ALMOST EVERY SIMPLY TYPED LAMBDA TERM HAS A LONG BETA-REDUCTION - - PowerPoint PPT Presentation

ALMOST EVERY SIMPLY TYPED LAMBDA TERM HAS A LONG BETA-REDUCTION SEQUENCE

RYOMA SIN’YA NAOKI KOBAYASHI KAZUYUKI ASADA TAKESHI TSUKADA

(THE UNIVERSITY OF TOKYO)

λ

MOTIVATION

• A simply-typed term can have a very long β-reduction

sequence.

• k-EXP in the size of terms of order k [Beckmann 2001].

0-EXP(n) = n (m + 1)-EXP(n) = 2m-EXP(n)

• How many terms have such long β-reduction

sequences?

MOTIVATION

e.g.

• A simply-typed term can have a very long β-reduction

sequence.

• k-EXP in the size of terms of order k [Beckmann 2001].
• How many terms have such long β-reduction

sequences?

where Twice = λf.λx.f(f x)

(Twice)n Twice · · · Twice | {z }

k−2 times

(λx.bxx)((λx.x)c)

• The work has been motivated by quantitative analysis
• f the complexity of higher-order model checking

(HOMC).

SIDE REMARK

• Input : tree automaton and λY-term t.


Output : YES if accepts the infinite tree
 represented by t, NO otherwise.
 Complexity: k-EXPTIME-complete for order-k λY-terms.

• We want to (dis)prove: HOMC can be efficiently

solved for almost every input.

HIGHER-ORDER MODEL CHECKING

A A [Ong 2006]

RELATED WORK

• Quantitative analysis of untyped terms:
• Almost every λ-term is strongly normalizing (SN), but

almost every SK-combinatory term is not SN 
 [David et al. 2009].

• Almost every de Bruijn λ-term is not SN

[Bendkowski et al. 2015].

• Empirical results: almost every λ-term is not β-normal,

untypable [Grygiel-Lescanne 2013].

RELATED WORK

• Quantitative analysis of untyped terms:
• Almost every λ-term is strongly normalizing (SN), but

almost every SK-combinatory term is not SN 
 [David et al. 2009].

• Almost every de Bruijn λ-term is not SN

[Bendkowski et al. 2015].

• Empirical results: almost every λ-term is not β-normal,

untypable [Grygiel-Lescanne 2013].

• Quantitative analysis of typed terms: little is known.

OUTLINE

λ

• Introduction
• Our result
• Proof of our result
• Conclusion

For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

k ≤ ι

OUR RESULT

: the set of α-equivalence classes of size-n 
 terms such that:

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ

Λα

n(k, ι, ξ)

(1) the order is at most .

k

For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

k ≤ ι

OUR RESULT

: the set of α-equivalence classes of size-n 
 terms such that:

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ

Λα

n(k, ι, ξ)

(1) the order is at most .

k

the maximum length of
 β-reduction sequences of t.

For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

k ≤ ι

OUR RESULT

: the set of α-equivalence classes of size-n 
 terms such that:

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ

Λα

n(k, ι, ξ)

(1) the order is at most .

k

the maximum length of
 β-reduction sequences of t.

For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

k ≤ ι

OUR RESULT

: the set of α-equivalence classes of size-n 
 terms such that:

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ Almost every term of size n and order at most k has a β-reduction sequence of length (k-2)-EXP(n).

Λα

n(k, ι, ξ)

(1) the order is at most .

k

the maximum length of
 β-reduction sequences of t.

OUR RESULT

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ

For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

(1) the order is at most .

k

k ≤ ι

: the set of α-equivalence classes of size-n 
 terms such that:

Λα

n(k, ι, ξ)

NUMBER OF DISTINCT VARIABLES

#V(t)

• : the # of variables in t excluding unused

variables.

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α}

• For an α-equivalence class ,

[t]α

NUMBER OF DISTINCT VARIABLES

• For an α-equivalence class ,

[t]α Example

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α} #Vα([(λz.z)λy.x]α) = ?

#V(t)

• : the # of variables in t excluding unused

variables.

NUMBER OF DISTINCT VARIABLES

• For an α-equivalence class ,

[t]α Example

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α} #Vα([(λz.z)λy.x]α) = ?

#V(t)

• : the # of variables in t excluding unused

variables.

#V((λz.z)λy.x) = #{x, z} = 2

NUMBER OF DISTINCT VARIABLES

• For an α-equivalence class ,

[t]α Example

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α} #Vα([(λz.z)λy.x]α) = ?

#V(t)

• : the # of variables in t excluding unused

variables.

#V((λz.z)λy.x) = #{x, z} = 2

NUMBER OF DISTINCT VARIABLES

• For an α-equivalence class ,

[t]α Example

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α} #Vα([(λz.z)λy.x]α) = ?

#V(t)

• : the # of variables in t excluding unused

variables.

#V((λz.z)λy.x) = #{x, z} = 2 #V((λx.x)λy.x) = #{x} = 1

NUMBER OF DISTINCT VARIABLES

• For an α-equivalence class ,

[t]α Example

#Vα([t]α) , min{#V(t0) | t0 ∈ [t]α} #Vα([(λz.z)λy.x]α) = 1

#V(t)

• : the # of variables in t excluding unused

variables.

#V((λz.z)λy.x) = #{x, z} = 2 #V((λx.x)λy.x) = #{x} = 1

OUR RESULT

(2) the number of arguments (internal arity) is at most .

ι

(3) the number of distinct variables is at most .

ξ

(1) the order is at most .

k For and ,

k, ι, ξ ≥ 2

lim

n→∞

#{[t]α ∈ Λα

n(k, ι, ξ)} | β(t) ≥ (k − 2)-EXP(n)}

#Λα

n(k, ι, ξ)

= 1.

k ≤ ι

[t]α ∈ Λα

n(k, ι, ξ)

#Vα([t]α) ≤ ξ for every : the set of α-equivalence classes of size-n 
 terms such that:

Λα

n(k, ι, ξ)

OUTLINE

λ

• Introduction
• Our result
• Proof of our result
• Conclusion

OVERVIEW OF OUR PROOF

• Almost every term contains a certain “context”

that has a very long β-reduction sequence.

OVERVIEW OF OUR PROOF

• Almost every term contains a certain “context”

that has a very long β-reduction sequence.

OVERVIEW OF OUR PROOF

• Almost every term contains a certain “context”

that has a very long β-reduction sequence.

• Inspired by Infinite Monkey Theorem: for any word x,

almost every word contains x as a subword.

OUTLINE

λ

• Introduction
• Our result
• Proof of our result


• Conclusion
• Idea
• Infinite Monkey Theorem
• Decomposition of terms
• Sketch of the proof

PROOF IDEA

• 1. Parameterizing Infinite Monkey Theorem.
• 2. Extending (1) to λ-terms.
• 3. Constructing “explosive context” that generates

a long β-reduction sequence.

INFINITE MONKEY THEOREM

For any word over an alphabet A, u, v ∈ A∗. for some words

x v w , w = uxv

lim

n→∞

#{w 2 An | x v w} #An = 1.

x

INFINITE MONKEY THEOREM

For any word over an alphabet A, u, v ∈ A∗. for some words

x v w , w = uxv

lim

n→∞

#{w 2 An | x v w} #An = 1.

x

IDEA1: PARAMETERIZING INFINITE MONKEY THEOREM

For any family of words over A such that

(xn)n

lim

n→∞

#{w 2 An | xn v w} #An = 1.

log(2)(n) = log(log(n))

|xn| = dlog(2)(n)e,

IDEA2: EXTENDING IDEA1 TO TERMS

For any family of contexts such that

(Cn)n

if .

|Cn| = dlog(2)(n)e, C t ⇔

for some context and term .

C0 t = C0[C[t0]] t0

lim

n→∞

#{[t]α 2 Λα

n(k, ι, ξ)} | Cn t}

#Λα

n(k, ι, ξ)

= 1.

k, ι, ξ ≥ 2

IDEA3: CONSTRUCTING “EXPLOSIVE” CONTEXT

• For parameters n and k, we define the explosive

context of order-k as:

λx.

• (Twice)n Twice · · · Twice

| {z }

k−2 times

Dup(Id [ ])

• where Twice = λf.λx.f(f x)

Dup = λx.(λy.λz.y)xx and Id = λx.x

n

k

IDEA3: CONSTRUCTING “EXPLOSIVE” CONTEXT

• For parameters n and k, we define the explosive

context of order-k as:

n

k

β ⇣ ⌘ ≥ k-EXP(n)

• It has the following “explosive property”:

λx.

• (Twice)n Twice · · · Twice

| {z }

k−2 times

Dup(Id [ ])

• where Twice = λf.λx.f(f x)

Dup = λx.(λy.λz.y)xx and Id = λx.x

n

k

IDEA3: CONSTRUCTING “EXPLOSIVE” CONTEXT

t ) k-EXP(n)  β(t).

n

k

• It has the following “explosive property”:
• For parameters n and k, we define the explosive

context of order-k as:

λx.

• (Twice)n Twice · · · Twice

| {z }

k−2 times

Dup(Id [ ])

• where Twice = λf.λx.f(f x)

Dup = λx.(λy.λz.y)xx and Id = λx.x

n

k

HARVEST

For and ,

k, ι, ξ ≥ 2

k

dlog(2)(n)e

lim

n→∞

#{[t]α 2 Λα

n(k, ι, ξ)} |

t} #Λα

n(k, ι, ξ)

= 1.

k ≤ ι

HARVEST

⇓

A direct corollary of the explosive property:

t ) (k 2)-EXP(n)  β(t).

Almost every term of size n and order at most k has a β-reduction sequence of length (k-2)-EXP(n).

k

dlog(2)(n)e

For and ,

k, ι, ξ ≥ 2

k

dlog(2)(n)e

lim

n→∞

#{[t]α 2 Λα

n(k, ι, ξ)} |

t} #Λα

n(k, ι, ξ)

= 1.

k ≤ ι

PROOF IDEA

• 1. Parameterizing Infinite Monkey Theorem.
• 2. Extending (1) to λ-terms.
• 3. Constructing “explosive context” that generates

a long β-reduction sequence.

PROOF IDEA

• 1. Parameterizing Infinite Monkey Theorem.
• 2. Extending (1) to λ-terms.
• 3. Constructing “explosive context” that generates

a long β-reduction sequence. Most technical part

PROOF IDEA

• 1. Parameterizing Infinite Monkey Theorem.
• 2. Extending (1) to λ-terms.
• 3. Constructing “explosive context” that generates

a long β-reduction sequence. Most technical part We first give a proof of (1), because it clarify the overall structure of the proof of (2).

λ

http://en.wikipedia.org/wiki/ Infinite_monkey_theorem

OUTLINE

• Introduction
• Our result
• Proof of our result


• Conclusion
• Idea
• Infinite Monkey Theorem
• Decomposition of terms
• Sketch of the proof

For any word over an alphabet A,

x

lim

n→∞

#{w 2 An | x v w} #An = 1.

*

It suffice to show that:

#{w 2 An | x 6v w} #An

→ 0 (n → ∞)

PROOF OF MONKEY THEOREM FOR WORDS

PROOF OF MONKEY THEOREM FOR WORDS

*

?

→ 0 (n → ∞)

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

?

→ 0 (n → ∞)

Let ` = |x|, w ∈ An.

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

?

→ 0 (n → ∞)

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

|{z}

`

?

→ 0 (n → ∞)

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

|{z}

`

?

→ 0 (n → ∞)

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

|{z}

`

?

→ 0 (n → ∞)

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

|{z}

`

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

|{z}

`

(n mod `)<`

z}|{

?

→ 0 (n → ∞)

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

|{z}

`

#{w 2 An | x 6v w} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

|{z}

`

#{w 2 An | x 6v w} #An

 #{w 2 An | wi 6= x for all i  bn/`c} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

= ✓ 1 − 1 #A` ◆bn/`c

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

|{z}

`

#{w 2 An | x 6v w} #An

 #{w 2 An | wi 6= x for all i  bn/`c} #An

PROOF OF MONKEY THEOREM FOR WORDS

*

= ✓ 1 − 1 #A` ◆bn/`c

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

Let ` = |x|, w ∈ An.

|{z}

`

|{z}

`

#{w 2 An | x 6v w} #An

→ 0 (n → ∞)

*

 #{w 2 An | wi 6= x for all i  bn/`c} #An

• Previous proof is based on a “good” decomposition of words.
• This good decomposition is induced by the following

coproduct-product form:

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

|{z}

`

|{z}

`

An 3

cf.

DECOMPOSITION OF WORDS

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

• This point of view forms the basis of the later extensions.

DECOMPOSITION OF WORDS

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

∈

w

w0

&

(w1, w2, · · · , wbn/`c)

∈ ∈

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

|{z}

`

|{z}

`

An 3

cf.

DECOMPOSITION OF WORDS

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

∈

w

w0

&

(w1, w2, · · · , wbn/`c)

∈ ∈

Residual part (coproduct part)

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

|{z}

`

|{z}

`

An 3

cf.

DECOMPOSITION OF WORDS

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

∈

w

w0

&

(w1, w2, · · · , wbn/`c)

∈ ∈

Decomposed parts (product parts)

Residual part (coproduct part)

|{z}

`

(n mod `)<`

z}|{

w = w1 w2 · · · wbn/`c w0

|{z}

`

|{z}

`

An 3

cf.

*

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

cf.

(residual part) (decomposed parts)

Let ` = |x|.

#{w 2 An | x 6v w} #An

PROOF OF INFINITE MONKEY THEOREM (REVISED)

*

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

cf.

(residual part) (decomposed parts)

Let ` = |x|.

#{w 2 An | x 6v w} #An

= # a

w02A(n mod `)

Y

ibn/`c

A` #An

PROOF OF INFINITE MONKEY THEOREM (REVISED)

*

An ∼ = a

w02A(n mod `)

Y

ibn/`c

A`

cf.

(residual part) (decomposed parts)

Let ` = |x|.

#{w 2 An | x 6v w} #An

= # a

w02A(n mod `)

Y

ibn/`c

A` #An

≤ # a

w02A(n mod `)

Y

ibn/`c

• A` \ {x}
• #An

PROOF OF INFINITE MONKEY THEOREM (REVISED)

PROOF OF INFINITE MONKEY THEOREM (REVISED)

* Let ` = |x|.

#{w 2 An | x 6v w} #An

= # a

w02A(n mod `)

Y

ibn/`c

A` #An

≤ # a

w02A(n mod `)

Y

ibn/`c

• A` \ {x}
• #An

PROOF OF INFINITE MONKEY THEOREM (REVISED)

* Let ` = |x|.

#{w 2 An | x 6v w} #An

= # a

w02A(n mod `)

Y

ibn/`c

A` #An

≤ # a

w02A(n mod `)

Y

ibn/`c

• A` \ {x}
• #An

= ✓ 1 − 1 #A` ◆bn/`c

PROOF OF INFINITE MONKEY THEOREM (REVISED)

* Let ` = |x|.

#{w 2 An | x 6v w} #An

= # a

w02A(n mod `)

Y

ibn/`c

A` #An

≤ # a

w02A(n mod `)

Y

ibn/`c

• A` \ {x}
• #An

= ✓ 1 − 1 #A` ◆bn/`c

→ 0 (n → ∞)

*

For any family of words over A such that

(xn)n

lim

n→∞

#{w 2 An | xn v w} #An = 1.

|xn| = dlog(2)(n)e,

PROOF OF PARAMETERIZED INFINITE MONKEY THEOREM FOR WORDS

cf.

(residual part) (decomposed parts)

An ∼ = a

w02A(n mod dlog(2)(n)e)

Y

ibn/dlog(2)(n)ec

Adlog(2)(n)e

* #{w 2 An | xn 6v w}

#An  #{w 2 An | every decomposed part 6= x} #An

For any family of words over A such that

(xn)n

lim

n→∞

#{w 2 An | xn v w} #An = 1.

|xn| = dlog(2)(n)e,

PROOF OF PARAMETERIZED INFINITE MONKEY THEOREM FOR WORDS

* #{w 2 An | xn 6v w}

#An  #{w 2 An | every decomposed part 6= x} #An

→ 0 (n → ∞)

*

= ✓ 1 − 1 Adlog(2)(n)e ◆bn/dlog(2)(n)ec

CHALLENGE IN PROVING PARAMETERISED MONKEY THEOREM FOR TERMS

• How to obtain such a “good” decomposition for the

set of λ-terms ?

• Non-trivial since terms have various shapes:

Λα

n(k, ι, ξ)

OUTLINE

λ

• Introduction
• Our result
• Proof of our result


• Conclusion
• Idea
• Infinite Monkey Theorem
• Decomposition of terms
• Sketch of the proof

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

Example

DECOMPOSITION OF TERMS

λx.(λy.(λz.λx.z)(λx.(λx.y)λz.z))x

decomposition size m = 3

z y

x @

@

@

z

λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

Residual part (second-order context)

Example

ANALOGY BETWEEN THE DECOMPOSITION OF TERMS AND WORDS

z

y x

@ @

@

z λz

λx

λy λz

λx λx λx

7 !

Φ3

@

J K

J K J K λx λx

(λy.[ ])x

λz.λx.z

(λx.y)λz.z

abracadabra

＊ Decomposed part ＊ Residual part

7 ! abr aca dab ra

decompose

cf.

DECOMPOSITION LEMMA

For and ,

n ≥ m ≥ 2

Λα

n(k, ι, ξ) ∼

= a

E∈Bn

m

Y

i≤shn(E)

U m

E.i

k, ι, ξ ≥ 0

An ∼ = a

w2A(n mod m)

Y

ib n

m c

Am

cf.

DECOMPOSITION LEMMA

For and ,

n ≥ m ≥ 2

Λα

n(k, ι, ξ) ∼

= a

E∈Bn

m

Y

i≤shn(E)

U m

E.i

k, ι, ξ ≥ 0

some set of second-order contexts

An ∼ = a

w2A(n mod m)

Y

ib n

m c

Am

cf.

DECOMPOSITION LEMMA

For and ,

n ≥ m ≥ 2

Λα

n(k, ι, ξ) ∼

= a

E∈Bn

m

Y

i≤shn(E)

U m

E.i

k, ι, ξ ≥ 0

some set of second-order contexts the number of holes in E

J K

An ∼ = a

w2A(n mod m)

Y

ib n

m c

Am

cf.

DECOMPOSITION LEMMA

For and ,

n ≥ m ≥ 2

Λα

n(k, ι, ξ) ∼

= a

E∈Bn

m

Y

i≤shn(E)

U m

E.i

k, ι, ξ ≥ 0

some set of second-order contexts the set of “good” contexts that can be filled in the i-th hole of E. the number of holes in E

J K

An ∼ = a

w2A(n mod m)

Y

ib n

m c

Am

For and ,

Each decomposed part does NOT depend on the residual part w

Am

cf.

DECOMPOSITION LEMMA

n ≥ m ≥ 2

Λα

n(k, ι, ξ) ∼

= a

E∈Bn

m

Y

i≤shn(E)

U m

E.i

k, ι, ξ ≥ 0

An ∼ = a

w2A(n mod m)

Y

ib n

m c

Am

Each decomposed part DOES depend on the residual part E

(and also on the index i) U m

E.i

OUTLINE

λ

• Introduction
• Our result
• Proof of our result


• Conclusion
• Idea
• Infinite Monkey Theorem
• Decomposition of terms
• Sketch of the proof

For any family of contexts of such that

(Cn)n

if .

|Cn| = dlog(2)(n)e,

lim

n→∞

#{[t]α 2 Λα

n(k, ι, ξ)} | Cn t}

#Λα

n(k, ι, ξ)

= 1.

k, ι, ξ ≥ 2

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS Λα

n(k, ι, ξ)

*

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

It is suffice to show that

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

*

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

cf.

∈ [t]α

Λα

n(k, ι, ξ) ∼

= a

E∈Blog(2)(n)

n

Y

i≤shn(E)

U log(2)(n)

E.i

∈

Φlog(2)(n)

7

• ! E &

∈

(u1, u2, · · ·, ushn(E))

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

*

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

cf.

∈ [t]α

Λα

n(k, ι, ξ) ∼

= a

E∈Blog(2)(n)

n

Y

i≤shn(E)

U log(2)(n)

E.i

∈

Φlog(2)(n)

7

• ! E &

∈

(u1, u2, · · ·, ushn(E))

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

≤ ⇣ 1 − 1/cγ2dlog(2)(n)e⌘n/4dlog(2)(n)e

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

≤ ⇣ 1 − 1/cγ2dlog(2)(n)e⌘n/4dlog(2)(n)e

shn(E) n/4dlog(2)(n)e for any E 2 Bdlog(2)(n)e

n

Lemma

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

≤ ⇣ 1 − 1/cγ2dlog(2)(n)e⌘n/4dlog(2)(n)e

shn(E) n/4dlog(2)(n)e for any E 2 Bdlog(2)(n)e

n

Lemma #U dlog(2)(n)e

E.i

= O(cγ2dlog(2)(n)e) for some constants c and γ Lemma

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

≤ ⇣ 1 − 1/cγ2dlog(2)(n)e⌘n/4dlog(2)(n)e

= # a

E2Bdlog(2)(n)e

n

Y

ishn(E)

n ui 2 U dlog(2)(n)e

E.i

| Cn 6 ui

• #Λα

n(k, ι, ξ)

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

?

 #{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 ui for every i}

#Λα

n(k, ι, ξ)

*

≤ ⇣ 1 − 1/cγ2dlog(2)(n)e⌘n/4dlog(2)(n)e → 0 (n → ∞) *

For any family of contexts of such that

(Cn)n

if .

|Cn| = dlog(2)(n)e,

lim

n→∞

#{[t]α 2 Λα

n(k, ι, ξ)} | Cn t}

#Λα

n(k, ι, ξ)

= 1.

k, ι, ξ ≥ 2

PROOF OF PARAMETERISED INFINITE MONKEY THOREM FOR TERMS Λα

n(k, ι, ξ)

* *

#{[t]α 2 Λα

n(k, ι, ξ) | Cn 6 t}

#Λα

n(k, ι, ξ)

! 0 (n ! 1)

SUMMARY OF THE MAIN PROOF

the probability that a term has a β-reduction sequence of length (k-2)-EXP(n) the probability that holds

k

( Monkey Theorem)

*

( explosive property)

*

≥

t

[t]α ∈ → 1 (n → ∞)

dlog(2)(n)e

Λα

n(k, ι, ξ)

OUTLINE

λ

• Introduction
• Proof of our result
• Conclusion

FUTURE WORK

• Quantitative analysis of simply typed λ-terms in

different settings:

• with an unbounded number of variables.
• with recursion.
• Almost every terms of size n and order at most k has a


β-reduction sequence of length (k-2)-EXP(n).

• The core of our proof is a non-trivial extension of

well-known Infinite Monkey Theorem.

CONCLUSION