[PPT] - Compositional Z Confluence for Permutative Conversion Koji Nakazawa PowerPoint Presentation

SLIDE 1

Compositional Z

Confluence for Permutative Conversion

Koji Nakazawa (Nagoya U) joint work with Ken-etsu Fujita (Gunma U) Workshop on Mathematical Logic and its Application 2016.9 @ Kyoto

SLIDE 2

M } } ! ! M1 ! ! M2 } } ∃N

Confluence

SLIDE 3

M } } ! ! M1 ! ! M2 } } ∃N

Confluence

Uniqueness of result of computation

SLIDE 4

M } } ! ! M1 ! ! M2 } } ∃N

Confluence

Uniqueness of result of computation Consistency of equational theory

SLIDE 5

This talk

Brief history of confluence of λ-calculus
parallel reduction and Z theorem
Compositional Z: a new confluence proof
simpler proof of λ + permutation rules
Z property for Church-Rosser theorem
quantitative analysis

SLIDE 6

History of Confluence of λ

SLIDE 7

Terms
Reduction rules

λβ

M, N ::= x | λx.M | MN

(λx.M)N →β M[x := N]

SLIDE 8

History of Confluence of λβ

Church and Rosser (1936) “Some Properties of Conversion”
residuals of redexes
Tait and Martin-Löf (19??)
parallel reduction
Takahashi (1995) “Parallel Reduction in λ-Calculus”
maximum parallel reduction
Dehornoy and van Oostrom (2008)
Z theorem

SLIDE 9

Z theorem

[Dehornoy&van Oostrom 2008]

M N

M∗

N∗

If we find a mapping (·)* s.t. then the reduction system is confluent

SLIDE 10

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 11

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 12

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 13

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 14

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 15

Z theorem

[Dehornoy&van Oostrom 2008]

Z theorem

M N

M∗

N∗

M1

M2
M3
M4
· · ·

N M∗

1

M∗

2

M∗

3

· · ·

SLIDE 16

Confluence of λβ by Z

SLIDE 17

Confluence of λβ by Z

Takahashi's maximum parallel reduction is Z

x∗ = x (λx.M)∗ = λx.M∗ ((λx.M)N)∗ = M∗[x := N∗] (MN)∗ = M∗N∗ (M is not abst.)

SLIDE 18

Confluence of λβ by Z

Takahashi's maximum parallel reduction is Z

x∗ = x (λx.M)∗ = λx.M∗ ((λx.M)N)∗ = M∗[x := N∗] (MN)∗ = M∗N∗ (M is not abst.)

M →∗ M∗ M∗[x := N∗] →∗ (M[x := N])∗

(i) (ii)

Key lemmas

SLIDE 19

(λx.M)N

M[x := N]

(i)

M∗[x := N∗]

(ii)

(M[x := N])∗

Confluence of λβ by Z

Proof of the base case

M →∗ M∗ M∗[x := N∗] →∗ (M[x := N])∗

(i) (ii)

Key lemmas

SLIDE 20

(λx.M)N

M[x := N]

(i)

M∗[x := N∗]

(ii)

(M[x := N])∗

Confluence of λβ by Z

Proof of the base case

M →∗ M∗ M∗[x := N∗] →∗ (M[x := N])∗

(i) (ii)

Key lemmas

SLIDE 21

(λx.M)N

M[x := N]

(i)

M∗[x := N∗]

(ii)

(M[x := N])∗

Confluence of λβ by Z

Proof of the base case

M →∗ M∗ M∗[x := N∗] →∗ (M[x := N])∗

(i) (ii)

Key lemmas

SLIDE 22

Z for Permutative Conversion

SLIDE 23

Permutative conversion

for natural deduction with ∨ and ∃ [Prawitz 1965]
exchanges order of elimination rules
for normal proofs to have good properties

such as the subformula property

makes confluence proofs much harder

[Ando 2003]

SLIDE 24

Exchanging E-Rules

π

. . . . P Γ A1 A2 . . . . Q1 Γ, A1 B C . . . . Q2 Γ, A2 B C Γ B C (E∨) . . . . R Γ B Γ C (E→)

. . . . P Γ A1 A2 . . . . Q1 Γ, A1 B C . . . . R Γ B Γ, A1 C (E→) . . . . Q2 Γ, A2 B C . . . . R Γ B Γ, A1 C (E→) Γ C (E∨)

SLIDE 25

Exchanging E-Rules

π

. . . . P Γ A1 A2 . . . . Q1 Γ, A1 B C . . . . Q2 Γ, A2 B C Γ B C (E∨) . . . . R Γ B Γ C (E→)

. . . . P Γ A1 A2 . . . . Q1 Γ, A1 B C . . . . R Γ B Γ, A1 C (E→) . . . . Q2 Γ, A2 B C . . . . R Γ B Γ, A1 C (E→) Γ C (E∨)

P[x1.Q1R, x2.Q2R]

π

(case P with x1→Q1 | x2→Q2)R P[x1.Q1,x2.Q2]R =

SLIDE 26

Terms and eliminators
Reduction rules

λβπ

M, N ::= x | λx.M | ι1M | ι2M | Me e ::= M | [x1.N1, x2.N2] (λx.M)N →β M[x := N] (ιiM)[x1.N1, x2.N2] →β Ni[xi := M] M[x1.N1, x2.N2]e →π M[x1.N1e, x2.N2e]

SLIDE 27

Terms and eliminators
Reduction rules

λβπ

M, N ::= x | λx.M | ι1M | ι2M | Me e ::= M | [x1.N1, x2.N2] (λx.M)N →β M[x := N] (ιiM)[x1.N1, x2.N2] →β Ni[xi := M] M[x1.N1, x2.N2]e →π M[x1.N1e, x2.N2e]

uniform representation

f elimination for → and ∨

SLIDE 28

Terms and eliminators
Reduction rules

λβπ

permutative conversion

left associative (M[x1.N1,x2.N2])e

M, N ::= x | λx.M | ι1M | ι2M | Me e ::= M | [x1.N1, x2.N2] (λx.M)N →β M[x := N] (ιiM)[x1.N1, x2.N2] →β Ni[xi := M] M[x1.N1, x2.N2]e →π M[x1.N1e, x2.N2e]

uniform representation

f elimination for → and ∨

SLIDE 29

λβπ, for simplicity

M, N ::= x | λx.M | ιM | Me e ::= M | [x.N] (λx.M)N →β M[x := N] (ιM)[x.N] →β N[x := M] M[x.N]e →π M[x.Ne]

Terms and eliminators
Reduction rules

SLIDE 30

Where are difficulties?

Parallel reduction for π-reduction
Maximum complete development for

the combination of β- and π-reductions

SLIDE 31

Parallel reduction for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

SLIDE 32

Parallel reduction for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

SLIDE 33

Parallel reduction for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

these steps must be considered as

ne-step parallel red.

SLIDE 34

Parallel reduction for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

these steps must be considered as

ne-step parallel red.

We can avoid parallel reduction by Z

SLIDE 35

Z for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

M N

M∗

N∗

SLIDE 36

Z for π?

x[y.y][z.z][w.w]

π

π
x[y.y[z.z]][w.w]

π

x[y.y][z.z[w.w]]

π

x[y.y[z.z][w.w]]

π

x[y.y[z.z[w.w]]]

we have to do π completely

M N

M∗

N∗

SLIDE 37

A naïve definition

Z for βπ?

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise) (M[x.N])@e = M[x.N@e] M@e = Me (otherwise)

SLIDE 38

A naïve definition

Z for βπ?

is not Z

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise) (M[x.N])@e = M[x.N@e] M@e = Me (otherwise)

SLIDE 39

Z for βπ?

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise)

Monotonicity fails

(ι(x[y.y]))[z.z]w →π (ι(x[y.y]))[z.zw]

SLIDE 40

Z for βπ?

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise)

Monotonicity fails

(ι(x[y.y]))[z.z]w →π (ι(x[y.y]))[z.zw]

((ι(x[y.y]))[z.z]w)∗ = ((ι(x[y.y]))[z.z])∗@w = (x[y.y])@w = x[y.yw] ((ι(x[y.y]))[z.zw])∗ = (zw)∗[z := x[y.y]] = x[y.y]w

SLIDE 41

Z for βπ?

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise)

Monotonicity fails

(ι(x[y.y]))[z.z]w →π (ι(x[y.y]))[z.zw]

((ι(x[y.y]))[z.z]w)∗ = ((ι(x[y.y]))[z.z])∗@w = (x[y.y])@w = x[y.yw] ((ι(x[y.y]))[z.zw])∗ = (zw)∗[z := x[y.y]] = x[y.y]w permutation is applied to the result of β

SLIDE 42

Z for βπ?

x∗ = x (λx.M)∗ = λx.M∗ (ιM)∗ = ιM∗ ((λx.M)N)∗ = M∗[x := N∗] ((ιM)[x.N])∗ = N∗[x := M∗] (Me)∗ = M∗@e∗ (otherwise)

Monotonicity fails

(ι(x[y.y]))[z.z]w →π (ι(x[y.y]))[z.zw]

((ι(x[y.y]))[z.z]w)∗ = ((ι(x[y.y]))[z.z])∗@w = (x[y.y])@w = x[y.yw] ((ι(x[y.y]))[z.zw])∗ = (zw)∗[z := x[y.y]] = x[y.y]w permutation is applied to the result of β

We want to consider functions for π and β

separately (and adapt Z to their composition)

SLIDE 43

Compositional Z

SLIDE 44

Z and weak Z

M N

M∗

N∗

(·)* is Z for → iff

SLIDE 45

Z and weak Z

M N

M∗

N∗

(·)* is Z for → iff (·)* is weakly Z for → by →x iff

M N

x

M∗

x N∗

SLIDE 46

Compositional Z

[N&Fujita'15]

Let → = →1 ∪ →2

If mappings (·)1 and (·)2 satisfying following,

(·)1 is Z for →1
if M →1 N, then M2 →* N2
M1 →* M12 holds for any M
(·)12 is weakly Z for →2 by →

then the composition (·)12 is Z for →

SLIDE 47

Compositional Z

M

1

N

1

M

2

N

M1

1

N1

M12 N12 M12 N12

SLIDE 48

Z for 1 Compositional Z

M

1

N

1

M

2

N

M1

1

N1

M12 N12 M12 N12

SLIDE 49

weak Z for 2 Z for 1 Compositional Z

M

1

N

1

M

2

N

M1

1

N1

M12 N12 M12 N12

SLIDE 50

Confluence of βπ  by compositional Z

xP = x (λx.M)P = λx.MP (ιM)P = ιMP (Me)P = MP@eP xB = x (λx.M)B = λx.MB (ιM)B = ιMB ((λx.M)N)B = MB[x := NB] ((ιM)[x.N])B = NB[x := MB] (Me)B = MBeB (otherwise)

The mappings (·)P and (·)B satisfies the conditions

f the compositional Z for →π and →β

SLIDE 51

Applications

λ with permutative conversion π and β

SLIDE 52

Applications

λ with permutative conversion π and β λμ with permutative conversion πμ and β

SLIDE 53

Applications

λ with permutative conversion π and β λμ with permutative conversion πμ and β extensional λ η and β

SLIDE 54

Applications

λ with permutative conversion π and β λμ with permutative conversion πμ and β extensional λ η and β λ with explicit subst. x and β

subst. propagation

SLIDE 55

Applications

λ with permutative conversion π and β λμ with permutative conversion πμ and β extensional λ η and β λ with explicit subst. x and β

subst. propagation

Compositional Z enables us to prove confluence  by dividing reduction system into two parts

SLIDE 56

Church-Rosser via Z

SLIDE 57

M } } ! ! M1 ! ! M2 } } ∃N

Confluence vs Church-Rosser

Confluence Church-Rosser (CR)

M1 ! ! M2 } } ∃N

SLIDE 58

Confluence vs Church-Rosser

In many textbooks, CR is shown as a

corollary of confluence

In fact, they are equivalent in almost all of

rewriting systems

How can we prove CR directly?

SLIDE 59

Church-Rosser via Z

Suppose
(A,→) : an ARS
M*: a Z function on A, and Mn* = n-fold of M*
Cross-Point Theorem [Fujita 2016]
a constructive proof of CR

For M =A N, we can find a common reduct  decided by the numbers of → and ← in the conversion sequence

SLIDE 60

Main Lemma

r = # of → in M0 = Mn l = # of ← in M0 = Mn

M0 / M1 · · ·

/ Mn

t t Mr∗

SLIDE 61

Main Lemma

r = # of → in M0 = Mn l = # of ← in M0 = Mn

M0 / * * M1 · · ·

/ Mn

t t Mr∗ Ml∗

n

SLIDE 62

Cross-point theorem [Fujita'16]

#l[0,r] = # of ← in M0 … Mr

= # of → in Mr … Mn = #r[r,n]

r = # of → in M0 = Mn l = # of ← in M0 = Mn

M0 ( ( / · · · Mr

· · ·
/ Mn

v v M#l[0,r]∗

r

v v ( ( Mr∗ Ml∗

n

SLIDE 63

Quantitative analysis via Z [Fujita'16]

From a bound of steps in Z property,

we can give a bound of steps in CR

Main lemma: M =A N ⇒ N →Main(M=N) Mr*
where Main(M=N) is defined from

Rev, Mon and maximum term size in M=N

M N

M∗

N∗

M → N ⇒ N →Rev(|M|) M* M →n N ⇒ M* →Mon(|M|,n) N*

SLIDE 64

Quantitative analysis via Z

M0 ( ( / · · · Mr

· · ·
/ Mn

v v M#l[0,r]∗

r

M N

M∗

N∗

M → N ⇒ N →Rev(|M|) M* M →n N ⇒ M* →Mon(|M|,n) N* Main(Mr=M0) Main(Mr=Mn)

SLIDE 65

Quantitative analysis via Z

M0 ( ( / · · · Mr

· · ·
/ Mn

v v M#l[0,r]∗

r

M N

M∗

N∗

M → N ⇒ N →Rev(|M|) M* M →n N ⇒ M* →Mon(|M|,n) N*

For λβ, it is  non-elementary

Main(Mr=M0) Main(Mr=Mn)

SLIDE 66

Quantitative analysis via compositional Z [Fujita&N'16]

From bounds of steps in compositional Z (given below),

we can give a bound Main(M=N) in CR

M →1 N ⇒ N →Rev1(|M|) M1 M →Eval2(|M|) M2 M →2 N ⇒ N →Rev2(|M|) M12 M →n N ⇒ M12 →Mon(|M|,n) N12

M

2

/ N ⌅ ⌅ M12 / / N12 M

1

/ N

1

| | M1

1

/ / ✏✏ N1 M12 / / N12

SLIDE 67

Quantitative analysis via compositional Z

Main(M=N) is defined as   Main(M←N) = 1 Main(M→1N) = Rev1(|M|) + Eval2(|M1|) Main(M→2N) = Rev2(|M|) Main(M=P←Q) = Main(M=P) + 1 Main(M=P→1Q) = Mon(n,Main(M=P)) + Eval2(n) + Rev1(n) Main(M=P→2Q) = Mon(n,Main(M=P)) + Rev2(n) where n = maximum term size in M=N

SLIDE 68

Summary

Confluence of λ with permutative conversions

becomes much simpler with compositional Z

Compositional Z suggests (quasi-)modular proofs 
f confluence
Quantitative analysis for CR via Z

can be extended to compositional Z

K. Nakazawa and K. Fujita. Compositional Z: confluence proofs for permutative conversion.

Studia Logica, to appear.

K. Fujita and K. Nakazawa. Church-Rosser Theorem and Compositional Z-Property.

In Proceedings of 33rd JSSST, 2016

SLIDE 69

“Simpler” proofs?

SLIDE 70

“Simpler” proofs?

Easier to check?

SLIDE 71

“Simpler” proofs?

Easier to check?
Easier to apply other calculi?

SLIDE 72

“Simpler” proofs?

Easier to check?
Easier to apply other calculi?
Shorter formal proof? …depending on logical system

SLIDE 73

“Simpler” proofs?

Easier to check?
Easier to apply other calculi?
Shorter formal proof? …depending on logical system
Easier to formalize? …I believe so, but we should check it

SLIDE 74

“Simpler” proofs?

Easier to check?
Easier to apply other calculi?
Shorter formal proof? …depending on logical system
Easier to formalize? …I believe so, but we should check it

“I feel that the new proofs (…) are  more beautiful than those we started with,  and this is my actual motivation.” — [Pollack 1995]

SLIDE 75

A Classical Japanese Poem

composed by Sutoku-In (崇徳院) in 12th cent.

瀬を早み岩にせかるる滝川の  われても末に逢はむとぞ思ふ 

(direct translation)  A stream of the river separates into two streams after hitting the rock,  but it will become one stream again 

(that is,)  although if I love someone but we cannot be together in this life,  I can be together with her in the next life

Japanese-English Bilingual Corpus of Wikipedia's Kyoto Articles (National Institute of Information and Communications Technology)

SLIDE 76

A Classical Japanese Poem

composed by Sutoku-In (崇徳院) in 12th cent.

瀬を早み岩にせかるる滝川の  われても末に逢はむとぞ思ふ 

(direct translation)  A stream of the river separates into two streams after hitting the rock,  but it will become one stream again 

(that is,)  although if I love someone but we cannot be together in this life,  I can be together with her in the next life

Japanese-English Bilingual Corpus of Wikipedia's Kyoto Articles (National Institute of Information and Communications Technology)

confluence makes us happy!