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「言語の測度に基づく非正規性の証明技法」

言語の大きさを考えることで非正規性を示す技法を説明しています。PPL 2016での発表資料です。

- 言語の測度に基づく非正規性の証明技法

A NEW TECHNIQUE FOR PROVING

NON-REGULARITY BASED ON

THE MEASURE OF A LANGUAGE

Ryoma Sin’ya

Tokyo Institute of Technology,

Department of Mathematics and Computer Science. - 無限の猿定理

- Inﬁnite Monkey Theorem -

(a.k.a. Borge’s theorem)

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

2 - The main issue of the talk is the “inverse direction” of

Inﬁnite Monkey Theorem.

In the case of regular languages, Inﬁnite Monkey

Theorem states a necessary and sufﬁcient condition for

the notion of “almost sureness”.

3 - 記法

- Notation -

A : an alphabet (finite set of letters)

An : the set of all words over A of length n

[

A⇤ : the set of all words over A(A⇤ =

An)

n2N

A language over A is a subset of A⇤.

4 - For languages L and K, we define the

following three operations:

union: L ∪ K;

concatenation: LK = {vw | v ∈ L, w ∈ K};

Kleene star: L∗ =

Ln = {ε} ∪ L ∪ LL ∪ LLL ∪ · · · .

n∈N

The class of regular languages is the smallest

class that includes all finite languages and

closed under the above three operations.

5 - 言語の階層

- Language Hierarchy -

6 - 言語の階層

- Language Hierarchy -

よわい？

6 - 言語の階層

- Language Hierarchy -

よわい？

きれい！

6 - 測度と零壱定理

- Measure & Zero-One Theorem -

7 - For a language L over A, its probability

function is the fraction deﬁned by:

number of all words of length n in L

µn(L) =

number of all words of length n

|L \ An|

=

.

|An|

8 - For a language L over A, its probability

function is the fraction deﬁned by:

number of all words of length n in L

µn(L) =

number of all words of length n

|L \ An|

=

.

|An|

This is exactly the probability that a

randomly chosen word of length n belongs

to L.

8 - For a language L over A, its probability

function is the fraction deﬁned by:

number of all words of length n in L

µn(L) =

number of all words of length n

|L \ An|

=

.

|An|

The measure

µ(L) of a language L is the

limit of its probability function:

µ(L) = lim µn(L).

n!1

9 - Example

The full language is almost full, and the empty language is almost empty. That

is, the set of all words A∗ over A satisfies µ(A∗) = 1, and its complement ∅

satisfies µ(∅) = 0.

10 - Example

The full language is almost full, and the empty language is almost empty. That

is, the set of all words A∗ over A satisfies µ(A∗) = 1, and its complement ∅

satisfies µ(∅) = 0.

Consider aA∗ the set of all words which start with the letter a in A. Then the

following holds:

1

µn(aA∗) = |aAn−1| =

.

|An|

|A|

Hence µ((aA)∗) = 1/|A| holds and aA∗ is not zero-one if |A| ≥ 2.

10 - Example

The full language is almost full, and the empty language is almost empty. That

is, the set of all words A∗ over A satisfies µ(A∗) = 1, and its complement ∅

satisfies µ(∅) = 0.

Consider aA∗ the set of all words which start with the letter a in A. Then the

following holds:

1

µn(aA∗) = |aAn−1| =

.

|An|

|A|

Hence µ((aA)∗) = 1/|A| holds and aA∗ is not zero-one if |A| ≥ 2.

Consider (AA)∗ the set of all words with even length. Then:

1 if n is even,

µn((AA)∗) =

0 if n is odd.

Hence, its limit µ((AA)∗) does not exist.

10 - 禁句

- Forbidden Word -

A word w is forbidden for a language L

over A, if

A⇤wA⇤ \ L = ; holds.

(A⇤wA⇤ ✓ L)

More intuitively, w is a forbidden word of L

if and only if every words in L does not contain

w as a factor. - 無限の猿定理

- Inﬁnite Monkey Theorem -

(a.k.a. Borge’s theorem)

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

12 - 無限の猿定理

- Inﬁnite Monkey Theorem -

(a.k.a. Borge’s theorem)

Infinite Monkey Theorem (formal statement)

Let L be a language over A. If L contains a

language of the form , then L is almost full.

A⇤wA⇤

(i.e., A⇤wA⇤ ✓ L ) µ(L) = 1)

13 - 零壱定理

- Zero-One Theore -

Theorem [S. 2015]

Let L be a regular language. Then the following are

equivalent:

1. L is almost empty (i.e., )

µ(L) = 0.

2. L has a forbidden word. - 零壱定理

- Zero-One Theore -

Theorem [S. 2015]

Let L be a regular language. Then the following are

equivalent:

1. L is almost empty (i.e., )

µ(L) = 0.

2. L has a forbidden word.

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

) 9w 2 A⇤(A⇤wA⇤ \ L = ;)

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

) 9w 2 A⇤(A⇤wA⇤ \ L = ;)

, 9w 2 A⇤(A⇤wA⇤ ✓ L)

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

) 9w 2 A⇤(A⇤wA⇤ \ L = ;)

, 9w 2 A⇤(A⇤wA⇤ ✓ L)

) µ(L) = 1

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

) 9w 2 A⇤(A⇤wA⇤ \ L = ;)

, 9w 2 A⇤(A⇤wA⇤ ✓ L)

) µ(L) = 1 ) µ(L) = 1 − µ(L) = 0

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

(2) L has a forbidden word

) 9w 2 A⇤(A⇤wA⇤ \ L = ;)

, 9w 2 A⇤(A⇤wA⇤ ✓ L)

) µ(L) = 1 ) µ(L) = 1 − µ(L) = 0

, L is almost empty (1).

The implication (2) → (1) is nothing but the

well-known Inﬁnite Monkey Theorem. - 零壱定理

- Zero-One Theore -

Theorem [S. 2015]

Let L be a regular language. Then the following are

equivalent:

1. L is almost empty (i.e., )

µ(L) = 0.

2. L has a forbidden word.

The remarkable fact of this theorem is that

its converse (1) → (2) is also true! - 零壱定理

- Zero-One Theore -

Theorem [S. 2015] (complete version)

Let L be a regular language. Then the following are

equivalent:

1. L is almost empty or almost full

(µ(L) = 0)

(µ(L) = 1).

2. L or its complement has a forbidden word.

3. The syntactic monoid of L has a zero element.

4. The minimal automaton of L is zero.

5. L is recognised by a quasi-zero automata. - An Automata Theoretic Approach to the Zero-One Law for

Regular Languages: Algorithmic and Logical Aspects

Ryoma Sin’ya

Tokyo Institute of Technology.

shinya.r.aa@m.titech.ac.jp

´Ecole Nationale Sup´erieure des T´el´ecommunications.

rshinya@enst.fr

A zero-one language L is a regular language whose asymptotic probability converges to either zero

or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the

zero-one law if and only if its syntactic monoid has a zero element, by means of Eilenberg’s variety

theoretic approach. Our proof gives an effective automata characterisation of the zero-one law for

regular languages, and it leads to a linear time algorithm for testing whether a given regular language

is zero-one. In addition, we discuss the logical aspects of the zero-one law for regular languages.

For more details, see arxiv:1509.07209

18 - 非正規性の証明技法

- Technique for Proving Non-Regularity-

Zero Lemma (corollary of Zero-One Theorem)

Let L be a almost empty language over A. If L does

not have a forbidden word, then L is not regular. - 非正規性の証明技法

- Technique for Proving Non-Regularity-

Zero Lemma (corollary of Zero-One Theorem)

Let L be a almost empty language over A. If L does

not have a forbidden word, then L is not regular.

A new necessary condition of the regularity. - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

20 - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

⎧

⎨ |A|n/2 = 1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

20 - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

⎧

⎨ |A|n/2 = 1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

20 - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

⎧

⎨ |A|n/2 = 1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

8w 2 A⇤(wwr 2 P ).

20 - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

⎧

⎨ |A|n/2 = 1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

8w 2 A⇤(wwr 2 P ).

(i.e., P does not have a forbidden word)

20 - 回文

- Palindromes -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if A is singleton (|A| = 1), then P = A∗ and hence P is regular.

⎧

⎨ |A|n/2 = 1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

8w 2 A⇤(wwr 2 P ).

(i.e., P does not have a forbidden word)

P is not regular by Zero Lemma!

20 - 括弧の対応

- Dyck Language -

Recall that the Dyck language D over A = {[, ]} is the set of all balanced square

brackets:

D = {ε, [], [[]], [][], [[[]]], [[][]], [[]][], [][[]], [][][], . . .}.

Θ

1

if n is even,

µ

n3/2

n(D) =

0

if n is odd.

8w 2 A⇤ 9n, m 2 N([nw]m 2 D) .

(i.e., D does not have a forbidden word)

D is not regular by Zero Lemma!

21 - 素数

- Primes -

: the set of all prime numbers.

by Prime Number Theorem.

by Dirichlet's theorem

is not regular by Zero Lemma!

22 - Zero Lemma ~

• states a necessary condition for regular

languages.

• can be only applied to almost empty

languages.

• is useful, since the assumption “L is

almost empty” is often intuitively clear. - Zero Lemma ~

• states a necessary condition for regular

languages.

• can be only applied to almost empty

languages.

• is useful, since the assumption “L is

almost empty” is often intuitively clear.

However, even though “L is almost empty” is

often intuitively clear, proving it requires extra

work. - Proving “L is almost empty” requires the

asymptotic behaviour of the probability

function of L.

However, even though “L is almost empty” is

often intuitively clear, proving it requires extra

work. - Proving “L is almost empty” requires the

asymptotic behaviour of the probability

function of L.

Motivation: Can we ﬁnd a simple sufﬁcient

condition for the almost emptiness?

However, even though “L is almost empty” is

often intuitively clear, proving it requires extra

work. - 零測度の十分条件

- Sufﬁcient Condition for the Almost Emptiness-

25 - Idea: If no element has a neighbour element, the set

looks like sparse, e.g., is of measure zero.

28 - In order to formalise this idea, we have to

introduce some distance between words!

Idea: If no element has a neighbour element, the set

looks like sparse, e.g., is of measure zero.

29 - Hamming距離

- Hamming Distance -

Hamming distance is a distance between

words of same length.

30 - Hamming距離

- Hamming Distance -

The hamming distance between two

d(u, v)

words is the number of positions

u, v 2 An

at which corresponding letters are different:

d(u, v) = |{i 2 [0, n − 1] | ui 6= vi}|

where wi is the i-th later of w.

31 - Hamming距離

- Hamming Distance -

d(0001, 0000) = 1,

d(1111, 0000) = 4,

d(1101, 0110) = 3,

d(1001, 1111) = 2.

d(u, v) = |{i 2 [0, n − 1] | ui 6= vi}|

where wi is the i-th later of w.

32 - Hamming距離

- Hamming Distance -

d(u, v) = |{i 2 [0, n − 1] | ui 6= vi}|

where wi is the i-th later of w.

33 - Hamming距離

- Hamming Distance -

34 - For a word , its

w 2 An

distance-one neighbours

is defined by:

B(w)

B(w) = {u 2 An | d(w, u) 1}.

35 - For a word , its

w 2 An

distance-one neighbours

is defined by:

B(w)

B(w) = {u 2 An | d(w, u) 1}.

35 - For a word , its

w 2 An

distance-one neighbours

is defined by:

B(w)

B(w) = {u 2 An | d(w, u) 1}.

B(000) =

35 - For a word , its

w 2 An

distance-one neighbours

is defined by:

B(w)

B(w) = {u 2 An | d(w, u) 1}.

B(000) =

35 - For a word , its

w 2 An

distance-one neighbours

is defined by:

B(w)

B(w) = {u 2 An | d(w, u) 1}.

Note: the size of satisfies:

B(w)

|B(w)| = n(|A|

1) + 1

for every word w 2 An.

36 - 零測度の十分条件

- Sufﬁcient Condition for the Almost Emptiness-

Lemma 1

Let L be a language over A.

If the number of distance-one neighbours that are in

L is bounded by some constant for every sufficiently

large word, then L is almost empty.

37 - 零測度の十分条件

- Sufﬁcient Condition for the Almost Emptiness-

Lemma 1

Let L be a language over A.

If the number of distance-one neighbours that are in

L is bounded by some constant for every sufficiently

large word, then L is almost empty.

Namely, if L satisfies the following condition, then L

is almost empty:

9C, N 2 N 8n 2 N 8w 2 An n > N ) |B(w) \ L| C .

37 - 回文・再

- Palindromes (Revised) -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if⎧A| is

A| singleton

n/2

(|A| = 1), then P = A∗ and hence P is regular.

⎨

=

1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

38 - 回文・再

- Palindromes (Revised) -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if⎧A| is

A| singleton

n/2

(|A| = 1), then P = A∗ and hence P is regular.

⎨

=

1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

38 - 回文・再

- Palindromes (Revised) -

Recall that the set of all palindromes P over A is defined as follows:

P = {w ∈ A∗ | w = wr}.

Note that, if⎧A| is

A| singleton

n/2

(|A| = 1), then P = A∗ and hence P is regular.

⎨

=

1

if n is even,

µ

|A|n

|A|n/2

n(P ) = ⎩|A|×|A|(n−1)/2 =

1

if n is odd.

|A|n

|A|(n−1)/2

µ(P ) = 0 since the number of distance-one

neighbours that is in L is bounded by |A| for any

words.

38 - 回文・再

- Palindromes (Revised) -

Example: madamimadam 2 P

(Madam, I’m Adam)

39 - 回文・再

- Palindromes (Revised) -

badamimadam /

2 P

distance-one

Example: madamimadam 2 P

(Madam, I’m Adam)

39 - 回文・再

- Palindromes (Revised) -

badamimadam /

2 P madambmadam 2 P

distance-one

distance-one

Example: madamimadam 2 P

(Madam, I’m Adam)

39 - 回文・再

- Palindromes (Revised) -

badamimadam /

2 P madambmadam 2 P

distance-one

distance-one

Example: madamimadam 2 P

(Madam, I’m Adam)

We can obtain another palindrome, only if we

change the central letter “i ” to another letter.

39 - 回文・再

- Palindromes (Revised) -

badamimadam /

2 P madambmadam 2 P

distance-one

distance-one

Example: madamimadam 2 P

(Madam, I’m Adam)

We can obtain another palindrome, only if we

change the central letter “i ” to another letter.

µ(P ) = 0 since the number of distance-one

neighbours that is in L is bounded by |A| for any

words.

39 - The proof of Lemma 1 is not so difﬁcult.

It uses a result of Coding Theory (Cohen’s

theorem), please see my paper for the

details of Lemma 1.

40 - A language L over A is said to be a covering of An

[

if holds.

B(w) = An

w2L

We denote the minimal size of a covering of

An

by KA(n) = min{|L| | L is a covering of An}.

|L|

Theorem [Cohen et al. 1986]

For any alphabet A, there exists some constant C

such that: K

lim sup

A(n) ⇥ (n(|A| − 1) + 1) < C.

n!1

|An| - 課題

- Future Works -

42 - Our Lemma 1 is:

• a sufﬁcient condition of the almost

emptiness.

• general. It can be applied to any

language.

43 - Our Lemma 1 is:

• a sufﬁcient condition of the almost

emptiness.

• general. It can be applied to any

language.

• but not strong enough, we can not

prove the almost emptiness of Dyck

language by Lemma 1.

• We want to improve Lemma 1. Some

conjectures are written in my paper.

43 - 予想

- Conjecture -

問題 1. 2 つ以上の文字を含むアルファベット A 上の言語を L とする．L から定められる 2 つの関

数 f, g : N → N をそれぞれ

f (n) = max{|L ∩ BA(w, n)| | w ∈ L ∩ An},

g(n) = min{|L ∩ BA(w, n)| | w /

∈ L ∩ An}

で定義する．

この時，次が成り立つか？

(イ) f(n) が定数で上から抑えられ (f(n) ∈ O(1))，かつ g(n) が線形で下から抑えられる (g(n) ∈

Ω(n)) ならば L はほとんど空．

(ロ) limn→∞ f(n)/g(n) = 0 ならば L はほとんど空．

Dyck 言語は問題 1 の (イ) の具体例となっている．直感的には (ロ) は「多数派 (L の要素) の周り

には多数派が多く，少数派 (L の要素) の周りには少数派が少ない)」という状況を表している． - Thank you♪

Tokyo Tech Ofﬁcial Mascot: “Tech-chan”

(東工大公式マスコット：テックちゃん)

45 - Any questions

or comments?

Tokyo Tech Ofﬁcial Mascot: “Tech-chan”

(東工大公式マスコット：テックちゃん)

46