Katsura, Masashi

Kobayashi, Yuji

Tsuji, Kayoko

Kyoto University Research Information Repository

TitleOn Termination of One-Rule String RewritingSystems(Semigroups, Formal Languages and ComputerSystems)Author(s)Tsuji, Kayoko; Katsura, Masashi; Kobayashi, YujiCitation数理解析研究所講究録 (1996), 960: 85-87Issue Date1996-08URL http://hdl.handle.net/2433/60505RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn Termination of One-Rule String Rewriting $S$ ystem $\mathrm{s}$( ) (Kayoko Shikish.$\mathrm{i}\mathrm{m}.\mathrm{a},-$ Tsu..j $\mathrm{i}^{1}$ )(Masashi $\mathrm{K}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}$ )$2$(Yuji $\mathrm{K}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}^{3}$) $.$ .Let $\Sigma$ be a finite alphabet. The free monoid and the free semigroupgenerated by $\iota\Sigma$ are denoted by $\Sigma^{*}$ and $\Sigma\star$, respectively. Thelength of a word $x$ in $\Sigma^{*}$ is denoted by $\mathrm{I}x1$ . For ’ $x,$ $y‘\in\Sigma^{*}$ , we set$\mathrm{O}\mathrm{V}\mathrm{L}(x, y)=$ {$z\in\Sigma+|x=uz$, $y=zv$ for some $u,$ $v\in\Sigma+$ }. A rewritingsystem $R$ on $\Sigma$ is a subset of $\Sigma^{*}\cross\Sigma^{*}$ . An element $(l, r)$ in $R$ isdenoted by $larrow r$ . If $R$ contains only one element, $R$ is said to be $\mathrm{a}$ .one-rule rewriting system. A singfe steP reduct.ion relation $arrow$ inducedby $R$ is the following relation on $\Sigma^{*}$ : For any $x,$ $y\in\Sigma^{*}$ , $xarrow y$ $\mathrm{t}\mathrm{l}\mathrm{f}$and only if there exists.. $(l, r)\in R$ such that $x=ulv,$ . $y_{\wedge}=urv$ ,for some$u,$ $v\in\Sigma^{*}$ . $arrow*$ is the reflexive and transitive closure of $arrow$ . 1A rewriting system $R$ is said to be confluent if for any $w,$ $x,$ $y\in\Sigma^{*}$,$warrow^{*}x$ and $warrow^{*}y$ imply $xarrow^{*}z$ and $yarrow^{*}z$ for some $z\in\Sigma^{*}$ .$R$ is terminating (or noetherian) if there is no infinite sequence $x_{1},$ $x2,$ $\cdots$such $\mathrm{t}\mathrm{h}^{\backslash }\mathrm{a}\mathrm{t}$ $x_{1}arrow x_{2}$ $arrow\cdot\cdot:$ .$\mathrm{Y}^{\cdot}$.A $\mathrm{c},.\mathrm{o}\mathrm{n}.\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{t}\backslash$ and $\mathrm{t}.’.\mathrm{e},\mathrm{r},\min_{\mathrm{t}\sim’\backslash }..\mathrm{a}.\dot{\mathrm{t}},\mathrm{i}\mathrm{n}\mathrm{g}..\mathrm{r}_{\vee}\mathrm{e}\mathrm{w}\backslash \cdot,\vee\cdot.,.\mathrm{g}\sim \mathrm{r}_{}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{n}’\iota$ system..is said to be complete. .,$\mathrm{t}$ . .. .. $\cdot:.t.\cdot$ . $\cdot$ ., .It is not known whether the completeness is decidable for one-rulerewriting systems. Let $\cdot$ $R=\{larrow\gamma\}$ be a $\mathrm{o}\mathrm{n}\mathrm{e}\neg \mathrm{r}\mathrm{u}\mathrm{l}\mathrm{e}$ rewriting $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}_{\vee}$. If$r\in\Sigma^{*}l\Sigma^{*}$ then $R$ is always non-terminating. If 1 $l1\geq|\gamma|$ and $l\neq r$then $R$ is always terminating.${\rm Re}$ sult 1 [31 It is decidable whether or not a $\mathit{0}\acute{n}e$ -rule rewriting sys-$tem$ is confluen.${\rm Re}$ sult 2 [2] For a confluent one-rule $rew$ riting system $R=\{larrow r\}$with $1l1<1r1,$ $we$ can effectively construct a rewriting system $R’=\{l’arrow r$ ?such that:(1) $1l’[<1r’|$ and $\mathrm{O}\mathrm{V}\mathrm{L}(l^{r}, l^{\mathrm{f}})=\emptyset$.(2) $R’$ is terminating if and only if $R$ is terminating.960 1996 85-87 85Hence the completeness problem for one-rule systems is reduced to thetermination problem for one-rule systems $R.=_{\mathrm{J}}\{larrow r\}$ with $\mathrm{O}\mathrm{V}\mathrm{L}(l, l)=$$\emptyset$. It is not difficult to see that if $\mathrm{O}\mathrm{V}\mathrm{L}(r, l)=\emptyset$ or $\mathrm{O}\mathrm{V}\mathrm{L}(l, r)=\emptyset$ then$R$ is terminating. In this note,.we consider the case where $\mathrm{O}\mathrm{V}\mathrm{L}(r, l)$ $=$$\{p\}$, a singleton. $\backslash .$.$j_{-}$ $,\cdot\iota$ . ‘.For each $s\in \mathrm{O}\mathrm{V}\mathrm{L}(l, r)$ , we determine $\overline{s}\in\Sigma^{*}$ by $l=\overline{s}s$ . The deci-dability of the terminating problem for such one-rule systems is given asfollows.Theorem 1. Let $R=\{larrow r\}$ be $a$ one-rule rew riting system suchthat $\mathrm{O}\mathrm{V}\mathrm{L}(l, l)=\emptyset$ and $\mathrm{O}\mathrm{V}\mathrm{L}(r, l)=\{p\}$ . Let $l=px$, $r=y_{\overline{S}_{k}\overline{S}1}\ldots.p$,where $s_{1},$ $\ldots,$ $s_{k}\in \mathrm{O}\mathrm{V}\mathrm{L}(l, r)$ and $y\not\in\Sigma^{*}\overline{s}$ for any $s\in \mathrm{O}\mathrm{V}\mathrm{L}(l, r)$.(1) If there is a reduction of length $1r1^{2}$ starting $with$ $(y\overline{s}_{k}\cdots\overline{s}1)3$then $R$ is non-terminating.(2) Assum $e$ that the maxim $al$ length of $re\dot{d}ucti_{onS}$ starting $with$$(y\overline{s}k\ldots\overline{S}1)3$ is $N$ $with$ $N<|\gamma|2$ . If $1x1>1y1$ and there is a reduction oflength $2N+1$ starting $with$ $(y\overline{s}_{k}\cdots\overline{s}1)4$ then $R$ is non-term inatin$g$,otherwise, $R$ is terminating.The exact characterization$\mathrm{o}\mathrm{r}_{d}$.$\mathrm{n}\mathrm{o}.\mathrm{n}$-te$:.$,rmin $a\mathrm{t}\mathrm{i}\mathrm{n}.\cdot \mathrm{g}$ one-rule $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e},\mathrm{m}\mathrm{s}$ isgiven as follows.Theorem 2. Let $R=\{larrow r\}$ be $a$ one-rule rew riting system suchthat $\mathrm{O}\mathrm{V}\mathrm{L}(Ll)=\emptyset$ and $\mathrm{O}\mathrm{V}\mathrm{L}(r, l)=\{p\}$ . Then $R$ is non-terminating ifand only if one of the following conditions is satisfied. . .’..($k,$ $m,$ $n$ are positive integers. $x,$ $y,$ $z,$ $w\in\Sigma^{+}$ and $u,$ $v\in\Sigma^{*}.$ )(1) $l\in\Sigma^{*}\gamma\Sigma^{*}$ .(2) $r=s_{k}u\overline{s}_{k}\cdots\overline{s}_{1}p$ , $s_{1},$ $\ldots,$ $s_{k}\in \mathrm{O}\mathrm{V}\mathrm{L}(l,r\rangle\cap(xu)^{*}X$ .(3) $l=p(ux)^{n}$ , $r=(Xu)^{n+m}\overline{S}_{k}\cdots\cdot\overline{s}1P$ , $s_{1},$ $\ldots,$ $s_{k}\in \mathrm{O}\mathrm{V}\mathrm{L}(l,\Gamma)\cap(xu)^{*_{X}}$ .(4) $l=p(ux)^{n}$ , $r=(Xu)^{n+m}\overline{s}_{k1p}\ldots.\overline{s}$, $s_{1},$ $\ldots,$ $s_{k}\in \mathrm{O}\mathrm{V}\mathrm{L}(l,r)$ ,$s1,$ $\ldots,$ $S_{j-}\iota\in(xu)^{*}x$, $s_{j}\in(xu)^{i}x$, $1\leq j\leq k$, $1\leq i\leq 2m$ .(5) $l=px_{\mathcal{Y}}$, $\gamma=y(xyz)mX\overline{s}_{k}\cdots\overline{s}_{1p}$,$s_{1},$$\ldots,$$s_{k}\in \mathrm{O}\mathrm{V}\mathrm{L}(l,r)$ , $s_{1}=y(x_{\mathcal{Y}\mathrm{Z})^{m}}\backslash ,$ $xyz=wxy$ .(6) $l=xy((_{\mathrm{Z}\chi^{m}}k)^{m-1}\mathcal{Y}z\mathcal{Y})^{n+1}$, $r=_{\mathcal{Y}((_{\mathrm{Z}^{\chi^{m}}}y}k)^{m}-1\mathrm{Z}y)n+1z\chi mky$(7) $l=zxyx\mathcal{V}(Z^{2m+n+1}xy\chi v)^{m}x$, $pu=zx_{\mathcal{Y}^{\chi \mathcal{V}}}$,$r=xyxv(\mathrm{Z}^{2m+n+}X1yxv)^{m}xuZ^{2m}+np$ .(8) $l=pu((pu)k\mathrm{Z})^{2n}m+pu(((pu)kZ)^{2m}+np)m- 1_{X}$, $pu=zxyxv$.$r=xyxv\mathrm{Z}((pu)^{k}z)^{2}m+n- 1P^{u((()^{k_{Z)}}}P^{u}2m+nP)m- 1_{Xu}((pu)^{k_{\mathrm{Z})^{2m}}}+np$ .(9) $l=zx(\mathcal{Y}x)k-1(\mathcal{Y}z^{m+}n+1x(yx)k-\iota)^{m_{\mathcal{Y}}}X$,$r=X(_{\mathcal{Y}^{X})^{k1}(}- yz^{m}+n+1X(yx)k-1)^{m}yXyz^{m}+n+1X(_{\mathcal{Y}^{\chi)}}k-1$ .86References1 [11 R.V. Book, F. Otto, String-Rew riting Systems, Texts and Monographsin Computer Science, $\mathrm{s}_{\mathrm{P}^{\mathrm{r}\mathrm{i}\mathrm{n}}\mathrm{g}}\mathrm{e}\mathrm{r}$ -Vrlag (1993).$[2]$:K Shikishima-Tsuji, M. Katsura, Y. Kobayashi, Termination of one-$arrow$ rule string rewriting systems, submitted for publication.[31 $\mathrm{C}$ Wrathall, COnfluence of one-rule Thue systems, Word $\Psi uationS$and Related Topics, Lecture Notes in oemputer Science 572 (1992),237-2.46:$.\sim..\cdot..\cdot$ .. $\cdot$ $:,$ $=$1 Tenri University2 Kyoto Sangyo University3 Toho University87

On Termination of One-Rule String Rewriting Systems(Semigroups, Formal Languages and Computer Systems)

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On Termination of One-Rule String Rewriting Systems(Semigroups, Formal Languages and Computer Systems)

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