Introduction to Theoretical CS

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1 3/5/1 7: Theory of Computtion Introduction to Theoreticl CS Two fundmentl questions. Wht cn computer do? (the most fundmentl question) Wht cn computer do with limited resources? (more prcticl) Pentium IV running inux kernel.4. Generl pproch. Don't tlk out specific mchines or prolems. Consider miniml strct mchines (DFA nd clss of prolems it cn solve). Consider generl clsses of prolems (next section more complicted mchines). Some of these questions seem ridiculously generl, ut we will descrie well-developed theories thn cn ddress these. ou need to understnd these issues or you'll e t disdvntge when solving prcticl pplictions. Why ern Theory In theory... Deeper understnding of wht is computer nd computing. Foundtion of ll modern computers. Pure science. Philosophicl implictions. In prctice... Pttern serch in ioinformtics: theory of pttern mtching. Sequentil circuits: theory of finite stte utomt. Compilers: theory of context free grmmrs. Cryptogrphy: theory of computtionl complexity. Dt compression, sequence nlysis: theory of informtion. "In theory there is no difference etween theory nd prctice. In prctice there is." -ogi Berr Introduction to Computer Science oert Sedgewick nd Kevin Wyne 3 egulr Expressions nd DFAs * (****)* 1 0 Wht s the regulr expression here? * ** ** 1 0 Pttern Mtching Applictions Test if string mtches some pttern. Process nturl lnguge. Scn for virus signtures. Serch for informtion using Google. Access informtion in digitl lirries. etrieve informtion from exis/exis. Serch-nd-replce in word processors. Filter text (spm, etnny, Crnivore, mlwre). Vlidte dt-entry fields (dtes, emil, U, credit crd). Serch for mrkers in humn genome using POSITE ptterns. Prse text files. Compile Jv progrm. Crwl nd index the We. ed in dt stored in TO input file formt. Automticlly crete Jv documenttion from Jvdoc comments. Introduction to Computer Science oert Sedgewick nd Kevin Wyne Introduction to Computer Science oert Sedgewick nd Kevin Wyne 6 1

2 3/5/1 egulr Expressions: Bsic Opertions egulr Expressions: Exmples Pttern Mtching in Google egulr expression. ottion to specify set of strings. egulr expression. ottion is surprisingly expressive. Google. Supports * for full word wildcrd nd for union. Opertion egulr Expression es o Conctention every other string Wildcrd.u.u.u. cumulus jugulum succuus tumultuous Union every other string Closure * ( ) every other string Prentheses ε ()* 7 egulr Expression es o.* sp.* contins the trigrph sp * (****)* multiple of three s.*0... fifth to lst digits is 0 gcg (cgg gg)* ctg frgile X syndrome indictor rsperry crispred gcgcggctg gcgcggggctg suspce suspecies gcgcgg cggcggcggctg gcgcggctg Frgile X syndrome is common cuse of mentl retrdtion epets: norml rnge: 5-50; crrier 51-00; disrupts production of FM protein nd usully cuses retrdtion (FM fcilittes trnsltion of essentil neuronl As) 8 9 Generlized egulr Expressions egulr Expressions in Jv Solving the Pttern Mtch Prolem 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)*. Opertion egulr Expression es o cde de One or more (c)+de ccde cde cpitlized cmelcse Chrcter clsses [A-Z-z][-z]* Word 4illegl Exctly k [0-9]{5-[0-9]{ egtions [^eiou]{6 rhythm decde 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 need help solving crosswords? % jv E "..oo..oo." loodroot legl Jv identifier % jv E "[$_A-Z-z][$_A-Z-z0-9]*" ident13 vlid emil ddress (simplified) % jv E "[-z]+@([-z]+\.)+(edu com)" ogt@cs.princeton.edu need quotes to "escpe" the shell 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 or flse questions for simplicity

3 3/5/1 Deterministic Finite Stte Automton (DFA) Wht does this DFA do? 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 string if unleled rc leves lst stte. DFA It recognizes itstrings with numer of 1 s divisile y 3, so 1101, , The regulr expression is: 0* 0* 1 0* 1 0* 1 0* 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. 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 Descriing more complex lnguges: FAs Implementing Pttern Mtcher Appliction: Hrvester FAs nondeterministic finite utomton An FA is finite stte mchine where for ech pir of stte nd input symol there my e severl possile next sttes 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 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)" ogt@cs.princeton.edu emil vlidtor (simplified) pop@cs.princeton.edu wyne@cs.princeton.edu

4 3/5/1 Appliction: Hrvester Appliction: Prsing Dt File Fundmentl Questions Hrvest informtion from input strem. Use Pttern dt type to compile regulr expression to FA. Use Mtcher dt type to simulte FA. 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()); 19 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 0 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? 1 Summry Turing Mchine: Components Progrmmer. egulr expressions re powerful pttern mtching tool. Implement regulr expressions with finite stte 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. ou. Prcticl ppliction of core CS principles. 7.5: Turing Mchines Chllenge: Design simplest mchine tht is "s powerful" s conventionl computers. Aln Turing ( ) Aln Turing sought the most primitive model of computing device. Tpe. Stores input, output, nd intermedite results. One ritrrily long strip, divided into cells. Finite lphet of symols. 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. Write overwrites whtever symol is current. tpe tpe hed Wht s the key difference from DFAs? memory cn write to tpe nd move ck nd forth Introduction to Computer Science oert Sedgewick nd Kevin Wyne 4 4

5 3/5/1 Turing Mchine: Fetch, Execute 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. 4 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. 4 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. 4 Before # # x x x # # After # # x x x 1 x 1 0 # # # # x x x x x x # # Exmple: Equl umer of 0's nd 1's Turing Mchine Summry Aln Turing find 1 Gol: simplest mchine tht is "s powerful" s conventionl computers. Surprising Fct 1. Such mchines re very simple. Surprising Fct. Some prolems cnnot e solved y A computer. Aln Turing ( ). Fther of computer science. Computer Science s oel Prize is clled the Turing Awrd. find left end skip x ccept find 0 reject Universlity/computility Consequences. lecture Precursor to generl purpose progrmmle mchines. Exposes fundmentl limittions of ll computers. Enles us to study the physics nd universlity of computtion. o need to seek more powerful mchines! It ws not only mtter of strct mthemtics, not only ply of symols, for it involved thinking out wht people did in the physicl world. It ws ply of imgintion like tht of Einstein or von eumnn, douting the xioms rther thn mesuring effects. Wht he hd done ws to comine such nïve mechnistic picture of the mind with the precise logic of pure mthemtics. His mchines soon to e clled Turing mchines offered ridge, connection etween strct symols, nd the physicl world. -John Hodges # # # # Aln Turing (left) Elder rother (right)