The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

Size: px
Start display at page:

Download "The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS"

Transcription

1 The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their desk rd ut must OT write nything else until the strt of the exmintion period is nnouned. Answer ALL THR questions o lultors re permitted in this exmintion. Ditionries re not llowed with one exeption. Those whose first lnguge is not nglish my use stndrd trnsltion ditionry to trnslte etween tht lnguge nd nglish provided tht neither lnguge is the sujet of this exmintion. Sujet-speifi trnsltion diretories re not permitted. o eletroni devies ple of storing nd retrieving text, inluding eletroni ditionries, my e used. ote: ASWRS Turn Over

2 2 Question 1 The following questions re multiple hoie. There is t lest one orret nswer, ut there my e severl. To get ll the mrks you hve to list ll orret nswers nd none of the inorret ones. 1 mistke results in 3 mrks, 2 mistkes result in 1 mrk, 3 or more mistkes result in zero mrks. Answer: ote tht the nswer tht should e provided is just list of the orret lterntive(s). The dditionl explntions elow re just for your informtion. () Whih of the following sttements re orret? (i) The empty word ǫ is the only word in the empty lnguge. (ii) ǫ Σ, where Σ = {0,1} (iii) ǫ / (iv) The empty word ǫ does not elong to ny non-empty lnguge. (v) ǫ { i j i,j,i+j 42} (where = {0,1,2,...}) Answer: Corret: ii, v Inorret: i The empty lnguge ; i.e., the empty set, does not ontin ny words, not even the empty one. iii By definition, ǫ L for ny lnguge L, inluding the empty lnguge. iv {ǫ} is n exmple of non-empty lnguge tht does ontin the empty word ǫ. (5)

3 3 () Consider the following finite utomton A over Σ = {,,}:,,,,,,,, Whih of the following sttements out A re orret? (i) The utomton A is Deterministi Finite Automton (DFA). (ii) ǫ L(A) (iii) L(A) (iv) L(A) (v) The lnguge epted y the utomton A is ll words over Σ tht ontins the letters,,, in tht order. (5) Answer: Corret: iv, v Inorret: i The utomton is n FA sine more thn one trnsition sometimes re possile for some sttes nd lphet symols. ii ǫ / L(A) sine no strt stte is epting. iii / L(A) sine it is not possile to reh ny epting stte on this word. () Consider the following regulr expression: () () Whih of the following regulr expressions denote the sme lnguge s the ove regulr expression? (i) (++) (ii) (+) (++) (iii) (ǫ+) ( +) (iv) +() +() (v) ( ) ( ) (5) Answer: Corret: iii Inorret: i, ii, iv, v Turn Over

4 4 (d) Consider the following Context-Free Grmmr (CFG) G: S XX Y X X Y Y Y ǫ where S, X, Y re nonterminl symols, S is the strt symol, nd,, re terminl symols. Whih of the following sttements out the lnguge L(G) generted y G re orret? (i) ǫ L(G) (ii) L(G) (iii) L(G) (iv) L(G) = L 1 L 1 L 2 where L 1 = { i j i i,j, i > 0} nd L 2 = { i i } (where = {0,1,2,...}) (v) The following CFG G is equivlent to G ove; i.e., L(G ) = L(G): S XX X X Y Y Y ǫ Answer: Corret: i, ii, iv Inorret: (5) iii Sine the word ontins s nd s, the derivtion must egin S XX. The only possiility to derive the word from XX is to split it into two prts fter the lst, nd derive the first prt from the first X nd the lst prt from the seond X. But while n e derived from the first X, nnot e derived from the seond. v ot equivlent euse ǫ n now e derived from X, mening tht word like L(G ). However, / L(G).

5 5 (e) Whih of the following sttements out the Hlting Prolem re orret? (i) The Hlting Prolem is undeidle. (ii) The Hlting Prolem is semi-deidle. (iii) The Hlting Prolem is the prolem tht Turing Mhines n get stuk. (iv) There is no Turing Mhine tht deides the Hlting Prolem. (v) A Turing mhine tht never hlts hs got the Hlting Prolem. Answer: Corret: i, ii, iv (The semi-deidle lnguges re proper suset of the undeidle ones.) Inorret: iii, v (5) Turn Over

6 6 Question 2 () Given the following ondeterministi Finite Automton (FA) over the lphet Σ = {,, }, onstrut Deterministi Finite Automton (DFA) D() tht epts the sme lnguge s y pplying the suset onstrution:,, To sve work, onsider only the rehle prt of D(). Clerly show your lultions in stte-trnsition tle. Then drw the trnsition digrm for the resulting DFA D(). Do not forget to indite the initil stte nd the finl sttes oth in the trnsition tle nd the finl trnsition digrm. (12) Answer: δ D() {0} = A {0,1} = B {0,2} = C {0} = A {0,1} = B {0,1} = B {0,2,3} = D {0} = A {0,2} = C {0,1,3} = {0,2} = C {0} = A {0,2,3} = D {0,1,3} = {0,2} = C {0,4} = F {0,1,3} = {0,1} = B {0,2,3} = D {0,4} = F {0,4} = F {0,1} = B {0,2} = C {0,3} = G {0,3} = G {0,1} = B {0,2} = C {0,4} = F We n now drw the trnsition digrm for D(): A B C D F G

7 7 () Wht lnguge does the following Turing Mhine (TM) M ept? where M = (Q, Σ, Γ, δ,q 0, B, F) Q = {q 0,q 1,q 2,q 3,q 4 } Σ = {,,} Γ = {,,,x,b} F = {q 4 } δ(q 0,) = {(q 1,x,R)} δ(q 1,) = {(q 1,x,R)} δ(q 1,) = {(q 2,x,R)} δ(q 1,) = {(q 3,x,R)} δ(q 2,) = {(q 2,x,R)} δ(q 2,) = {(q 3,x,R)} δ(q 3,) = {(q 3,x,R)} δ(q 3,B) = {(q 4,B,R)} δ(q, x) = stop elsewhere Give preise, mthemtil hrteristion of the epted lnguge long with rief explntion of why the mhine epts this lnguge. (8) Answer: L(M) = { i j k i,j,k,i 1,k 1}. Alterntively, we n oserve tht the lnguge tully is regulr nd given y e.g. the regulr expression. The mhine only ever move right nd either stys in stte or moves to higher numered stte. The mhine strts in stte q 0, where only is epted, using the mhine to move to stte q 1. In q 1 the mhine n loop on, epting further ritrry numer of s, or it n ept or moving to q 2 or q 3, respetively. Both q 2 nd q 3 re looping sttes, llowing n ritrry numer of s nd s respetively to e epted. A single is required to move form q 2 to q 3, ut note tht q 3 lso n e rehed diretly from q 1 on single, mening tht the numer of s in n epted word n e 0. The words epted re thus those onsisting of 1 or more s, followed y 0 or more s, followed y 1 or more s. Turn Over

8 8 () In the ontext of Turing Mhines, explin wht it mens for lnguge to e: reursive, reursively-enumerle, deidle, undeidle. (5) Answer: A reursive lnguge is lnguge tht is epted y Turing Mhine tht lwys hlts. This is the sme s sying tht the lnguge is deidle (or tht the prolem represented y the lnguge is deidle). A reursively enumerle lnguge is lnguge tht is epted y Turing Mhine (ut not y ny Turing mhine tht neessrily hlts). A lnguge tht is not reursive is undeidle. The undeidle lnguges thus inludes the reursively enumerle (or semi-deidle) lnguges nd those lnguges whih re not even reursively-enumerle.

9 9 Question 3 () The following is ontext-free grmmr (CFG) for simple rithmeti expressions. For simpliity, we only onsider the numers 0, 1, nd 2: + () nd re nonterminls, is the strt symol, +,,, (, ), 0, 1, 2 re terminls. Show tht this grmmr is miguous. (5) Answer: To demonstrte miguity, demonstrte tht there is t lest one derivle word for whih there re two different leftmost or rightmost derivtions, or two different prse tress. For exmple, here re two different leftmost derivtions of the word 0+1+2: nd Alterntively, here re the two prse trees orresponding to the derivtions ove: nd Turn Over

10 10 () Construt n equivlent ut unmiguous version of the ontext-free grmmr for rithmeti expressions given in () ove y mking it reflet the following onventions regrding opertor preedene nd ssoitivity: Opertors Preedene Assoitivity highest right medium left + lowest left It should further e possile to use prentheses for grouping in the usul wy. (8) Answer: Strtify the grmmr so tht produtions for opertors refer to either produtions t the sme level or produtions t the next level up, nd use left-reursive produtions for left-ssoitive opertors nd right-reursive produtions for right-ssoitive opertors () 0 1 2

11 11 () The following Context-free grmmr(cfg) is immeditely left-reursive: S S X X XX YXd Y Y Ye Yf g S, X, nd Y re nonterminls,,,, d, e, f, nd g re terminls, nd S is the strt symol. Trnsform this grmmr into n equivlent right-reursive CFG. Stte the generl trnsformtion rule you re using nd show the min trnsformtion steps. (12) Answer: First identify the immeditely left-reursive non-terminls. Then group the produtions for eh suh non-terminl into two groups: one where eh RHS strts with the non-terminl in question, nd one where they don t: A Aα 1... Aα m A β 1... β n Then reple those produtions with new produtions for A nd produtions for A, where A is new nme, s follows: A β 1 A... β n A A α 1 A... α m A ǫ There re two immeditely left-reursive non-terminls in the given grmmr: X nd Y. The grmmr is essentilly lredy grouped s required. Applying the ove trnsformtion rule to oth the X nd Y produtions yields: S S X X YXdX YX X XX ǫ Y gy Y ey fy ǫ nd

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS The University of Nottinghm SCHOOL OF COMPUTER SCIENCE LEVEL 2 MODULE, SPRING SEMESTER 2016 2017 LNGUGES ND COMPUTTION NSWERS Time llowed TWO hours Cndidtes my complete the front cover of their nswer ook

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

Assignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages

Assignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd

More information

The University of Nottingham

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

More information

Nondeterministic Automata vs Deterministic Automata

Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

More information

Formal languages, automata, and theory of computation

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

NON-DETERMINISTIC FSA

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

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30 Tlen en Automten Test 1, Mon 7 th Dec, 2015 15h45 17h30 This test consists of four exercises over 5 pges. Explin your pproch, nd write your nswer to ech exercise on seprte pge. You cn score mximum of 100

More information

Closure Properties of Regular Languages

Closure Properties of Regular Languages Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

= state, a = reading and q j

= 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

Harvard University Computer Science 121 Midterm October 23, 2012

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

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

Nondeterministic Finite Automata

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

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

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

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

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

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

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

Let's start with an example:

Let's start with an example: Finite Automt Let's strt with n exmple: Here you see leled circles tht re sttes, nd leled rrows tht re trnsitions. One of the sttes is mrked "strt". One of the sttes hs doule circle; this is terminl stte

More information

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:

More information

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz University of Southern Cliforni Computer Siene Deprtment Compiler Design Spring 7 Lexil Anlysis Smple Exerises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sienes Institute 47 Admirlty Wy, Suite

More information

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

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

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

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 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)

More information

Worked out examples Finite Automata

Worked out examples Finite Automata Worked out exmples Finite Automt Exmple Design Finite Stte Automton which reds inry string nd ccepts only those tht end with. Since we re in the topic of Non Deterministic Finite Automt (NFA), we will

More information

CS241 Week 6 Tutorial Solutions

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.

More information

Compiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz

Compiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite

More information

Regular languages refresher

Regular languages refresher Regulr lnguges refresher 1 Regulr lnguges refresher Forml lnguges Alphet = finite set of letters Word = sequene of letter Lnguge = set of words Regulr lnguges defined equivlently y Regulr expressions Finite-stte

More information

Context-Free Grammars and Languages

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

CSCI565 - Compiler Design

CSCI565 - Compiler Design CSCI565 - Compiler Deign Spring 6 Due Dte: Fe. 5, 6 t : PM in Cl Prolem [ point]: Regulr Expreion nd Finite Automt Develop regulr expreion (RE) tht detet the longet tring over the lphet {-} with the following

More information

Chapter 2 Finite Automata

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

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

SWEN 224 Formal Foundations of Programming WITH ANSWERS

SWEN 224 Formal Foundations of Programming WITH ANSWERS T E W H A R E W Ā N A N G A O T E Ū P O K O O T E I K A A M Ā U I VUW V I C T O R I A UNIVERSITY OF WELLINGTON Time Allowed: 3 Hours EXAMINATIONS 2011 END-OF-YEAR SWEN 224 Forml Foundtions of Progrmming

More information

Lecture 6 Regular Grammars

Lecture 6 Regular Grammars Lecture 6 Regulr Grmmrs COT 4420 Theory of Computtion Section 3.3 Grmmr A grmmr G is defined s qudruple G = (V, T, S, P) V is finite set of vribles T is finite set of terminl symbols S V is specil vrible

More information

First Midterm Examination

First Midterm Examination Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does

More information

Convert the NFA into DFA

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

2.4 Theoretical Foundations

2.4 Theoretical Foundations 2 Progrmming Lnguge Syntx 2.4 Theoretil Fountions As note in the min text, snners n prsers re se on the finite utomt n pushown utomt tht form the ottom two levels of the Chomsky lnguge hierrhy. At eh level

More information

@#? Text Search ] { "!" Nondeterministic Finite Automata. Transformation NFA to DFA and Simulation of NFA. Text Search Using Automata

@#? Text Search ] { ! Nondeterministic Finite Automata. Transformation NFA to DFA and Simulation of NFA. Text Search Using Automata g Text Serh @#? ~ Mrko Berezovský Rdek Mřík PAL 0 Nondeterministi Finite Automt n Trnsformtion NFA to DFA nd Simultion of NFA f Text Serh Using Automt A B R Power of Nondeterministi Approh u j Regulr Expression

More information

Finite State Automata and Determinisation

Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

More information

Exercises Chapter 1. Exercise 1.1. Let Σ be an alphabet. Prove wv = w + v for all strings w and v.

Exercises Chapter 1. Exercise 1.1. Let Σ be an alphabet. Prove wv = w + v for all strings w and v. 1 Exercises Chpter 1 Exercise 1.1. Let Σ e n lphet. Prove wv = w + v for ll strings w nd v. Prove # (wv) = # (w)+# (v) for every symol Σ nd every string w,v Σ. Exercise 1.2. Let w 1,w 2,...,w k e k strings,

More information

I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines

I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines CI 3104 /Winter 2011: Introduction to Forml Lnguges Chter 13: Grmmticl Formt Chter 13: Grmmticl Formt I. Theory of Automt II. Theory of Forml Lnguges III. Theory of Turing Mchines Dr. Neji Zgui CI3104-W11

More information

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes CSE 260-002: Exm 3-ANSWERS, Spring 20 ime: 50 minutes Nme: his exm hs 4 pges nd 0 prolems totling 00 points. his exm is closed ook nd closed notes.. Wrshll s lgorithm for trnsitive closure computtion is

More information

Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15

Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15 Models of Comput:on Lecture #8 Chpter 7 con:nued Any lnguge tht e defined y regulr expression, finite utomton, or trnsi:on grph cn e defined y ll three methods We prove this y showing tht ny lnguge defined

More information

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014 CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA

More information

Prefix-Free Regular-Expression Matching

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

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

FABER Formal Languages, Automata and Models of Computation

FABER Formal Languages, Automata and Models of Computation DVA337 FABER Forml Lnguges, Automt nd Models of Computtion Lecture 5 chool of Innovtion, Design nd Engineering Mälrdlen University 2015 1 Recp of lecture 4 y definition suset construction DFA NFA stte

More information

Thoery of Automata CS402

Thoery of Automata CS402 Thoery of Automt C402 Theory of Automt Tle of contents: Lecture N0. 1... 4 ummry... 4 Wht does utomt men?... 4 Introduction to lnguges... 4 Alphets... 4 trings... 4 Defining Lnguges... 5 Lecture N0. 2...

More information

3 Regular expressions

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

Converting Regular Expressions to Discrete Finite Automata: A Tutorial

Converting Regular Expressions to Discrete Finite Automata: A Tutorial Converting Regulr Expressions to Discrete Finite Automt: A Tutoril Dvid Christinsen 2013-01-03 This is tutoril on how to convert regulr expressions to nondeterministic finite utomt (NFA) nd how to convert

More information

1 Nondeterministic Finite Automata

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

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw

More information

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

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

System Validation (IN4387) November 2, 2012, 14:00-17:00

System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

More information

Table of contents: Lecture N Summary... 3 What does automata mean?... 3 Introduction to languages... 3 Alphabets... 3 Strings...

Table of contents: Lecture N Summary... 3 What does automata mean?... 3 Introduction to languages... 3 Alphabets... 3 Strings... Tle of contents: Lecture N0.... 3 ummry... 3 Wht does utomt men?... 3 Introduction to lnguges... 3 Alphets... 3 trings... 3 Defining Lnguges... 4 Lecture N0. 2... 7 ummry... 7 Kleene tr Closure... 7 Recursive

More information

Formal Languages and Automata

Formal Languages and Automata Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University

More information

Deterministic Finite Automata

Deterministic Finite Automata Finite Automt Deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion Sciences Version: fll 2016 J. Rot Version: fll 2016 Tlen en Automten 1 / 21 Outline Finite Automt Finite

More information

CS375: Logic and Theory of Computing

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

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.

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

Regular expressions, Finite Automata, transition graphs are all the same!!

Regular expressions, Finite Automata, transition graphs are all the same!! CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1 Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more

More information

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny

More information

Petri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering

Petri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering Petri Nets Ree Alreht Seminr: Automt Theory Chir of Softwre Engeneering Overview 1. Motivtion: Why not just using finite utomt for everything? Wht re Petri Nets nd when do we use them? 2. Introdution:

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

Lexical Analysis Finite Automate

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

Nondeterminism and Nodeterministic Automata

Nondeterminism and Nodeterministic Automata Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely

More information

Arrow s Impossibility Theorem

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

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9. Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.

More information

Name Ima Sample ASU ID

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

Tutorial Automata and formal Languages

Tutorial Automata and formal Languages Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we

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

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

Bottom-Up Parsing. Canonical Collection of LR(0) items. Part II

Bottom-Up Parsing. Canonical Collection of LR(0) items. Part II 2 ottom-up Prsing Prt II 1 Cnonil Colletion of LR(0) items CC_LR(0)_I items(g :ugmented_grmmr){ C = {CLOURE({ })} ; repet{ foreh(i C) foreh(grmmr symol X) if(goto(i,x) && GOTO(I,X) C) C = C {GOTO(I,X)};

More information

Finite Automata-cont d

Finite Automata-cont d Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww

More information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic

More information

CS 330 Formal Methods and Models

CS 330 Formal Methods and Models CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 2 1. Prove ((( p q) q) p) is tutology () (3pts) y truth tle. p q p q

More information

Review for the Midterm

Review for the Midterm Review for the Midterm Stephen A. Edwrds Columi University Fll 2018 The Midterm Structure of Compiler Scnning Lnguges nd Regulr Expressions NFAs Trnslting REs into NFAs: Thompson s Construction Building

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

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

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

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

Languages and Computation (G52LAC) Lecture notes Spring 2018

Languages and Computation (G52LAC) Lecture notes Spring 2018 Lnguges nd Computtion (G52LAC) Leture notes Spring 28 Thorsten Altenkirh, Vennzio Cprett, nd Henrik Nilsson Contents Ferury 2, 28 Introdution 4. Exmple: Vlid Jv progrms................... 4.2 Exmple: The

More information

CSE 401 Compilers. Today s Agenda

CSE 401 Compilers. Today s Agenda CSE 401 Compilers Leture 3: Regulr Expressions & Snning, on?nued Mihel Ringenurg Tody s Agend Lst?me we reviewed lnguges nd grmmrs, nd riefly strted disussing regulr expressions. Tody I ll restrt the regulr

More information

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton 25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q

More information

Running an NFA & the subset algorithm (NFA->DFA) CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/

Running an NFA & the subset algorithm (NFA->DFA) CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/ Running n NFA & the suset lgorithm (NFA->DFA) CS 350 Fll 2018 gilry.org/lsses/fll2018/s350/ 1 NFAs operte y simultneously exploring ll pths nd epting if ny pth termintes t n ept stte.!2 Try n exmple: L

More information

Revision Sheet. (a) Give a regular expression for each of the following languages:

Revision Sheet. (a) Give a regular expression for each of the following languages: Theoreticl Computer Science (Bridging Course) Dr. G. D. Tipldi F. Bonirdi Winter Semester 2014/2015 Revision Sheet University of Freiurg Deprtment of Computer Science Question 1 (Finite Automt, 8 + 6 points)

More information

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions

CS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p

More information

Automata and Languages

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

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

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

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.

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

Fundamentals of Computer Science

Fundamentals of Computer Science Fundmentls of Computer Science Chpter 3: NFA nd DFA equivlence Regulr expressions Henrik Björklund Umeå University Jnury 23, 2014 NFA nd DFA equivlence As we shll see, it turns out tht NFA nd DFA re equivlent,

More information

CHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA)

CHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA) Finite Automt (FA or DFA) CHAPTER Regulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, equivlence of NFAs DFAs, closure under regulr

More information

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

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

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting. ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s

More information

CISC 4090 Theory of Computation

CISC 4090 Theory of Computation 9/6/28 Stereotypicl computer CISC 49 Theory of Computtion Finite stte mchines & Regulr lnguges Professor Dniel Leeds dleeds@fordhm.edu JMH 332 Centrl processing unit (CPU) performs ll the instructions

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

CSC 473 Automata, Grammars & Languages 11/9/10

CSC 473 Automata, Grammars & Languages 11/9/10 CSC 473 utomt, Grmmrs & Lnguges 11/9/10 utomt, Grmmrs nd Lnguges Discourse 06 Decidbility nd Undecidbility Decidble Problems for Regulr Lnguges Theorem 4.1: (embership/cceptnce Prob. for DFs) = {, w is

More information

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

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

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

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