# Automata and Regular Languages

Save this PDF as:

Size: px
Start display at page:

Download "Automata and Regular Languages"

## Transcription

1 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, n riefly esrie the others t the en. First, we tke rief look t the fountion in sequentil igitl iruits. A new element is e tht n hol its. This pility to pture n rememer vlues gretly inreses wht igitl systems re ple of oing. 9.. Finite Stte Mhines* The igitl iruit shown to the left elow is lle flip-flop. The symol stns for not. The iruit is esrie y the simultneous system of oolen equtions in the enter, whose truth tle is on the right. S R P Q P = S + Q Q = R + P S R P Q In the se tht R = S = the system reues to P = Q Q = P, whih hs two solutions s the truth tle shows. A physil iruit must somehow selet one of these solutions, n it oes so y seleting the solution tht ws present in the previous instnt. In other wors, the flip-flop will hol P s n Q s vlues

2 2 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES inefinitely, until it is fore to hnge y pulse on S, mking P = n Q =, or pulse on R, mking P = n Q =. Simultneous pulses on P n Q n fore the flip-flop into metstle solution, P Q, ut eventully, it will snp to vli stte. Aitionl iruitry is e to minimize this vulnerility. A loke flipflop hs one input presenting the it to e hel. The CLK input is use to ontrol when the next it is pture. DATA CLK D Q We n think of ny igitil system s n instne of finite-stte mhine in n f m out stte k Q S D k next stte k CLK In this rhiteture, Blok f represents purely omintionl oolen system of n + k inputs n m + k outputs. Comintionl tht eh output of f is just oolen omintion of some or ll of the n + k inputs. Blok S represents n rry of simple storge elements, eh ple of storing one it, or, inefinitely. Think of the signl CLK s kin of eletroni metronome, going tik-tok-tik-tok-. On tik, the k one-it storge elements simultneously pture the vlue presente on its input n hols it until the next tik. The perio etween CLK tiks is long enough to llow the iruitry to eletronilly stilize, so tht there re no re onitions where n input is hnging when the tik ours. Copyright 22 Steven D. Johnson Deemer 3, 22

3 9.2. AUTOMATA 3 out O O O 2 O 3 O 4 O i+ = f(s, I ) i i in I I I 2 I 3 I 4 s S S S 2 S 3 S 4 S i+ = ns(s i, I i) s S S 2 S 3 S 4 S 5 CLK 9.2 Automt A finite-stte utomoton (FA) moels the tions of n ielize omputing mhine. Before formlizing this onept, let us look t n exmple epite y the igrm W W X Y X Y,,,, Z Z,,, U,,, The noes of the FA form grph whose eges re lelle y letters of n lphet. The noes S = {W, X, Y, Z} re lle the sttes of the FA n the lphet is A = {,,, }. The FA strts in stte s = W, s inite y the rrow. It is presente with n input wor, w A whih it onsumes one letter t time. If there is n ege (s, s ) lelle with the next input letter, the FA mkes trnsition into stte s onsuming the letter. The proess of mking trnsition n onsuming letter repets until () the input wor is empty, or the FA rehes stte with no trnsition for the next input letter. Copyright 22 Steven D. Johnson Deemer 3, 22

4 4 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES When input wor is empty n the FA is in n epting stte, inite y oule-irle, we sy tht the FA epts w. remrk. Termintion onition () is not stritly neessry. As epite to the right ove, one n lwys eges n trnsition lelle with more thn one letter represents more thn one trnsitions to the trnsistion reltion for missing letters, tking the utomton to e stte tht onsumes ll the remining input letters, n from whih there is no pth to n epting stte. The result is finite-utomton tht eptes extly the sme wors. Exmple 9. For exmple, suppose this FA is presente with the wor the pth it follows is W Y W Y W X X X Z We ll suh pth tre of the FA. It is ustomry to omit the stte nmes, s in to ssert tht tre exists, in this se n epting tre, for. We my t times revite this further to something like 9.2. Formliztion Let us egin to formlize the ie of finite-stte utomton. Definition 9. Let A e n lphet. A finite-stte utomton, or FA over lphet A is struture S, s, F, T onsisting of finite set S is of sttes, single intitl or strting stte s S non-empty suset F S of finl or epting sttes, n A trnsition reltion T (S A) S. Next we efine how FA exeutes. Definition 9.2 Given finite-stte utomoton, FA = S, s, F, T over lphet A n wor w =... n A, we sy FA epts w if () w = n s F, or Copyright 22 Steven D. Johnson Deemer 3, 22

5 9.2. AUTOMATA 5 () there is pth, P = s, s,..., s n in T suh tht s is the strting stte, s n F is n epting stte, n for eh suessive pir (s i, s i+ ), i < n, there is pir ((s i, i ), s i+ ) T. Definition 9.3 Given finite-stte utomton, FA over lphet A the lnguge L(FA) A is the set of ll wors epte y FA Noneterminism The trnsition reltion of Definition 9. rises t lest one importnt question: Wht oes it men when the trnsition reltion is not funtion? When T is funtion, or prtil funtion, it is uniqely etermine wht tion is tken on eh input letter. Suppose to the ontrry tht our FA llows two istint trnsitions from stte on the sme ltter. For instne, elow we hve e seon trnsition uner the letter from stte W. W X Y Z The efinition of eptne still hols; it sks only whether tre exists. However, the question of how tht pth is selete is less ler. How woul physil mehnism eie whih trnsition to tke? Woul it hve to preit wht input letters it will see in the future? If so, oes this pility mke the utomt more powerful in the sense tht they epts roer lss of lnguges? Definition 9.4 A eterministi finite-stte utomton (DFA) is one whose trnsition reltion is (prtil) funtion. A non-eterminsiti finite-stte utomton (NFA) is one whose trnsition reltion is not funtion, tht is, n FA with multiple istit trnsitions from some stte for the sme letter. Copyright 22 Steven D. Johnson Deemer 3, 22

6 6 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES 9.3 Composing Automt 9.3. Prllel Composition s s s s2 s s3 s4 L = {inry wors tht en with } L 2 = {inry wors tht strt n en with the sme letter} s4 s2 s s s3 s s2 s s4 s3 L L 2 prout FA S = S S 2 s = ( s, s 2 ) F = (F S 2 ) (F 2 S ) (For L L 2, F = F F 2 ) T = {((s, s 2 ), (, 2 )), (s, s 2)) ((s, ), s ) T n((s 2, 2 ), s 2) T 2 } Copyright 22 Steven D. Johnson Deemer 3, 22

7 9.3. COMPOSING AUTOMATA 7 Remove unrehle (lue) n reunnt (re) sttes s s2 s s4 L L 2 Copyright 22 Steven D. Johnson Deemer 3, 22

8 8 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Empty Trnsitions Definition 9.5 An empty trnsition is trnsition lelle y. A DFA or NFA with n empty trnsition my noneterminstilly move to the trget stte without onsuming the next input letter. Equivlently, on my think of the input wor s hving one or more s etween ny two letters. Exmple 9.2 s s s s2 s s3 s4 L = {wors tht en with } L 2 = {wors tht strt n en with the sme letter} Contention s s t t t2 t3 t4 L ˆL 2 = {uˆv u L n v L 2 } Copyright 22 Steven D. Johnson Deemer 3, 22

9 9.3. COMPOSING AUTOMATA 9 Proposition 9. Given ny FA with empty trnsitions, A there is n utomton A tht epts the sme lnguge Removing Noneterminism (!?) v s t s t t2 t3 t4 NFA for L L 2 Copyright 22 Steven D. Johnson Deemer 3, 22

10 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Remove -trnsitions e f g h.6 Power utomton {} {,,h} {,e} {,f}, {e,,h} {,g} {,h} {g,,f}.6 Copyright 22 Steven D. Johnson Deemer 3, 22

11 9.4. REGULAR EXPRESSIONS 9.4 Regulr Expressions Regulr expressions re lnguge for esriing regulr lnguges. Definition 9.6 For given lphet A, the lnguge rexp of regulr expressions n their interprettion s lnguges over A re efine: rexp ( A {,,,, (, ) } ) rexp ::= L[ ] = A R: rexp P(A ) L[ ] = {} L[ x ] = {x} for x A rexp L[ r ] = {u n u L[ r, n N ] rexp rexp L[ r r 2 ] = {uˆv u L[ r, v L[ r 2 ] rexp rexp L[ r r 2 ] = L[ r ] L[ r 2 ] ( rexp ) L[ ( r ) ] = L[ r ] Copyright 22 Steven D. Johnson Deemer 3, 22

12 2 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES Theorem 9.2 There is finite-stte utomton tht epts the lnguge speifie y ny regulr expression. Proof. The proof is y inution on rexp. In eh se n FA is reursively onstrute. The onstrutions re shown informlly elow. : : x A: x r : A 2 A r r 2 : A 2 A A 2 r r 2 : Copyright 22 Steven D. Johnson Deemer 3, 22

13 9.4. REGULAR EXPRESSIONS 3 Theorem 9.3 The lnguge esrie y ny finite-stte utomton n e speifie y regulr expression. proof ie. There re numer of wys to reue FA to regulr expression (Google finite-utomton to regulr expression ). One wy is to perform stte reution on utomt over rexp. For instne, one reution rule might trnslte to * As reution ontinues some of the sttes eome unrehle n n e isre. The utomton ultimtely reues to L(A) n the rexp lelling the single remining ege speifies the lnguge epte y the originl utomton. Like other onstrutions, suh s removing noneterministi trnsitions, the reution of n utomton to regulr expression my involve explosion in the size of the resulting expression. Exmple 9.3 Let A = {,,, }. Is it possile to use regulr expressions to speify lnguge of wors in whih no letter ours twie in suession? Solution The expression elow ws ontriute y Ahijit Mhl, who Copyright 22 Steven D. Johnson Deemer 3, 22

14 4 CHAPTER 9. AUTOMATA AND REGULAR LANGUAGES generte it using the JFLAP visuliztion tool ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) ( ( )() ( ) ( ( )() ( ))( ( )() ( )) ( ( )() ( )) ) remrk. Though vli (proly), this is not something humn oul write, or even omprehen, without it of thought. If you exmine the formul, you will egin to see some repete ptterns. We n ientify some of these suformuls n efine lnguges to go with them, in orer to uil hierrhy. The suformul ( )() ( ) esries lnguge over the lphet {, } in whih neither nor ours twie in suession in ny wor. Nme this lnguge L L = ( )() ( ) The sulnguges L n L some wors tht ontin s n some tht ontin s. L = ( )() ( ) L = ( )() ( ) Using these lnguges, the lrge formul ove reues to L ( L )L L ( L ( L )L ( L ))(L L L ( L )) (L L L L ) This formul omines the sulnguges, ing in some more s n s where neee. You n hek tht wors in this lnguge stisfy the property of no suessive ournes,, or ; ut it is muh more iffiult to etermine whether this formul works for ll suh wors. Copyright 22 Steven D. Johnson Deemer 3, 22

15 9.5. IS THAT ALL THERE IS? Is Tht All There Is? Proposition 9.4 (Pumping Lemm) Let L e regulr lnguge. Then there is whole numer p tht epens only on L suh tht every wor w L of length t lest p is of the form w = xyz where () y > () xy p () for ll i N, xy i z L. Exmple 9.4 {u n v n n N} is not regulr. Copyright 22 Steven D. Johnson Deemer 3, 22

### = 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

More information

### 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)

More information

### 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

More information

### Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs

Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if

More information

### 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

More information

### 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

More information

### 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}.

More information

### Prefix-Free Regular-Expression Matching

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 Pttern Mthing Given pttern P nd text T, find ll sustrings

More information

### 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

More information

### CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014

S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown

More information

### 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,

More information

### CS 360 Exam 2 Fall 2014 Name

CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output

More information

### 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

More information

### 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

More information

### 80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES. 2.6 Finite State Automata With Output: Transducers

80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES 2.6 Finite Stte Automt With Output: Trnsducers So fr, we hve only considered utomt tht recognize lnguges, i.e., utomt tht do not produce ny output on ny input

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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)

More information

### 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

More information

### 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

More information

### I 3 2 = I I 4 = 2A

ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

### CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University

CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted

More information

### 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}

More information

### Convert the NFA into DFA

Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:

More information

### Lecture 11 Binary Decision Diagrams (BDDs)

C 474A/57A Computer-Aie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, K-mp, iruit, et

More information

### 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:

More information

### 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

More information

### 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)

More information

### Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51

Non Deterministic Automt Linz: Nondeterministic Finite Accepters, pge 51 1 Nondeterministic Finite Accepter (NFA) Alphbet ={} q 1 q2 q 0 q 3 2 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices

More information

### 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

More information

### 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

More information

### The DOACROSS statement

The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows prouer-onsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete

More information

### 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

More information

### 1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

### NFA DFA Example 3 CMSC 330: Organization of Programming Languages. Equivalence of DFAs and NFAs. Equivalence of DFAs and NFAs (cont.

NFA DFA Exmple 3 CMSC 330: Orgniztion of Progrmming Lnguges NFA {B,D,E {A,E {C,D {E Finite Automt, con't. R = { {A,E, {B,D,E, {C,D, {E 2 Equivlence of DFAs nd NFAs Any string from {A to either {D or {CD

More information

### Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}

Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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.

More information

### 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,

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### Solutions to Problem Set #1

CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### Arrow s Impossibility Theorem

Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties

More information

### Symbolic Automata for Static Specification Mining

Symoli Automt for Stti Speifition Mining Hil Peleg 1, Shron Shohm, Ern Yhv, n Hongseok Yng 1 Tel Aviv University, Isrel Tel Aviv-Yffo Aemi College, Isrel University of Ofor, UK Tehnion, Isrel Astrt. We

More information

### Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

### 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.

More information

### Let s divide up the interval [ ab, ] into n subintervals with the same length, so we have

III. INTEGRATION Eonomists seem muh more intereste in mrginl effets n ifferentition thn in integrtion. Integrtion is importnt for fining the epete vlue n vrine of rnom vriles, whih is use in eonometris

More information

### EE 108A Lecture 2 (c) W. J. Dally and P. Levis 2

EE08A Leture 2: Comintionl Logi Design EE 08A Leture 2 () 2005-2008 W. J. Dlly n P. Levis Announements Prof. Levis will hve no offie hours on Friy, Jn 8. Ls n setions hve een ssigne - see the we pge Register

More information

### Today s Topics Automata and Languages

Tody s Topics Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition

More information

### Automata and Languages

Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Tody s Topics DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition

More information

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

### 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;

More information

### 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

More information

### 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,

More information

### for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

### APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

### Algebra 2 Semester 1 Practice Final

Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

### 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

More information

### 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

More information

### The Word Problem in Quandles

The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given

More information

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

### 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

More information

### QUADRATIC EQUATION. Contents

QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

### 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

More information

### 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

More information

### 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

More information

### MCH T 111 Handout Triangle Review Page 1 of 3

Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

### Identifying and Classifying 2-D Shapes

Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte

More information

### PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

### 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

More information

### 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.............................

More information

### Section 2.3. Matrix Inverses

Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

### 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

More information

### 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,

More information

### Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

More information

### ( x) ( ) takes at the right end of each interval to approximate its value on that

III. INTEGRATION Economists seem much more intereste in mrginl effects n ifferentition thn in integrtion. Integrtion is importnt for fining the expecte vlue n vrince of rnom vriles, which is use in econometrics

More information

### 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

More information

### Lexical Analysis, II Comp 412

COMP 412 FALL 2017 Lexicl Anlysis, II Comp 412 source code IR Front End OpMmizer Bck End IR trget code Copyright 2017, Keith D. Cooper & Lind Torczon, ll rights reserved. Students enrolled in Comp 412

More information

### Situation Calculus. Situation Calculus Building Blocks. Sheila McIlraith, CSC384, University of Toronto, Winter Situations Fluents Actions

Plnning gent: single gent or multi-gent Stte: complete or Incomplete (logicl/probbilistic) stte of the worl n/or gent s stte of knowlege ctions: worl-ltering n/or knowlege-ltering (e.g. sensing) eterministic

More information

### Review of Gaussian Quadrature method

Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

### ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

### y = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is...

. Liner Equtions in Two Vriles C h p t e r t G l n e. Generl form of liner eqution in two vriles is x + y + 0, where 0. When we onsier system of two liner equtions in two vriles, then suh equtions re lle

More information