Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers. Mehryar Mohri Courant Institute and Google Research
|
|
- Clarissa Armstrong
- 5 years ago
- Views:
Transcription
1 Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Mehryr Mohri Cournt Institute nd Google Reserch
2 Preliminries Finite lphet Σ, empty string. Set of ll strings over : Σ Σ (free monoid). Length of string x Σ : x. Mirror imge or reverse of string x = x 1 x n : x R = x n x 1. A lnguge L: suset of Σ. Mehryr Mohri - Speech Recognition pge 2 Cournt Institute, NYU
3 Rtionl Opertions Rtionl opertions over lnguges: union: lso denoted L 1 + L 2, conctention: closure: L 1 L 2 = {x Σ : x L 1 x L 2 }. L 1 L 2 = {x = uv Σ : u L 1 v L 2 }. L = n=0 L n, where L n = L L. n Mehryr Mohri - Speech Recognition pge 3 Cournt Institute, NYU
4 Regulr or Rtionl Lnguges Definition: closure under rtionl opertions of Σ. Thus, Rt(Σ ) is the smllest suset L of 2 Σ verifying L ; x Σ, {x} L ; L 1,L 2 L,L 1 L 2 L,L 1 L 2 L,L 1 L. Exmples of regulr lnguges over Σ={,, c} : Σ, ( + ) c, n c, ( +( + c) ) c. Mehryr Mohri - Speech Recognition pge 4 Cournt Institute, NYU
5 Finite Automt Definition: finite utomton A over the lphet is 4-tuple (Q, I, F, E) where Q is finite set of sttes, I Q set of initil sttes, F Q set of finl sttes, nd E multiset of trnsitions which re elements of Q (Σ {}) Q. pth in n utomton π A =(Q, I, F, E) element of E. pth from stte in I to stte in is n F is clled n ccepting pth. Lnguge L(A) ccepted y A: set of strings leling ccepting pths. Σ Mehryr Mohri - Speech Recognition pge 5 Cournt Institute, NYU
6 Finite Automt - Exmple Mehryr Mohri - Speech Recognition pge 6 Cournt Institute, NYU
7 Finite Automt - Some Properties Trim: ny stte lies on some ccepting pth. Unmiguous: no two ccepting pths hve the sme lel. Deterministic: unique initil stte, two trnsitions leving the sme stte hve different lels. Complete: t lest one outgoing trnsition leled with ny lphet element t ny stte. Acyclic: no pth with cycle. Mehryr Mohri - Speech Recognition pge 7 Cournt Institute, NYU
8 Normlized Automt Definition: finite utomton is normlized if it hs unique initil stte with no incoming trnsition. it hs unique finl stte with no outgoing trnsition. i A f Mehryr Mohri - Speech Recognition pge 8 Cournt Institute, NYU
9 Elementry Normlized Automton Definition: normlized utomton ccepting n element Σ {} constructed s follows. 0 1 Mehryr Mohri - Speech Recognition pge 9 Cournt Institute, NYU
10 Normlized Automt: Union Construction: the union of two normlized utomt is normlized utomton constructed s follows. i 1 A 1 1 f i f i 2 A 2 2 f Mehryr Mohri - Speech Recognition pge 10 Cournt Institute, NYU
11 Normlized Automt: Conctention Construction: the conctention of two normlized utomt is normlized utomton constructed s follows. i1 A f 1 1 i f 2 A 2 2 Mehryr Mohri - Speech Recognition pge 11 Cournt Institute, NYU
12 Normlized Automt: Closure Construction: the closure of normlized utomton is normlized utomton constructed s follows. i 0 i A f f 0 Mehryr Mohri - Speech Recognition pge 12 Cournt Institute, NYU
13 Normlized Automt - Properties Construction properties: ech rtionl opertion require creting t most two sttes. ech stte hs t most two outgoing trnsitions. the complexity of ech opertion is liner. Mehryr Mohri - Speech Recognition pge 13 Cournt Institute, NYU
14 Thompson s Construction Proposition: let r e regulr expression over the lphet Σ. Then, there exists normlized utomton A with t most 2 r sttes representing r. Proof: (Thompson, 1968) liner-time context-free prser to prse regulr expression. construction of normlized utomton strting from elementry expressions nd following opertions of the tree. Mehryr Mohri - Speech Recognition pge 14 Cournt Institute, NYU
15 Thompson s Construction - Exmple ε 4 ε 5 ε ε 1 2 ε 3 ε 6 ε 0 ε 7 c 8 ε 9 Normlized utomton for regulr expression + c. Mehryr Mohri - Speech Recognition pge 15 Cournt Institute, NYU
16 Regulr Lnguges nd Finite Automt Theorem: A lnguge is regulr iff it cn e ccepted y finite utomton. Proof: Let for A =(Q, I, F, E) e finite utomton. (i, j, k) [1, Q ] [1, Q ] [0, Q ] L(A) = is thus regulr. i I,f F X Q if Mehryr Mohri - Speech Recognition pge 16 define Xij 0 is regulr for ll (i, j) since E is finite. y recurrence Xij k for ll (i, j, k) since (Kleene, 1956) X k ij = {i q 1 q 2... q n j : n 0,q i k}. X k+1 ij = X k ij + Xk i,k+1 (Xk k+1,k+1 ) X k k+1,j. Cournt Institute, NYU
17 Regulr Lnguges nd Finite Automt Proof: the converse holds y Thompson s construction. Notes: more generl theorem (Schützenerger, 1961) holds for weighted utomt. not ll lnguges re regulr, e.g., L = { n n : n N} is not regulr. Let A e n utomton. If L L(A), then for lrge enough n, n n corresponds to pth with cycle: n n = p u q, p u q L(A), which implies L(A) = L. Mehryr Mohri - Speech Recognition pge 17 Cournt Institute, NYU
18 Left Syntctic Congruence Definition: for ny lnguge L Σ, the left syntctic congruence is the equivlence reltion defined y u L v u 1 L = v 1 L, where for ny u Σ, u 1 L is defined y u 1 L = {w : uw L}. u 1 L of L with respect to u nd denoted L. is sometimes clled the prtil derivtive u Mehryr Mohri - Speech Recognition pge 18 Cournt Institute, NYU
19 Regulr Lnguges - Chrcteriztion Theorem: lnguge L is regulr iff the set of is finite ( hs finite index). L Proof: let utomton ccepting A =(Q, I, F, E) L e trim deterministic (existence seen lter). let δ the prtil trnsition function. Then, urv δ(i, u) =δ(i, v). u 1 L lso defines n eq. reltion with index Q. since δ(i, u) =δ(i, v) u 1 L = v 1 L, the index of L is t most Q, thus finite. Mehryr Mohri - Speech Recognition pge 19 Cournt Institute, NYU
20 Regulr Lnguges - Chrcteriztion Proof: conversely, if the set of the utomton Q = {u 1 L: u Σ } ; i = 1 L = L, I = {i} ; F = {u 1 L: u L} ; since ccepts exctly L. u 1 L A =(Q, I, F, E) is finite, then defined y ; is well defined nd E = {(u 1 L,, (u) 1 L): u Σ } u 1 L = v 1 L (u) 1 L =(v) 1 L Mehryr Mohri - Speech Recognition pge 20 Cournt Institute, NYU
21 Illustrtion Miniml deterministic utomton for ( + ) : L -1 L () -1 L Mehryr Mohri - Speech Recognition pge 21 Cournt Institute, NYU
22 ε-removl Theorem: ny finite utomton dmits n equivlent utomton with no ε- trnsition. A =(Q, I, F, E) Proof: for ny stte q Q, let [q] denote the set of sttes reched from q y pths leled with. Define A =(Q,I,F,E ) y Q = {[q]: q Q}, I = [q], F = {[q]: [q] F = }. q I E = {([p],,[q]) : (p,,q ) E,p [p],q [q]}. Mehryr Mohri - Speech Recognition pge 22 Cournt Institute, NYU
23 ε-removl - Illustrtion {0, 1} {0, 2} {0, 1, 3} {0} Mehryr Mohri - Speech Recognition pge 23 Cournt Institute, NYU
24 Determiniztion Theorem: ny utomton A =(Q, I, F, E) without -trnsitions dmits n equivlent deterministic utomton. Proof: Suset construction: A =(Q,I,F,E ) Q =2 Q. I = {s Q : s I = }. F = {s Q : s F = }. E = {(s,, s ): (q,, q ) E,q s, q s }. with Mehryr Mohri - Speech Recognition pge 24 Cournt Institute, NYU
25 Determiniztion - Illustrtion {0} {1} {1, 2} {2} {0, 1} Mehryr Mohri - Speech Recognition pge 25 Cournt Institute, NYU
26 Completion Theorem: ny deterministic utomton dmits n equivlent complete deterministic utomton. Proof: constructive, see exmple Mehryr Mohri - Speech Recognition pge 26 Cournt Institute, NYU
27 Complementtion Theorem: let A =(Q, I, F, E) e deterministic utomton, then there exists deterministic utomton ccepting L(A). Proof: y previous theorem, we cn ssume A complete. The utomton otined from A y mking non-finl sttes finl nd finl sttes non-finl exctly ccepts L(A). B =(Σ,Q,I,Q F, E) Mehryr Mohri - Speech Recognition pge 27 Cournt Institute, NYU
28 Complementtion - Ilustrtion Mehryr Mohri - Speech Recognition pge 28 Cournt Institute, NYU
29 Regulr Lnguges - Properties Theorem: regulr lnguges re closed under rtionl opertions, intersection, complementtion, reversl, morphism, inverse morphism, nd quotient with ny set. Proof: closure under rtionl opertions holds y definition. intersection: use De Morgn s lw. complementtion: use lgorithm. others: lgorithms nd equivlence reltion. Mehryr Mohri - Speech Recognition pge 29 Cournt Institute, NYU
30 Rtionl Reltions Definition: closure under rtionl opertions of the monoid Σ, where Σ nd re finite lphets, denoted y Rt(Σ ). exmples: (, ), (, ) (, )+(, ). Mehryr Mohri - Speech Recognition pge 30 Cournt Institute, NYU
31 Rtionl Reltions - Chrcteriztion Theorem: R Rt(Σ ) is rtionl reltion iff there exists regulr lnguge L (Σ ) such tht R = {(π Σ (x),π (x)) : x L} where is the projection of over nd π the projection over. π Σ (Σ ) Σ Proof: use surjective morphism π :(Σ ) (Σ ) x (π Σ (x),π (x)). (Nivt, 1968) Mehryr Mohri - Speech Recognition pge 31 Cournt Institute, NYU
32 Trnsductions Definition: function trnsduction from Σ to. reltion ssocite to τ : τ :Σ 2 is clled R(τ) ={(x, y) Σ : y τ(x)}. trnsduction ssocited to reltion: x Σ,τ(x) ={y :(x, y) R}. rtionl trnsductions: trnsductions with rtionl reltions. Mehryr Mohri - Speech Recognition pge 32 Cournt Institute, NYU
33 Finite-Stte Trnsducers Definition: finite-stte trnsducer T over the lphets Σ nd is 4-tuple where Q is finite set of sttes, I Q set of initil sttes, F Q set of finl sttes, nd E multiset of trnsitions which re elements of Q (Σ {}) ( {}) Q. T defines reltion vi the pir of input nd output lels of its ccepting pths, R(T )={(x, y) Σ : I x:y F }. Mehryr Mohri - Speech Recognition pge 33 Cournt Institute, NYU
34 Rtionl Reltions nd Trnsducers Theorem: trnsduction is rtionl iff it cn e relized y finite-stte trnsducer. Proof: Nivt s theorem comined with Kleene s theorem, nd construction of normlized trnsducer from finite-stte trnsducer. Mehryr Mohri - Speech Recognition pge 34 Cournt Institute, NYU
35 References Kleene, S. C Representtion of events in nerve nets nd finite utomt. Automt Studies. Nivt, Murice Trnsductions des lngges de Chomsky. Annles 18, Institut Fourier. Schützenerger, Mrcel~Pul On the definition of fmily of utomt. Informtion nd Control, 4 Thompson, K Regulr expression serch lgorithm. Comm. ACM, 11. Mehryr Mohri - Speech Recognition pge 35 Cournt Institute, NYU
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 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 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 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 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 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 informationA tutorial on sequential functions
A tutoril on sequentil functions Jen-Éric Pin LIAFA, CNRS nd University Pris 7 30 Jnury 2006, CWI, Amsterdm Outline (1) Sequentil functions (2) A chrcteriztion of sequentil trnsducers (3) Miniml sequentil
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 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 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 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 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 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 informationNFAs continued, Closure Properties of Regular Languages
Algorithms & Models of Computtion CS/ECE 374, Fll 2017 NFAs continued, Closure Properties of Regulr Lnguges Lecture 5 Tuesdy, Septemer 12, 2017 Sriel Hr-Peled (UIUC) CS374 1 Fll 2017 1 / 31 Regulr Lnguges,
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 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 informationDeterministic 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 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 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 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 informationRegular 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 informationCS375: Logic and Theory of Computing
CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tble of Contents: Week 1: Preliminries (set lgebr, reltions, functions) (red Chpters 1-4) Weeks
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 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 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 informationNFAs continued, Closure Properties of Regular Languages
lgorithms & Models of omputtion S/EE 374, Spring 209 NFs continued, losure Properties of Regulr Lnguges Lecture 5 Tuesdy, Jnury 29, 209 Regulr Lnguges, DFs, NFs Lnguges ccepted y DFs, NFs, nd regulr expressions
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 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 informationClosure 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 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 information80 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 informationa,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 informationChapter 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 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 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 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 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 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 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 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 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 informationFABER 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 informationLecture 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 informationCompiler 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 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 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 information1.3 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 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 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 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 informationLexical 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 informationPART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA
PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA RIGHT LINEAR LANGUAGES. Right Liner Grmmr: Rules of the form: A α B, A α A,B V N, α V T + Left Liner Grmmr: Rules of the form: A Bα, A α A,B V N, α V T
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 informationNFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun:
CMPU 240 Lnguge Theory nd Computtion Spring 2019 NFAs nd Regulr Expressions Lst clss: Introduced nondeterministic finite utomt with -trnsitions Tody: Prove n NFA- is no more powerful thn n NFA Introduce
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 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 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 informationɛ-closure, Kleene s Theorem,
DEGefW5wiGH2XgYMEzUKjEmtCDUsRQ4d 1 A nice pper relevnt to this course is titled The Glory of the Pst 2 NICTA Resercher, Adjunct t the Austrlin Ntionl University nd Griffith University ɛ-closure, Kleene
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 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 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 informationAgenda. Agenda. Regular Expressions. Examples of Regular Expressions. Regular Expressions (crash course) Computational Linguistics 1
Agend CMSC/LING 723, LBSC 744 Kristy Hollingshed Seitz Institute for Advnced Computer Studies University of Mrylnd HW0 questions? Due Thursdy before clss! When in doubt, keep it simple... Lecture 2: 6
More informationTable 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 informationContext-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 informationLet'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 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 informationHomework 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 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 informationThoery 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 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 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 informationNormal Forms for Context-free Grammars
Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S
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 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 informationIn-depth introduction to main models, concepts of theory of computation:
CMPSCI601: Introduction Lecture 1 In-depth introduction to min models, concepts of theory of computtion: Computility: wht cn e computed in principle Logic: how cn we express our requirements Complexity:
More informationLexical Analysis Part III
Lexicl Anlysis Prt III Chpter 3: Finite Automt Slides dpted from : Roert vn Engelen, Florid Stte University Alex Aiken, Stnford University Design of Lexicl Anlyzer Genertor Trnslte regulr expressions to
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 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 informationNondeterministic Biautomata and Their Descriptional Complexity
Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle
More informationA Unified Construction of the Glushkov, Follow, and Antimirov Automata
A Unified Construction of the Glushkov, Follow, nd Antimirov Automt Cyril Alluzen nd Mehryr Mohri Cournt Institute of Mthemticl Sciences 251 Mercer Street, New York, NY 10012, USA {lluzen,mohri}@cs.nyu.edu
More informationCS375: 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 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 informationCS103B 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 informationCONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM
Logicl Methods in Computer Science Vol. 2 (2:6) 2006, pp. 1 24 www.lmcs-online.org Sumitted Jn. 31, 2005 Pulished Jul. 19, 2006 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM ARNAUD CARAYOL
More informationScanner. Specifying patterns. Specifying patterns. Operations on languages. A scanner must recognize the units of syntax Some parts are easy:
Scnner Specifying ptterns source code tokens scnner prser IR A scnner must recognize the units of syntx Some prts re esy: errors mps chrcters into tokens the sic unit of syntx x = x + y; ecomes
More informationHybrid Control and Switched Systems. Lecture #2 How to describe a hybrid system? Formal models for hybrid system
Hyrid Control nd Switched Systems Lecture #2 How to descrie hyrid system? Forml models for hyrid system João P. Hespnh University of Cliforni t Snt Brr Summry. Forml models for hyrid systems: Finite utomt
More informationGeneral Algorithms for Testing the Ambiguity of Finite Automata and the Double-Tape Ambiguity of Finite-State Transducers
Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny Generl Algorithms for Testing the Amiguity of Finite Automt nd the Doule-Tpe Amiguity of Finite-Stte Trnsducers
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 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 informationFormal Language and Automata Theory (CS21004)
Forml Lnguge nd Automt Forml Lnguge nd Automt Theory (CS21004) Khrgpur Khrgpur Khrgpur Forml Lnguge nd Automt Tle of Contents Forml Lnguge nd Automt Khrgpur 1 2 3 Khrgpur Forml Lnguge nd Automt Forml Lnguge
More informationCMSC 330: Organization of Programming Languages. DFAs, and NFAs, and Regexps (Oh my!)
CMSC 330: Orgniztion of Progrmming Lnguges DFAs, nd NFAs, nd Regexps (Oh my!) CMSC330 Spring 2018 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All
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 informationLanguages & Automata
Lnguges & Automt Dr. Lim Nughton Lnguges A lnguge is sed on n lphet which is finite set of smols such s {, } or {, } or {,..., z}. If Σ is n lphet, string over Σ is finite sequence of letters from Σ, (strings
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 informationGeneral Algorithms for Testing the Ambiguity of Finite Automata
Generl Algorithms for Testing the Amiguity of Finite Automt Cyril Alluzen 1,, Mehryr Mohri 2,1, nd Ashish Rstogi 1, 1 Google Reserch, 76 Ninth Avenue, New York, NY 10011. 2 Cournt Institute of Mthemticl
More information3 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 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 information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014
CS125 Lecture 12 Fll 2014 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 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 informationBACHELOR THESIS Star height
BACHELOR THESIS Tomáš Svood Str height Deprtment of Alger Supervisor of the chelor thesis: Study progrmme: Study rnch: doc. Štěpán Holu, Ph.D. Mthemtics Mthemticl Methods of Informtion Security Prgue 217
More information