! What can a computer do? ! What can a computer do with limited resources? ! Don't talk about specific machines or problems.

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1 Introduction to Theoreticl CS ecture 18: Theory of Computtion Two fundmentl questions.! Wht cn computer do?! Wht cn computer do with limited resources? Generl pproch. Pentium IV running inux kernel.4.! Don't tlk out specific mchines or prolems.! Consider miniml strct mchines.! Consider generl clsses of prolems. COS16: Generl Computer Science Why ern Theory In theory...! Deeper understnding of wht is computer nd computing.! Foundtion of ll modern computers.! Pure science.! Philosophicl implictions. In prctice...! We serch: theory of pttern mtching.! Sequentil circuits: theory of finite stte utomt.! Compilers: theory of context free grmmrs.! Cryptogrphy: theory of computtionl complexity.! Dt compression: theory of informtion. "In theory there is no difference etween theory nd prctice. In prctice there is." -ogi Berr egulr Expressions nd DFAs * (****)* 3 4

edu/~cos16 Why ern Theory In theory...! Deeper understnding of wht is computer nd computing.! Foundtion of ll modern computers.! Pure science.! Philosophicl implictions. In prctice.

2 Pttern Mtching Applictions egulr Expressions: Bsic Opertions Test if string mtches some pttern.! Process nturl lnguge. egulr expression. ottion to specify set of strings.! Scn for virus signtures.! Serch for informtion using Google.! Access informtion in digitl lirries. Opertion egulr Expression es o! etrieve informtion from exis/exis. Conctention every other string! Serch-nd-replce in word processors.! Filter text (spm, etnny, Crnivore, mlwre). Wildcrd.u.u.u. cumulus jugulum succuus tumultuous! Vlidte dt-entry fields (dtes, emil, U, credit crd).! Serch for mrkers in humn genome using POSITE ptterns. Union every other string Prse text files. Closure *! Compile Jv progrm.! Crwl nd index the We.! ed in dt stored in TO input file formt.! Automticlly crete Jv documenttion from Jvdoc comments. Prentheses ( ) ()* every other string! 5 6 egulr Expressions: Exmples Generlized egulr Expressions egulr expression. ottion is surprisingly expressive. egulr Expression es o egulr expressions re stndrd progrmmer's tool.! Built in to Jv, Perl, Unix, Python,....! Additionl opertions typiclly dded for convenience.! Ex: [-e]+ is shorthnd for ( c d e)( c d e)*..* sp.* contins the trigrph sp rsperry crispred suspce suspecies * (****)* multiple of three s.*0... fifth to lst digit is Opertion One or more Chrcter clsses egulr Expression (c)+de [A-Z-z][-z]* es cde ccde cpitlized Word de cde o cmelcse 4illegl gcg (cgg gg)* ctg frgile X syndrome indictor gcgctg gcgcggctg gcgcggggctg gcgcgg cggcggcggctg gcgcggctg Exctly k [0-9]{5-[0-9]{ egtions [^eiou]{6 rhythm decde 7 8

Serch-nd-replce in word processors.! Filter text (spm, etnny, Crnivore, mlwre). Wildcrd.u.u.u. cumulus jugulum succuus tumultuous! Vlidte dt-entry fields (dtes, emil, U, credit crd).

3 egulr Expressions in Jv Solving the Pttern Mtch Prolem Vlidity checking. Is input in the set descried y the re? pulic clss Vlidte { pulic sttic void min(string[] rgs) { String re = rgs[0]; String input = rgs[1]; System.out.println(input.mtches(re)); powerful string lirry method egulr expressions re concise wy to descrie ptterns.! How would you implement String.mtches?! Hrdwre: uild deterministic finite stte utomton (DFA).! Softwre: simulte DFA. DFA: simple mchine tht solves the pttern mtch prolem.! Different mchine for ech pttern.! Accepts or rejects string specified on input tpe.! Focus on true or flse questions for simplicity. need help solving crosswords? % jv Vlidte "..oo..oo." loodroot true legl Jv identifier % jv Vlidte "[$_A-Z-z][$_A-Z-z0-9]*" ident13 true vlid emil ddress (simplified) % jv Vlidte "[-z]+@([-z]+\\.)+(edu com)" doug@cs.princeton.edu true need quotes to "escpe" the shell 9 10 Deterministic Finite Stte Automton (DFA) Theory of DFAs nd Es Simple mchine with sttes.! Begin in strt stte.! ed first input symol.! Move to new stte, depending on current stte nd input symol.! epet until lst input symol red.! Accept or reject string depending on lst stte. E. Concise wy to descrie set of strings. DFA. Mchine to recognize whether given string is in given set. Dulity: for ny DFA, there exists regulr expression to descrie the sme set of strings; for ny regulr expression, there exists DFA tht recognizes the sme set. * (****)* DFA multiple of 3 's multiple of 3 's Input Prcticl consequence of dulity proof: to mtch regulr expression ptterns, (i) uild DFA nd (ii) simulte DFA on input string. 11 1

mtches(re)); powerful string lirry method egulr expressions re concise wy to descrie ptterns.! How would you implement String.mtches?! Hrdwre: uild deterministic finite stte utomton (DFA).

4 Implementing Pttern Mtcher Appliction: Hrvester Prolem: given regulr expression, crete progrm tht tests whether given input is in set of strings descried. Step 1: uild the DFA.! A compiler!! See COS 6 or COS 30. Step : simulte it with given input. Esy. Hrvest informtion from input strem.! Hrvest ptterns from DA. % jv Hrvester "gcg(cgg gg)*ctg" chromosomex.txt gcgcggcggcggcggcggctg gcgctg gcgctg gcgcggcggcggggcggggcggctg Stte stte = strt; while (!ChrStdIn.isEmpty()) { chr c = ChrStdIn.redChr(); stte = stte.next(c); System.out.println(stte.ccept());! Hrvest emil ddresses from we for spm cmpign. % jv Hrvester "[-z]+@([-z]+\\.)+(edu com net tv)" doug@cs.princeton.edu emil vlidtor (simplified) dgi@cs.princeton.edu mon@cs.princeton.edu Appliction: Hrvester Appliction: Prsing Dt File Hrvest informtion from input strem.! Use Pttern dt type to compile regulr expression to FA.! Use Mtcher dt type to simulte FA.! (FA is fncy ut equivlent vriety of DFA) import jv.util.regex.pttern; import jv.util.regex.mtcher; pulic clss Hrvester { pulic sttic void min(string[] rgs) { String re = rgs[0]; In in = new In(rgs[1]); String input = in.redall(); Pttern pttern = Pttern.compile(re); Mtcher mtcher = pttern.mtcher(input); while (mtcher.find()) { System.out.println(mtcher.group()); 15 Ex: prsing n CBI genome dt file. OCUS AC p DA liner HTG 13-OV-003 DEFIITIO Ornithorhynchus ntinus clone CM1-393H9, ACCESSIO AC VESIO AC GI: KEWODS HTG; HTGS_PHASE; HTGS_DAFT. SOUCE Ornithorhynchus ntinus (pltypus) OIGI 1 tgttttct ttgccgtgc tgttttttcc cggtttttc gtcggtgtt ggggccc 61 gtgttctgt ttgtttttg ctgccgt gctgctcgt gtctctgc tgcgct // comment 11 gccgcggg gtgcc gtttgtgtg ctgt gggctgt ttcttct ggtgcg ccccccgct tgtcgc ttctttgt tg // String re = "[ ]*[0-9]+([ctg ]*).*"; Pttern pttern = Pttern.compile(re); In in = new In(filenme); String line; while ((line = in.redine())!= null) { Mtcher mtcher = pttern.mtcher(line); if (mtcher.find()) { extrct the E prt in prentheses String s = mtcher.group(1).replceall(" ", ""); // do something with s replce this E with this string 16

$Hrvest emil ddresses from we for spm cmpign. % jv Hrvester "[-z]+@([-z]+\\.)+(edu com net tv)" http://www.princeton.edu/~cos16 doug@cs.princeton.edu emil vlidtor (simplified) dgi@cs.princeton.edu mon@cs.$

5 imittions of DFA Fundmentl Questions o DFA cn recognize the lnguge of ll it strings with n equl numer of 0's nd 1's.! Suppose n -stte DFA cn recognize this lnguge.! Consider following input: ! DFA must ccept this string. +1 0's +1 1's! Some stte x is revisited during first +1 0's since only sttes x x! Mchine would ccept sme string without intervening 0's Which lnguges CAOT e descried y ny E?! Bit strings with equl numer of 0s nd 1s.! Deciml strings tht represent prime numers.! Genomic strings tht re Wtson-Crick complemented plindromes.! Mny more.... How cn we extend Es to descrie richer sets of strings?! Context free grmmr (e.g., Jv). eference: Q. How cn we mke simple mchines more powerful? Q. Are there ny limits on wht kinds of prolems mchines cn solve?! This string doesn't hve n equl numer of 0's nd 1's Summry Progrmmer.! egulr expressions re powerful pttern mtching tool.! Implement regulr expressions with finite stte mchines. Turing Mchines Theoreticin.! egulr expression is compct description of set of strings.! DFA is n strct mchine tht solves pttern mtch prolem for regulr expressions.! DFAs nd regulr expressions hve limittions. Chllenge: Design simplest mchine tht is "s powerful" s conventionl computers. Vritions! es (ccept) nd o (reject) sttes sometimes drwn differently! Terminology: Deterministic Finite Stte Automton (DFA), Finite Stte Mchine (FSM), Finite Stte Automton (FSA) re the sme! DFA s cn hve output, specified on the rcs or in the sttes These my not hve explicit es nd o sttes Aln Turing ( ) 19 0

Mchine would ccept sme string without intervening 0's. 000011111111 Which lnguges CAOT e descried y ny E?! Bit strings with equl numer of 0s nd 1s.! Deciml strings tht represent prime numers.

6 Turing Mchine: Components Turing Mchine: Fetch, Execute Aln Turing sought the most primitive model of computing device. Tpe.! Stores input, output, nd intermedite results.! One ritrrily long strip, divided into cells. tpe hed! Finite lphet of symols. tpe Sttes.! Finite numer of possile mchine configurtions.! Determines wht mchine does nd which wy tpe hed moves. Stte trnsition digrm.! Ex. if in stte nd input symol is 1 then: overwrite the 1 with x, move to stte 0, move tpe hed to left. Tpe hed.! Points to one cell of tpe.! eds symol from ctive cell.! Writes symol to ctive cell.! Moves left or right one cell t time Before # # x x x # # 3 Turing Mchine: Fetch, Execute Turing Mchine: Initiliztion nd Termintion Sttes.! Finite numer of possile mchine configurtions.! Determines wht mchine does nd which wy tpe hed moves. Stte trnsition digrm.! Ex. if in stte nd input symol is 1 then: overwrite the 1 with x, move to stte 0, move tpe hed to left Initiliztion.! Set input on some portion of tpe.! Set tpe hed. # # # #! Set initil stte. Termintion.! Stop if enter yes, no, or hlt stte.! Infinite loop possile After # # x x x 1x 1 0 # # # # x x x x x x # # 4 5

Stte trnsition digrm.! Ex. if in stte nd input symol is 1 then: overwrite the 1 with x, move to stte 0, move tpe hed to left. Tpe hed.! Points to one cell of tpe.! eds symol from ctive cell.

7 Exmple: Equl umer of 0's nd 1's Turing Mchine Summry find 1 Gol: simplest mchine tht is "s powerful" s conventionl computers. Surprising Fct 1. Such mchines re very simple: TM is enough! Surprising Fct. Some prolems cnnot e solved y A computer. skip x ccept reject Consequences.! Precursor to generl purpose progrmmle mchines. next lecture find left end! Exposes fundmentl limittions of ll computers.! Enles us to study the physics nd universlity of computtion.! o need to seek more powerful mchines! find 0 # # # # Vritions! Insted of just recognizing strings, TM s cn produce output: the contents of the tpe! Insted of nd sttes, TM s cn hve plin Hlt stte 6 7

! Precursor to generl purpose progrmmle mchines. next lecture find left end! Exposes fundmentl limittions of ll computers.

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge