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2013/7/12に東工大で行なった「2013上半期オフライン論文読み/紹介し会」での発表資料です． http://partake.in/events/7289fbce-7b7d-4d6f-9b1d-0946803f881e ＊このスライドには正規表現は全く出て来ません Thue-Morse sequenceの紹介だけでも面白いと思います．
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Mix-Automatic Sequences (LATA 2013)
2013上半期オフライン論文読み/紹介し会
新屋良磨@東工大D1
(2013/7/12)
読み手の紹介しんやりょうま@sinya8282
東工大首藤研D1
首藤研のテーマはP2Pや分散システム
僕は形式言語(オートマトン)やってます
正規表現が好きです
Mix-Automatic Sequences (LATA 2013)
2013上半期オフライン論文読み/紹介し会新屋良磨@東工大D1
Automatic Sequences?無限列 “Sequence” に対するクラス
形式言語理論は有限長文字列の(無限)
集合，つまり“言語”が元々の対象
Automatic sequenceは無限長文字列に対する興味から始まった．
発表の流れ
Introduction to !ue-Morse seqeunce.
Introduction to Automatic Sequences
Automatic Sequences and Zip-Specifications (LICS’12)
Mix-Automatic Sequences (LATA’13)
Introduction to Thue-Morse sequence
Axel Thue (1863–1922)
5 / 55
Marston Morse (1892–1977)
6 / 55
Thue-Morse Seqeunce Axel Thue (1863–1922)
5 / 55
Marston Morse (1892–1977)
6 / 55
Axel !ue(1863-1922)
Maston Morse(1892-1977)
Thue-Morse SeqeunceDefinition (1)
!ue-Morse sequence is defined as
Thue-Morse SeqeunceDefinition (1)
!ue-Morse sequence is defined as
Chapter 1
The Thue–Morse Word
This chapter introduces the Thue-Morse word and presents several equiva-lent characterizations. We end with a novel application of the Thue-Morseword to construct magic squares.
1.1 The Thue–Morse word
Recall that a binary word is a word over the alphabet {0, 1}.
Definition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary wordt : N ! {0, 1} defined recursively by: t0 = 0; and for n " 0, t2n = tn andt2n+1 = tn, where a = 1 # a for a $ {0, 1}. (See Figure 1.1.)
t = t0 t1 t2 t3 · · · tm · · · t2m t2m+1 · · ·= 0 1 1 0 · · · a · · · a a · · · .
Figure 1.1: The Thue-Morse word t. Here a $ {0, 1}.
Example. Here are the first forty letters of the Thue–Morse word,
t = 0110100110010110100101100110100110010110 · · ·
Our first characterization of the Thue-Morse word is in terms of binaryexpansions of nonnegative integers. For every n $ N, let d2(n) denote thesum of the digits in the binary expansion of n.
Proposition 1.2. For all n $ N, we have tn = d2(n) mod 2.
83
Thue-Morse SeqeunceDefinition (1)
!ue-Morse sequence is defined as
Chapter 1
The Thue–Morse Word
This chapter introduces the Thue-Morse word and presents several equiva-lent characterizations. We end with a novel application of the Thue-Morseword to construct magic squares.
1.1 The Thue–Morse word
Recall that a binary word is a word over the alphabet {0, 1}.
Definition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary wordt : N ! {0, 1} defined recursively by: t0 = 0; and for n " 0, t2n = tn andt2n+1 = tn, where a = 1 # a for a $ {0, 1}. (See Figure 1.1.)
t = t0 t1 t2 t3 · · · tm · · · t2m t2m+1 · · ·= 0 1 1 0 · · · a · · · a a · · · .
Figure 1.1: The Thue-Morse word t. Here a $ {0, 1}.
Example. Here are the first forty letters of the Thue–Morse word,
t = 0110100110010110100101100110100110010110 · · ·
Our first characterization of the Thue-Morse word is in terms of binaryexpansions of nonnegative integers. For every n $ N, let d2(n) denote thesum of the digits in the binary expansion of n.
Proposition 1.2. For all n $ N, we have tn = d2(n) mod 2.
83
Chapter 1
The Thue–Morse Word
This chapter introduces the Thue-Morse word and presents several equiva-lent characterizations. We end with a novel application of the Thue-Morseword to construct magic squares.
1.1 The Thue–Morse word
Recall that a binary word is a word over the alphabet {0, 1}.
Definition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary wordt : N ! {0, 1} defined recursively by: t0 = 0; and for n " 0, t2n = tn andt2n+1 = tn, where a = 1 # a for a $ {0, 1}. (See Figure 1.1.)
t = t0 t1 t2 t3 · · · tm · · · t2m t2m+1 · · ·= 0 1 1 0 · · · a · · · a a · · · .
Figure 1.1: The Thue-Morse word t. Here a $ {0, 1}.
Example. Here are the first forty letters of the Thue–Morse word,
t = 0110100110010110100101100110100110010110 · · ·
Our first characterization of the Thue-Morse word is in terms of binaryexpansions of nonnegative integers. For every n $ N, let d2(n) denote thesum of the digits in the binary expansion of n.
Proposition 1.2. For all n $ N, we have tn = d2(n) mod 2.
83
Thue-Morse SeqeunceDefinition (2)
!en, !ue-Morse sequence is defined as
Definition (3)For every , let denote the sum of the digits in the binary expansion of .!en, !ue-Morse sequence is defined as
Thue-Morse Seqeunce
ホワイトボードで！
他にも定義は一杯(らしい)．
「円周率 やネイピア数 並の普遍的存在」
Thue-Morse Seqeunce
by Je#rey Shallit (Shallit先生)
Sequenceにおける「繰り返し」
DefinitionA square is a string of the form for some string .
Sequenceにおける「繰り返し」
DefinitionA square is a string of the form for some string .
A word is square-free if it contains no subword that is square.
Sequenceにおける「繰り返し」
DefinitionA square is a string of the form for some string .
A word is square-free if it contains no subword that is square.
An overlap is a string of the form for some
string and some single letter .
Sequenceにおける「繰り返し」
DefinitionA square is a string of the form for some string .
A word is square-free if it contains no subword that is square.
An overlap is a string of the form for some
string and some single letter .
A word is overlap-free if it contains no subword that is overlap.
Sequenceにおける「繰り返し」
DefinitionA square is a string of the form for some string .
A word is square-free if it contains no subword that is square.
An overlap is a string of the form for some
string and some single letter .
A word is overlap-free if it contains no subword that is overlap.
（証明は省略．結構ややこしい）!ue-Morse sequence is overlap-free.
Fact
Sequenceにおける「繰り返し」
ホワイトボードで！
!eorem!ere are no square-free binary strings of length
Sequenceにおける「繰り返し」
Sequenceにおける「繰り返し」
文字が2種類の場合は，square-freeなsequenceは存在しない．
では文字が3種類の場合は？
Sequenceにおける「繰り返し」
文字が2種類の場合は，square-freeなsequenceは存在しない．
では文字が3種類の場合は？
「square-freeでなるべく長い文字列を生成するバックトラックベースのプログラムを動かすとどうも止まらないっぽい．」らしい
QuestionDoes there exists a square-free sequence
over the alphabet
Avoidability in words という分野の芽吹き
QuestionDoes there exists a square-free sequence
over the alphabet Axel Thue (1863–1922)
5 / 55
Axel !ue
存在する．!ue-Morse sequence を使って構成できる．
Avoidability in words という分野の芽吹き
ホワイトボードで！
!eorem (!ue)
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
For n # 1, define cn to be the number of 1’s between the n-thand (n + 1)-th occurence of 0 in the Thue-Morse sequence t.Then the seqeunce c = 210201 · · · is a square-free sequenceover the alphabet !3.
1
Avoidability in words という分野の芽吹き
Avoidability in words という分野の芽吹き
最近でも色々activeに研究されてるらしい．
詳しくはShallit先生による解説スライド “!e Ubiquitous !ue-Morse Sequence” をチェック！
https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
おまけ: Thue-Morse sequenceと級数84 CHAPTER 1. THE THUE–MORSE WORD
Proof. Note that d2 satisfies the following recurrence relations: d2(0) = 0;d2(2n) = d2(n); and d2(2n + 1) = d2(n) + 1. Since d2(n) mod 2 satisfies thesame recurrences defining tn, we have tn = d2(n) mod 2.
Exercise 1.1. If t = t0t1t2 · · · is the Thue-Morse word, show that
!
n!0
(!1)tnxn = (1 ! x)(1 ! x2)(1 ! x4)(1 ! x8) · · · .
Exercise 1.2 ([AS1999]). Let t = t0t1t2 · · · be the Thue-Morse word andlet sn = (!1)tn for n " 0. Compute the following.
"1
2
#s0"
3
4
#s1"
5
6
#s2
· · ·"
2i + 1
2i + 2
#si
· · · .
(Hint: Let P =$
n!0
%2n+12n+2
&sn
, Q =$
n!1
%2n
2n+1
&sn
. Show that PQ = Q2P .)
1.2 The Thue–Morse morphism
Definition 1.3. The Thue-Morse morphism is the map µ : {0, 1}" #{0, 1}" defined by µ(0) = 01 and µ(1) = 10.
The Thue-Morse morphism µ is an example of a 2-uniform morphism:a morphism ! of words over an alphabet A is a k-uniform morphism if!(a) is a word of length k for all a $ A. Chapter 2.1 will have more to sayabout k-uniform morphisms.
If s is an infinite word over the alphabet {0, 1}, then let s be the imageof s under the endomorphism defined by 0 %# 1 and 1 %# 0. This morphismis often called the exchange morphism. Note that µ(s) = µ(s) for anyfinite or infinite word s over {0, 1}.
Proposition 1.4. The Thue-Morse word t is a fixed point of the Thue-Morse morphism µ, i.e., µ(t) = t. Moreover, t and t are the only fixedpoints of µ.
Proof. Suppose s is a binary word. Since µ maps each a $ {0, 1} to aa, itfollows that (µ(s))2n = sn and (µ(s))2n+1 = sn for all n " 0. So if µ(s) = s,then s2n = sn and s2n+1 = sn. If s0 = 0, then s = t; and if s0 = 1, thens = t. Therefore, t and t are the only fixed points of µ.
The above result characterizes the Thue-Morse word as the infinite bi-nary word beginning with 0 that is a fixed point of µ. Defining infinite words
Definition(3)から成り立つことが自明．
数列
は収束するか？
!ue-Morse sequenceを使って解く
おまけ: Thue-Morse sequenceと級数
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · ·
where sn = (!1)tn .
1
先ほどの数列の極限は
と!ue-Morse sequenceによる級数で表現可能．
と置いて， について求める．
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
として P について求める．
1
おまけ: Thue-Morse sequenceと級数
おまけ: Thue-Morse sequenceと級数
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
おまけ: Thue-Morse sequenceと級数
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
よって となり ．
おまけ: Thue-Morse sequenceと級数
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
よって となり ．
ところでこの
は何？有理数？無理数？
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
おまけ: Thue-Morse sequenceと級数
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
よって となり ．
ところでこの
は何？有理数？無理数？
!!
n"0
"2n + 1
2n + 2
#sn
=
"1
2
#s0 "3
4
#s1
· · ·"
2n + 1
2n + 2
#sn
· · · (1)
where sn = (!1)tn .
P = ("), Q =!!
n"0
"2n
2n + 1
#sn
.
PQ =1
2
!!
n"0
"n
n + 1
#sn
=1
2
!!
n"0
"2n + 1
2n + 2
#s2n+1 !!
n"1
"2n
2n + 1
#sn
=1
2· Q
P.
1
＿人人人人人人人人＿＞ 解けたら25＄ ＜‾Y^Y^Y^Y^Y^Y^Y‾
by Shallit先生
おまけ: Thue-Morse sequenceとチェス
FIDEの公式ルール (50手ルール): 以下の条件を満たすとき、どちらか一方のプレーヤーの要求によりゲームはその場でドローとなる。・過去50手の間、白・黒ともにポーンが動かず、またどの駒も取られていないとき。・（自分の手番の場合は）これから指す自分の着手の結果、上の条件が満たされるとき。 スコアシートにその手をあらかじめ記入し、確かにその手を指す意思があることを示さ なければならない。
チェスには「無限手数ゲーム」を防ぐためのルールがいくつかある．
チェスには「無限手数ゲーム」を防ぐためのルールがいくつかある．
FIDEの公式ルール (千日手):
相手の手で同一局面が3回生じたとき、または自分の次の手で同一局面が3回生じるときに引き分けとなる。ただし自動的に引き分けになるのではなく、自分の手番の時に指摘しなければならない。公式戦では、審判員（アービター）に申し立てる必要がある。
おまけ: Thue-Morse sequenceとチェス
実質的には50手ルールだけがあれば無限手数ゲームは起きない(引き分け要求した場合)．
千日手ルールだけの場合でも無限手数ゲームは起きない(引き分け要求した場合)．
千日手ルールを「同手順が3回連続したら」に緩めた場合は？
おまけ: Thue-Morse sequenceとチェス
The Thue-Morse Sequence and Chess
Consider the following alternative rule: a draw occurs if the samesequence of moves occurs twice in succession and is immediatelyfollowed by the first move of a third repetition.Can an infinite game of chess occur under this rule?
The question was answered by Max Euwe, the Dutch chess master(and world champion from 1935–1937) in 1929.
Figure: Max Euwe (1901–1981)
49 / 55
可能だよ！!ue-Morse sequence を使って構成できるよ！
Max Euwe (1901-1981)
おまけ: Thue-Morse sequenceとチェス
1935-1937 年度チェス世界王者
千日手ルールを「同手順が3回連続したら」に緩めた場合は？
おまけ: Thue-Morse sequenceとチェス
Chapter 1
The Thue–Morse Word
This chapter introduces the Thue-Morse word and presents several equiva-lent characterizations. We end with a novel application of the Thue-Morseword to construct magic squares.
1.1 The Thue–Morse word
Recall that a binary word is a word over the alphabet {0, 1}.
Definition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary wordt : N ! {0, 1} defined recursively by: t0 = 0; and for n " 0, t2n = tn andt2n+1 = tn, where a = 1 # a for a $ {0, 1}. (See Figure 1.1.)
t = t0 t1 t2 t3 · · · tm · · · t2m t2m+1 · · ·= 0 1 1 0 · · · a · · · a a · · · .
Figure 1.1: The Thue-Morse word t. Here a $ {0, 1}.
Example. Here are the first forty letters of the Thue–Morse word,
t = 0110100110010110100101100110100110010110 · · ·
Our first characterization of the Thue-Morse word is in terms of binaryexpansions of nonnegative integers. For every n $ N, let d2(n) denote thesum of the digits in the binary expansion of n.
Proposition 1.2. For all n $ N, we have tn = d2(n) mod 2.
83
!ue-Morse sequence
に対して， の時
OTHER SEMI-OPEN GAMES They start: 1. e2-e4 XABCDEFGH 8rsnlwqkvlntr( 7zppzppzppzpp' 6-+-+-+-+& 5+-+-+-+-% 4-+-+P+-+$ 3+-+-+-+-# 2PzPPzP-zPPzP" 1tRNvLQmKLsNR! Xabcdefgh
WHITE SAYS: These openings are not so popular because they're not so good. Whichever one you play I know how to gain an advantage.
BLACK SAYS: My opening's just as good as anything else. Because it's not so popular I have less to learn and you probably won't know very much about it.
Nb1-c3 Nb8-c6 Nc3-b1 Nc6-b8
の時Ng1-f3 Ng8-f6 Nf3-g1 Nf6-g8
と進めれば同一手順を3回以上繰り返さずに無限手数ゲームが可能！
おまけ: Thue-Morse sequenceとチェス
Chapter 1
The Thue–Morse Word
This chapter introduces the Thue-Morse word and presents several equiva-lent characterizations. We end with a novel application of the Thue-Morseword to construct magic squares.
1.1 The Thue–Morse word
Recall that a binary word is a word over the alphabet {0, 1}.
Definition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary wordt : N ! {0, 1} defined recursively by: t0 = 0; and for n " 0, t2n = tn andt2n+1 = tn, where a = 1 # a for a $ {0, 1}. (See Figure 1.1.)
t = t0 t1 t2 t3 · · · tm · · · t2m t2m+1 · · ·= 0 1 1 0 · · · a · · · a a · · · .
Figure 1.1: The Thue-Morse word t. Here a $ {0, 1}.
Example. Here are the first forty letters of the Thue–Morse word,
t = 0110100110010110100101100110100110010110 · · ·
Our first characterization of the Thue-Morse word is in terms of binaryexpansions of nonnegative integers. For every n $ N, let d2(n) denote thesum of the digits in the binary expansion of n.
Proposition 1.2. For all n $ N, we have tn = d2(n) mod 2.
83
!ue-Morse sequence
に対して， の時
OTHER SEMI-OPEN GAMES They start: 1. e2-e4 XABCDEFGH 8rsnlwqkvlntr( 7zppzppzppzpp' 6-+-+-+-+& 5+-+-+-+-% 4-+-+P+-+$ 3+-+-+-+-# 2PzPPzP-zPPzP" 1tRNvLQmKLsNR! Xabcdefgh
WHITE SAYS: These openings are not so popular because they're not so good. Whichever one you play I know how to gain an advantage.
BLACK SAYS: My opening's just as good as anything else. Because it's not so popular I have less to learn and you probably won't know very much about it.
Nb1-c3 Nb8-c6 Nc3-b1 Nc6-b8
の時Ng1-f3 Ng8-f6 Nf3-g1 Nf6-g8
と進めれば同一手順を3回以上繰り返さずに無限手数ゲームが可能！
!ue-Morse sequence はcube-free という性質を使っている(証明略)．
遍在する Thue-Morse sequence
この章で扱った !ue-Morse sequence の紹介は主に Shallit 先生による解説スライド
“!e Ubiquitous !ue-Morse Sequence” から．https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
遍在する Thue-Morse sequence
この章で扱った !ue-Morse sequence の紹介は主に Shallit 先生による解説スライド
“!e Ubiquitous !ue-Morse Sequence” から．
ここからようやくAutomatic Sequences の話
https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
Introduction to Thue-Morse sequence
Automatic Sequences?無限列 “Sequence” に対するクラス
形式言語理論は有限長文字列の(無限)
集合，つまり“言語”が元々の対象
Automatic sequenceは無限長文字列に対する興味から始まった．
形式言語理論における言語の階層
RegularContext free
Context sensitive
Recursively enumerable
形式言語理論における言語の階層
Automaton
Pushdownautomaton
Linear-boundedTuring machine
Turing machine
形式言語理論における言語の階層
Automaton
Pushdownautomaton
Linear-boundedTuring machine
Turing machine
「言語には計算モデルが色々ある．Sequenceは？」
DefinitionSequenceを特徴付ける: k-morphic
A morphism is a function satisfying for all
A morphism is prolongable on if there exists a letter such that for some
In this case, the infinite sequence
is the unique infinite fixed point of starting with
DefinitionSequenceを特徴付ける: k-morphic
A morphism is a function satisfying for all
A morphism is prolongable on if there exists a letter such that for some
In this case, the infinite sequence
is the unique infinite fixed point of starting with
このように，ある morphism の不動点となる sequence を morphic sequence と呼ぶ．
DefinitionSequenceを特徴付ける: k-morphic
A morphism is k-uniform if for all
An infinite sequence is k-morphic if there exists ak-uniform morphism that has as a fixed point.
!ue-Morse sequence は 2-morphic.
となるmorphismに対し，
DefinitionSequenceを特徴付ける: k-automatic
An infinite sequence is k-automatic if there exists ak-DFAO such that for all the output of the automatonwhen reading the word is ,with the base-k expansion of
ホワイトボードで！DFAOとか説明が面倒なので
Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
Sequenceを特徴付ける: k-automatic
!ue-Morse sequence は 2-automatic.Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
Sequenceを特徴付ける: k-automatic
!ue-Morse sequence は 2-automatic.Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
Sequenceを特徴付ける: k-automatic
!ue-Morse sequence は 2-automatic.Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
Sequenceを特徴付ける: k-automatic
!ue-Morse sequence は 2-automatic.Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
注意: この資料ではDFAOは常に右から左に文字列を読み進める！！
k-automatic = k-morphic
ホワイトボードで！
!eorem (Cobham)Let . !en a sequence is k-automatic if and only if is k-morphic.
Automatic Sequences andZip Specificatinos (LICS’12)
Automatic Sequences and Zip-SpecificationsClemens Grabmayer
Utrecht University, Dept. of PhilosophyJanskerkhof 13a, 3512 BL Utrecht, The Netherlands
Email: [email protected]
Jorg EndrullisVU University Amsterdam, Dept. of Computer Science
De Boelelaan 1081a, 1081 HV Amsterdam, The NetherlandsEmail: [email protected]
Dimitri HendriksVU University Amsterdam, Dept. of Computer Science
De Boelelaan 1081a, 1081 HV Amsterdam, The NetherlandsEmail: [email protected]
Jan Willem KlopVU University Amsterdam, Dept. of Computer Science
De Boelelaan 1081a, 1081 HV Amsterdam, The NetherlandsEmail: [email protected]
Lawrence S. MossIndiana University, Dept. of Mathematics
831 East Third Street, Bloomington, IN 47405-7106 USAEmail: [email protected]
Abstract—We consider infinite sequences of symbols, alsoknown as streams, and the decidability question for equality ofstreams defined in a restricted format. (Some formats lead to un-decidable equivalence problems.) This restricted format consistsof prefixing a symbol at the head of a stream, of the streamfunction ‘zip’, and recursion variables. Here ‘zip’ interleavesthe elements of two streams alternatingly. The celebrated Thue–Morse sequence is obtained by the succinct ‘zip-specification’
M = 0 : X X = 1 : zip(X,Y) Y = 0 : zip(Y,X)
The main results are as follows. We establish decidabilityof equivalence of zip-specifications, by employing bisimilar-ity of observation graphs based on a suitably chosen coba-sis. Furthermore, our analysis, based on term rewriting andcoalgebraic techniques, reveals an intimate connection betweenzip-specifications and automatic sequences. This leads to a newand simple characterization of automatic sequences. The study ofzip-specifications is placed in a wider perspective by employingobservation graphs in a dynamic logic setting, yielding yetanother alternative characterization of automatic sequences.
By the first characterization result, zip-specifications can beperceived as a term rewriting syntax for automatic sequences.For streams � the following are equivalent: (a) � can be specifiedusing zip; (b) � is 2-automatic; and (c) � has a finite observationgraph using the cobasis hhd, even, oddi. Here even and odd aredefined by even(a : s) = a : odd(s), and odd(a : s) = even(s).The generalization to zip-k specifications (with zip-k interleavingk streams) and to k-automaticity is straightforward.
As a natural extension of the class of automatic sequences, wealso consider ‘zip-mix’ specifications that use zips of differentarities in one specification. The corresponding notion of automa-ton employs a state-dependent input-alphabet, with a numberrepresentation (n)A = dm . . . d0 where the base of digit di isdetermined by the automaton A on input di�1 . . . d0.
Finally we show that equivalence is undecidable for a simpleextension of the zip-mix format with projections analogous toeven and odd. But it remains open whether zip-mix specificationswithout the extension have a decidable equivalence problem.
Index Terms—Automatic sequences, term rewriting, coalgebra,dynamic logic.
I. INTRODUCTION
Infinite sequences of symbols, also called ‘streams’, are aplayground of common interest for logic, computer science(functional programming, formal languages, combinatorics oninfinite words), mathematics (numerations and number theory,fractals) and physics (signal processing). For logic and theo-retical computer science this interest focuses in particular onunique solvability of systems of recursion equations definingstreams, expressivity of specification formats, and productivity(does a stream specification indeed unfold to its intendedinfinite result without stagnation). In addition, there is the‘infinitary word problem’: when do two stream specificationsover a first-order signature define the same stream? And, is thatquestion decidable? If not, what is the logical complexity?
Against this general background, we can now situate theactual content of this paper. In the landscape of streams thereare some well-known families, with automatic sequences [2]as a prominent family, including members such as the Thue–Morse sequence [1]. Such sequences are defined in first-ordersignature that includes some basic stream functions such ashd (head), tl (tail), ‘:’ (prefixing a symbol to an infinitestream), even, odd; all these are familiar from any functionalprogramming language.
One stream function in particular is frequently used instream specifications. This is the zip function, that ‘zips’ theelements of two streams in alternating order, starting withthe first stream. Now there is an elegant definition of theThue–Morse sequence M using only this function zip, nextto prefixing an element, and of course recursion variables:
M = 0 : X X = 1 : zip(X,Y) Y = 0 : zip(Y,X) (1)
For general term rewrite systems, stream equality is easilyseen to be undecidable [18], just as most interesting prop-erties of streams. But by adopting some restrictions in thedefinitional format, decidability may hold.
Zip-k specification
DefinitionFor , the function is defined by the following rewriting rule:
!us interleaves its argument sequences:
ホワイトボードで！Specificationの説明が面倒なので
Zip specification and Thue-Morse sequence!ue-Morse sequence は zip-2 specified
という zip-2 specification について，開始記号Mから!ue-Morse sequenceが生成される．
Zip specification and Thue-Morse sequence!ue-Morse sequence は zip-2 specified
という zip-2 specification について，開始記号Mから!ue-Morse sequenceが生成される．
Zip specification and Thue-Morse sequence!ue-Morse sequence は zip-2 specified
という zip-2 specification について，開始記号Mから!ue-Morse sequenceが生成される．
Zip specification and Thue-Morse sequence!ue-Morse sequence は zip-2 specified
という zip-2 specification について，開始記号Mから!ue-Morse sequenceが生成される．
Zip specification and Thue-Morse sequence!ue-Morse sequence は zip-2 specified
という zip-2 specification について，開始記号Mから!ue-Morse sequenceが生成される．
Unziping zip-2 に対する destructor: even と odd．
Unziping zip-2 に対する destructor: even と odd．
これをzipに対して使うと:
のように “unzip” できる．
zip-k specification = k-automatic
“Automatic Sequences and Zip Specifications” (LICS’12)での成果(簡略化して紹介)．
!eoremA sequence is k-automatic if and only if iszip-k specified.
証明にはObservation-graph なるものを使う説明が面倒くさいのでホワイトボードで！
paperfolding: (even/odd)-observation graph of zip-spec
(even/odd)-observation graph
Fold /^odd Tyrol /^
Peaks /^
Valleys /_
even
even
odd
even, odd
even, odd
Folds = zip(Tyrol,Folds)
Tyrol = zip(Peaks,Valleys)
Peaks = ^ : Peaks
Valleys = _ : Valleys
Folds !! ^ :^ :_ :^ :^ :_ :_ :^ :^ :^ :_ :_ :^ :_ :_ :^ . . .
証明にはObservation-graph なるものを使う説明が面倒くさいのでホワイトボードで！
zip-k specification = k-automatic
LICS’12 の論文．
arxiv版は証明とかでページ数が多い(32p)．
coalgebra とか cobasisとか bisimulation とか出てくる(困惑)．
このzip-k specificationを一般化したものが次章のMix-Automatic Sequencesに繋がる！
“Automatic Sequences and Zip-Specifications”
Mix-Automatic Sequences(LATA’13)
Mix-Automatic Sequences
Jorg Endrullis1, Clemens Grabmayer2, and Dimitri Hendriks1
1 VU University Amsterdam, The Netherlands2 Utrecht University, The Netherlands
Abstract. Mix-automatic sequences form a proper extension of theclass of automatic sequences, and arise from a generalization of finitestate automata where the input alphabet is state-dependent. In this pa-per we compare the class of mix-automatic sequences with the class ofmorphic sequences. For every polynomial ! we construct a mix-automaticsequence whose subword complexity exceeds !. This stands in contrastto automatic and morphic sequences which are known to have at mostquadratic subword complexity. We then adapt the notion of k-kernels toobtain a characterization of mix-automatic sequences, and employ thisnotion to construct morphic sequences that are not mix-automatic.
1 Introduction
Automatic sequences [1] were introduced by Cobham [4] in 1972, and have sincebeen been studied extensively. A sequence w : N ! ! over a finite alphabet !is automatic if it can be realized by a finite automaton that, for some k " 2,takes the base-k expansion (n)k of a number n # N as input and outputs the n-thletter of w; in this case w is called k-automatic. For multiplicatively independentk and ", k-automaticity and "-automaticity are almost separated notions; e.g., ifa sequence is both 2-automatic and 3-automatic, then it is ultimately periodic.
Therefore it is natural to study also nonstandard numeration systems, andthe classes of automatic sequences they give rise to. Rigo [10] and Rigo andMaes [11] study ‘abstract numeration systems’ based on the ‘shortlex’ order onan infinite regular language, induced by an order on the alphabet. With thisconcept they precisely capture the class of morphic sequences.
We introduce dynamic radix numeration systems which are obtained as a nat-ural generalization from another variation of the standard base-k representation:the mixed radix numeration systems [8] in which the base used only depends onthe position of a digit. In dynamic radix numeration systems the base used maydepend on the input digits read so far. Sequences realized by finite automatathat take dynamic radix input we call mix-automatic.
We first consider an example of a 2-automatic sequence, the celebrated Thue–Morse sequence, and explain how it is generated by the automaton in Figure 1.The automaton has states {q0, q1}, initial state q0, input alphabet {0, 1} andoutput alphabet {a, b}. The output letter assigned to q0 is a and to q1 is b(indicated by state/output in the states of the automaton). The n-th letter ofthe sequence is the output of the automaton when reading (n)2, the base-2
A.-H. Dediu, C. Martın-Vide, and B. Truthe (Eds.): LATA 2013, LNCS 7810, pp. 262–274, 2013.c! Springer-Verlag Berlin Heidelberg 2013
zip-k specification はk引数のzipのみを使う． (kは固定)
「1つの specification に任意引数の zip を使えるようにしたらどうなる？」 → Zip-mix specification
Zip-k specification の一般化
zip-k specification には DFAOが対応した． では zip-mix specificationには何が対応？
Zip-mix specification and mix-DFAO
ホワイトボードで！mix-DFAOとか説明が面倒なので
Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
mix-DFAOでは dynamic numeration system という特殊な記数法を使う！
Mix-Automatic Sequences 263
q0/a q1/b
0 1
1
0
Fig. 1. DFAO generating the Thue–Morse sequence abbabaabbaababba · · ·
expansion of n. For example, for input (3)2 = 11 the automaton ends in state q0with output a, and for input (4)2 = 100 in state q1 with output b.
The automaton of Figure 1 is called a deterministic finite-state automaton withoutput (DFAO). For k ! 2, a k-DFAO is an automaton over the input alphabetN<k = {0, 1, . . . , k " 1}. An infinite sequence w # !! is called k-automatic ifthere exists a k-DFAO such that for every n # N the output of the automatonwhen reading the word (n)k # N!
<k is w(n), with (n)k the base-k expansion of n.
Mix-Automatic Sequences. The class of automatic sequences is well-known tohave good closure properties; for example, it is closed under shifts (prependingletters or removing prefixes), and taking arithmetic subsequences. The class ofmix-automatic sequences extends the class of automatic sequences, has all theseclosure properties, and additionally is closed under k-shu!ing, for all k ! 2.
Mix-automatic sequences are defined via mix-DFAOs, automata that general-ize k-DFAOs by allowing that the alphabet of the symbol to be processed nextdepends on the current state. Let us consider the example of a mix-DFAO shownin Figure 2. The state q0 has two outgoing edges, reflecting the input alphabet{0, 1}, while q1 has three outgoing edges, reflecting the input alphabet {0, 1, 2}.
q0/a q1/b
0 10, 1
2
Fig. 2. An example of a mix-DFAO
Dynamic Radix Numeration Systems. Clearly, the numeration system used forthe input of mix-DFAOs cannot be the standard base-k representation. Instead,in the number representation that we let these automata operate on, the base foreach digit is determined by the lower-significance digits that have already beenread.Thus we let the automata read from the least to the most significant digit(i.e., we let the reading direction be from right to left). We write (n)M for thenumber representation of n that serves as input for the automaton M . For Mthe automaton from Figure 2, the representations of the first eight numbers are
(0)M = " (2)M = 1202 (4)M = 120202 (6)M = 131202
(1)M = 12 (3)M = 1312 (5)M = 2312 (7)M = 130312
where a subscript b (not part of the number representation) in db indicates thebase employed for d. Let us explain this at the example (17)M = 12022312.
264 J. Endrullis, C. Grabmayer, and D. Hendriks
Knowing the base for each digit, we can reconstruct the value of the representa-tion as follows: 17 = 1 ·2 ·3 ·2+0 ·3 ·2+2 ·2+1 where each digit is multiplied withthe product of the bases of the lower digits. Given just the representation 1021,the base of each of the digits is determined by the input alphabet of the state ofthe automaton reading the digit. The states q0 and q1 ofM have input alphabets{0, 1} and {0, 1, 2} and thus expect the input in base 2 and 3, respectively. Whenreading 1021 (right to left) the automaton M visits the states q0, q1, q0, q0 andq1. Annotating the input digits with the state of the automaton when readingthe digit, we obtain 1q00q02q11q0 , and taking into account the bases expected bythese states, yields 12022312.
We emphasize that, given a mix-DFAO M , every n ! N has a unique rep-resentation (n)M = dm · · · d0 (without leading zeros). This representation canbe computed as follows. Assume that we have determined the value of the dig-its di!1 · · · d0 with corresponding bases bi!1 · · · b0. The base bi of digit di isdetermined by the input alphabet of the state of the automaton after read-ing di!1 · · · d0 (right to left), and digit di is the remainder of the division ofn"
!0"j<i dj(bj!1 · · · b1 · b0) by bi.
Every mix-DFAO M gives rise to a mix-automatic sequence w ! !! bydefining for every n ! N, w(n) as the output of M when reading (n)M .
Zip-Specifications. In [6] it has been shown that k-automatic sequences are pre-cisely the class of sequences definable by zip-k specifications, that is, systems ofrecursion equations {X1 = t1, . . . , Xn = tn} with terms ti built from the syntax
t ::= Xi | a : t | zipk(t, . . . , t) (1 # i # n, a ! !)
Semantically, the term notation a : t indicates the concatenation of a letter with asequence, and the k-ary symbol zipk stands for the function of type ("!)k $ "!
that zips (or interleaves or shu!es) its k argument sequences, and is defined by
zipk(w0, . . . , wk!1)(kn+ i) = wi(n) (0 # i < k)
Operationally, zipk can be defined by the rewrite rule
zipk(x : t0, t1, . . . , tk!1) $ x : zipk(t1, . . . , tk!1, t0) (1)
The zip operation on finite words is known in the literature as perfect shu!e [2].An example of a zip-2 specification corresponding to the 2-DFAO from Fig. 1 is
M = a : Q1 Q0 = a : zip2(Q0,Q1) Q1 = b : zip2(Q1,Q0) (2)
The Thue–Morse sequence is the unique solution for the variable M in this spec-ification, or, from a rewriting perspective, it is the infinite normal form of M inthe rewrite system consisting of (1) and (2), orienting the equations from left toright. For further details we refer to [6].
The introduction of mix-automatic sequences was motivated by the charac-terization of k-automatic sequences as the class of sequences that can defined by
zip-mix specified = mix-automatic sequence
実はここまで “Automatic Sequences and Zip Specifications” (LICS’12)での成果．
!eoremA sequence is mix-automatic if and only if iszip-mix specified.
では，“Mix-Automatic Sequences” (LATA’13)での成果は？
Mix-Automatic Sequences での成果は三つMix-automatic sequence に対して 「mix-kernelが有限」 という別の特徴づけをした(超マニアック)
Mix-Automatic Sequences での成果は三つMix-automatic sequence に対して 「mix-kernelが有限」 という別の特徴づけをした(超マニアック)
任意の多項式 f に対して subword complexity が Ω(f(n)) の mix-automatic sequence となる具体的構成法を与えた．系として「morphic sequence でない mix-automatic sequence」の存在を示した．
Mix-Automatic Sequences での成果は三つMix-automatic sequence に対して 「mix-kernelが有限」 という別の特徴づけをした(超マニアック)
任意の多項式 f に対して subword complexity が Ω(f(n)) の mix-automatic sequence となる具体的構成法を与えた．系として「morphic sequence でない mix-automatic sequence」の存在を示した．
「mix-automatic sequence でない morphic sequence」 の存在(構成法)を示した．
Mix-Automatic Sequences での成果は三つ
任意の多項式 f に対して subword complexity が Ω(f(n)) の mix-automatic sequence となる具体的構成法を与えた．系として「morphic sequence でない mix-automatic sequence」の存在を示した．
僕が疲れてない∧皆が疲れてない∧時間がある ⇒ ホワイトボードで！
Mix-Automatic Sequences での成果は三つ
「mix-automatic sequence でない morphic sequence」 の存在(構成法)を示した．
僕が疲れてない∧皆が疲れてない∧時間がある ⇒ ホワイトボードで！
形式言語理論における言語の階層(再)
RegularContext free
Context sensitive
Recursively enumerable
Sequenceの階層
k-automatic sequence,
k-morphic sequencezip-k specified sequence,morphic
sequencemix
automaticsequence
論文で紹介されてる未解決問題
Mix-Automatic Sequences 273
sequences are generated by DFAOs with state-dependent input alphabets. Theseautomata read number representations dndn!1 · · · d0 where the base of a digitdk depends on the value of the lower-significance digits dk!1 · · · d0.
The results of this paper can be summarized as follows:
(i) A characterization of mix-automatic sequences via a generalization of theconcept of k-kernel (by which automatic sequences can be characterized).
(ii) For every polynomial ! there is a mix-automatic sequence whose subwordcomplexity exceeds !. As a consequence there are mix-automatic sequencesthat are not morphic, since morphic sequences have quadratic subwordcomplexity at most.
(iii) A morphic sequence that is not mix-automatic, showing that the class ofmorphic sequences is not contained in the class of mix-automatic sequences.
All of these concepts are very recent, and many interesting questions remain.We highlight three particularly intriguing, and challenging questions:
(1) (J.-P. Allouche) Characterize the intersection of mix-automatic and morphicsequences. (Note that at least all automatic sequences are in.)
(2) Is the following problem decidable: Given two mix-DFAOs, do they generatethe same sequence?
(3) Can Cobham’s Theorem (below) be generalized to mix-automatic sequences?
Cobham’s Theorem ([3]). Let k, " ! 2 be multiplicatively independent (i.e.,ka "= "b, for all a, b > 0), and let w # #! be both k- and "-automatic. Then w isultimately periodic.
In order to generalize this theorem to mix-automatic sequences, one could lookfor a suitable notion of multiplicative independence for base determiners. Recallthat base determiners are themselves finite automata with output.
References
1. Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generaliza-tions. Cambridge University Press, New York (2003)
2. Berstel, J.: Transductions and Context-Free Languages. Teubner (1979)3. Cobham, A.: On the Base-Dependence of Sets of Numbers Recognizable by Finite
Automata. Mathematical Systems Theory 3(2), 186–192 (1969)4. Cobham, A.: Uniform Tag Sequences. Theory of Computing Systems 6, 164–192
(1972)5. Ehrenfeucht, A., Lee, K.P., Rozenberg, G.: Subword Complexity of Various Classes
of Deterministic Languages without Interaction. Theoretical Computer Science 1,59–75 (1975)
6. Grabmayer, C., Endrullis, J., Hendriks, D., Klop, J.W., Moss, L.S.: AutomaticSequences and Zip-Specifications. In: Proc. Symp. on Logic in Computer Science(LICS 2012), pp. 335–344. IEEE Computer Society (2012)
