Katsura, Masashi

Kunimochi, Yoshiyuki

Tanaka, Genjiro

Kyoto University Research Information Repository

TitleRemarks on extractable submonoids (Languages,Computations, and Algorithms in Algebraic Systems)Author(s)Tanaka, Genjiro; Kunimochi, Yoshiyuki; Katsura, MasashiCitation数理解析研究所講究録 (2009), 1655: 106-110Issue Date2009-06URL http://hdl.handle.net/2433/140852RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRemarks on extractable submonoidsGenjiro Tanaka, Yoshiyuki KunimochiDept. of Computer Science, Shizuoka Institute of Science and Technology,Fukuroi-shi, 437-8555 Japan.Masashi KatsuraDept. of Mathematic, Kyoto Sangyo University,Kita-Ku, Kyoto, 603-5555 Japan.Key words: code, extractable code, bifix code, uniform code, free submonoid, free monoid.Abstract. This paper deals with extractable codes. The base of free submonoid of a free monoidis called a code. The code $C$ with the property that $z,$ $xzy\in C’$ implies $xy\in C^{*}$ is called anextractable code. Such $C$ is a bifix code, and for example, appears as a Petri net code or a groupcode. In this paper we examine basic properties of extractable codes.1. INTRODUCTIONLet $A$ be an alphabet, $A^{*}$ the free monoid over $A$ , and 1 the empty word. Let $A^{+}=A^{*}-\{1\}$ . Aword $v\in A^{*}$ is a right $factor\langle resp$ . left factor) of a word $u\in A^{*}$ if there is a word $w\in A^{*}$ such that$u=wv(resp. u=vw)$ . If $v$ is a right factor of $u$ , we write $v<_{S}u$ . Similarly, we write $v<_{p}u$ if $v$is a left factor of $u$ . For a word $w\in A^{*}$ and a letter $x\in A$ we let $|w|_{x}$ denote the number of $x$ in$w$ . The length $|w|$ of $w$ is the number of letters in $w$ . Therefore $A^{n}=\{w\in A^{*}||w|=n\},$ $n\geq 1$ .Alph$(w)$ is the set of all letters occurring at least once in $w$ .A non-empty subset $C$ of $A+$ is said to be a code if for $x_{1},$ $\ldots,$ $x_{p},$ $y_{1},$ $\ldots,$ $y_{q}\in C,$ $p,$ $q\geq 1$ ,$x_{1}\cdots x_{p}=y_{1}\cdots y_{q}\Rightarrow p=q,$ $x_{1}=y_{1},$ $\ldots,$ $x_{p}=y_{p}$ .A subset $M$ of $A^{*}$ is a submonoid of $A^{*}$ if $M^{2}\subseteq M$ and $1\in M$ . Every submonoid $M$ of a freemonoid has a unique minimal set of generators $C=(M-\{1\})-(M-\{1\})^{2}$ . $C$ is called the baseof $M$ . A submonoid $M$ is right unitary in $A^{*}$ if for all $u,$ $v\in A^{*}$ ,$u,$ $uv\in M\Rightarrow v\in M$ .$M$ is called left unitary in $A^{*}$ if it satisfies the dual condition. A submonoid $M$ is biunitary if itis both left and right unitary. Let $M$ be a submonoid of a free monoid $A^{*}$ , and $C$ its base. If$CA^{+}\cap C=\emptyset$ , $($ resp. $A^{+}C\cap C=\emptyset)$ , then $C$ is called a prefix (resp. suffix) code over A. $C$ is calleda bifix code if it is a prefix and suffix code. A submonoid $M$ of $A^{*}$ is right unitary (resp. biunitary)if and only if its minimal set of generators is a prefix code (resp. bifix code) $([1, p.46],[3, p.108])$ .1655 2009 106-110 106Let $C$ be a nonempty subset of $A^{*}$ . If $|x|=|y|$ for all $x,$ $y\in C$ , then $C$ is a bifix code. We callsuch a code a uniform code. The uniform code $A^{n},$ $n\geq 1$ , is called a full uniform code.Let $M$ be a submonoid of $A^{*}$ . If the condition $z,xzy\in M$ implies $xy\in M$ , then $M$ is called anextractable submonoid of $A^{*}$ .If a submonoid $N$ is extractable, then $u,$ $1uv\in C^{*}$ implies lv $=v\in C^{*}$ . Similarly $v,$ $uv\in C^{*}$implies $u\in C^{*}$ . Consequently $M$ is biunitary. Therefore its minimal set of generators $C$ is a bifixcode.Deflnition 1. Let $C\subset A^{*}$ be a code. If C’ is extractable, then $C$ is called an extractable code.1. FUNDMENTAL PROPERTIES OF EXTRACTABLE CODESLet $\mathcal{A}=(Q, A, \delta, 1, F)$ be an automaton, where $Q,$ $A,$ $\delta$ : $Q\cross Aarrow Q,$ $1$ , and $F$ , are the stateset, the input set, the next-state function, the initial state, and the final set of $\mathcal{A}$ , respectively(For basic concepts of automata, refer to [4] or [1]). If for any $(p, q)\in QxQ$ there exists some$w\in A^{*}$ such that $\delta(p, w)=q$, then $\mathcal{A}$ is called transitive. If $\mathcal{A}$ has a fixed point $s$ , and if for evcry$p,$ $q\in Q,p\neq s$ , there exists $w\in A^{*}$ such that $\delta(p, w)=q$ , then $\mathcal{A}$ is called $0$ -transitive. If anautomaton $\mathcal{A}$ is either transitive or 0-transitive, then $\mathcal{A}$ is called a $[0]$ -transitive automaton.Let $L$ be a subset of $A^{*}$ . For each $x\in A^{*}$ , we define the set of all right contexts of $x$ with respectto $L$ by$C\sigma nt_{L}^{(r)}(x)=\{w\in A^{*}|xw\in L\}$ .The right principal congruence $P_{L}^{(r)}$ of $L$ is defined by $(x, y)\in P_{L}^{(r)}$ if and only if $Cmt_{L}^{(r)}(x)=$$Cont_{L}^{(r)}(y)$ . Let $u\in A^{*}$ . By $[u]_{L}$ we denote the $P_{L}^{(r)}$ -class of $u$ by $[u]_{L}$ or simply by $[u]$ . That is$[u]_{L}=\{v|Ccmt_{C}^{(r)}.(v)=Cont_{C^{r}}^{(r)}(u), v\in A^{*}\}$ .We denote by $[w_{\phi}]$ the class of $P_{C^{*}}^{(r)}$ consisting of all words $w\in A^{*}$ such that $wA^{*}\cap C^{*}=\phi$.Let $C$ be a prefic code. We defined the automaton $\mathcal{A}(C^{*})=(A^{*}/P_{C}^{(r)}, A, \delta, [1], \{[1]\})$, where$\delta([w], x)=[wx]$ for $[w]\in A^{*}/P_{C^{r}}^{(r)}$ and $x\in A^{*}$ . Then the automaton $\mathcal{A}(C")$ , is minimal and$[0]$ -transitive.Let $\mathcal{A}=(Q, A, \delta, 1, \{1\})$ be a $[0]$ -transitive minimal automaton recognizing $C^{*}$ for some prefixcode $C$ . For each $p\in Q$ we put$W_{p}=\{w\in A^{*}|\delta(p, w)=1\}$ .Define the congruence $\rho$ on $\mathcal{A}$ is by$p,$ $q\in Q,$ $p\rho q\Leftrightarrow W_{p}=W_{q}$ .Then $\rho$ is the equality(See [2, p.215]).107Proposition 1. Let $C$ be a prefix code, and let $\mathcal{A}=(Q, A, \delta, 1, \{1\})$ be a $[0]$ -transitive au-tomaton recognizing $C^{*}$ . The following conditions are equivalent:(1) $C$ is extractable.(2) $W_{\delta(p,z)}\subset W_{p}$ for all $p\in Q,$ $z\in C^{*}$ .Corollary 2. Let $C$ be a prefix code, and let $A=(Q, A, \delta, 1, \{1\})$ be a $[0]$ -transitive minimalautomaton recognizing $C^{*}$ . If there exists some $z_{1},$ $z_{2}\in C$“ such that $\delta(p, z_{1})=q$ and $\delta(q,z_{2})=p$for some $p,$ $q\in Q,$ $p\neq q$ , then $C$ is not extractable.Let $C$ be a prefix code, and let $\mathcal{A}(C^{*})=(A^{*}/P_{C}^{(r)}, A,\delta, [1], \{[1]\})$ be the minimal automaton of$C$“. Then, for $[x]\in A^{*}/P_{C}^{(r)}$ we have$u\in W_{[x]}\Leftrightarrow\delta([x], u)=[1]\Leftrightarrow xu\in C^{*}\Leftrightarrow u\in Cont_{C^{r}}^{(r)}(x)$ .That is, $W_{[x]}=Cont_{C^{*}}^{(r)}(x)$ . Therefore we have the following rewriting of Proposition 1.Proposition 3. Let $C$ be a prefix code. Then the following two condition are equivalent:(1) $C$ is extractable.(2) $Cont_{C}^{(r)}(xz)\subset Cont_{C}^{(r)}.(x)$ for all $x\in A^{r}$ and $z\in C$ .Corollary 4. Let $C\subset A^{*}$ be a prefix code. If there exists some $z_{1},$ $z_{2}\in$ C’ and some$[x],$ $[y]\in A"/P_{C*}^{(r)},$ $[x]\neq[y]$ , such that $[xz_{1}]=[y]$ and $[yz_{2}]=[x]$ , then $C^{*}$ is not extractable.Corollary 5. Let $C\subset A^{*}$ be a prefix code. If either $[xz]=[x]$ or $[xz]=[w_{\phi}]$ for all $[x]\in A^{*}/P_{C}^{(r)}$and $z\in C^{*}$ , then $C$ is extractable.Proposition 6. Let $C\subset A^{*}$ be a prefix code such that $C\sigma nt_{c}^{(r)}(x)\cap Cont_{C^{r}}^{(r)}(y)=\phi$ for anydistinct $[x],$ $[y]\in A^{*}/P_{C}^{(r)}$ . Then the following two conditions are equivalent(1) C’ is an extractable code.(2) Either $[xz]=[x]$ or $[xz]=[w_{\phi}]$ for any $x\in A^{*}$ and $z\in C$ .2. EXTRACTABLE REFLECTIVE CODETwo words $x,$ $y$ are called conjugate if there exists words $u,$ $v$ such that $x=uv,$ $y=vu.$ theconjugacy relation is an equivalence relation. By $d(x)$ we denote the class of $x$ of this equivalencerelation. Let $C\subset A^{*}$ be a code. If $uv\in C$ implies $vu\in C$ , then $C$ is called rejflective. It is obviousthat a reflective $C$ is a union of conjugacy classes of words. Note that $C$“ is not necessarily areflective language.A code $C\subset A^{+}$ is called infix if for all$x,$ $y,$ $z\in A,$ $xzy\in C\Rightarrow x=y=1$ .108Proposition 7. The reflective code $C$ is an infix code.Example 1. (1) The set $C=\{ab^{2}, bab, b^{2}a, a^{3}b, a^{2}ba, aba^{2}, ba^{3}\}$ is a union of two coujugacyclasses of words in $\{a, b\}^{*}$ . Sinoe $C$ is a prefix set, $C$ is an infix code.(2) Let $B\subset A,$ $B\neq\emptyset$ , and $n,$ $k(k\leq n)$ be a positive integer. We let $|w|B$ denote the number ofletters of $w$ which are in $B$ . Then $U=\{w\in A^{n}||w|_{B}=k\}$ is an extractable code. Since $uv\in U$imples $vu\in U,$ $U$ is reflective.Proposition 8. Let $C\subset A^{*}$ be a reflective code. The following two conditions are equivaJent.(1) $C$ is extractable.(2) For any $[x],$ $[y]\in A^{*}/P_{C^{r}}^{(r)}$ ,$[x]\neq[y]\Rightarrow Cont_{C}^{(r)}(x)\cap C\sigma nt_{C^{*}}^{(r)}(y)=\phi$ .Let $L\subset A^{*}$ and $n\geq 1$ . We set$L^{(n)}=\{w^{n}|w\in L\}$ .If $D$ is a bifix code, then $D^{(n)}$ is a bifix code.Proposition 9. If $C$ is a reflective code, then $C^{(n)},$ $n\geq 1$ , is a reflective code.Proposition 10. Let $D$ be a bifix code, and let $C=D^{(n)},$ $n\geq 2$ . Then, for $u,$ $v\in C(A^{+})^{-1}$ ,$C\sigma nt_{C}^{(r)}(u)\cap Cont_{C^{*}}^{(r)}(v)\neq\phi$ $\Rightarrow$ $[u]_{C^{r}}=[v]_{C^{r}}$ .$\mathbb{R}om$ Proposition 10 and Proposition 8 we haveCorollary 11. If $C$ is a reflective code, then $C^{(n)},$ $n\geq 2$ , is an extractable reflective code.3. EXTRACRTABLE UNIFORM CODESIn this section we examine extractable uniform codes.Proposition 12. Let $C\subset A^{*}$ be a extractable code. Then $C\cap A^{n}$ is an extractable code.Proposition 13. Let $M$ be an extractable submonoid of $A^{*}$ . Then $M\cap A^{n},$ $n\geq 1$ , is anextractable code.109Example 2. Let $S$ be a monoid and $H$ an extractable submonoid. Let $\varphi$ : $A^{*}arrow S$ be asurjective morphism. Then $\varphi^{-1}(H)$ is an extractable submonoid of $A^{*}$ .Corollary 14. Let $G$ be a group and $H$ a normal subgroup of $G$ . Let $\varphi$ : $A^{*}arrow G$ be a surjectivemorphism. Then $\varphi^{-1}(H)\cap A^{n}$ is an extractable reflective code.Proposition 15. Let $C\subset A^{+}$ be a finite code with $A=alph(C)$ . Then $C$ is a maximalextractable code if and only if $C=A^{n}$ for some $n\geq 1$ .Proposition 16. Let $D\subset A^{m},$ $m\geq 1$ , be a uniform code, then $D^{(n)},$ $n\geq 2$ , is extractable.Proposition 17. Let $w=(uv)^{n}u,$ $u,$ $v\in A^{*},$ $n\geq 2$ , and let $C$ be a conjugacy class of $w$ .(1) If $u=1$ and $v\in A^{+}$ , then $C$ is extractable.(2) If $u,$ $v\in A+$ , then $C$ is not extractable.Proposition 18. Let $w\in A^{+}$ , and let $C$ be a conjugacy class of $w$ .(1) If $w=uv,$ $u,v\in A^{+}$ and Alph $(u)\cap Alph(v)=\phi$ . Then $C$ is extractable.(2) If $w=(uvu)^{n}(uv)^{m},$ $u,$ $v\in A^{+},$ $m\geq 1,$ $n\geq 0$ and Alph$(u)\cap Alph(v)=\phi$ . Then $C$ is notextractable.Lemma 19. Let $x,y,u,$ $v\in A^{+}$ with $|u|,$ $|v|<|x|=|y|$ . Then the following conditions hold.(1) $x^{2}y=ux^{2}v\Rightarrow y=uv$ ($uv=x$ is not necessarily true).(1’) $xy^{2}=uy^{2}v\Rightarrow x=uv$ ($uv=y$ is not necessarily tme).(2) $x^{2}y=uxyv\Rightarrow x=y=uv$ ($uv$ is the power of some primitive word).(2’) $xy^{2}=uyxv\Rightarrow x=y=uv$ ($uv$ is the power of some primitive word).(3) $x^{2}y=uyxv\Rightarrow x=y=uv$ .(3’) $xy^{2}=uyxv\Rightarrow x=y=uv$ .(4) $x^{2}y=uy^{2}v\Rightarrow y=uv$ ( $x$ and $y$ are conjugate).(4’) $xy^{2}=ux^{2}v\Rightarrow x=uv(x$ and $y$ are $coi\iota|ugate)$ .Proposition 20. Let $x,$ $y\in A^{*}$ with $|x|=|y|>0$ and $C=\{x^{2}, xy, yx, y^{2}\}$ . Then $C$ isextractable.References[1] Berstel, J., and Perrin.D.: Theory of Codes. Academic Press, 1985[2] Lallement, G.: Semigroup and Combinatorial Applications. Wiley. 1979.[4] Tanaka. G.; “Limited codes associated with Petri nets”, to be submitted.[5] Tanaka. G., and Kunimochi. Y.: “Initial literal shuﬄes of uniform codes”, to be submitted.110

Remarks on extractable submonoids (Languages, Computations, and Algorithms in Algebraic Systems)

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Remarks on extractable submonoids (Languages, Computations, and Algorithms in Algebraic Systems)

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