Lecture T4: Pattern Matching

Transcription

1 Introduction to Theoreticl CS Lecture T4: Pttern Mtching Two fundmentl questions. Wht cn computer do? How fst cn it do it? Generl pproch. Don t tlk bout specific mchines or problems. Consider miniml bstrct mchines. Consider generl clsses of relted problems. 2 Tody. Simplest type of mchine tht is still interesting = FSA. Clss of (pttern mtching) problems tht it cn solve. Future lectures. More complicted mchines nd problems. 3/29/ Copyright 2, Kevin Wyne T.8 Why Lern Theory In theory... Deeper understnding of wht is computer nd computing. Foundtion of ll modern computers. Pure science. Philosophicl implictions. In prctice... Web serch: theory of pttern mtching. Sequentil circuit: theory of finite stte utomt. Compilers: theory of context free grmmr. Cryptogrphy: theory of complexity. Web Serch Exmple Stndrd Web serch for +censorship +net might yield million hits, ordered s follows: Observtion: not mny useful pges. 3/29/ Copyright 2, Kevin Wyne T.9 3/29/ Copyright 2, Kevin Wyne T.

2 3/29/ Copyright 2, Kevin Wyne T. Web Serch Exmple Stndrd Web serch for +censorship +net might yield million hits, ordered s follows: Jon Kleinberg s clever lgorithm produces "uthorittive" pges without humn fine-tuning: (Electronic Frontier Foundtion) (Center for Democrcy nd Technology) (Voters Telecommunictions Wtch) (Americn Civil Liberties Union) Unix Tools Unix. A lrge number of simple tools. Some fundmentl pttern mtching tools. egrep, wk, sed, more, emcs, perl Useful for vriety of pplictions including Web serch. Not C progrmming, though tools re s powerful. Directly relted to fundmentls tenets of computer science. Bsed on theoreticl principles. Involves computing eigenvectors of Web mtrix! Use google.com 3/29/ Copyright 2, Kevin Wyne T.2 egrep Crossword Puzzle or Scrbble Too Hrd? Generl regulr expressions pttern mtching. Acts s filter. Sends lines from stdin to stdout tht "mtch" rgument string. Elementry Exmples %egrep beth clsslist 2/Condliffe/Elizbeth/3/condlife 3/Dnher/Elizbeth/8/ednher 3/Smythe/Elizbeth/6/esmythe 3/Bethke/Kristen/3/kbethke % egrep /3/ clsslist 3/Mrin/Anthony/3/mrin 3/Prker/Andrew/3/prker... 3/Weiss/Jcob/3/weiss Find ll lines in file clsslist with substring beth List ll people in precept 3. %egrep zeuglodon mobydick.txt rechristened the monster zeuglodon nd in his 3/29/ Copyright 2, Kevin Wyne T.3 /usr/dict/words is list of (25,43) words in dictionry. More Exmples % egrep hh /usr/dict/words bechhed highhnded Two consecutive h s. withheld withhold % egrep u.u.u /usr/dict/words cumulus % egrep..oo..oo /usr/dict/words bloodroot schoolbook Why not "cookbook"? schoolroom % egrep -c..oo..oo /usr/dict/words 3 A dot mtches ny single chrcter count number of mtches 3/29/ Copyright 2, Kevin Wyne T.4

3 3/29/ Copyright 2, Kevin Wyne T.5 % mn egrep Excerpts From "mn egrep" Unix Nme: egrep - serch file using full regulr expressions Syntx: egrep [option...] expression [file...] Description: Serch the input files (stndrd input defult) for lines mtching pttern. Normlly, ech line found is copied to the stndrd output. Tke cre when using specil chrcters in the expression becuse they re lso meningful to the Shell. It is sfest to enclose the entire expression rgument in single quotes. Options: -c Produces count of mtching lines only. -i Considers upper nd lowercse letter identicl. -n Precedes ech mtching line with its line number. -v Displys ll lines tht do not mtch. Grep Pttern Conventions Conventions for egrep: c ny non-specil chrcter mtches itself. ny single chrcter r* zero or more occurrence of r (r) grouping r r2 logicl OR [...] ny chrcter in [-z] [^...] ny chrcter not in [-z] ^ beginning of line $ end of line Restrictions: Lines re limited to 256 chrs; longer lines re truncted. 3/29/ Copyright 2, Kevin Wyne T.6 Still More Exmples Pttern Mtching Alterntives in Unix Unix % egrep n(ie ei)ther /usr/dict/words neither % egrep ctg(tc)*gct humn.dt ggtctggctggc % egrep ctg(tc)*gct student.dt ttctgtctctcgctttc % egrep ^y.(..)*y$ /usr/dict/words yesterdy % grep -v [eiou] /usr/dict/words grep... rhythm syzygy Do spell checking by specifying wht you know. Strts nd ends with y, odd number of chrcters. Find ll words with no vowels nd 6 or more letters. Pttern mtching. grep, egrep more Substitution editing. emcs, ex sed: filter, line by line substitution sed s/pples/ornges/g file.txt Pttern mtching lnguges. wk, perl Mtching, substitution, pttern mnipultion, vribles, numeric cpbilities, control nd logic. 3/29/ Copyright 2, Kevin Wyne T.7 3/29/ Copyright 2, Kevin Wyne T.8

4 3/29/ Copyright 2, Kevin Wyne T.9 Regulr Expressions Regulr Expressions Specifying "pttern" for grep cn be complex. ^[^eiou]*[^eiou]*e[^eiou]*i[^eiou]*o[^eiou]*u[^eiou]*! fcetious Wht kinds of ptterns cn be specified? Mtch ll lines contining n even number? Mtch ll lines contining prime number? Which spects re essentil? Unix egrep regulr expressions re useful. But more complex thn theoreticl minimum. Rules for creting regulr expressions (RE s): or symbols () grouping b conctention + b logicl OR use + insted of * closure ( or more replictions) where nd b re regulr expressions. Exmples: ()* ε = empty string ε,,,,... ( + )*,,,,,... (****)* ε,,,,... 3/29/ Copyright 2, Kevin Wyne T.2 Forml Lnguges Why Study Forml Lnguges? An lphbet is finite set of symbols. Binry lphbet = {, } Lower-cse lphbet = {, b, c, d,..., y, z} Genetic lphbet = {, c, t, g} Cn cst ny computtion s lnguge recognition problem. Is x = 23,536,48,273 prime number?! L = {2, 3, 5, 7,, 3, 7,... }! Is x in lnguge L? A string is finite sequence of symbols in the lphbet. is string in the binry lphbet. tigers is string in the lower-cse lphbet. cctgct is string in the genetic lphbet. A forml lnguge is n (unordered) set of strings in n lphbet. Cn hve infinitely mny strings. Exmples: {,,,,,,...} {,,,,,...} 3/29/ Copyright 2, Kevin Wyne T.2 3/29/ Copyright 2, Kevin Wyne T.22

5 3/29/ Copyright 2, Kevin Wyne T.23 Regulr Lnguges Mchines Every RE describes lnguge (the set of ll strings tht mtch). (***)* describes the lnguge of ll bit strings with n even number of s: {,,,,...} Regulr lnguge. Any lnguge tht cn be described by RE. Which lnguges re regulr? (ll but one of the following) Cn cst ny computtion s lnguge recognition problem. Is x = 23,536,48,273 prime number? L = {2, 3, 5, 7,, 3, 7,... } Is x in lnguge L? Strt by trying to understnd simple lnguges. Build mchine to recognize regulr lnguges. All bit strings tht: Exmple Begin with nd end with. Hve multiple of 3 s. Hve more s thn s. Hve no consecutive s. 3/29/ Copyright 2, Kevin Wyne T.25 Finite Stte Automt C Code for ()* FSA Simple mchine with N sttes. Strt in stte. Red n input bit. Move to new stte depends on input bit nd current stte Stop when lst bit red. yes if end in ccept stte(s) no otherwise Yes lso clled ccepted or recognized inputs from lnguge. FSA to right ccepts ll bit strings of the form ()* Rejects ll others. strt stte 2 3 ccept stte fs.c int min(void) { int c, stte = START_STATE; while ((c = getchr())!= EOF) { if (stte == && c == ) stte = 2; if (stte == && c == ) stte = ; if (stte == && c == ) stte = 3; if (stte == && c == ) stte = 2; if (stte == 2 && c == ) stte = 2; if (stte == 2 && c == ) stte = 2; if (stte == 3 && c == ) stte = 2; if (stte == 3 && c == ) stte = ; } if (stte == ACCEPT_STATE) printf("accepted\n"); else printf("rejected\n"); return ; 3/29/ Copyright 2, Kevin Wyne T.27 3/29/ Copyright 2, Kevin Wyne T.28

6 3/29/ Copyright 2, Kevin Wyne T.29 Better C Code for FSA #include <stdio.h> #define STATES 4 #define ALPHABET_SIZE 2 #define START_STATE #define ACCEPT_STATE 3 fs.c int min(void) { int c, stte = START_STATE int trnsition[states][alphabet_size] = { {2, }, {3, 2}, {2, 2}, {2, } }; while ((c = getchr())!= EOF) if (c >= && c < + ALPHABET_SIZE) stte = trnsition[stte][c - ]; if (stte == ACCEPT_STATE) printf("accepted\n"); else printf("rejected\n"); return ; } A Second Exmple Consider the following two stte FSA. Wht bit strings does it ccept? Yes:,,,, ll bit strings with n odd number of s. No:,,,, ll bit strings with n even number of s. Recognizes sme lnguge s (***)* (**) even # of s exctly one 3/29/ Copyright 2, Kevin Wyne T.3 A Third Exmple More Applictions Build n FSA tht ccepts ll strings tht contin ct s substring. tgctg cctg cgt Finish building: c cgt φ c c c t ct gt cgt g 3/29/ Copyright 2, Kevin Wyne T.35 3/29/ Copyright 2, Kevin Wyne T.36

7 3/29/ Copyright 2, Kevin Wyne T.37 Dulity Between FSA s nd RE s Observtion: every FSA we crete genertes regulr lnguge. (the inputs ccepted cn be specified by regulr expression) Is this lwys the cse?! Yes. It s no coincidence. Wht bout the OTHER wy round? Suppose I give you regulr expression. Cn you lwys crete FSA tht ccepts precisely the sme strings?! Yes. I don t see why? FSA re simple mchines. Limittions of FSA N sttes cn t remember more thn N things. Some lnguges require remembering more thn N things. No FSA cn recognize the lnguge of ll bit strings with n equl number of s nd s. A wrmup exercise: Sty tuned: see Lecture T2. If xyz ccepted then so is xyz 3/29/ Copyright 2, Kevin Wyne T.38 Limittions of FSA No FSA cn recognize the lnguge of ll bit strings with n equl number of s nd s. Suppose n N-stte FSA cn recognize this lnguge. Consider following input: FSA must ccept this string. N+ s N+ s Some stte x is revisited during first N+ s since only N sttes. x x Limittions of Regulr Lnguges Consequence: there re lnguges tht re not regulr. No FSA cn recognize the lnguge of ll bit strings with n equl number of s nd s. We climed tht FSA s re equivlent to regulr expressions, i.e., for ny regulr lnguge, there is FSA tht ccepts precisely those strings the lnguge of ll strings ccepted by ny specific FSA is regulr Hence, the lnguge bove cnnot be regulr. Mchine would ccept sme string without intervening s. "FSA s cn t count." This string doesn t hve n equl number of s nd s. 3/29/ Copyright 2, Kevin Wyne T.39 3/29/ Copyright 2, Kevin Wyne T.4

8 3/29/ Copyright 2, Kevin Wyne T.4 Looking Ahed A Fourth Exmple Tody. Defined simple bstrct mchine = FSA. Cpble of pttern mtching. Incpble of "counting." Need to consider more powerful mchines. Hmm. Which will we run out of first? FSA to decide if input (convert binry to deciml) is divisible by 3? 2 Future lectures. Define n bstrct mchine. Understnd how it works nd wht it cn do. Find things it cn t do. Define more powerful mchine. Repet until we run out of problems or mchines. is strt nd ccept stte Wht bit strings does it ccept? Yes: (3 ), (6 ), (9 ), (2 ), (5 ), integers whose binry representtion is divisible by 3. No:,,,,, integers not divisible by 3. 3/29/ Copyright 2, Kevin Wyne T.42 A Fourth Exmple An Appliction: Bounce Filter FSA to decide if input (convert binry to deciml) is divisible by 3? How does it work? Stte : input so fr is divisible by 3. Stte : input hs reminder upon division by 3. Stte 2: input hs reminder 2 upon division by 3. Trnsition exmple. Input (2) ends in stte. If next bit is then sty in stte. Adding to lst bit is sme s multiplying number by 2. Remins divisible by 3. 2 Bounce filter: remove isolted s nd s in input. Input: Output (one-bit dely): - x/y: if input is x, then chnge stte nd OUTPUT y / / / / / / / 2 3 / 3/29/ Copyright 2, Kevin Wyne T.43 3/29/ Copyright 2, Kevin Wyne T.44

9 3/29/ Copyright 2, Kevin Wyne T.45 An Appliction: Bounce Filter Bounce filter: remove isolted s nd s in input. Input: Output (one-bit dely): - x/y: if input is x, then chnge stte nd OUTPUT y Stte interprettions. : t lest two consecutive s : sequence of s followed by 2: t lest two consecutive s 3: sequence of s followed by