

• M

sub

→ M′ ≡ λx.B M′ hyb → M′′ N

hyb

→ N′ M N

hyb

→ M′′ N′ Subsidiary Hybrid

17 / 1

Standard Hybridisation Operator

x

sub

→ x x

hyb

→ x B

sub

→ B′ λx.B

sub

→ λx.B′ B

hyb

→ B′ λx.B

hyb

→ λx.B′

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

sub

→ S M N

sub

→ S

• M

sub

→ λx.B N

sub

→ N′ [N′/x]B

hyb

→ S M N

hyb

→ S

• M

sub

→ M′ ≡ λx.B N

sub

→ N′ M N

sub

→ M′ N′

• M

sub

→ M′ ≡ λx.B M′ hyb → M′′ N

hyb

→ N′ M N

hyb

→ M′′ N′ Subsidiary Hybrid

17 / 1

Strict Face with Open terms

◮ Cbv is not correct with respect to λV :

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Strict Face with Open terms

◮ Cbv is not correct with respect to λV :

(λx.B) (y z)

cbv

→1 [(y z)/x]B

18 / 1

Strict Face with Open terms

◮ Cbv is not correct with respect to λV :

(λx.B) (y z)

cbv

→1 [(y z)/x]B (y z) is a stuck term, not a value!

18 / 1

Strict Face with Open terms

◮ Cbv is not correct with respect to λV :

(λx.B) (y z)

cbv

→1 [(y z)/x]B (y z) is a stuck term, not a value!

◮ Do we really need open terms?

18 / 1

Strict Face with Open terms

◮ Cbv is not correct with respect to λV :

(λx.B) (y z)

cbv

→1 [(y z)/x]B (y z) is a stuck term, not a value!

◮ Do we really need open terms?

◮ Open terms may occur as subterms of well-formed (closed) terms

when performing full reduction.

18 / 1

Rule Template for Open Terms

var x → x

19 / 1

Rule Template for Open Terms

var x → x abs

B

la

→ B′ λx.B → λx.B′

19 / 1

Rule Template for Open Terms

var x → x abs

B

la

→ B′ λx.B → λx.B′

red M → λx.B

N

ar1

→ N′ ∈ Val [N′/x]B → B′ M N → B′

19 / 1

Rule Template for Open Terms

var x → x abs

B

la

→ B′ λx.B → λx.B′

red M → λx.B

N

ar1

→ N′ ∈ Val [N′/x]B → B′ M N → B′

nred M → λx.B

N

ar1

→ N′ ∈ Val M N → M′ N′

19 / 1

Rule Template for Open Terms

var x → x abs

B

la

→ B′ λx.B → λx.B′

red M → λx.B

N

ar1

→ N′ ∈ Val [N′/x]B → B′ M N → B′

nred M → λx.B

N

ar1

→ N′ ∈ Val M N → M′ N′

app M → M′ ≡ λx.B

N

ar2

→ N′ M N → M′ N′

19 / 1

Cube and Hybridisation for Open Terms

Extending the cube and the standard hybridisation operator with the new rule is trivial.

20 / 1

Strict Normal Order Strategy

Standard normalising strategy in λV :

var

x

snor

→ x

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Strict Normal Order Strategy

Standard normalising strategy in λV :

var

x

snor

→ x

abs

B

snor

→ B′ λx.B

snor

→ λx.B′

21 / 1

Strict Normal Order Strategy

Standard normalising strategy in λV :

var

x

snor

→ x

abs

B

snor

→ B′ λx.B

snor

→ λx.B′

red M

cbvO

→ λx.B N

cbvO

→ N′ ∈ Val [N′/x]B

snor

→ B′ M N

snor

→ B′

21 / 1

Strict Normal Order Strategy

Standard normalising strategy in λV :

var

x

snor

→ x

abs

B

snor

→ B′ λx.B

snor

→ λx.B′

red M

cbvO

→ λx.B N

cbvO

→ N′ ∈ Val [N′/x]B

snor

→ B′ M N

snor

→ B′

nred M

cbvO

→ λx.B N

cbvO

→ N′ ∈ Val M′ snor → M′′ N′ snor → N′′ M N

snor

→ M′′ N′′

21 / 1

Strict Normal Order Strategy

Standard normalising strategy in λV :

var

x

snor

→ x

abs

B

snor

→ B′ λx.B

snor

→ λx.B′

red M

cbvO

→ λx.B N

cbvO

→ N′ ∈ Val [N′/x]B

snor

→ B′ M N

snor

→ B′

nred M

cbvO

→ λx.B N

cbvO

→ N′ ∈ Val M′ snor → M′′ N′ snor → N′′ M N

snor

→ M′′ N′′

app M

cbvO

→ M′ ≡ λx.B M′ snor → M′′ N

snor

→ N′ M N

snor

→ M′ N′

21 / 1

Previous work

◮ Plotkin 1974: accounts for standardisation and confluence, but not

for full reduction

22 / 1

Previous work

◮ Plotkin 1974: accounts for standardisation and confluence, but not

for full reduction

◮ Sestoft 2002: accounts for full reduction, but doesn’t account for

standardisation.

22 / 1

Previous work

◮ Plotkin 1974: accounts for standardisation and confluence, but not

for full reduction

◮ Sestoft 2002: accounts for full reduction, but doesn’t account for

standardisation.

◮ Leroy and Gregoire 2002: accounts for full reduction and

standardisation, but doesn’t account for confluence in λV .

22 / 1

Conclusions

◮ Strict normal order: accounts for full reduction, standardisation and

confluence in λV .

23 / 1

Conclusions

◮ Strict normal order: accounts for full reduction, standardisation and

confluence in λV .

◮ Beta Cube and hybridisation to articulate strategies space.

23 / 1

Conclusions

◮ Strict normal order: accounts for full reduction, standardisation and

confluence in λV .

◮ Beta Cube and hybridisation to articulate strategies space. ◮ Plotkin’s meta-theory leads the improvements in cube and

hybridisation.

23 / 1