Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1
|
|
- Brian Fowler
- 5 years ago
- Views:
Transcription
1 Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1
2 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2
3 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 λ Σ- q3 Σ- q 1 q 2 q 1 q 2 q 1 q 2 A B C Fll 2018 Costs Busch - RPI 3
4 Alphbet ={} Two choices q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 4
5 Alphbet ={} Two choices q q2 1 No trnsition q 0 q 3 No trnsition Fll 2018 Costs Busch - RPI 5
6 First Choice q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 6
7 First Choice q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 7
8 First Choice All input is consumed q q2 1 ccept q 0 q 3 Fll 2018 Costs Busch - RPI 8
9 Second Choice q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 9
10 q 0 Second Choice Input cnnot be consumed q q2 1 q 3 Automton Hlts reject Fll 2018 Costs Busch - RPI 10
11 An NFA ccepts string: if there is computtion of the NFA tht ccepts the string i.e., ll the input string is processed nd the utomton is in n ccepting stte Fll 2018 Costs Busch - RPI 11
12 is ccepted by the NFA: ccept q q1 2 q1 q2 q 0 q 3 becuse this computtion ccepts Fll 2018 Costs Busch - RPI 12 q 0 q 3 reject this computtion is ignored
13 Rejection exmple q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 13
14 First Choice reject q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 14
15 Second Choice 1 q q2 q 0 q 3 Fll 2018 Costs Busch - RPI 15
16 Second Choice 1 q q2 q 0 q 3 reject Fll 2018 Costs Busch - RPI 16
17 Another Rejection exmple q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 17
18 First Choice q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 18
19 First Choice Input cnnot be consumed q q2 1 reject q 0 q 3 Automton hlts Fll 2018 Costs Busch - RPI 19
20 Second Choice q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 20
21 q 0 Second Choice Input cnnot be consumed q q2 1 q 3 Automton hlts reject Fll 2018 Costs Busch - RPI 21
22 An NFA rejects string: if there is no computtion of the NFA tht ccepts the string. For ech computtion: All the input is consumed nd the utomton is in non finl stte OR The input cnnot be consumed Fll 2018 Costs Busch - RPI 22
23 is rejected by the NFA: q q1 2 reject q1 2 q q 0 q 3 reject q 0 q 3 All possible computtions led to rejection Fll 2018 Costs Busch - RPI 23
24 is rejected by the NFA: reject q q1 2 q1 q2 q 0 q 3 q 0 q 3 reject All possible computtions led to rejection Fll 2018 Costs Busch - RPI 24
25 Lnguge ccepted: L {} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 25
26 NFA: Mouse Mze Fll 2018 Costs Busch - RPI 26
27 Lmbd Trnsitions q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 27
28 q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 28
29 q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 29
30 input tpe hed does not move q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 30
31 ll input is consumed q0 ccept q1 q 2 q3 String is ccepted Fll 2018 Costs Busch - RPI 31
32 Rejection Exmple q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 32
33 q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 33
34 (red hed doesn t move) q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 34
35 Input cnnot be consumed Automton hlts reject q0 q1 q 2 q3 String is rejected Fll 2018 Costs Busch - RPI 35
36 Lnguge ccepted: L {} q0 q1 q 2 q3 Fll 2018 Costs Busch - RPI 36
37 Another NFA Exmple q b q 2 q0 1 q 3 Fll 2018 Costs Busch - RPI 37
38 b q b q 2 q0 1 q 3 Fll 2018 Costs Busch - RPI 38
39 b q b q 2 0 q 1 q 3 Fll 2018 Costs Busch - RPI 39
40 b ccept q q1 b 0 q 2 q 3 Fll 2018 Costs Busch - RPI 40
41 Another String b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 41
42 b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 42
43 b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 43
44 b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 44
45 b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 45
46 b b q q b 1 q2 q3 0 Fll 2018 Costs Busch - RPI 46
47 b b q q b 1 q2 q3 0 ccept Fll 2018 Costs Busch - RPI 47
48 Lnguge ccepted L b, bb, bbb,... b q b q 2 q0 1 q 3 Fll 2018 Costs Busch - RPI 48
49 Another NFA Exmple Wht is the lnguge ccepted?? 0 q0 1 1 q 0,1 q2 Fll 2018 Costs Busch - RPI 49
50 Lnguge ccepted 0 q0 1 1 q 0,1 q2 (redundnt stte) Fll 2018 Costs Busch - RPI 50
51 Remrks: The symbol never ppers on the input tpe Simple utomt: M 1 q 0 M 2 q 0 L(M 1 ) ={} (M ) = {λ} L 2 Fll 2018 Costs Busch - RPI 51
52 NFAs re interesting becuse we cn express lnguges esier thn DFAs NFA M 1 DFA M 2 q 0 q1 q 2 q 0 q 1 L( M1) = { } L( M2) = { } Fll 2018 Costs Busch - RPI 52
53 Forml Definition of NFAs M Q,,, q0, F Q : Set of sttes, i.e. q, q q 0 1, 2 Input plhbet, i.e., b : : Trnsition function q 0 : Initil stte F : Accepting sttes Fll 2018 Costs Busch - RPI 53
54 Trnsition Function q x q, q,,, 1 2 q k x q 1 resulting sttes with q x x q 1 following one trnsition with symbol x q k Fll 2018 Costs Busch - RPI 54
55 q, q q0 1 1 q 0,1 q2 Fll 2018 Costs Busch - RPI 55
56 ( q1,0) { q0, q2} 0 q q 0,1 q2 1 Fll 2018 Costs Busch - RPI 56
57 ( q, ) { q } q 0 1 q 0,1 q2 1 Fll 2018 Costs Busch - RPI 57
58 ( q,1) 2 q q 0,1 q2 1 Fll 2018 Costs Busch - RPI 58
59 Extended Trnsition Function * Sme with but pplied on strings * q, q 0 1 q 4 q 5 q 0 q 1 b q 2 q 3 Fll 2018 Costs Busch - RPI 59
60 * q, q, q q 4 q 5 q 0 q 1 b q 2 q 3 Fll 2018 Costs Busch - RPI 60
61 * q, b q, q, q q 4 q 5 q 0 q 1 b q 2 q 3 Fll 2018 Costs Busch - RPI 61
62 Specil cse: for ny stte q q * q, Fll 2018 Costs Busch - RPI 62
63 Fll 2018 Costs Busch - RPI 63 w q q i j, * : there is wlk from to with lbel i q j q w i q j q w w k k i q j q In generl
64 The Lnguge of n NFA The lnguge ccepted by M is: M M w w L,... 1, 2 w n where * ( q 0, w m ) { q i,..., q k,, q j } nd there is some q k F (ccepting stte) Fll 2018 Costs Busch - RPI 64
65 w m L M * ( q, w 0 m ) w m q i q0 w m qk q k F w m q j Fll 2018 Costs Busch - RPI 65
66 F q 0,q 5 q 4 q 5 q 0 q 1 b q 2 q 3 * q, q, q F L(M) Fll 2018 Costs Busch - RPI 66
67 F q 0,q 5 q 4 q 5 q 0 q 1 b q 2 q 3 q, b q, q, q * b LM F Fll 2018 Costs Busch - RPI 67
68 F q 0,q 5 q 4 q 5 q 0 q 1 b q 2 q 3 * q, b q, q bl(m ) F Fll 2018 Costs Busch - RPI 68
69 F q 0,q 5 q 4 q 5 q 0 q 1 b q 2 q 3 * q 0, b q b LM 1 F Fll 2018 Costs Busch - RPI 69
70 q 4 q 5 q 0 q 1 b q 2 q 3 L M b* b * { } Fll 2018 Costs Busch - RPI 70
71 NFAs ccept the Regulr Lnguges Fll 2018 Costs Busch - RPI 71
72 Equivlence of Mchines Definition: Mchine M1 is equivlent to mchine M2 if LM 1 L M 2 Fll 2018 Costs Busch - RPI 72
73 Exmple of equivlent mchines LM 1 {10} * NFA 0 q q M 1 LM 2 {10} * q q q2 1 Fll 2018 Costs Busch - RPI 73 0 DFA 0 M 2 0,1
74 0 q1 1 q2 Fll 2018 Costs Busch - RPI 74
75 Theorem: Lnguges ccepted by NFAs Regulr Lnguges Lnguges ccepted by DFAs NFAs nd DFAs hve the sme computtion power, ccept the sme set of lnguges Fll 2018 Costs Busch - RPI 75
76 Proof: we only need to show Lnguges ccepted by NFAs Lnguges ccepted by NFAs AND Regulr Lnguges Regulr Lnguges Fll 2018 Costs Busch - RPI 76
77 Proof-Step 1 Lnguges ccepted by NFAs Regulr Lnguges Every DFA is trivilly n NFA Any lnguge L ccepted by DFA is lso ccepted by n NFA Fll 2018 Costs Busch - RPI 77
78 Proof-Step 2 Lnguges ccepted by NFAs Regulr Lnguges Any NFA cn be converted to n equivlent DFA Any lnguge L is lso ccepted by DFA ccepted by n NFA Fll 2018 Costs Busch - RPI 78
79 NFA M Conversion NFA to DFA q 0 q1 q2 b DFA M q 0 Fll 2018 Costs Busch - RPI 79
80 NFA M * ( q0, ) { q1, q2 } q 0 q1 q2 b DFA M q 0 q 1,q 2 Fll 2018 Costs Busch - RPI 80
81 NFA M * ( q 0, b) q 0 q1 q2 b empty set DFA M q 0 q 1,q 2 b trp stte Fll 2018 Costs Busch - RPI 81
82 NFA DFA M M q 0 q1 q2 b q 0 q 1,q 2 b * ( q1, ) { q1, q2 } * ( q 2, ) union q 1,q 2 Fll 2018 Costs Busch - RPI 82
83 NFA M q 0 q1 q2 b * ( q1, b) { q0} * q, b) { q } ( 2 0 union q 0 DFA M b q 0 q 1,q 2 b Fll 2018 Costs Busch - RPI 83
84 NFA M q 0 q1 q2 b DFA M b q 0 q 1,q 2 b,b trp stte Fll 2018 Costs Busch - RPI 84
85 NFA M END OF CONSTRUCTION q 0 q1 q2 b q F 1 DFA M b q 0 q 1,q 2 b q q F 1, 2,b Fll 2018 Costs Busch - RPI 85
86 Generl Conversion Procedure Input: n NFA M Output: n equivlent DFA with LM L(M ) M Fll 2018 Costs Busch - RPI 86
87 The NFA hs sttes q, q, q, The DFA hs sttes from the power set,,,,,,,..., q q q q q q q Fll 2018 Costs Busch - RPI 87
88 Conversion Procedure Steps step 1. Initil stte of NFA: q 0 Initil stte of DFA: q 0 Fll 2018 Costs Busch - RPI 88
89 Exmple NFA M q 0 q1 q2 b DFA M q 0 Fll 2018 Costs Busch - RPI 89
90 Fll 2018 Costs Busch - RPI For every DFA s stte compute in the NFA dd trnsition to DFA },...,, { m j i q q q q q q m j i, *..., *, * },...,, { n l k q q q },...,, { },,...,, { n l k m j i q q q q q q Union step
91 Exmple NFA M *( q0, ) { q1, q2} q 0 q1 q2 b DFA M q, q q 0 1, q 0 q 1,q 2 2 Fll 2018 Costs Busch - RPI 91
92 step 3. Repet Step 2 for every stte in DFA nd symbols in lphbet until no more sttes cn be dded in the DFA Fll 2018 Costs Busch - RPI 92
93 Exmple NFA M q 0 q1 q2 b DFA M b q 0 q 1,q 2 b,b Fll 2018 Costs Busch - RPI 93
94 step 4. For ny DFA stte { qi, q j,..., qm} if some q j is ccepting stte in NFA Then, { qi, q j,..., qm} is ccepting stte in DFA Fll 2018 Costs Busch - RPI 94
95 Exmple NFA M q 0 q1 q2 b q F 1 DFA M b q 0 q 1,q 2 b q q F 1, 2,b Fll 2018 Costs Busch - RPI 95
96 Lemm: If we convert NFA M to DFA then the two utomt re equivlent: LM LM M Proof: We only need to show: LM LM AND LM LM Fll 2018 Costs Busch - RPI 96
97 First we show: LM LM We only need to prove: wl(m ) wl(m ) Fll 2018 Costs Busch - RPI 97
98 NFA Consider wl(m ) q0 w q f symbols w 1 2 k q0 1 2 k q f Fll 2018 Costs Busch - RPI 98
99 symbol q i i q j denotes possible sub-pth like qi symbol i q j Fll 2018 Costs Busch - RPI 99
100 We will show tht if wl(m ) NFA M : q0 1 2 w 1 2 k k q f then DFA M : { q 0 } stte lbel 1 2 w L(M ) Fll 2018 Costs Busch - RPI 100 k {, } q f stte lbel
101 More generlly, we will show tht if in : M NFA (rbitrry string) M : q0 v qi 2 q j n ql n qm then DFA M : { q 0 } 1 2 {, } q i n { q, } { q, } {, } j l q m Fll 2018 Costs Busch - RPI 101
102 Proof by induction on v v Induction Bsis: v 1 1 NFA M : q 1 0 q i DFA M : { q 0 } 1 {, } q i is true by construction of M Fll 2018 Costs Busch - RPI 102
103 Induction hypothesis: 1 v k v 1 2 k NFA M Suppose tht the following hold : q0 1 2 qi q j qc k qd DFA M : { q 0 } 1 2 k { q, } { q, } { q, } {, } i j c q d Fll 2018 Costs Busch - RPI 103
104 Induction Step: v k 1 NFA M : q0 v 1 2 qi 1 2 k k1 k1 v Then this is true by construction of M q j qc k qd v k1 q e v DFA M : { q 0 } 1 2 k { q, } { q, } { q, } {, } i j c q d k1 {, } q e Fll 2018 Costs Busch - RPI 104 v
105 Therefore if wl(m ) w 1 2 k NFA M : q0 1 2 k q f then DFA M : { q 0 } 1 2 k {, } q f w L(M ) Fll 2018 Costs Busch - RPI 105
106 We hve shown: LM LM With similr proof we cn show: LM LM Therefore: LM LM END OF LEMMA PROOF Fll 2018 Costs Busch - RPI 106
Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51
Non Deterministic Automt Linz: Nondeterministic Finite Accepters, pge 51 1 Nondeterministic Finite Accepter (NFA) Alphbet ={} q 1 q2 q 0 q 3 2 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices
More informationNon-Deterministic Finite Automata
Non-Deterministic Finite Automt http://users.comlb.ox.c.uk/luke. ong/teching/moc/nf2up.pdf 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q1 q2 2 Alphbet ={} Two choices q1 q2 3 Alphbet ={} Two choices
More informationNon Deterministic Automata. Formal Languages and Automata - Yonsei CS 1
Non Deterministic Automt Forml Lnguges nd Automt - Yonsei CS 1 Nondeterministic Finite Accepter (NFA) We llow set of possible moves insted of A unique move. Alphbet = {} Two choices q 1 q2 Forml Lnguges
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 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 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 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 informationCS:4330 Theory of Computation Spring Regular Languages. Equivalences between Finite automata and REs. Haniel Barbosa
CS:4330 Theory of Computtion Spring 208 Regulr Lnguges Equivlences between Finite utomt nd REs Hniel Brbos Redings for this lecture Chpter of [Sipser 996], 3rd edition. Section.3. Finite utomt nd regulr
More information5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata
CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton
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 informationLecture 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 informationRecursively Enumerable and Recursive. Languages
Recursively Enumerble nd Recursive nguges 1 Recll Definition (clss 19.pdf) Definition 10.4, inz, 6 th, pge 279 et S be set of strings. An enumertion procedure for Turing Mchine tht genertes ll strings
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 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 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
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 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 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 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 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 informationNondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA
Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn
More 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 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 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 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 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 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 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 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 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 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 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 informationCSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not
More informationAutomata 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 informationinput tape head moves current state
CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter
More 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 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 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 informationDeterministic Finite-State Automata
Deterministic Finite-Stte Automt Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 12 August 2013 Outline 1 Introduction 2 Exmple DFA 1 DFA for Odd number of
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 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 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 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 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 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 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 informationToday s Topics Automata and Languages
Tody s Topics Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Tody s Topics DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition
More 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 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 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 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 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 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 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 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 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 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 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 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 informationTuring Machines Part One
Turing Mchines Prt One Wht problems cn we solve with computer? Regulr Lnguges CFLs Lnguges recognizble by ny fesible computing mchine All Lnguges Tht sme drwing, to scle. All Lnguges The Problem Finite
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 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 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 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 informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed
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 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 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 informationProbabilistic Model Checking Michaelmas Term Dr. Dave Parker. Department of Computer Science University of Oxford
Probbilistic Model Checking Michelms Term 2011 Dr. Dve Prker Deprtment of Computer Science University of Oxford Long-run properties Lst lecture: regulr sfety properties e.g. messge filure never occurs
More informationThis lecture covers Chapter 8 of HMU: Properties of CFLs
This lecture covers Chpter 8 of HMU: Properties of CFLs Turing Mchine Extensions of Turing Mchines Restrictions of Turing Mchines Additionl Reding: Chpter 8 of HMU. Turing Mchine: Informl Definition B
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY. FLAC (15-453) - Spring L. Blum
15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY THE PUMPING LEMMA FOR REGULAR LANGUAGES nd REGULAR EXPRESSIONS TUESDAY Jn 21 WHICH OF THESE ARE REGULAR? B = {0 n 1 n n 0} C = { w w hs equl numer of
More informationSpeech 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 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 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 informationSection: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.
Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM
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 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 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 informationTuring Machines Part One
Turing Mchines Prt One Hello Hello Condensed Condensed Slide Slide Reders! Reders! Tody s Tody s lecture lecture consists consists lmost lmost exclusively exclusively of of nimtions nimtions of of Turing
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 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 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 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 informationFirst 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 informationFor convenience, we rewrite m2 s m2 = m m m ; where m is repeted m times. Since xyz = m m m nd jxyj»m, we hve tht the string y is substring of the fir
CSCI 2400 Models of Computtion, Section 3 Solutions to Homework 4 Problem 1. ll the solutions below refer to the Pumping Lemm of Theorem 4.8, pge 119. () L = f n b l k : k n + lg Let's ssume for contrdiction
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 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 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 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 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 informationIntroduction to ω-autamata
Fridy 25 th Jnury, 2013 Outline From finite word utomt ω-regulr lnguge ω-utomt Nondeterministic Models Deterministic Models Two Lower Bounds Conclusion Discussion Synthesis Preliminry From finite word
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 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 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 informationChapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)
C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,
More 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 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 informationClosure Properties of Regular Languages
of Regulr Lnguges Dr. Neil T. Dntm CSCI-561, Colordo School of Mines Fll 2018 Dntm (Mines CSCI-561) Closure Properties of Regulr Lnguges Fll 2018 1 / 50 Outline Introduction Closure Properties Stte Minimiztion
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 information