Drawing Finite State Machines in L A TEX and TikZ A Tutorial

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Lenard Osborne
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1 Drwing Finite Stte Mchines in L A TEX nd TikZ A Tutoril Styki Sikdr nd Dvid Ching ssikdr@nd.edu Version 3 Jnury 7, 208 Introduction L A TEX (pronounced ly-tek) is n open-source, multipltform document preprtion system for producing professionl-looking documents....it is prticulrly suited to producing long, structured documents, nd is very good t typesetting equtions [University of Edinurgh Informtion Services, 204]. The cpilities of the system re gretly enhnced with the help of ntive nd third-prty pckges. TikZ 2 is pckge for drwing ll kinds of grphics. This tutoril introduces the reder to L A TEX nd the TikZ pckge, prticulrly for drwing stte digrms of finite utomt. 2 Setting up L A TEX To proceed with the tutoril, working L A TEX setup is necessry. You my instll it loclly on your mchine, ut the simplest thing to do is use ShreL A TEX (shreltex.com). If you sign up using your nd.edu ddress, you ll get unlimited privte project. For further informtion regrding setup, visit ltex-project.org/get/. 3 The premle Every L A TEX document strts with premle. To mke our utomt look like the ones in the textook [Sipser, 202], use the following premle: \documentclss{rticle} % Wht kind of document this is \usepckge{tikz} % Import the tikz pckge \usetikzlirry{utomt} % Import lirry for drwing utomt \usetikzlirry{positioning} %...positioning nodes \usetikzlirry{rrows} %...customizing rrows \tikzset{node distnce=2.5cm, % Minimum distnce etween two nodes. Chnge if necessry. every stte/.style={ % Sets the properties for ech stte semithick, fill=gry!0}, initil text={}, % No lel on strt rrow doule distnce=2pt, % Adjust ppernce of ccept sttes every edge/.style={ % Sets the properties for ech trnsition The Comprehensive TEX Archive Network (CTAN) is the centrl plce for ll kinds of mteril round TEX. https: // 2

2 drw, ->,>=stelth, uto, semithick}} \let\epsilon\vrepsilon % Mkes edges directed with old rrowheds After the premle comes the content: \egin{document} % Content goes here \end{document} 4 Bsics While there re mny tutorils online, I suggest two: University of Edinurgh Informtion Services [204] nd Here re some symols often used in this course: 5 Drwing utomt symol control sequence usul mening Σ \Sigm lphet Γ \Gmm nother lphet ε \vrepsilon empty string \circ conctention # \texttt{\#} mrker symol $ \texttt{\$} mrker symol \textvisilespce lnk symol { } \{ \} delimiters for sets \emptyset empty set \neq is not equl to \in is n element of / \notin is not n element of \suseteq is suset of \rightrrow (vrious menings) δ \delt trnsition function α \lph regulr expression ^\st Kleene str Let s strt off with simple DFA from Sipser [202] (Figure.6). The forml description of the DFA is: where δ is given y: M = ({q, q 2, q 3 }, {0, }, δ, q, {q 2 }), 0 q q q 2 q 2 q 3 q 2 q 3 q 2 q 2 Below is the code tht genertes the stte digrm of M. 2

3 0 0 q q 2 q 3 0, \node[stte, initil] (q) {$q_$}; \node[stte, ccepting, right of=q] (q2) {$q_2$}; \node[stte, right of=q2] (q3) {$q_3$}; \drw (q) edge[loop ove] node {\tt 0} (q); \drw (q) edge node {\tt } (q2); \drw (q2) edge[loop ove] node {\tt } (q2); \drw (q2) edge[end left] node {\tt 0} (q3); \drw (q3) edge[end left] node {{\tt 0}, {\tt }} (q2); Below, we ll go through this exmple piece y piece. 5. The tikzpicture environment Inside the document, ech TikZ digrm must reside in tikzpicture environment: % tikz code goes here 5.2 Nodes Let s strt off y drwing the nodes. Nodes cn e positioned either mnully or reltive to other nodes. Reltive plcement is often much esier. q q 2 q 3 \node[stte, initil] (q) {$q_$}; \node[stte, ccepting, right of=q] (q2) {$q_2$}; \node[stte, right of=q2] (q3) {$q_3$}; The generl form of the \node commnd is: \node[<options>] (<nme>) t (<x>,<y>) {<lel>}; The <options>, (<nme>), nd t (<x>,<y>) re ll optionl, ut the {<lel>} is required. Options The options (for finite utomt) re: stte: lwys give this option to drw node s stte initil: specifies the strt stte ccepting: specifies n ccept stte Note tht the size of node depends on the length of its lel; to force minimum size (sy, inch), use minimum size=in. Nme The nme of node is the nme y which you refer to the node, when positioning other nodes reltive to it or when drwing edges into or out of it. 3

4 Position You specify the solute position of node using t (<x>,<y>) where <x> nd <y> re coordintes (<x> coordintes go to the right; <y> coordintes go up). Or you cn specify reltive position using left of=<nme>, right of=<nme>, ove of=<nme>, elow of=<nme>. There s lso ove left of=<nme>, etc. 3 The positioning lirry which we hve lredy imported provides some further options. xshift=x, yshift=y: Gives mnul control of the node positions fter reltive plcement. Eg: \node[stte, right of=q, xshift=cm] (q2) {$q_2$}; Lel This cn e nything you wnt. Typiclly you will surround it with dollr signs to use mth mode. 5.3 Edges Once the sttes re ll in plce, let s strt dding the trnsitions, tht is, the edges etween the sttes. The \drw commnd cn e used to drw the edges etween the lredy creted nodes (sttes). q q 2 q 3 \node[stte, initil] (q) {$q_$}; \node[stte, ccepting, right of=q] (q2) {$q_2$}; \node[stte, right of=q2] (q3) {$q_3$}; \drw (q) edge node {\tt } (q2); The generl syntx is s follows: \drw (<source node>) edge[<options>] node {<lel>} (<dest node>); Source nd destintion nodes not their lels. Note tht <source node> nd <dest node> re the nmes of the nodes, Options The <options> modify the ppernce of the edge. For edges tht strt nd end in the sme node (self-loops), you must loop ove, loop ove, loop left, or loop right. By defult, the edges re stright, so to prevent overlps use end left or end right. To modify the plcement of the edge lel, use ove or elow. Lel This cn e nything you wnt. Note tht Sipser uses typewriter font for symols, so you proly wnt to write {\tt 0} or \texttt{0}. Shorthnd Multiple edges cn e drwn with the sme drw commnd, like so: \drw (q) edge[loop ove] node {\tt 0} (q) edge node {\tt } (q2) \drw (q2) edge[loop ove] node {\tt } (q2) edge[end left] node {\tt 0} (q3) \drw (q3) edge[end left] node {{\tt 0}, {\tt }} (q2); 3 Techniclly, these options re deprected, ut we find them useful nywy. See questions/9386/difference-etween-right-of-nd-right-of-in-pgf-tikz. 4

5 6 More exmples As nother exmple, let s drw NFA (Sipser, Figure.42). where δ is given y: N = ({, 2, 3}, {, }, δ,, {}), ε {} {2} {3} 2 {2, 3} {3} {} 3 {} {} {} Below is the code tht genertes the stte digrm of N. ε 2 3, \node[stte, initil, ccepting] () t (.5,2.6) {$$}; \node[stte] (2) t (0,0) {$2$}; \node[stte] (3) t (3,0) {$3$}; \drw () edge node{\tt } (2) edge[end left=5] node {$\epsilon$} (3) (2) edge[loop left] node{\tt } (2) edge[elow] node{{\tt }, {\tt }} (3) (3) edge[end left=5] node{\tt } (); Our finl exmple is the stte digrm of the DFA equivlent to the NFA N : D 2 = ({, {}, {2}, {3}, {, 2}, {, 3}, {2, 3}, {, 2, 3}}, {, }, δ, {, 3}, {{}, {, 2}, {, 3}, {, 2, 3}}), where δ is given y {} {2} {2} {2, 3} {3} {3} {, 3} {, 2} {2, 3} {2, 3} {, 3} {, 3} {2} {2, 3} {, 2, 3} {3} {, 2, 3} {, 2, 3} {2, 3} Below is the code tht genertes the stte digrm., {} {2} {, 2}, {3} {, 3} {2, 3} {, 2, 3} 5

6 \tikzset{every stte/.ppend style={rectngle, rounded corners}} \node[stte] (emp) {$\emptyset$}; \node[stte, ccepting, right of=emp] () {$\{\}$}; \node[stte, right of=] (2) {$\{2\}$}; \node[stte, ccepting, right of=2] (2) {$\{, 2\}$}; \node[stte, elow of=emp] (3) {$\{3\}$}; \node[stte, initil, initil where=ove, ccepting, right of=3] (3) {$\{, 3\}$}; \node[stte, right of=3] (23) {$\{2, 3\}$}; \node[stte, ccepting, right of=23] (23) {$\{, 2, 3\}$}; \drw (emp) edge[loop left] node {{\tt }, {\tt }} (emp) () edge[ove] node {\tt } (emp) () edge node {\tt } (2) (2) edge node {\tt } (23) (2) edge[ove] node {\tt } (3) (2) edge[uto=right,ner strt] node {{\tt }, {\tt }} (23) (3) edge node {\tt } (emp) (3) edge node {\tt } (3) (3) edge[loop right] node {\tt } (3) (3) edge node {\tt } (2) (23) edge[end left,ove] node {\tt } (23) (23) edge[end left] node {\tt } (3) (23) edge[loop ove] node {\tt } (23) (23) edge[end left,elow] node {\tt } (23); References Michel Sipser. Introduction to the Theory of Computtion. Cengge Lerning, 3rd edition, 202. University of Edinurgh Informtion Services. LTeX for eginners. skills/documents/3722/ pdf,

9.3. Regular Languages

9.3. REGULAR LANGUAGES 139 9.3. Regulr Lnguges 9.3.1. Properties of Regulr Lnguges. Recll tht regulr lnguge is the lnguge ssocited to regulr grmmr, i.e., grmmr G = (N, T, P, σ) in which every production