Prefix-Free Regular-Expression Matching

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15

2 Pttern Mthing Given pttern P nd text T, find ll sustrings of T tht re in P. P = 1: string pttern mthing [BM, KMP] P = k: keyword pttern mthing [AC] P is regulr expression: regulr-expression pttern mthing!!! Prefix-Free Regulr-Expression Mthing p.2/15

3 Overview Bsi Notions Relted Work Regulr-Expression Mthing Infix-Free Regulr-Expression Mthing Prefix-Free Regulr-Expression Mthing Determine whether or not L(E) is prefix-free Conlusions Prefix-Free Regulr-Expression Mthing p.3/15

4 Bsi Notions An utomton A is speified y tuple (Q, Σ,δ,s,F); Q finite set of sttes Σ finite lphet δ Q Σ Q s Q strt stte F Q set of finl sttes λ = the null-string symol A = Q + δ E = the numer of hrter ppernes in given regulr expression E Prefix-Free Regulr-Expression Mthing p.4/15

5 Bsi Notions Given trnsition (p,,q) in δ p hs n out-trnsition q hs n in-trnsition p is soure stte of q q is trget stte of p A to e non-returning if the strt stte of A does not hve ny in-trnsitions A to e non-exiting if finl stte of A does not hve ny out-trnsitions p q Prefix-Free Regulr-Expression Mthing p.4/15

6 Bsi Notions Given two strings x nd y over Σ, we sy x is prefix of y if there exists z Σ suh tht xz = y. x is n infix of y if there exists u,v Σ suh tht uxv = y; we often ll x sustring of y. Prefix-Free Regulr-Expression Mthing p.4/15

7 Bsi Notions We define lnguge L to e prefix-free if no string in L is prefix of ny other strings in L. infix-free if no string in L is n infix of ny other strings in L. Prefix-Free Regulr-Expression Mthing p.4/15

8 Relted Work Given regulr expression E nd text T, The memership prolem: We n determine whether or not T L(E) in O(mn) time [Thompson] The deision prolem: We n determine whether or not there is sustring of T tht is in L(E)) in O(mn) time [Aho] or in O(m log n) time [Myers] The reognition prolem: We n report ll end positions of mthing sustrings of T in O(mn) time [Aho] or in O(m log n) time [Myers] The identifition prolem: We n report ll (strt, end) positions of mthing sustrings of T in O(mn log n) time [Myers et l.] Prefix-Free Regulr-Expression Mthing p.5/15

9 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

10 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

11 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

12 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

13 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

14 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

15 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

16 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

17 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

18 The Memership Prolem E = ( + ) nd T = Prefix-Free Regulr-Expression Mthing p.6/15

19 The Reognition Prolem Given E over Σ, we prepend Σ to E; thus, llowing mthing to egin t ny position in T. Σ ( + ) Σ Prefix-Free Regulr-Expression Mthing p.7/15

20 The Reognition Prolem Given E over Σ, we prepend Σ to E; thus, llowing mthing to egin t ny position in T. ExpressionMthing (A, T) Q = null({s}) if f Q then output λ for j=1 to n Q = null(goto(q,w j )) if f Q then output j null(q) omputes ll sttes in A tht n e rehed from stte in the set Q of sttes y null trnsitions goto(q,w j ) gives ll sttes tht n e rehed from stte in Q y trnsition with w j, the urrent input hrter Prefix-Free Regulr-Expression Mthing p.7/15

21 The Reognition Prolem Given E over Σ, we prepend Σ to E; thus, llowing mthing to egin t ny position in T. E = ( + ) T Given regulr expression E nd text T, we n find ll end positions of mthing sustrings of T in O(mn) worst-se time using O(m) spe [Crohemore nd Hnrt]. Prefix-Free Regulr-Expression Mthing p.7/15

22 The Identifition Prolem Given regulr expression E nd text T, we n identify ll mthing sustrings of T tht elong to L(E) in O(mn 2 ) worst-se time using O(m) spe. Note tht the lgorithm of Myers et l. tkes O(mn log n) time using O(m log n) spe. Prefix-Free Regulr-Expression Mthing p.8/15

23 Infix-Free Regulr-Expression Mthing L IN L PRE L REG T Given n infix-free regulr expression E nd text T, we n identify ll mthing sustrings of T tht elong to L(E) in O(mn) worst-se time using O(m) spe. Prefix-Free Regulr-Expression Mthing p.9/15

24 Prefix-Free Regulr-Expression Mthing L IN L PRE L REG If E is infix-free, we hve n O(mn) running time lgorithm If E is (norml) regulr expression, we hve n O(mn 2 ) running time lgorithm If E is prefix-free, there re t most n mthing sustrings of T tht elong to L(E), where n is the size of T Prefix-Free Regulr-Expression Mthing p.10/15

25 Prefix-Free Regulr-Expression Mthing Given prefix-free regulr expression E nd text T, we find ll end positions of mthing sustrings of T in O(mn) time. T Let P = {p 1,p 2,...,p k } e the set of end positions of mthing sustrings for k n Construt the Thompson utomton A = (Q, Σ,δ,s,f ) for E R Sn T R = w n w 1 strting from the lst position p k in P to find the orresponding strt position Prefix-Free Regulr-Expression Mthing p.10/15

26 Prefix-Free Regulr-Expression Mthing T Q 15 For urrent input position i in T R, Q 15 is set of sttes suh tht there is pth from s to eh stte in Q 15 tht spells out w 15 w 14 w i. We keep reding T R until we meet f. Prefix-Free Regulr-Expression Mthing p.10/15

27 Prefix-Free Regulr-Expression Mthing T Q 13 Q 15 Prefix-Free Regulr-Expression Mthing p.10/15

28 Prefix-Free Regulr-Expression Mthing T Q 10 Q 13 Q 15 Prefix-Free Regulr-Expression Mthing p.10/15

29 Prefix-Free Regulr-Expression Mthing T Q 9 Q 10 Q 13 Q 15 In the worst-se, there re k suh sets of sttes nd we need O(km) time for eh hrter of T to updte these k sets. Thus, the totl running time is O(mn 2 ) in the worst-se sine k is t most n. Prefix-Free Regulr-Expression Mthing p.10/15

30 Prefix-Free Regulr-Expression Mthing If stte r in A is rehed from two different sttes p nd q, where p Q i nd q Q j, when reding hrter w h in EM, where h i < j, then oth pths from p nd q vi r nnot reh f y reding ny prefix of the remining input in EM. p Q i, q Q j T R j i h Q i Q j p r q Prefix-Free Regulr-Expression Mthing p.11/15

31 Prefix-Free Regulr-Expression Mthing If stte r in A is rehed from two different sttes p nd q, where p Q i nd q Q j, when reding hrter w h in EM, where h i < j, then oth pths from p nd q vi r nnot reh f y reding ny prefix of the remining input in EM. Eh stte in A ppers in t most one rehle set Any two sets of rehle sttes re disjoint We need t most O(m) time to updte ll sets of rehle sttes simultneously t eh step Given prefix-free regulr expression E nd text T, we n identify ll mthing sustrings of T tht elong to L(E) in O(mn) worst-se time using O(m) spe. Prefix-Free Regulr-Expression Mthing p.11/15

32 Prefix-Freeness An FA A is prefix-free if L(A) is prefix-free A DFA A is prefix-free if it is non-exiting Wht out the NFA se? Prefix-Free Regulr-Expression Mthing p.12/15

33 Prefix-Freeness An FA A is prefix-free if L(A) is prefix-free A DFA A is prefix-free if it is non-exiting Wht out the NFA se? If n NFA A is prefix-free, then A must e non-exiting However, the reverse does not hold Prefix-Free Regulr-Expression Mthing p.12/15

34 Prefix-Freeness An FA A is prefix-free if L(A) is prefix-free A DFA A is prefix-free if it is non-exiting Wht out the NFA se? If n NFA A is prefix-free, then A must e non-exiting However, the reverse does not hold s f Prefix-Free Regulr-Expression Mthing p.12/15

35 Stte-Pir Grph Given finite-stte utomton A = (Q, Σ, δ, s, f), we define the stte-pir grph G A = (V,E), where V is set of nodes nd E is set of edges, s follows: V = {(i,j) q i nd q j Q} nd E = {((i,j),, (x,y)) (q i,,q x ) nd (q j,,q y ) δ nd Σ} ,2 1,1 3,3 4,6 4,4 5,5 6,6 5,7 7,7 Prefix-Free Regulr-Expression Mthing p.13/15

36 Stte-Pir Grph & Prefix-Freeness CPM 2005 Given finite-stte utomton A, L(A) is prefix-free if nd only if there is no pth from (1, 1) to (m,j), for ny j m, in G A ,2 1,1 3,3 4,6 4,4 5,5 6,6 5,7 7,7 Prefix-Free Regulr-Expression Mthing p.14/15

37 Stte-Pir Grph & Prefix-Freeness CPM 2005 Given finite-stte utomton A, L(A) is prefix-free if nd only if there is no pth from (1, 1) to (m,j), for ny j m, in G A Given finite-stte utomton A = (Q, Σ, δ, s, f), we n determine whether or not L(A) is prefix-free in O( Q 2 + δ 2 ) worst-se time Let G A = (V, E) e the stte-pir grph of A V = Q 2 Let δ i denote the set of out-trnsitions from stte q i in A δ = m i=1 δ i, where m = Q node (i, j) in G A n hve t most δ i δ j out-trnsitions E = m i,j=1 δ i δ j δ 2 Prefix-Free Regulr-Expression Mthing p.14/15

38 Stte-Pir Grph & Prefix-Freeness CPM 2005 Given finite-stte utomton A, L(A) is prefix-free if nd only if there is no pth from (1, 1) to (m,j), for ny j m, in G A Given finite-stte utomton A = (Q, Σ, δ, s, f), we n determine whether or not L(A) is prefix-free in O( Q 2 + δ 2 ) worst-se time Given regulr expression E, we n determine whether or not L(E) is prefix-free in O( E 2 ) worst-se time Construt the Thompson utomton for E Q = δ = O( E ) Prefix-Free Regulr-Expression Mthing p.14/15

39 Conlusions Solve the prefix-free regulr-expression mthing prolem in O(mn) time using O(m) spe sed on the Thompson utomt Determine whether or not L(A) is prefix-free for given NFA A in polynomil time sed on stte-pir grphs Prefix-Free Regulr-Expression Mthing p.15/15

Nondeterministic Finite Automata

Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

= state, a = reading and q j

4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those

CS375: Logic and Theory of Computing

CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1

Chpter 1, Prt 1 Regulr Lnguges CSC527, Chpter 1, Prt 1 c 2012 Mitsunori Ogihr 1 Finite Automt A finite utomton is system for processing ny finite sequence of symols, where the symols re chosen from finite

NON-DETERMINISTIC FSA

Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

CHAPTER 1 Regular Languages. Contents

Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

State Complexity of Union and Intersection of Binary Suffix-Free Languages

Stte Complexity of Union nd Intersetion of Binry Suffix-Free Lnguges Glin Jirásková nd Pvol Olejár Slovk Ademy of Sienes nd Šfárik University, Košie 0000 1111 0000 1111 Glin Jirásková nd Pvol Olejár Binry

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

Nondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA

Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18

Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re

1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

Harvard University Computer Science 121 Midterm October 23, 2012

Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

Formal languages, automata, and theory of computation

Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm

Lexical Analysis Finite Automate

Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion

CS241 Week 6 Tutorial Solutions

241 Week 6 Tutoril olutions Lnguges: nning & ontext-free Grmmrs Winter 2018 1 nning Exerises 1. 0x0x0xd HEXINT 0x0 I x0xd 2. 0xend--- HEXINT 0xe I nd ER -- MINU - 3. 1234-120x INT 1234 INT -120 I x 4.

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.

Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

Chapter 2 Finite Automata

Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More

Non-deterministic Finite Automata

Non-deterministic Finite Automt From Regulr Expressions to NFA- Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2

CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

State Minimization for DFAs

Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

input tape head moves current state

CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute

Victor Admchik Dnny Sletor Gret Theoreticl Ides In Computer Science CS 5-25 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn

Prefix-Free Subsets of Regular Languages and Descriptional Complexity

Prefix-Free Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek Jurj Šeej DCFS 2015 Prefix-Free Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek, Jurj Šeej 1/22 Outline Mximl

CS 330 Formal Methods and Models

CS 330 Forml Methods nd Models Dn Richrds, section 003, George Mson University, Fll 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 7 1. Prove (p q) (p q), () (5pts) using truth tles. p q

Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

Automata and Languages

Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers

Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.

1.3 Regular Expressions

56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

Automata and Regular Languages

Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level,

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

Normal Forms for Context-free Grammars

Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S

On Determinisation of History-Deterministic Automata.

On Deterministion of History-Deterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YR-ICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil

CS S-12 Turing Machine Modifications 1. When we added a stack to NFA to get a PDA, we increased computational power

CS411-2015S-12 Turing Mchine Modifictions 1 12-0: Extending Turing Mchines When we dded stck to NFA to get PDA, we incresed computtionl power Cn we do the sme thing for Turing Mchines? Tht is, cn we dd

3 Regular expressions

3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

Nondeterministic Biautomata and Their Descriptional Complexity

Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle

INTRODUCTION TO AUTOMATA THEORY

Chpter 3 INTRODUCTION TO AUTOMATA THEORY In this hpter we stuy the most si strt moel of omputtion. This moel els with mhines tht hve finite memory pity. Setion 3. els with mhines tht operte eterministilly

Name Ima Sample ASU ID

Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

Probabilistic Model Checking Michaelmas Term Dr. Dave Parker. Department of Computer Science University of Oxford

Probbilistic Model Checking Michelms Term 2011 Dr. Dve Prker Deprtment of Computer Science University of Oxford Long-run properties Lst lecture: regulr sfety properties e.g. messge filure never occurs

Context-Free Grammars and Languages

Context-Free Grmmrs nd Lnguges (Bsed on Hopcroft, Motwni nd Ullmn (2007) & Cohen (1997)) Introduction Consider n exmple sentence: A smll ct ets the fish English grmmr hs rules for constructing sentences;

CS 330 Formal Methods and Models

CS 0 Forml Methods nd Models Dn Richrds, George Mson University, Fll 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 8 1. Prove q (q p) p q p () (4pts) with truth tle. p q p q p (q p) p q

Periodic string comparison

Periodi string omprison Alexnder Tiskin Deprtment of Computer Siene University of Wrwik http://www.ds.wrwik..uk/~tiskin Alexnder Tiskin (Wrwik) Periodi string omprison 1 / 51 1 Introdution 2 Semi-lol string

General Suffix Automaton Construction Algorithm and Space Bounds

Generl Suffix Automton Constrution Algorithm nd Spe Bounds Mehryr Mohri,, Pedro Moreno, Eugene Weinstein, Cournt Institute of Mthemtil Sienes 251 Merer Street, New York, NY 10012. Google Reserh 76 Ninth

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

Chapter 4 State-Space Planning

Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

Section: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.

Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM

Formal Methods in Software Engineering

Forml Methods in Softwre Engineering Lecture 09 orgniztionl issues Prof. Dr. Joel Greenyer Decemer 9, 2014 Written Exm The written exm will tke plce on Mrch 4 th, 2015 The exm will tke 60 minutes nd strt

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

Lecture 9: LTL and Büchi Automata

Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

Solutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and non-accepting state in M.

Solution Prolem Set 2 Prolem.4 () Let M denote the DFA contructed y wpping the ccept nd non-ccepting tte in M. For ny tring w B, w will e ccepted y M, tht i, fter conuming the tring w, M will e in n ccepting

On-Line Construction. of Suffix Trees. Overview. Suffix Trees. Notations. goo. Suffix tries

On-Line Cnstrutin Overview Suffix tries f Suffix Trees E. Ukknen On-line nstrutin f suffix tries in qudrti time Suffix trees On-line nstrutin f suffix trees in liner time Applitins 1 2 Suffix Trees A suffix

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

Domino Recognizability of Triangular Picture Languages

Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 Domino Recognizility of ringulr icture Lnguges V. Devi Rjselvi Reserch Scholr Sthym University Chenni 600 9. Klyni Hed of

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

CS 267: Automated Verification. Lecture 8: Automata Theoretic Model Checking. Instructor: Tevfik Bultan

CS 267: Automted Verifiction Lecture 8: Automt Theoretic Model Checking Instructor: Tevfik Bultn LTL Properties Büchi utomt [Vrdi nd Wolper LICS 86] Büchi utomt: Finite stte utomt tht ccept infinite strings

Where did dynamic programming come from?

Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/1526-5463-2002-50-01-0048.pdf

Good-for-Games Automata versus Deterministic Automata.

Good-for-Gmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt

Last time: introduced our first computational model the DFA.

Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody

Design and Analysis of Distributed Interacting Systems

Design nd Anlysis of Distriuted Intercting Systems Lecture 6 LTL Model Checking Prof. Dr. Joel Greenyer My 16, 2013 Some Book References (1) C. Bier, J.-P. Ktoen: Principles of Model Checking. The MIT

Recursively Enumerable and Recursive. Languages

Recursively Enumerble nd Recursive nguges 1 Recll Definition (clss 19.pdf) Definition 10.4, inz, 6 th, pge 279 et S be set of strings. An enumertion procedure for Turing Mchine tht genertes ll strings

Semantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4

Semnti nlysis SI 16 Priniples of Progrmming Lnguges Fulty of omputer Siene Dlhousie University Winter 2012 Reding: hpter 4 Motivtion Soure progrm (hrter strem) Snner (lexil nlysis) Front end Prse tree

Agenda. Agenda. Regular Expressions. Examples of Regular Expressions. Regular Expressions (crash course) Computational Linguistics 1

Agend CMSC/LING 723, LBSC 744 Kristy Hollingshed Seitz Institute for Advnced Computer Studies University of Mrylnd HW0 questions? Due Thursdy before clss! When in doubt, keep it simple... Lecture 2: 6

The University of Nottingham

The University of Nottinghm SCHOOL OF COMPUTR SCINC AND INFORMATION TCHNOLOGY A LVL 1 MODUL, SPRING SMSTR 2004-2005 MACHINS AND THIR LANGUAGS Time llowed TWO hours Cndidtes must NOT strt writing their

6.5 Improper integrals

Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

On NFA reductions. N6A 5B7, London, Ontario, CANADA ilie 2 Department of Computer Science, University of Chile

On NFA reductions Lucin Ilie 1,, Gonzlo Nvrro 2, nd Sheng Yu 1, 1 Deprtment of Computer Science, University of Western Ontrio N6A 5B7, London, Ontrio, CANADA ilie syu@csd.uwo.c 2 Deprtment of Computer

Automatic Synthesis of New Behaviors from a Library of Available Behaviors

Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u

LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon

LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N

Software Engineering using Formal Methods

Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,

References. Theory of Computation. Theory of Computation. Introduction. Alexandre Duret-Lutz

References Theory of Computtion Alexndre Duret-Lutz dl@lrde.epit.fr Septemer 10, 2010 Introduction to the Theory of Computtion (Michel Sipser, 2005). Lecture notes from Pierre Wolper's course t http://www.montefiore.ulg.c.e/~pw/cours/clc.html

This lecture covers Chapter 8 of HMU: Properties of CFLs

This lecture covers Chpter 8 of HMU: Properties of CFLs Turing Mchine Extensions of Turing Mchines Restrictions of Turing Mchines Additionl Reding: Chpter 8 of HMU. Turing Mchine: Informl Definition B

STRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada

STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the

ON THE DETERMINIZATION OF WEIGHTED FINITE AUTOMATA

To pper in SIAM Journl on Computing c SIAM 000 ON THE DETERMINIZATION OF WEIGHTED FINITE AUTOMATA ADAM L. BUCHSBAUM, RAFFAELE GIANCARLO, AND JEFFERY R. WESTBROOK Astrct. We study the prolem of constructing

i 1 i 2 i 3... i p o 1 o 2 AUTOMATON q 1, q 2,,q n ... o q Model of a automaton Characteristics of automaton:

Definition of n Automton:-An Automton is defined s system tht preforms certin functions without humn intervention. it ccepts rw mteril nd energy s input nd converts them into the finl product under the

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

Dynamic Fully-Compressed Suffix Trees

Motivtion Dynmic FCST s Conclusions Dynmic Fully-Compressed Suffix Trees Luís M. S. Russo Gonzlo Nvrro Arlindo L. Oliveir INESC-ID/IST {lsr,ml}@lgos.inesc-id.pt Dept. of Computer Science, University of

Arrow s Impossibility Theorem

Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

Formal Languages and Automata Theory. D. Goswami and K. V. Krishna

Forml Lnguges nd Automt Theory D. Goswmi nd K. V. Krishn Novemer 5, 2010 Contents 1 Mthemticl Preliminries 3 2 Forml Lnguges 4 2.1 Strings............................... 5 2.2 Lnguges.............................

On Decentralized Observability of Discrete Event Systems

2011 50th IEEE Conference on Decision nd Control nd Europen Control Conference (CDC-ECC) Orlndo, FL, USA, Decemer 12-15, 2011 On Decentrlized Oservility of Discrete Event Systems M.P. Csino, A. Giu, C.

8 Automata and formal languages. 8.1 Formal languages

8 Automt nd forml lnguges This exposition ws developed y Clemens Gröpl nd Knut Reinert. It is sed on the following references, ll of which re recommended reding: 1. Uwe Schöning: Theoretische Informtik

Algorithm Design and Analysis

Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing

Models of Computation: Automata and Processes. J.C.M. Baeten

Models of Computtion: Automt nd Processes J.C.M. Beten Jnury 4, 2010 ii Prefce Computer science is the study of discrete ehviour of intercting informtion processing gents. Here, ehviour is the centrl notion.

CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

CSCI FOUNDATIONS OF COMPUTER SCIENCE

1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not