Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
|
|
- Johnathan Long
- 5 years ago
- Views:
Transcription
1 Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due Exerise 2. Let A e the following finite utomton: strt, () Trnsform the utomton A into n equivlent regulr expression, then trnsform this expression into n NFA (with ε-trnsitions), remove the ε-trnsitions, determinize the utomton nd then minimize it using the lgorithms from the leture. Chek tht the resulting utomton is isomorphi to the originl one. () Use JFLAP ( to perform the sme trnsformtions. Is there ny differene? () Let us denote Σ = {, }. Then L(A) = Σ \ Σ Σ. Prove tht A is miniml y proving tht no equivlene reltion on Σ of index t most 2 n stisfy x y w(xw L(A) yw L(A)) (nd thus onluding y Nerode theorem); omputing the index of L(A) (nd if it is 3 then onluding y Myhill-Nerode theorem); performing the Bisim lgorithm for minimiztion (nd heking the result is isomorphi to the originl utomton). Exerise 2.2 For n N let msf(n) e the lnguge of ll words over {, } whih represent n w.r.t. to the most signifint it first representtion where n ritrry numer of leding zeros is llowed. For exmple: msf(3) = L( ) nd msf() = L( ). Similrly, let lsf(n) denote the lnguge of ll lest signifint it first representtions of n with n ritrry numer of following zeros, e.g.: lsf(6) = L( ) nd lsf() = L( ). () Construt nd ompre the miniml DFAs representing ll even nturl numers n N w.r.t. the unry enoding (i.e., n n ), the msf enoding, nd the lsf enoding. () Consider the following FA A over the lphet {,,, }: strt
2 W.r.t. the msf enoding, we my interpret ny word w {,,, } s pir of nturl numers (X(w), Y (w)) N N. Exmple : (Underlined letters orrespond to Y (w).) w = () k () k ( k, k ) (3, ) = (X(w), Y (w)) Find onstnts, Z suh tht X(w) + Y (w) = for ll w L(A). () Construt the miniml DFA representing the lnguge {w {, } msf (w) is divisile y 3}. Solution: () Unry enoding: MSBF enoding: strt strt LSBF enoding:, strt, () We lel the sttes s follows: strt 2 We show y indution on the length of word w Σ tht for δ(, w) = q we hve 3Y (w) X(w) = q. Oviously, this will then imply tht 3Y (w) X(w) = for ll w L(A). Let l denote the length of word w (w.r.t. the lphet {,,, }). l = : We hve w = ε nd oth δ(, ε) = nd X(w) = = Y (w). l l + : We my write w s u with, {, }. Note tht we then hve X(w) = 2X(u) + nd Y (w) = 2Y (w) +. Assume tht δ(, w) =: q is defined, otherwise there is nothing to show. Then lso δ(, u) =: q is defined. By indution hypothesis we hve 3Y (u) X(u) = q. Assume q = q = : Then =, = hs to hold, i.e., X(w) = 2X(u) nd Y (w) = 2Y (u). Hene, 3Y (w) X(w) = 3 2Y (u) 2X(u) = 2(3Y (u) X(u)) = 2 q = 2 = = q.
3 Assume q = q =. Then =, =, nd X(w) = 2X(u) + nd Y (w) = 2Y (u) + susequently follow, leding to: 3Y (w) X(w) = 3 (2Y (u) + ) (2X(u) + ) = 2(3Y (u) X(u)) + 2 = 2 q + 2 = 2 ( ) + 2 = = q. Similrly, the remining ses follow. One n even show tht A epts extly those words w with 3Y (w) X(w) =. For this, let w = u e word suh tht δ(, u) is defined, ut δ(, w) is undefined, i.e., is the rejeting stte. Assume δ(, u) =. Then either = or =. We only onsider the se =, s the se = is quite similr. Then y our preeding result we hve 3Y (u) X(u) =, whih leds to 3Y (w) X(w) = 3(2Y (u) + ) 2X(u) = 3. Assume we dd letter xy to w. In order to lulte 3Y (wxy) X(wxy) we doule the vlue 3Y (w) X(w) nd then dd numer from {,, 2, 3}. Hene, 5 = 2(3Y (w) X(w)) 3Y (wxy) X(wxy) 2(3Y (w) X(w)) + 3 = 9, i.e., no mtter how we extend w y some word u {,,, }, we never will e le to otin 3Y (wu) X(wu). Similrly, one shows for the other ses of δ(, u) tht ll words leding to the rejeting stte nnot stisfy the liner eqution 3y x =. () The ide is tht the stte rehed fter reding the word u orresponds to the reminder of the numer represented y u when dividing y 3. We therefore tke s sttes {,, 2} with the initil stte nd define This yields the utomton: δ(q, ) = 2q + (mod 3). strt 2 Oviously, this utomton hs to e miniml s two different sttes enode two different reminder lsses. Note tht we lso otin this utomton from the one of () y forgetting (projeting) the y-omponent. Exerise 2.3 In the leture you hve seen the definition of the Myhill-Nerode reltion L for given lnguge L Σ : For two words x, y Σ we write x L y iff xv L yv L holds for ll v Σ. () Determine the equivlene lsses of L w.r.t. the following lnguges over Σ = {, }: L := L(( + ) ), L 2 := L(() ), L 3 := {w {, } the numer of ourrenes of nd in w is the sme }. L 4 := { n n n n }. () In the definition of L we ompre two given words y ppending ll possile words. Insted of ppending, we my lso prepend. Consider therefore the inry reltion L on Σ defined y For two words x, y Σ we write x L y iff ux L uy L holds for ll u Σ. Determine the equivlene lsses of L nd L2. Show tht L is regulr iff L hs only finitely mny equivlene lsses. Is the numer of equivlene lsses of L equl to the one of L? () Finlly, we my ompre two words x, y lso y ppending nd prepending ritrry words, i.e., define the reltion L s follows:
4 For two words x, y Σ we write x L y iff uxv L uzv L holds for ll u, v Σ. For x Σ let [x] L denote the equivlene lss of x w.r.t. L, i.e., [x] L = {t Σ x L y}. We write Σ / L for the set of equivlene lsses. How does L relte to L, resp. L? Determine the equivlene lsses of L2. Show tht the following multiplition on Σ / L is well-defined, ssoitive nd hs [ε] L s its neutrl element: Remrk : Σ / L is lled the syntti monoid of L. [w] L [w ] L := [ww ] L. Solution: () Reminder : For L Σ, the reltion L on Σ ws defined y x L y iff z Σ : xz L yz L. Let A = (Q, Σ, δ, q I, F ) e DFA. Set P (q) = {x Σ δ(q I, x) = q} for q Q. Oviously, for x, y P (q) we hve for ny word v Σ tht δ(q initil, xv) = δ(δ(q initil, x), v) = δ(q, v) = δ(δ(q initil, y), v) = δ(q initil, yv). s the utomton is deterministi, i.e., x, y P (q) x L y. Hene, the prtition {P (q) q Q} is refinement of Σ / L. So, if A is lso miniml, then we even hve x, y P (q) x L y. Hene, for lulting the equvilene lsses of regulr lnguge L it suffies to onstrut miniml DFA for L nd otin from it the lnguges P (q). L : q strt q q 3, q 2 P (q ) = L(( + ) ), P (q ) = L(( + ) ), P (q 2 ) = L(( + ) ) nd P (q 3 ) = L(( + ) ( + )). L 2 : q strt q q 2 P (q ) = L(() ), P (q ) = L(() )) nd P (q 2 ) = L(() ( + )). L 3 :,
5 q q 2 strt q q 3 q 4 It s left to the reder to determine the equivlene lsses from the DFA. L 4 : Although L 4 is ontext-sensitive, ut neither ontext-free nor regulr, the sme ide s for the preeding lnguges n e used, leit one hs to onsider n utomton with infinite numer of sttes (the rejeting stte isn t shown here): strt One esily heks tht ny two sttes re inequivlent. We otin the equivlene lsses: { k } for k, { k l } for k > l, { k+l k+l l l } for k, nd the omplement of the union of these. () For w Σ let w R e its reverse, i.e., the word we otin when reding w from right to left, e.g., () R =. For L Σ we then set L R := {w R w L}. It is well-known tht L is regulr if nd only if L R is regulr (exerise!). Consider the definition of L now: x L y iff u Σ ux L uy L iff u Σ (ux) R L R (uy) R L R iff u Σ x R u R L R y R u R L R iff v Σ x R v L R y R v L R iff v Σ x R v L R y R v L R iff x R L R y R. The ruil point here is tht u R is ny possile word s we onsider ll u Σ whih llows us to reple u R nd u y v. This mens tht we n otin the equivlene lsses of L from those of L R y simply reversing the memers of the equivlene lsses, nd we lredy know how to otin the equivlene lsses of L R (see ()). In oth ses, L nd L 2, L R i = L i, so we hve lredy otined the sought equivlene lsses in ().
6 It lso immeditely follows tht L is finite iff L is regulr, s L = L R nd we know tht (i) L is regulr iff L R is regulr, nd (ii) L R is regulr iff L R is finite. Finlly, we show tht in generl L nd L do not need to hve the sme numer of lsses: Let L Σ onsist of those words whose third letter is n. It is esy to see tht miniml DFA for L onsists of 4 + sttes; four sttes re required to ount how mny letters hve lredy een red, while the fifth stte is the rejeting stte. On the other hnd, L R onsists of ll words whose third lst letter is n. A orresponding miniml DFA hs to rememer the lst seen within window of three letters, leding to the equivlene lsses [ε], [], [], [], [], [], [], []: strt ε () Oviously, x L y implies oth x L y nd x L y. But in generl, L hs more equivlene lsses thn L or L. Consider for exmple L : We hve oth ε L nd ε L, ut ε L s ε L, ut () L. (See lso exerise 3.2.) In the se of L 2 one esily heks x L2 y iff x L2 y It remins to show tht the defined multiplition on Σ / L is independent of the representtive tken from the equivlene lss. Choose x, y, x, y Σ s.t. x L x nd y L y. We hve to show tht xy L x y : x L x implies xy L x y. (Set v = yv in the definition with v Σ ). y L y implies x y L x y. (Set u = u x in the definition with u Σ.) As L is equivlene reltion, it is trnsitive, nd we otin xy L x y. Exerise 2.4 Let Σ e n lphet. The set S Σ of str-free expression over Σ is indutively defined: S := Σ {ε, } S k+ := {(φ + ψ), (φ ψ), φ, (φ ψ) φ, ψ S k } S k S Σ := k N Sk. The lnguge L(ρ) represented y str-free expression ρ S Σ is defined s expeted: if ρ = L(φ) L(ψ) if ρ = (φ + ψ) L(ρ) := {ρ} if ρ Σ {ε} L(φ) L(ψ) if ρ = (φ ψ) Σ \ L(φ) if ρ = φ L(φ) L(ψ) if ρ = (φ ψ). To void prentheses, we ssume tht ontention hs the highest priority, followed y intersetion, then ddition. Some exmples: L( ) = Σ \ L( ) = Σ, L( ( + )) = {}. () Show tht for every str-free expression φ S Σ it holds tht L(φ) is regulr lnguges over Σ. () Give str-free expression for the regulr lnguge L(() ) for Σ = {, }. Remrk : There is no str-free expression for the lnguge L(() )!
7 Solution: () We lim tht Note tht L( ) = Σ = L(( + ) ). So we my write: L( ( + ) )! = L(()() ). L( + ) = L(( + ) ) \ L(( + ) ( + )( + ) ). This follows from strightforwrd indution on the term struture using tht regulr lnguges re losed under union, intersetion nd omplement. L(( + ) ) \ L(( + ) ( + )( + ) ) L(()() ): Consider w = () k for some k. Clerly, w L(( + ) ). Further, neither nor pper s suwords in w, so w L(Σ ( + )Σ ). L(( + ) ) \ L(( + ) ( + )( + ) ) L(()() ): Choose some w L(( + ) ) \ L(( + ) ( + )( + ) ). (Note tht this lnguge is not empty s it ontins the word.) Oviously, w ε. One now shows y indution on the length of w, tht the letters nd hve to lternte in w nd tht w hs to e of even length (s it strts with n, ut ends with ). So, w L(()() ). It follows tht L(() ) = L(ε + ( + ) ).
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 informationMinimal 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= 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 informationNondeterministic 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 informationCS 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 informationLecture 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 informationRegular 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 informationNON-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 informationFormal 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 informationRegular 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 informationNondeterministic 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 informationMyhill-Nerode Theorem
Overview Myhill-Nerode Theorem Correspondence etween DA s nd MN reltions Cnonicl DA for L Computing cnonicl DFA Myhill-Nerode Theorem Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute
More informationCMPSCI 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 informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationFinite 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 informationMore on automata. Michael George. March 24 April 7, 2014
More on utomt Michel George Mrch 24 April 7, 2014 1 Automt constructions Now tht we hve forml model of mchine, it is useful to mke some generl constructions. 1.1 DFA Union / Product construction Suppose
More informationState 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 informationTheory of Computation Regular Languages
Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring 2012 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of
More informationTheory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38
Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationCSCI 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 informationAUTOMATA 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 informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationNFA 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 informationConvert 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 informationCS 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 informationFundamentals 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 informationAssignment 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 informationTalen 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 informationCMSC 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 informationDFA minimisation using the Myhill-Nerode theorem
DFA minimistion using the Myhill-Nerode theorem Johnn Högerg Lrs Lrsson Astrct The Myhill-Nerode theorem is n importnt chrcteristion of regulr lnguges, nd it lso hs mny prcticl implictions. In this chpter,
More informationChapter 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 informationBisimulation, Games & Hennessy Milner logic
Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory
More informationDiscrete 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 informationTypes 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 informationHomework 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 information11.1 Finite Automata. CS125 Lecture 11 Fall Motivation: TMs without a tape: maybe we can at least fully understand such a simple model?
CS125 Lecture 11 Fll 2016 11.1 Finite Automt Motivtion: TMs without tpe: mybe we cn t lest fully understnd such simple model? Algorithms (e.g. string mtching) Computing with very limited memory Forml verifiction
More information1. 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 informationWorked 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 informationChapter 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 informationTypes 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 informationPrefix-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 informationIntermediate 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 informationFinite 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 informationSystem 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 information2.4 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 informationFinite 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 informationNon 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 information12.1 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 informationDesigning 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 informationKENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A
More informationSolutions 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 informationCS 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 informationConverting 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 informationNon-deterministic Finite Automata
Non-deterministic Finite Automt Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd T. vn Lrhoven Institute for Computing nd Informtion Sciences Intelligent
More information1 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 informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationHarvard 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 information5. (±±) 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 informationState 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 informationTutorial 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 informationDescriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata
Desriptionl Complexity of Non-Unry Self-Verifying Symmetri Differene Automt Lurette Mris 1,2 nd Lynette vn Zijl 1 1 Deprtment of Computer Siene, Stellenosh University, South Afri 2 Merk Institute, CSIR,
More informationCHAPTER 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 informationNon-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 informationGrammar. 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 informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationCISC 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 information1.4 Nonregular Languages
74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll
More information1 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 informationFinite-State Automata: Recap
Finite-Stte Automt: Recp Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 09 August 2016 Outline 1 Introduction 2 Forml Definitions nd Nottion 3 Closure under
More informationCHAPTER 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 informationNon-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1
Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet
More informationHybrid Systems Modeling, Analysis and Control
Hyrid Systems Modeling, Anlysis nd Control Rdu Grosu Vienn University of Tehnology Leture 5 Finite Automt s Liner Systems Oservility, Rehility nd More Miniml DFA re Not Miniml NFA (Arnold, Diky nd Nivt
More informationAnatomy 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 informationHomework 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 informationCS241 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 informationAlgorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:
Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer
More informationFinite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh
Finite Automt Informtics 2A: Lecture 3 Mry Cryn School of Informtics University of Edinburgh mcryn@inf.ed.c.uk 21 September 2018 1 / 30 Lnguges nd Automt Wht is lnguge? Finite utomt: recp Some forml definitions
More informationFormal 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 informationCS415 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 informationAutomata 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 informationParse 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 informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationKleene 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 information1 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 informationProject 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 informationNondeterminism 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 informationSome Theory of Computation Exercises Week 1
Some Theory of Computtion Exercises Week 1 Section 1 Deterministic Finite Automt Question 1.3 d d d d u q 1 q 2 q 3 q 4 q 5 d u u u u Question 1.4 Prt c - {w w hs even s nd one or two s} First we sk whether
More informationJava II Finite Automata I
Jv II Finite Automt I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz Finite Automt I p.1/13 Processing Regulr Expressions We lredy lerned out Jv s regulr expression
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationCS 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 informationGNFA GNFA GNFA GNFA GNFA
DFA RE NFA DFA -NFA REX GNFA Definition GNFA A generlize noneterministic finite utomton (GNFA) is grph whose eges re lele y regulr expressions, with unique strt stte with in-egree, n unique finl stte with
More informationRunning 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 informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 22 September 2017 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationExercises 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 informationCS 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 informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationCompression of Palindromes and Regularity.
Compression of Plinromes n Regulrity. Kyoko Shikishim-Tsuji Center for Lierl Arts Eution n Reserh Tenri University 1 Introution In [1], property of likstrem t t view of tse is isusse n it is shown tht
More informationCONTROLLABILITY and observability are the central
1 Complexity of Infiml Oservle Superlnguges Tomáš Msopust Astrt The infiml prefix-losed, ontrollle nd oservle superlnguge plys n essentil role in the reltionship etween ontrollility, oservility nd o-oservility
More informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
More information