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


 Neal Casey
 1 years ago
 Views:
Transcription
1 CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University
2 First Progrmming Project Instruction Scheduling Project hs een posted Due dtes: code Ferury 25, report Mrch 4  strt erly  prolem is not totlly defined; you my run into situtions where you will need time to redesign your solution  the report is 40% of the grde, code is 60%  need to follow clling conventions since we will use utomtic testing frmework  we minly provide help for C/C++ implementtions  not group project! 2
3 Constructing Scnner Review source code specifictions Scnner Scnner Genertor prts of speech & words Tles nd code The scnner is the first stge in the front end Specifictions cn e expressed using regulr expressions Build tles nd code from DFA 3
4 Review: Gol We show how to construct finite stte utomton to recognize ny RE Overview: Direct construction of nondeterministic finite utomton (NFA) to recognize given RE Requires trnsitions to comine regulr suexpressions Construct deterministic finite utomton (DFA) to simulte the NFA Use setofsttes construction Minimize the numer of sttes Hopcroft stte minimiztion lgorithm Generte the scnner code Additionl specifictions needed for detils 4
5 Nondeterministic Finite Automt Ech RE corresponds to deterministic finite utomton (DFA) My e hrd to directly construct the right DFA Wht out n RE such s ( ) *? S 0 S 1 S 2 S 3 S 4 This is little different S 0 hs trnsition on S 1 hs two trnsitions on This is nondeterministic finite utomton (NFA) 5
6 Nondeterministic Finite Automt An NFA ccepts string x iff pth though the trnsition grph from s 0 to finl stte such tht the edge lels spell x Trnsitions on consume no input To run the NFA, strt in s 0 nd guess the right trnsition t ech step Alwys guess correctly If some sequence of correct guesses ccepts x then ccept Why study NFAs? They re the key to utomte the RE DFA construction We cn pste together NFAs with trnsitions NFA NFA ecomes n NFA 6
7 Reltionship etween NFAs nd DFAs DFA is specil cse of n NFA DFA hs no trnsitions DFA s trnsition function is singlevlued Sme rules will work DFA cn e simulted with n NFA Oviously NFA cn e simulted with DFA Simulte sets of possile sttes Possile exponentil lowup in the stte spce Still, one stte per chrcter in the input strem (less ovious) 7
8 Automting Scnner Construction To convert specifiction into code: 1 Write down the RE for the input lnguge 2 Build ig NFA 3 Build the DFA tht simultes the NFA 4 Systemticlly shrink the DFA 5 Turn it into code Scnner genertors Lex nd Flex work long these lines Algorithms re wellknown nd wellunderstood Key issue is interfce to prser You could uild one in weekend! (define ll prts of speech) 8
9 Review: Automting Scnner Construction RE NFA (Thompson s construction) Build n NFA for ech term Comine them with moves NFA DFA (suset construction) Build the simultion DFA Miniml DFA Hopcroft s lgorithm The Cycle of Constructions RE NFA DFA miniml DFA DFA RE (Not prt of the scnner construction) All pirs, ll pths prolem Tke the union of ll pths from s 0 to n ccepting stte 9
10 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for 10
11 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for 11
12 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 1 S 2 S 3 S 4 NFA for 12
13 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 1 S 2 S 0 S 3 S 4 S 5 NFA for 13
14 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 1 S 2 S 0 S 5 S 3 S 4 NFA for 14
15 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 0 S 1 S 2 S 3 S 4 NFA for S 5 S 1 S 3 NFA for * 15
16 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 0 S 1 S 2 S 3 S 4 NFA for S 5 S 1 S 3 NFA for * 16
17 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 0 S 1 S 2 S 3 S 4 NFA for S 5 S 3 S 4 NFA for * 17
18 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 0 S 1 S 2 S 3 S 4 NFA for S 5 NFA for * S 3 S 4 18
19 RE NFA using Thompson s Construction Key ide NFA pttern for ech symol nd ech opertor Ech NFA hs single strt nd ccept stte Join them with moves in precedence order S 3 S 4 NFA for NFA for S 0 S 1 S 2 S 3 S 4 NFA for S 5 NFA for * S 3 S 4 Ken Thompson, CACM,
20 Exmple of Thompson s Construction Let s try ( c ) * 1.,, & c c 2. c S 1 S 2 S 0 S 5 c S 3 S 4 3. ( c ) * S 2 S 3 S 6 S 7 c S 4 S 5 20
21 Exmple of Thompson s Construction Let s try ( c ) * 1.,, & c c 2. c S 1 S 2 S 0 S 5 c S 3 S 4 3. ( c ) * S 2 S 3 S 6 S 7 S 4 c S 5 21
22 Exmple of Thompson s Construction (con t) 4. ( c ) * S 4 S 5 S 2 S 3 S 8 S 9 S 6 c S 7 Of course, humn would design something simpler... c But, we cn utomte production of the more complex one... 22
23 Review: Automting Scnner Construction RE NFA (Thompson s construction) Build n NFA for ech term Comine them with moves NFA DFA (suset construction) Build the simultion DFA Miniml DFA Hopcroft s lgorithm The Cycle of Constructions RE NFA DFA miniml DFA DFA RE (Not prt of the scnner construction) All pirs, ll pths prolem Tke the union of ll pths from s 0 to n ccepting stte 23
24 NFA DFA with Suset Construction Need to uild simultion of the NFA Two key functions Move(s i, ) is set of sttes rechle from s i y closure(s i ) is set of sttes rechle from s i y The lgorithm: Strt stte derived from s 0 of the NFA Tke its closure S 0 = closure(s 0 ) Tke the imge of S 0, move(s 0, ) for ech Σ, nd tke its closure, dd it to the stte set S For ech stte S, compute move(s, ) for ech Σ, nd tke its closure Iterte until no more sttes re dded Sounds more complex thn it is 24
25 NFA DFA with Suset Construction The lgorithm: s 0 closure(q 0 ) dd s 0 to S while ( S is still chnging ) for ech s i S for ech Σ s? closure(move(s i,)) if ( s? S ) then dd s? to S s s j T[ s i,] s j else T[ s i,] s? Let s think out why this works 25
26 NFA DFA with Suset Construction The lgorithm: s 0 closure(q 0 ) dd s 0 to S while ( S is still chnging ) for ech s i S for ech Σ s? closure(move(s i,)) if ( s? S ) then dd s? to S s s j T[ s i,] s j else T[ s i,] s? The lgorithm hlts: 1. S contins no duplictes (test efore dding) 2. 2 Q is finite 3. while loop dds to S, ut does not remove from S (monotone) the loop hlts S contins ll the rechle NFA sttes It tries ech symol in ech s i. It uilds every possile NFA configurtion. S nd T form the DFA Let s think out why this works 26
27 NFA DFA with Suset Construction Exmple of fixedpoint computtion Monotone construction of some finite set Hlts when it stops dding to the set Proofs of hlting & correctness re similr These computtions rise in mny contexts Other fixedpoint computtions Cnonicl construction of sets of LR(1) items Quite similr to the suset construction Clssic dtflow nlysis Solving sets of simultneous set equtions DFA minimiztion lgorithm (coming up!) We will see mny more fixedpoint computtions 27
28 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1 28
29 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 none none 29
30 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 s 1 q 1, q 2, q 3, q 4, q 6, q 9 none none 30
31 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 s 1 q 1, q 2, q 3, none q 4, q 6, q 9 none none 31
32 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 none none s 1 q 1, q 2, q 3, q 4, q 6, q 9 none q 5 q 7 32
33 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 none s 1 q 1, q 2, q 3, none q 5, q 8, q 9, q 4, q 6, q 9 q 3, q 4, q 6 none q 7, q 8, q 9, q 3, q 4, q 6 33
34 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, q 4, q 6, q 9 none s 1 q 1, q 2, q 3, none q 5, q 8, q 9, q 4, q 6, q 9 q 3, q 4, q 6 s 2 q 5, q 8, q 9, q 3, q 4, q 6 none q 7, q 8, q 9, q 3, q 4, q 6 34
35 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, none none q 4, q 6, q 9 s 1 q 1, q 2, q 3, q 4, q 6, q 9 none q 5, q 8, q 9, q 3, q 4, q 6 q 7, q 8, q 9, q 3, q 4, q 6 s 2 q 5, q 8, q 9, q 3, q 4, q 6 none s 2 s 3 35
36 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, none none q 4, q 6, q 9 s 1 q 1, q 2, q 3, q 4, q 6, q 9 none q 5, q 8, q 9, q 3, q 4, q 6 q 7, q 8, q 9, q 3, q 4, q 6 s 2 q 5, q 8, q 9, none s 2 s 3 q 3, q 4, q 6 s 3 q 7, q 8, q 9, q 3, q 4, q 6 36
37 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, none none q 4, q 6, q 9 s 1 q 1, q 2, q 3, none q 5, q 8, q 9, q 7, q 8, q 9, q 4, q 6, q 9 q 3, q 4, q 6 q 3, q 4, q 6 s 2 q 5, q 8, q 9, none s 2 s 3 q 3, q 4, q 6 s 3 q 7, q 8, q 9, none s 2 s 3 q 3, q 4, q 6 37
38 NFA DFA with Suset Construction ( c )* : q 0 q 1 q 4 q 5 q q 3 q 2 8 q 9 q 6 c q 7 Applying the suset construction: closure(move(s,*)) NFA sttes c s 0 q 0 q 1, q 2, q 3, none none q 4, q 6, q 9 s 1 q 1, q 2, q 3, none q 5, q 8, q 9, q 7, q 8, q 9, q 4, q 6, q 9 q 3, q 4, q 6 q 3, q 4, q 6 s 2 q 5, q 8, q 9, none s 2 s 3 q 3, q 4, q 6 s 3 q 7, q 8, q 9, none s 2 s 3 q 3, q 4, q 6 Finl sttes 38
39 NFA DFA with Suset Construction The DFA for ( c ) * s 0 s 1 c s 2 s 3 c c δ c s 0 s s 1  s 2 s 3 s 2  s 2 s 3 s 3  s 2 s 3 Ends up smller thn the NFA All trnsitions re deterministic 39
40 Automting Scnner Construction RE NFA (Thompson s construction) Build n NFA for ech term The Cycle of Constructions Comine them with moves NFA DFA (suset construction) Build the simultion DFA Miniml DFA Hopcroft s lgorithm RE NFA DFA miniml DFA DFA RE (not relly prt of scnner construction) All pirs, ll pths prolem Union together pths from s 0 to finl stte 40
41 DFA Minimiztion The Big Picture Discover sets of equivlent sttes Represent ech such set with just one stte 41
42 DFA Minimiztion The Big Picture Discover sets of equivlent sttes Represent ech such set with just one stte Two sttes re equivlent if nd only if: Σ, trnsitions on led to equivlent sttes (DFA) trnsitions to distinct sets sttes must e in distinct sets 42
43 DFA Minimiztion The Big Picture Discover sets of equivlent sttes Represent ech such set with just one stte Two sttes re equivlent if nd only if: Σ, trnsitions on led to equivlent sttes (DFA) trnsitions to distinct sets sttes must e in distinct sets A prtition P of S Ech stte s S is in exctly one set p i P The lgorithm itertively prtitions the DFA s sttes 43
44 DFA Minimiztion Detils of the lgorithm Group sttes into mximl size sets, optimisticlly Itertively sudivide those sets, s needed Sttes tht remin grouped together re equivlent Initil prtition, P 0, hs two sets: {F} & {QF} (D =(Q,Σ,δ,q 0,F)) Splitting set ( prtitioning set y ) Assume q, & q s, nd δ(q,) = q x, & δ(q,) = q y If q x & q y re not in the sme set, then s must e split q hs trnsition on, q does not splits s 44
45 DFA Minimiztion The lgorithm P { F, {QF}} while ( P is still chnging) T { } for ech set S P T T split(s) P T split(s): for ech Σ if splits S into S 1, S 2, then return {S 1, S 2, } else return S Why does this work? Prtition P 2 Q Strt off with 2 susets of Q {F} nd {QF} While loop tkes P i P i+1 y splitting 1 or more sets P i+1 is t lest one step closer to the prtition with Q sets Mximum of Q splits Note tht Prtitions re never comined 45
46 DFA Minimiztion The lgorithm P { F, {QF}} while ( P is still chnging) T { } for ech set S P T T split(s) P T split(s): for ech Σ if splits S into S 1, S 2, then return {S 1, S 2, } else return S Why does this work? Prtition P 2 Q Strt off with 2 susets of Q {F} nd {QF} While loop tkes P i P i+1 y splitting 1 or more sets P i+1 is t lest one step closer to the prtition with Q sets Mximum of Q splits Note tht Prtitions re never comined This is fixedpoint lgorithm! 46
47 Bck to our DFA Minimiztion exmple Then, pply the minimiztion lgorithm Split on Current Prtition c P 0 { s 1, s 2, s 3 } {s 0 } none none none s 0 s 1 c s 2 c finl sttes s 3 c To produce the miniml DFA s 0 s 1 c We oserved tht humn would design simpler utomton thn Thompson s construction & the suset construction did. Minimizing tht DFA produces the one tht humn would design! 47
48 Arevited Register Specifiction Strt with regulr expression r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 The Cycle of Constructions RE NFA DFA miniml DFA 48
49 Arevited Register Specifiction Thompson s construction produces r 0 r 1 s 0 r 2 r 8 s f r 9 The Cycle of Constructions RE NFA DFA miniml DFA 49
50 Arevited Register Specifiction The suset construction uilds s 0 r 0 s f s f 1 s f8 s f2 s f9 This is DFA, ut it hs lot of sttes The Cycle of Constructions RE NFA DFA miniml DFA 50
51 Arevited Register Specifiction The DFA minimiztion lgorithm uilds s 0 r 0,1,2,3,4, 5,6,7,8,9 s f This looks like wht skilled compiler writer would do! The Cycle of Constructions RE NFA DFA miniml DFA 51
52 Homework nd Next clss Syntx Anlysis Red EC:
Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More 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. 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 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 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 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 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 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 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 informationNondeterministic Finite Automata
Nondeterministic Finite Automt From Regulr Expressions to NFA Eliminting nondeterminism Rdoud University Nijmegen Nondeterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion
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, 14:1018:30 Techer: Dniel Hedin, phone 021107052 The exm
More informationCHAPTER 1 Regular Languages. Contents
Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Nondeterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr
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 525 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn
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 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 informationAutomata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ωautomata. Ralf Huuck.
Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ωautomt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent
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 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 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 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 informationCS 311 Homework 3 due 16:30, Thursday, 14 th October 2010
CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w
More 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 informationHarvard University Computer Science 121 Midterm October 23, 2012
Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closedook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is
More informationGrammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages
5//6 Grmmr Automt nd Lnguges Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive
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 14) Weeks
More informationSpeech Recognition Lecture 2: Finite Automata and FiniteState Transducers
Speech Recognition Lecture 2: Finite Automt nd FiniteStte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.
More informationOn Determinisation of HistoryDeterministic Automata.
On Deterministion of HistoryDeterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YRICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil
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 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 informationName Ima Sample ASU ID
Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:359:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to
More 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 informationCS S12 Turing Machine Modifications 1. When we added a stack to NFA to get a PDA, we increased computational power
CS4112015S12 Turing Mchine Modifictions 1 120: Extending Turing Mchines When we dded stck to NFA to get PDA, we incresed computtionl power Cn we do the sme thing for Turing Mchines? Tht is, cn we dd
More 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 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 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 informationOverview HC9. Parsing: TopDown & LL(1) ContextFree Grammars (1) Introduction. CFGs (3) ContextFree Grammars (2) Vertalerbouw HC 9: Ch.
Overview H9 Vertlerouw H 9: Prsing: opdown & LL(1) do 3 mei 2001 56 heo Ruys h. 8  Prsing 8.1 ontextfree Grmmrs 8.2 opdown Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough
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 Longrun properties Lst lecture: regulr sfety properties e.g. messge filure never occurs
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 informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationContextFree Grammars and Languages
ContextFree 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 informationFormal Methods in Software Engineering
Forml Methods in Softwre Engineering Lecture 09 orgniztionl issues Prof. Dr. Joel Greenyer Decemer 9, 2014 Written Exm The written exm will tke plce on Mrch 4 th, 2015 The exm will tke 60 minutes nd strt
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 informationCS 267: Automated Verification. Lecture 8: Automata Theoretic Model Checking. Instructor: Tevfik Bultan
CS 267: Automted Verifiction Lecture 8: Automt Theoretic Model Checking Instructor: Tevfik Bultn LTL Properties Büchi utomt [Vrdi nd Wolper LICS 86] Büchi utomt: Finite stte utomt tht ccept infinite strings
More informationCS 330 Formal Methods and Models
CS 0 Forml Methods nd Models Dn Richrds, George Mson University, Fll 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 8 1. Prove q (q p) p q p () (4pts) with truth tle. p q p q p (q p) p q
More informationNormal Forms for Contextfree Grammars
Norml Forms for Contextfree Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S
More informationCS 310 (sec 20)  Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20)  Winter 2003  Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More informationPrefixFree RegularExpression Matching
PrefixFree RegulrExpression Mthing YoSu Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST PrefixFree RegulrExpression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings
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 informationGoodforGames Automata versus Deterministic Automata.
GoodforGmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt
More informationSoftware Engineering using Formal Methods
Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive
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 informationChapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
More informationThe University of Nottingham
The University of Nottinghm SCHOOL OF COMPUTR SCINC AND INFORMATION TCHNOLOGY A LVL 1 MODUL, SPRING SMSTR 20042005 MACHINS AND THIR LANGUAGS Time llowed TWO hours Cndidtes must NOT strt writing their
More informationDesign and Analysis of Distributed Interacting Systems
Design nd Anlysis of Distriuted Intercting Systems Lecture 6 LTL Model Checking Prof. Dr. Joel Greenyer My 16, 2013 Some Book References (1) C. Bier, J.P. Ktoen: Principles of Model Checking. The MIT
More informationNondeterministic Biautomata and Their Descriptional Complexity
Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik JustusLieigUniversität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle
More informationThe Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem
Simpler & More Generl Minimiztion for Weighted FiniteStte Automt Json Eisner Johns Hopkins University My 28, 2003 HLTNAACL First hlf of tlk is setup  revies pst ork. Second hlf gives outline of the
More informationOn the Relative Succinctness of Nondeterministic Büchi and cobüchi Word Automata
On the Reltive Succinctness of Nondeterministic Büchi nd cobüchi Word Automt Benjmin Aminof, Orn Kupfermn, nd Omer Lev Herew University, School of Engineering nd Computer Science, Jeruslem 91904, Isrel
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationChapter 4 StateSpace Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 StteSpe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationThe Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higherorder Pushdown Automata
The Cucl Hierrchy of Infinite Grphs in Terms of Logic nd Higherorder Pushdown Automt Arnud Cryol 1 nd Stefn Wöhrle 2 1 IRISA Rennes, Frnce rnud.cryol@iris.fr 2 Lehrstuhl für Informtik 7 RWTH Achen, Germny
More informationDesigning Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work
More informationCompilers. Lexical analysis. Yannis Smaragdakis, U. Athens (original slides by Sam
Compilers Lecture 3 Lexical analysis Yannis Smaragdakis, U. Athens (original slides by Sam Guyer@Tufts) Big picture Source code Front End IR Back End Machine code Errors Front end responsibilities Check
More informationexpression simply by forming an OR of the ANDs of all input variables for which the output is
2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output
More informationHow Deterministic are GoodForGames Automata?
How Deterministic re GoodForGmes Automt? Udi Boker 1, Orn Kupfermn 2, nd Mich l Skrzypczk 3 1 Interdisciplinry Center, Herzliy, Isrel 2 The Herew University, Isrel 3 University of Wrsw, Polnd Astrct
More informationResources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations
Introduction: Binding Prt of 4lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding
More informationPrefixFree Subsets of Regular Languages and Descriptional Complexity
PrefixFree Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek Jurj Šeej DCFS 2015 PrefixFree Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek, Jurj Šeej 1/22 Outline Mximl
More informationPreviously on GLT. Basic technologies. Lexical analysis. Basic technologies ASF+SDF. Concepts of programming languages
Prevously on GLT Generc Lnguge Technology: Bsc technologes Pro.dr. Mrk vn den Brnd ASF+SDF syntx descrptons semntc descrptons: type checkng nlyss trnsormtons t so to s Concepts o progrmmng lnguges syntctc
More informationWhere did dynamic programming come from?
Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/15265463200250010048.pdf
More informationOn NFA reductions. N6A 5B7, London, Ontario, CANADA ilie 2 Department of Computer Science, University of Chile
On NFA reductions Lucin Ilie 1,, Gonzlo Nvrro 2, nd Sheng Yu 1, 1 Deprtment of Computer Science, University of Western Ontrio N6A 5B7, London, Ontrio, CANADA ilie syu@csd.uwo.c 2 Deprtment of Computer
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 informationRegular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15
Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll
More informationCover Automata for Finite Languages
Cover Automt for Finite Lnguges Michël Cdilhc Technicl Report n o 0504, June 2005 revision 681 Astrct. Although regulr lnguges comined with finite utomt re widely used nd studied, mny pplictions only use
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 nonfinl.
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 informationComplementing Büchi Automata with a Subsettuple Construction
DEPARTEMENT D INFORMATIQUE DEPARTEMENT FÜR INFORMATIK Bd de Pérolles 90 CH1700 Friourg www.unifr.ch/informtics WORKING PAPER Complementing Büchi Automt with Susettuple Construction J. Allred & U. UltesNitsche
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationState Complexity of Union and Intersection of Binary SuffixFree Languages
Stte Complexity of Union nd Intersetion of Binry SuffixFree 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 informationi 1 i 2 i 3... i p o 1 o 2 AUTOMATON q 1, q 2,,q n ... o q Model of a automaton Characteristics of automaton:
Definition of n Automton:An Automton is defined s system tht preforms certin functions without humn intervention. it ccepts rw mteril nd energy s input nd converts them into the finl product under the
More informationCDM Memoryless Machines
CDM Memoryless Mchines Zero Spce Klus Sutner Crnegie Mellon Universlity 2 Finite Stte Mchines Fll 27 3 DFA Decision Problems Where Are We? 3 Turing Mchines 4 We hve description of bstrct computbility in
More informationModels of Computation: Automata and Processes. J.C.M. Baeten
Models of Computtion: Automt nd Processes J.C.M. Beten Jnury 4, 2010 ii Prefce Computer science is the study of discrete ehviour of intercting informtion processing gents. Here, ehviour is the centrl notion.
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 informationReferences. Theory of Computation. Theory of Computation. Introduction. Alexandre DuretLutz
References Theory of Computtion Alexndre DuretLutz dl@lrde.epit.fr Septemer 10, 2010 Introduction to the Theory of Computtion (Michel Sipser, 2005). Lecture notes from Pierre Wolper's course t http://www.montefiore.ulg.c.e/~pw/cours/clc.html
More informationHow to simulate Turing machines by invertible onedimensional cellular automata
How to simulte Turing mchines by invertible onedimensionl cellulr utomt JenChristophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex
More informationHyperminimisation of weighted finite automata
Hyperminimistion of weighted finite utomt Dniel Quernheim nd ndres Mletti Institute for Nturl Lnguge Processing, University of Stuttgrt @ims.unistuttgrt.de ugust 18, 2011. Mletti nd D. Quernheim
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationSolutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and nonaccepting state in M.
Solution Prolem Set 2 Prolem.4 () Let M denote the DFA contructed y wpping the ccept nd nonccepting tte in M. For ny tring w B, w will e ccepted y M, tht i, fter conuming the tring w, M will e in n ccepting
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 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 informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationGlobal Types for Dynamic Checking of Protocol Conformance of MultiAgent Systems
Globl Types for Dynmic Checking of Protocol Conformnce of MultiAgent Systems (Extended Abstrct) Dvide Ancon, Mtteo Brbieri, nd Vivin Mscrdi DIBRIS, University of Genov, Itly emil: dvide@disi.unige.it,
More informationLearning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus
Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection,
More informationLast time: introduced our first computational model the DFA.
Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody
More informationA Bounded Incremental Test Generation Algorithm for Finite State Machines
A Bounded Incrementl Test Genertion Algorithm for Finite Stte Mchines Zoltán Pp 1, Mhdevn Surmnim 2, Gáor Kovács 3, Gáor Árpád Németh3 1 Ericsson Telecomm. Hungry, H1117 Budpest, Irinyi J. u. 420, Hungry
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More information