Fujioka, Kaoru

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TitleA note on picture insertion systems (New Streams ofComputation Theory and Algorithms)Author(s)Fujioka, KaoruCitation数理解析研究所講究録 (2014), 1894: 28-31Issue Date2014-05URL http://hdl.handle.net/2433/195843RightType Departmental Bulletin PaperTextversionpublisherKyoto University2013 LA [7]A note on picture insertion systemsKaoru Fujioka *1 Introduction picture $p$ in $\tau**,$ $|p|=(m, n)$ denotes the size ofthe picture $p$ with $m.=\ell_{1}(p)$ and $n=\ell_{2}(p)$ .Insertion and deletion systems are computing The row and column concatenations are denotedmodels based on the field of molecular biology. Sev-$p\ominus q$ and $p$, respectively, and defined if $p$ anderal proposals have been made for generating two-$q$ have the same number of columns (resp. lows).dimensional languages based on insertion and dele-$p^{k\ominus}$ (resp. $p^{k\fcircle}$ ) is the vertical (horizontal) juxta-tion with replicative transposition operation. position of $k’ sp.$In this paper, we focus on insertion operations A tiling system [3] is a tuple $\mathcal{T}=(\Sigma, \Gamma, \theta, \pi)$ ,and extend an insertion system from one dimen- where $\Sigma$ and $\Gamma$ are alphabets, $\theta$ is a finite set of tilession ( $ID$ ) to $2D$ then introduce a picture insertion over the alphabet $\Gamma$ , and $\pi$ : $\Gammaarrow\Sigma$ is a projection.system to generate picture languages. The picture Let $TS$ be the class of picture languages generatedinsertion operation introduced in this paper relates by tiling systems.to the insertion operations in one dimensions of theform $(u, x, v)$ to produce a string $auxv\beta$ from agiven string $\alpha uv\beta$ with context $uv$ by inserting a3 Picture insertion systemsstring $x[2]$ . We also present some examples andresults concerning picture insertion systems. Definition 1$A$ picture insertion system is a tuple$\gamma=(T, P, A)$ , where $T$ is an alphabet, $P$ is a finiteset of picture insertion rules, and $A$ is a finite set2 Preliminaries of pictures over $T.$ $P$ may contain the followingthree types of picture insertion rules:In this section, we introduce notation and basicdefinitions that are necessary for this paper. The . $R$-type: $(u, w, v)$ , where $\ell_{1}(u)=l_{1}(v)=P_{1}(w)$ .basic notions and definitions in formal languagetheory are found in [4]. $\cdot$ $C$-type: $[Matrix]$ where $l_{2}(u)=\ell_{2}(v)=\ell_{2}(w)$ .For an alphabet $T$ , a picture $p$ is a two-dimensional rectangular array of elements of $T.$$T^{**}$ is the set of all pictures over $T.$ $A$ picture . $RC$-type: $[Matrix]$ , wherelanguage over $T$ is a subset of $T^{**}.$For a picture $p\in\tau**$ , let $\ell_{1}(p)$ (resp. $\ell_{2}(p)$ ) be $\ell_{1}(u)=l_{1}(w_{1})=l_{1}(v), \ell_{1}(w_{2})=l_{1}(w_{3})=$the number of rows (resp. columns) of $p$ . For a $\ell_{1}(w_{4}), \ell_{1}(x)=\ell_{1}(w_{5})=\ell_{1}(y)$ ,$\ell_{2}(u)=\ell_{2}(w_{2})=\ell_{2}(x), \ell_{2}(w_{1})=\ell_{2}(w_{3})=$$\ell_{2}(w_{5}), \ell_{2}(v)=\ell_{2}(w_{4})=P_{2}(y)$ ,’Department of Environmental Science, InternationalCollege of Arts and Sciences, Fukuoka Women’s University and $w_{3}\neq\lambda.$1894 2014 28-31 28Intuitively, $R$-type (resp. $C$-type) rule means aninsertion rule in row (resp. column), that is, thepicture $w$ is inserted in between the pictures $u$ and$v$ . An $RC$-type rule is intend to insert the pic-tures $w_{i}(1\leq i\leq 5)$ into the picture consisting of$u,v,x$ , and $y$ . We break up the rectangle into sub-pictures $u,v,x$ , and $y$ and secure the cross-shapedspace, then insert those pictures.We show how to apply insertion rules in the fol-lowing definition.Definition 2 For pictures $p_{1},$ $p_{2}$ in $\tau**$ , we saythat $p_{1}$ denves $p_{2}$ in one step if. there is an $R$-type rule $(u, w, v)$ with $u,$ $v,$ $w\in$$T^{m*}$ for $m\geq 1$ such that $p_{1}=\alpha$and $p_{2}=\alpha$ with $\alpha,$ $\beta\in T^{m*}.$We write $p_{1}arrow Rp_{2}$ . In a graphical represen-tation, it means$arrow R$. there $u$ a C- Type rule $(\begin{array}{l}uwv\end{array})$ with $u,$ $v,$ $w\in$$T^{*n}$ for $n\geq 1$ such that $p_{1}=\alpha\ominus u\ominus v\ominus\beta$ and$p_{2}=\alpha\ominus u\ominus w\ominus v\ominus\beta$ with $\alpha,$ $\beta\in T^{*n}$ . Wewrite $p_{1}arrow cp_{2}$ . In a graphical representation,We write $p_{1}arrow RCp_{2}$ . In a graphical repre-sentation, it meansel $\Rightarrow$If there is no confusion, we write $arrow$ instead of$arrow R,$ $arrow c$ , and $arrow RC$ . The reflexive and transitiveclosure of $arrow$ $(resp. arrow R, arrow c)$ is defined as $arrow*$$(resp. arrow*R, arrow^{*}c)$ . The transitive closure of $arrow$ (resp.$arrow R,$ $arrow c)$ is denoted by $arrow+(resp. arrow+R, arrow+c)$ .With $arrow R,$ $arrow c$ , and $arrow RC$ we introduce a stan-dard derivation denoted by $\Rightarrow$ in the following def-tnition.Definition 3 For pictures $p_{1}$ and $p_{2},$ $p_{1}\Rightarrow p_{2}$ isdefined in the following three cases:1. lUsing $R$-type rulesl. pictures $p_{1}$ and $p_{2}$ satisfy$p_{1}=(\alpha_{1}$ and $p_{2}=$$(\alpha_{1}$ , where foreach $1\leq i\leq n,$- there is a derivation $\alpha_{i}$ $arrow*R$$\alpha_{i}$- there are $l_{a},$ $l_{b},$ $l_{w}\geq 0$ such that $\ell_{2}(\alpha_{i})=$$l_{a}, l_{2}(\beta_{i})=l_{b}, \ell_{2}(w_{i})=l_{w},$. there is no picture $p’$ in $T^{**}$ such that $p_{1}arrow+R$$p’arrow+_{p_{2}}R.$In a graphical representation, it means$\Rightarrow$. there is an $RC$-Type rule $(\begin{array}{lll}u w_{1} vw_{2} w_{3} w_{4}x w_{5} y\end{array})$such that $p_{1}=(u(Dv)\ominus(x$ and $p_{2}=$$(u$.2. [Using $C$-type rulesl. pictures $p_{1}$ and $p_{2}$ satisfy$p_{1}=(\alpha_{1}\ominus\beta_{1})$ and$p_{2}=(\alpha_{1}\ominus w_{1}\ominus\beta_{1})$ $CD(\alpha_{n}\ominus w_{n}\ominus\beta_{n})$ ,where for each $1\leq i\leq n,$29- there is a $der^{v}\iota$vation $\alpha_{i}\ominus\beta_{i}arrow*c^{\alpha_{i}}\ominus w_{i}\ominus$$\beta_{i},$- there are $1_{a},$ $l_{b},$ $l_{w}\geq 0$ such that $\ell_{1}(\alpha_{i})=$$l_{a}, \ell_{1}(\beta_{i})=l_{b}, l_{1}(w_{i})=l_{w},$. there is no picture $p’$ in $T^{**}$ such that $p_{1}arrow+c$$p’arrow+c^{p_{2}}.$In a graphical representation, it means$\Rightarrow$3. lUsing an $RC$-type rule]. there is an $RC$-type rule $(\begin{array}{lll}u w_{1} vw_{2} w_{3} w_{4}x w_{5} y\end{array}),$$\circ$ pictures $p_{1}$ and $p_{2}$ satisfy $p_{1}=(q_{1}$$(q_{3}$ and $p_{2}$ $=$ $(q_{1}$$(z_{2}$ ,. the lower nght comer (resp. lower left, uppermght, upper left) of $q_{1}$ $(resp. q_{2}, q_{3}, q_{4})$ is $u$$(resp. v, x, y)$ ,. $z_{1}$ $(resp. z_{2}, z_{3}, z_{4})$ is inserted by $R$ -type (resp.$C$-type, $C$-type, $R$-type) rules.In a graphical representation, it means$\Rightarrow$Intuitively, the standard derivation $\Rightarrow$ is thesmallest unit to applied to a picture by applyingpicture insertion rules. The reflexive and transitiveclosure of $\Rightarrow$ is defined as $\Rightarrow^{*}.$A picture language generated by $\gamma=(T, P, A)$is defined as $L(\gamma)$ $=$ $\{w$ $\in$ $\tau**$ $s$ $\Rightarrow^{*}$$w$ , for some $s\in A$ }.A picture insertion system $\gamma=(T, P, A)$ is said tobe of weight $(i,j;k, l)$ if the number of rows (resp.columns) for context checking picture is not morethan $i$ (resp. $j$ ), and the number of rows (resp.columns) for inserted picture is not more than $k$(resp. $l$ ).For $i,j,$ $k,$ $l\geq 0$ , let $INS_{k,l}^{ij}$ be the class of picturelanguages generated by picture insertion systems ofweight $(i’,j’;k’, l’)$ with $i’\leq i,$ $j’\leq j,$ $k’\leq k$ , and$l’\leq l$ . If some of the parameters $i,$ $j,$ $k,$ $l$ are notbounded, we $use*$ in place of the symbols for thoseparameters.Example 1 Consider a picture insertion system$\gamma=(T, P, A)$ , where $T=\{a, b\},$ $P=\{(\lambda, ab, \lambda)\},$$A=\{\lambda\}$ . The picture language generated by $\gamma$ isviewed as a Dyck’s string language.As shown in Example 1, picture insertion systemsare $2D$ generalizations of insertion systems in $1D$cases. We slightly note that Dyck language is notregular (in $1D$ sense).Example 2 Consider a picture insertion system$\gamma=(T, P, A)$ , where $T=\{a, b\},$ $P=\{(\lambda, \S_{a}^{b} , \lambda)$ ,$(\begin{array}{l}\lambda_{)}\theta_{a}^{b}\lambda\end{array})\},$ $A=\{\lambda\}.$The followings are some of the pictures gener-ated by $\gamma;\lambda,$ $\theta_{a}^{b}$ $\theta_{a}^{b}\theta_{a}^{b}$ $8k_{a}^{bb}$ $8_{a} \oint^{b}\#$ $\beta_{a}^{\mathfrak{g}}aa$$\theta_{a}^{b}\theta_{a}^{b}\theta_{a}^{b}$ $\theta\%_{aaa}^{abbb}$ $\theta k_{a}^{bb}\theta_{a}^{b}$For example, the picture $ccaad\#_{\mathcal{C}}^{a}{\}$ is derived intwo ways as follows: $\lambda\Rightarrow$ $\ ^{b}$ $\Rightarrow$ $\theta_{a}^{b}\theta_{a}^{b}$ $\Rightarrow$$\theta k_{a}^{bb}\theta_{a}^{b},$ $\lambda\Rightarrow$ $g_{a}^{b}$ $\Rightarrow$ $\oint g_{aa}^{bb}$ $\Rightarrow$ $\theta k_{a}^{bb}\theta_{a}^{b}$Example 3 Consider a picture insertion system$\gamma=(T, P, A)$ , where $T=\{a, b\},$ $P=\{(b, b, \lambda)$ ,30$(\begin{array}{l}bb\lambda\end{array})$ $(\begin{array}{lll}a b bb a bb b a\end{array})\},$ $A=\{\theta_{a}^{b} \}.$A derivation in $\gamma$ proceeds as follows:$\theta_{a}^{b}$ $\Rightarrow$ $q_{8_{a}}^{b}\mathfrak{g}$ $\Rightarrow$ $\Vert_{\theta_{\theta_{a}}^{\mathfrak{g}\S}}^{b}$ $\Rightarrow\cdots.$The derivation proceeds deterministically usingthe $RC$-type rule. $A$ language generated by $\gamma$ is aset of squares whose positions in the main diagonalare covered by $a$ and the remaining ones are coveredby $b.$Lemma 1 There is a picture language which can-not be generated by any picture insertion systems.Proof Consider a picture language defined by$(a^{2n+1})^{n\ominus}\ominus(a^{n}ba^{n})\ominus(a^{2n+1})^{n\ominus}$ for $n\geq 1.$The claim can be proved by contradiction. $\square$Lemma 2 There is a picture insertion system $\gamma$such that $L(\gamma)w$ not generated by a tiling system.Proof Consider a picture insertion system $\gamma=$$(T, \{(\lambda, aa\S \lambda)\}, \{\lambda\})$ with $T=\{a, b\}$ . Fkom thedefinition of $\gamma$ , a picture $p$ in $L(\gamma)$ satisfies that thenumber of $a$ in $p$ is equivalent to that of $b.$Suppose that there is a tiling system $\mathcal{T}$ $=$$(T, \Gamma, \theta, \pi)$ such that $L(\gamma)=L(\mathcal{T})$ , where $\Gamma$ is afinite alphabet, $\theta$ is a finite set of tiles over $\Gamma$ , and$\pi$ : $\Gammaarrow T$ is a projection. Then we can generate acontradiction. $\square$Lemma 3 For any $i,$ $j\geq 0,$ $INS_{**}^{i,j}$ is incompara-ble with $TS.$Proof As an example, for the class of picture in-sertion systems, we consider $INS_{**}^{0,0}.$From Lemma 2, we can prove that there is a pic-ture language $L(\gamma)$ in $INS_{**}^{0,0}$ but not in $TS.$Consider a tiling system $\mathcal{T}=(\{a, b\}, \{a, b\}, \theta, \pi)$ ,where $\theta=$ $\{ 8_{a}^{b} ab\# \},$ $\pi$ : $\{a, b\}arrow\{a, b\}$ isan identity projection such that $\pi(x)=x$ with$x\in\{a, b\}$ . The followings are some examples ofpictures in $L(\mathcal{T})$ ; $\theta_{a}^{b}$ $8_{a}^{b}\#$ $\oint_{a}^{b}(\#$ $a\S^{b}\# 8_{a},$ $\cdots$Suppose that there is a picture insertion system$\gamma$ such that $L(\gamma)=L(\mathcal{T})$ , then we can generate acontradiction.Similarly, for the case of $INS_{**}^{i,j}$ with $i,$ $j\geq 0,$the claim can be proved. $\square$4 Concluding RemarksIn this paper, we introduced picture insertionsystems which generate two-dimensional languages.As considered in $1D$ case, picture insertion-deletion systems can be defined in which we canuse not only picture insertion operations but alsodeletion operations.Using insertion systems together with some mor-phisms, representation theorems are shown in $1D$case [1]. Those representation might be possiblein $2D$ case. Furthermore, to compare with cellularautomaton is also our future work.References[1] K. Fujioka. Morphic characterizations of lan-guages in Chomsky hierarchy with insertionand locality. Information and Computation209, pp.397-408, 2011.[2] G. $P\dot{a}un$ , G. Rozenberg, and A. Salomaa.DNA Computing; new computing paradigms.Springer, 1998.[3] S.C. Reghizzi and M. Pradella. Tile rewritinggrammars and picture languages. TheoreticalComputer Science, 340, pp.257-272, 2005.[4] G. Rozenberg and A. Salomaa, Eds. Handbookof Formal Languages. Springer, 1997.31

A note on picture insertion systems (New Streams of Computation Theory and Algorithms)

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A note on picture insertion systems (New Streams of Computation Theory and Algorithms)

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