"Mix Automatic Sequences"(LATA'13) の紹介
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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'13) の紹介
- 4. 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．
- 5. 発表の流れ Introduction to Thue-Morse seqeunce. Introduction to Automatic Sequences Automatic Sequences and Zip-Specifications (LICS’12) Mix-Automatic Sequences (LATA’13)
- 6. Introduction to Thue-Morse sequence Thue (1863–1922) 5 / 55 Marston Morse (1892–1977)
- 7. Thue-Morse Seqeunce Thue (1863–1922) 5 / 55 Marston Morse (1892–1977) Axel Thue (1863-1922) Maston Morse (1892-1977)
- 8. Thue-Morse Seqeunce Definition (1) Thue-Morse sequence is defined as
- 9. Thue-Morse Seqeunce Definition (1) Thue-Morse sequence is defined as to construct magic squares. The Thue–Morse word that a binary word is a word over the alphabet {0, 1}. ition 1.1. The Thue-Morse word t = t0t1t2 · · · is the bina → {0, 1} defined recursively by: t0 = 0; and for n ≥ 0, t2n = = ¯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}.
- 10. Thue-Morse Seqeunce Definition (1) Thue-Morse sequence is defined as to construct magic squares. The Thue–Morse word that a binary word is a word over the alphabet {0, 1}. ition 1.1. The Thue-Morse word t = t0t1t2 · · · is the bina → {0, 1} defined recursively by: t0 = 0; and for n ≥ 0, t2n = = ¯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}. 1 The Thue–Morse word call that a binary word is a word over the alphabet {0, 1}. finition 1.1. The Thue-Morse word t = t0t1t2 · · · is the binary N → {0, 1} defined recursively by: t0 = 0; and for n ≥ 0, t2n = +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}. ample. Here are the first forty letters of the Thue–Morse word, t = 0110100110010110100101100110100110010110 · · ·
- 11. Thue-Morse Seqeunce Definition (2) Then, Thue-Morse sequence is defined as
- 12. Definition (3) For every , let denote the sum of the digits in the binary expansion of . Then, Thue-Morse sequence is defined as Thue-Morse Seqeunce ホワイトボードで！
- 13. 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 Jeffrey Shallit (Shallit先生)
- 14. Sequenceにおける「繰り返し」 Definition A square is a string of the form for some string .
- 15. Sequenceにおける「繰り返し」 Definition A square is a string of the form for some string . A word is square-free if it contains no subword that is square.
- 16. Sequenceにおける「繰り返し」 Definition A 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 .
- 17. Sequenceにおける「繰り返し」 Definition A 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.
- 18. Sequenceにおける「繰り返し」 Definition A 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） Thue-Morse sequence is overlap-free. Fact
- 19. Sequenceにおける「繰り返し」 ホワイトボードで！ Theorem There are no square-free binary strings of length
- 23. Question Does there exists a square-free sequence over the alphabet Avoidability in words という分野の芽吹き
- 24. Question Does there exists a square-free sequence over the alphabet Axel Thue (1863–1922) 5 / 55 Axel Thue 存在する． Thue-Morse sequence を 使って構成できる． Avoidability in words という分野の芽吹き
- 25. ホワイトボードで！ Theorem (Thue) 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-th and (n + 1)-th occurence of 0 in the Thue-Morse sequence t. Then the seqeunce c = 210201 · · · is a square-free sequence over the alphabet Σ3. Avoidability in words という分野の芽吹き
- 26. 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 “The Ubiquitous Thue-Morse Sequence” を チェック！ https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
- 27. おまけ: Thue-Morse sequenceと級数 CHAPTER 1. THE THUE–MORSE W f. Note that d2 satisfies the following recurrence relations: d2(0) n) = d2(n); and d2(2n + 1) = d2(n) + 1. Since d2(n) mod 2 satisfie recurrences defining tn, we have tn = d2(n) mod 2. rcise 1.1. If t = t0t1t2 · · · is the Thue-Morse word, show that n≥0 (−1)tn xn = (1 − x)(1 − x2 )(1 − x4 )(1 − x8 ) · · · . rcise 1.2 ([AS1999]). Let t = t0t1t2 · · · be the Thue-Morse word n = (−1)tn for n ≥ 0. Compute the following. 1 2 s0 3 4 s1 5 6 s2 · · · 2i + 1 2i + 2 si · · · . Definition(3)から成り立つことが自明．
- 28. 数列 は収束するか？ Thue-Morse sequenceを使って解く おまけ: Thue-Morse sequenceと級数
- 29. ∞ n≥0 2n + 1 2n + 2 sn = 1 2 s0 3 4 s1 · · · 2n + 1 2n + 2 sn · · · where sn = (−1)tn . 先ほどの数列の極限は とThue-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 · · · where sn = (−1)tn . P = (∗), Q = ∞ n≥0 2n 2n + 1 sn . て求める． おまけ: Thue-Morse sequenceと級数
- 30. おまけ: Thue-Morse sequenceと級数 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 .
- 31. おまけ: Thue-Morse sequenceと級数 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 . よって となり ．
- 32. おまけ: Thue-Morse sequenceと級数 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 . 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 · 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 .
- 33. おまけ: Thue-Morse sequenceと級数 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 . 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 · 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 . ＿人人人人人人人人＿ ＞ 解けたら25＄ ＜ ￣Y^Y^Y^Y^Y^Y^Y￣ by Shallit先生
- 34. 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．
- 37. Can an infinite game of chess occur under this The question was answered by Max Euwe, the (and world champion from 1935–1937) in 192 Figure: Max Euwe (1901–19 可能だよ！ Thue-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？
- 38. おまけ: Thue-Morse sequenceとチェス = 0 1 1 0 · · · a · · · a ¯a · · · . Figure 1.1: The Thue-Morse word t. Here a ∈ {0, 1}. mple. 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 b ansions of nonnegative integers. For every n ∈ N, let d2(n) denot of the digits in the binary expansion of n. position 1.2. For all n ∈ N, we have tn = d2(n) mod 2. 83 Thue-Morse sequence に対して， の時 OTHER SEMI-OPEN GAM They start: 1. e2-e4 XABCDEFGH 8rsnlwqkvlntr( 7zppzppzppzpp' 6-+-+-+-+& 5+-+-+-+-% 4-+-+P+-+$ 3+-+-+-+-# 2PzPPzP-zPPzP" 1tRNvLQmKLsNR! Xabcdefgh WHITE SAYS: 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！
- 39. おまけ: Thue-Morse sequenceとチェス = 0 1 1 0 · · · a · · · a ¯a · · · . Figure 1.1: The Thue-Morse word t. Here a ∈ {0, 1}. mple. 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 b ansions of nonnegative integers. For every n ∈ N, let d2(n) denot of the digits in the binary expansion of n. position 1.2. For all n ∈ N, we have tn = d2(n) mod 2. 83 Thue-Morse sequence に対して， の時 OTHER SEMI-OPEN GAM They start: 1. e2-e4 XABCDEFGH 8rsnlwqkvlntr( 7zppzppzppzpp' 6-+-+-+-+& 5+-+-+-+-% 4-+-+P+-+$ 3+-+-+-+-# 2PzPPzP-zPPzP" 1tRNvLQmKLsNR! Xabcdefgh WHITE SAYS: 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！ Thue-Morse sequence は cube-free という性質を 使っている(証明略)．
- 40. 遍在する Thue-Morse sequence この章で扱った Thue-Morse sequence の紹介 は主に Shallit 先生による解説スライド “The Ubiquitous Thue-Morse Sequence” から． https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
- 41. 遍在する Thue-Morse sequence この章で扱った Thue-Morse sequence の紹介 は主に Shallit 先生による解説スライド “The Ubiquitous Thue-Morse Sequence” から． ここからようやくAutomatic Sequences の話 https://cs.uwaterloo.ca/~shallit/Talks/green3.pdf
- 43. 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．
- 47. Definition Sequence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
- 48. Definition Sequence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．
- 49. Definition Sequence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 a k-uniform morphism that has as a fixed point. Thue-Morse sequence は 2-morphic. となるmorphismに対し，
- 50. Definition Sequenceを特徴付ける: k-automatic An infinite sequence is k-automatic if there exists a k-DFAO such that for all the output of the automaton when 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 ating the Thue–Morse sequence abbabaabbaababba· · ·
- 51. Sequenceを特徴付ける: k-automatic Thue-Morse sequence は 2-automatic. Mix-Automatic Sequences q0/a q1/b 0 1 1 0 1. DFAO generating the Thue–Morse sequence abbabaabbaababba· · · of n. For example, for input (3)2 = 11 the automaton ends in sta ut a, and for input (4)2 = 100 in state q1 with output b. tomaton of Figure 1 is called a deterministic finite-state automaton FAO). For k ≥ 2, a k-DFAO is an automaton over the input alph 0, 1, . . ., k − 1}. An infinite sequence w ∈ ∆ω is called k-automat
- 52. Sequenceを特徴付ける: k-automatic Thue-Morse sequence は 2-automatic. Mix-Automatic Sequences q0/a q1/b 0 1 1 0 1. DFAO generating the Thue–Morse sequence abbabaabbaababba· · · of n. For example, for input (3)2 = 11 the automaton ends in sta ut a, and for input (4)2 = 100 in state q1 with output b. tomaton of Figure 1 is called a deterministic finite-state automaton FAO). For k ≥ 2, a k-DFAO is an automaton over the input alph 0, 1, . . ., k − 1}. An infinite sequence w ∈ ∆ω is called k-automat
- 53. Sequenceを特徴付ける: k-automatic Thue-Morse sequence は 2-automatic. Mix-Automatic Sequences q0/a q1/b 0 1 1 0 1. DFAO generating the Thue–Morse sequence abbabaabbaababba· · · of n. For example, for input (3)2 = 11 the automaton ends in sta ut a, and for input (4)2 = 100 in state q1 with output b. tomaton of Figure 1 is called a deterministic finite-state automaton FAO). For k ≥ 2, a k-DFAO is an automaton over the input alph 0, 1, . . ., k − 1}. An infinite sequence w ∈ ∆ω is called k-automat
- 54. Sequenceを特徴付ける: k-automatic Thue-Morse sequence は 2-automatic. Mix-Automatic Sequences q0/a q1/b 0 1 1 0 1. DFAO generating the Thue–Morse sequence abbabaabbaababba· · · of n. For example, for input (3)2 = 11 the automaton ends in sta ut a, and for input (4)2 = 100 in state q1 with output b. tomaton of Figure 1 is called a deterministic finite-state automaton FAO). For k ≥ 2, a k-DFAO is an automaton over the input alph 0, 1, . . ., k − 1}. An infinite sequence w ∈ ∆ω is called k-automat 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！！
- 55. k-automatic = k-morphic ホワイトボードで！ Theorem (Cobham) Let . Then a sequence is k-automatic if and only if is k-morphic.
- 56. Automatic Sequences and Zip Specificatinos (LICS’12)
- 57. Automatic Sequences and Zip-Specifications Clemens Grabmayer Utrecht University, Dept. of Philosophy Janskerkhof 13a, 3512 BL Utrecht, The Netherlands Email: clemens@phil.uu.nl J¨org Endrullis VU University Amsterdam, Dept. of Computer Science De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands Email: j.endrullis@vu.nl Dimitri Hendriks VU University Amsterdam, Dept. of Computer Science De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands Email: r.d.a.hendriks@vu.nl Jan Willem Klop VU University Amsterdam, Dept. of Computer Science De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands Email: j.w.klop@vu.nl Lawrence S. Moss Indiana University, Dept. of Mathematics 831 East Third Street, Bloomington, IN 47405-7106 USA Email: lsm@cs.indiana.edu Abstract—We consider infinite sequences of symbols, also known as streams, and the decidability question for equality of streams defined in a restricted format. (Some formats lead to un- decidable equivalence problems.) This restricted format consists of prefixing a symbol at the head of a stream, of the stream function ‘zip’, and recursion variables. Here ‘zip’ interleaves the elements of two streams alternatingly. The celebrated Thue– I. INTRODUCTION Infinite sequences of symbols, also called ‘streams’, are a playground of common interest for logic, computer science (functional programming, formal languages, combinatorics on infinite words), mathematics (numerations and number theory,
- 58. Zip-k specification Definition For , the function is defined by the following rewriting rule: Thus interleaves its argument sequences: ホワイトボードで！ Specificationの説明が面倒なので
- 59. Zip specification and Thue-Morse sequence Thue-Morse sequence は zip-2 specified という zip-2 specification について，開始記号 MからThue-Morse sequenceが生成される．
- 60. Zip specification and Thue-Morse sequence Thue-Morse sequence は zip-2 specified という zip-2 specification について，開始記号 MからThue-Morse sequenceが生成される．
- 61. Zip specification and Thue-Morse sequence Thue-Morse sequence は zip-2 specified という zip-2 specification について，開始記号 MからThue-Morse sequenceが生成される．
- 62. Zip specification and Thue-Morse sequence Thue-Morse sequence は zip-2 specified という zip-2 specification について，開始記号 MからThue-Morse sequenceが生成される．
- 63. Zip specification and Thue-Morse sequence Thue-Morse sequence は zip-2 specified という zip-2 specification について，開始記号 MからThue-Morse sequenceが生成される．
- 64. Unziping zip-2 に対する destructor: even と odd．
- 65. 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．
- 66. zip-k specification = k-automatic “Automatic Sequences and Zip Specifications” (LICS’12)での成果(簡略化して紹介)． Theorem A sequence is k-automatic if and only if is zip-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！
- 67. 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
- 68. 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”
- 70. Mix-Automatic Sequences J¨org Endrullis1 , Clemens Grabmayer2 , and Dimitri Hendriks1 1 VU University Amsterdam, The Netherlands 2 Utrecht University, The Netherlands Abstract. Mix-automatic sequences form a proper extension of the class of automatic sequences, and arise from a generalization of finite state automata where the input alphabet is state-dependent. In this pa- per we compare the class of mix-automatic sequences with the class of
- 71. 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？
- 72. 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 e state q0 has two outgoing edges, reflecting the inp has three outgoing edges, reflecting the input alpha q0/a q1/b 0 1 0, 1 2 Fig. 2. An example of a mix-DFAO Numeration Systems. Clearly, the numeration sys -DFAOs cannot be the standard base-k representat epresentation that we let these automata operate on mix-DFAOでは dynamic numeration system という特殊な記数法を使う！ numeration system used for ase-k representation. Instead, mata operate on, the base for digits that have already been to the most significant digit left). We write (n)M for the or the automaton M. For M of the first eight numbers are 0202 (6)M = 131202 12 (7)M = 130312 sentation) in db indicates the example (17)M = 12022312. 264 J. Endrullis, C. Grabmayer, and D. Hendriks Knowing the base for each digit, we can reconstru tion as follows: 17 = 1·2·3·2+0·3·2+2·2+1 wher the product of the bases of the lower digits. Given the base of each of the digits is determined by the the automaton reading the digit. The states q0 and {0, 1} and {0, 1, 2} and thus expect the input in ba
- 73. zip-mix specified = mix-automatic sequence 実はここまで “Automatic Sequences and Zip Specifications” (LICS’12)での成果． Theorem A sequence is mix-automatic if and only if is zip-mix specified. では，“Mix-Automatic Sequences” (LATA’13) での成果は？
- 74. 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)
- 75. 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．
- 76. 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．
- 77. 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！
- 78. 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！
- 80. Sequenceの階層 k-automatic sequence, k-morphic sequence zip-k specified sequence,morphic sequence mix automatic sequence
- 81. 論文で紹介されてる未解決問題 (ii) For every polynomial ϕ there is a mix-automatic sequence whose subword complexity exceeds ϕ. As a consequence there are mix-automatic sequences that are not morphic, since morphic sequences have quadratic subword complexity at most. (iii) A morphic sequence that is not mix-automatic, showing that the class of morphic 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 morphic sequences. (Note that at least all automatic sequences are in.) (2) Is the following problem decidable: Given two mix-DFAOs, do they generate the 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 ∈ ∆ω be both k- and -automatic. Then w is ultimately periodic. In order to generalize this theorem to mix-automatic sequences, one could look for a suitable notion of multiplicative independence for base determiners. Recall that base determiners are themselves finite automata with output.