Chapter 5. Push-down Automata. By Dr Zalmiyah Zakaria
|
|
- Derek Hodge
- 7 years ago
- Views:
Transcription
1 Chpter 5 Push-down Automt By Dr Zlmiyh Zkri
2 Push-Down Automt Regulr expressions re string genertors they tell us how to generte ll strings in lnguge L. Finite Automt (DFA, NFA) re string cceptors they tell us if specific string w is in L. CFGs re string genertors Are there string cceptors for Context-Free lnguges? YES! Push-down utomt Sept2011 Theory of Computer Science 2
3 Push-Down Automt DFAs ccept regulr lnguges. Now, we wnt to design mchines similr to DFAs tht will ccept context-free lnguges. These mchines will need to e more powerful. To hndle lnguge like { n n n 0}, the mchine needs to rememer the numer of s. To do this, we use stck. A push-down utomton (PDA) is essentilly n NFA with stck.
4 Push-Down Automt A PDA hs some memory, ut not generl purpose, rndom-ccess memory. It uses stck, which hs two opertions Push nd Pop. Using the stck, it llows us to: Count Mtch strings Sept2011 Theory of Computer Science 4
5 Building PDA stte control push $ red 0 push x red 1 pop x red 1 pop x pop $ L = {0 n 1 n : n 1} Rememer ech 0 y pushing x onto the stck When we see 1, we pop n x from the stck We wnt to ccept when we hit the stck ottom $ = specil mrker for ottom
6 A PDA in ction stte control push $ red 0 push x red 1 pop x red 1 pop x pop $ input L = {0 n 1 n : n 1} $ stck x x x
7 Nottion for PDAs push $ red 0 push x red 1 pop x red 1 pop x pop $ q 0 q 1 q 2 q 3, / $ 1, x /, $ / 0, / x 1, x / red, pop / push
8 Definition of PDA A pushdown utomton (PDA) is sextuple (Q,,,, q 0, F) where: Q is finite set of sttes; is the input lphet; is the stck lphet q 0 in Q is the strt stte; F Q is set of finl sttes; is the trnsition function : Q ( {}) ( {}) susets of Q ( {}) PDA ccepts w if we end up in finl stte with n empty stck
9 Exmple q 0 q 1 q 2 q 3, / $ 0, / x 1, x /, $ / 1, x / = {0, 1} = {$, x} (q 0,, ) = {(q 1, $)} (q 0,, $) = (q 0,, x) = (q 0, 0, ) =... : Q ( {}) ( {}) susets of Q ( {}) stte input symol pop symol stte push symol
10 The lnguge of PDA A PDA is nondeterministic Multiple trnsitions on sme pop/input llowed Trnsitions my ut do not hve to push or pop The lnguge of PDA is the set of ll strings in S* tht cn led the PDA to n ccepting stte
11 Trnsitions Let ((p,, β), (q, γ)) Δ e trnsition. It mens tht we Move from stte p. Red from the tpe, Pop the string β from the stck, Move to stte q, Push string γ onto the stck. The first three (p,, β), re input. The lst two (q, γ) re output.
12 Trnsitions We will drw it s or If t q 0 (p), with next input symol () nd top of stck x (), then cn consume (), pop x (), push y () onto stck nd move to q 1 (q)
13 Pushing nd Popping When we push β, we push the symols of β s we red them right to left. When we push the string c, we push c, then push, then push. When we pop γ, we pop the symols of γ s we red them from left to right (reverse order). When we pop the string c, we pop, then pop, then pop c.
14 Pushing nd Popping Thus, if we push the string c nd then pop it, we will get ck c, not c. If we wnted to reverse the order, we would use three seprte trnsitions: Push Push Push c
15 Configurtions A configurtion fully descries the current stte of the PDA. The current stte p. The remining input w. The current stck contents. Thus, configurtion is triple (p, w, ) (K, *, * ).
16 Computtions A configurtion (p, w, ) yields configurtion (p', w', ') in one step, denoted (p, w, ) (p', w', '), if there is trnsition ((p,, ), (p', )) Δ such tht w = w', =, nd ' = for some *. The reflexive, trnsitive closure of is denoted *.
17 Accepting Strings After processing the string on the tpe, The PDA is in either finl or nonfinl stte, nd The stck is either empty or not empty. The input string is ccepted if The ending stte is finl stte, nd The stck is empty. Tht is, the string w * is ccepted if (s, w, e) * (f, e, e) for some f F.
18 Accepting Strings One my define cceptnce y finl stte only. The input is ccepted if nd only if the lst stte is finl stte, regrdless of whether the stck is empty. One my define cceptnce y empty stck only. The input is ccepted if nd only if the stck is empty once the input is processed, regrdless of which stte the PDA is in.
19 Exmple of PDA Run the following PDA on the input string.
20 Exmple of PDA The steps in the processing re (s,, e) (p,, A) (p,, AA) (p,, AAA) (p,, AA) (p,, A) (q, e, e).
21 The Lnguge of PDA The lnguge of PDA A is L(A) = {w * A ccepts w}. Wht is the lnguge of the PDA in the previous exmple?
22 Exmple of PDA Wht is the lnguge of the following PDA?
23 Exmples of PDAs Let = {, }. Design PDA tht ccepts the lnguge {wcw R w * }. Design PDA tht ccepts the lnguge {ww R w * }.
24 Exmples of PDAs Design PDA whose lnguge is { m n 0 m < n}. Design PDA whose lnguge is { m n 0 n < m}.
25 Exmples of PDAs Design PDA whose lnguge is { m n c n d m m 0, n 0}. Design PDA whose lnguge is { m m c n d n m 0, n 0}. Design PDA whose lnguge is { m n c p d q m + n = p + q}. Design PDA whose lnguge is { m n c k m = n or m = k}.
26 Pushdown Automt PDAs 27
27 Pushdown Automton -- PDA Input String Stck Sttes 28
28 Initil Stck Symol Stck Stck stck hed $ z top ottom specil symol Appers t time 0 29
29 Input symol The Sttes Pop symol Push symol 30
30 , c q 1 q 2 input stck e h $ top Replce c e h $ 31
31 , c q 1 q 2 input stck c top Push e h $ e h $ 32
32 , q 1 q 2 input stck top Pop e h $ e h $ 33
33 , q 1 q 2 input stck e h $ top No Chnge e h $ 34
34 Empty Stck input, $ q 1 q 2 stck $ top Pop empty The utomton HALTS No possile trnsition fter q 2 35
35 A Possile Trnsition input, $ q 1 q 2 stck $ top Pop 36
36 Non-Determinism PDAs re non-deterministic Allowed non-deterministic trnsitions, c q 2 q 1, c q 1 q 2, c q 3 -trnsition 37
37 Exmple PDA PDA M, L(M) = { n n : n 0},,,,, $ $ q 0 q 1 q 2 q 3 38
38 Bsic Ide: L(M) = { n n : n 0} 1. Push the s on the stck 2. Mtch the s on input with s on stck,, 3. Mtch found,,, $ $ q 0 q 1 q 2 q 3 39
39 Execution Exmple: Input Time 0 $ Stck current stte,,,,, $ $ q q 1 q 2 q
40 Input Time 1 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 41
41 Input Time 2 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 42
42 Input Time 3 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 43
43 Input Time 4 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 44
44 Input Time 5 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 45
45 Input Time 6 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 46
46 Input Time 7 $ Stck,,,,, $ $ q 0 q 1 q 2 q 3 47
47 Input Time 1 $ Stck,, ccept,,, $ $ q 0 q 1 q 2 q 3 48
48 A string is ccepted if there is computtion such tht: All the input is consumed AND The lst stte is n ccepting stte At the end of the computtion, we do not cre out the stck contents (the stck cn e empty t the lst stte) 49
49 The input string is ccepted y the PDA:,,,,, $ $ q 0 q 1 q 2 q 3 50
50 In generl, L = { n n : n 0} is the lnguge ccepted y the PDA:,,,,, $ $ q 0 q 1 q 2 q 3 51
51 Rejection Exmple: Input Time 0 $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 52
52 Rejection Exmple: Input Time 1 $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 53
53 Rejection Exmple: Input Time 2 $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 54
54 Rejection Exmple: Input Time 3 $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 55
55 Rejection Exmple: Input Time 4 $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 56
56 Rejection Exmple: Input Time 4 reject $ Stck current stte,, q 0,, $ $, q 1 q 2 q 3 57
57 The input string is rejected y the PDA:,,,,, $ $ q 0 q 1 q 2 q 3 58
58 A string is rejected if there is no computtion such tht: All the input is consumed AND The lst stte is n ccept stte At the end of the computtion, we do not cre out the stck contents 59
59 Another PDA exmple L (M) = {vv R : v {,}*} PDA M,,,, q 0,, $ $ q 1 q 2 60
60 Bsic Ide: L (M) = {vv R : v {,}*} 1. Push v on stck 2. Guess middle 3. Mtch v R on input with v on stck of input,, 4. Mtch,, found q 0,, $ $ q 1 q 2 61
61 Execution Exmple: Input Time 0 $,, Stck,, q 0,, $ $ q 1 q 2 62
62 Input Time 1 $,, Stck,, q 0,, $ $ q 1 q 2 63
63 Time 2 Input $,, Stck,, q 0,, $ $ q 1 q 2 64
64 Time 3 Input Guess the middle of string $,, Stck,, q 0,, $ $ q 1 q 2 65
65 Time 4 Input $,, Stck,, q 0,, $ $ q 1 q 2 66
66 Input Time 5 $,, Stck,, q 0,, $ $ q q
67 Input Time 6 $,, Stck,, Accept q 0, q 1, $ $ q 2 68
68 Rejection Exmple: Input Time 0 $,, Stck,, q 0,, $ $ q 1 q 2 69
69 Input Time 1 $,, Stck,, q 0,, $ $ q 1 q 2 70
70 Input Time 2 $,, Stck,, q 0,, $ $ q 1 q 2 71
71 Input Time 3 Guess the middle of string $,, Stck,, q 0,, $ $ q 1 q 2 72
72 Input Time 4 $,, Stck,, q 0,, $ $ q 1 q 2 73
73 Input Time 5 There is no possile trnsition. Input is not consumed $,, Stck,, q 0,, $ $ q q
74 Another computtion on sme string: Input Time 0 $,, Stck,, q 0,, $ $ q 1 q 2 75
75 Input Time 1 $,, Stck,, q 0,, $ $ q 1 q 2 76
76 Input Time 2 $,, Stck,, q 0,, $ $ q 1 q 2 77
77 Input Time 3 $,, Stck,, q 0,, $ $ q 1 q 2 78
78 Input Time 4 $,, Stck,, q 0,, $ $ q 1 q 2 79
79 Input Time 5 No finl stte is reched $,, Stck,, q 0,, $ $ q 1 q 2 80
80 There is no computtion tht ccepts string L(M),,,, q 0,, $ $ q 1 q 2 81
81 Another PDA exmple L (M) = {w {,}*: in every prefix v, n (v) n (v)} PDA M,, q 0 82
82 Execution Exmple: Time 0 Input,,, $ $ Stck q 0 83
83 Time 1 Input,,, $ $ Stck q 0 84
84 Time 2 Input,,, $ $ Stck q 0 85
85 Time 3 Input,,, $ $ Stck Accept q 0 86
86 Rejection Exmple: Time 0 Input,,, $ $ Stck q 0 87
87 Time 1 Input,,, $ $ Stck q 0 88
88 Time 2 Input,,, $ $ Stck q 0 89
89 Time 3 Input,,, $ Stck q 0 90
90 Time 4 Input,,, $ Stck q 0 Hlt nd Reject 91
91 Pushing Strings Input symol Pop symol Push string, w q 1 q 2 92
92 Exmple:, cdf q 1 q 2 input stck d top f h Push h e e $ $ c pushed string 93
93 Another PDA exmple L (M) = {w {,}*: n (w) = n (w)} PDA M, $ 0$, 0 00,1, $ 1$, 1 11, 0, $ $ q 1 q 2 94
94 Execution Exmple: Input Time 0, $ 0$, 0 00,1, $ 1$, 1 11, 0 $ Stck current stte, $ $ q 1 q 2 95
95 Input Time 1, $ 0$, $ 1$, 0 00, 1 11,1, 0 0 $ Stck, $ $ q 1 q 2 96
96 Input Time 3, $ 0$, 0 00,1, $ 1$, 1 11, 0 0 $ Stck, $ $ q 1 q 2 97
97 Input Time 4, $ 0$, 0 00,1, $ 1$, 1 11, 0 1 $ Stck, $ $ q 1 q 2 98
98 Input Time 5, $ 0$, 0 00,1, $ 1$, 1 11, $ Stck, $ $ q 1 q 2 99
99 Input Time 6, $ 0$, 0 00,1, $ 1$, 1 11, $ Stck, $ $ q 1 q 2 100
100 Input Time 7, $ 0$, 0 00,1, $ 1$, 1 11, 0 1 $ Stck, $ $ q 1 q 2 101
101 Input Time 8, $ 0$, 0 00,1, $ 1$, 1 11, 0 Accept $ Stck, $ $ q 1 q 2 102
102 Formlities for PDAs 103
103 , w q 1 q 2 Trnsition function: ( 2 q1,, ) {( q, w)} 104
104 , w q 2 q 1, w q 3 Trnsition function: ( 3 q1,, ) {( q2, w), ( q, w)} 105
105 Forml Definition Pushdown Automton (PDA) M ( Q, Σ, Γ, δ, q0, z, F) Finl Sttes sttes Input lphet Stck lphet Trnsition function Initil stte Stck strt symol 106
106 Instntneous Description ( q, u, s) Current stte Remining input Current stck contents 107
107 Exmple: Time 4: Instntneous Description ( q 1,, $) Input $,, Stck,,, $ $ q 0 q 1 q 2 q 3 108
One Minute To Learn Programming: Finite Automata
Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
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 informationA Visual and Interactive Input abb Automata. Theory Course with JFLAP 4.0
Strt Puse Step Noninverted Tree A Visul nd Interctive Input Automt String ccepted! 5 nodes generted. Theory Course with JFLAP 4.0 q0 even 's, even 's q2 even 's, odd 's q1 odd 's, even 's q3 odd 's, odd
More informationflex Regular Expressions and Lexical Scanning Regular Expressions and flex Examples on Alphabet A = {a,b} (Standard) Regular Expressions on Alphabet A
flex Regulr Expressions nd Lexicl Scnning Using flex to Build Scnner flex genertes lexicl scnners: progrms tht discover tokens. Tokens re the smllest meningful units of progrm (or other string). flex is
More informationFORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES
FORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES Introduction This compendium contins exercises out regulr lnguges for the course Forml Lnguges, Automt nd Theory of Computtion
More informationUnambiguous Recognizable Two-dimensional Languages
Unmbiguous Recognizble Two-dimensionl Lnguges Mrcell Anselmo, Dor Gimmrresi, Mri Mdoni, Antonio Restivo (Univ. of Slerno, Univ. Rom Tor Vergt, Univ. of Ctni, Univ. of Plermo) W2DL, My 26 REC fmily I REC
More informationRegular Languages and Finite Automata
N Lecture Notes on Regulr Lnguges nd Finite Automt for Prt IA of the Computer Science Tripos Mrcelo Fiore Cmbridge University Computer Lbortory First Edition 1998. Revised 1999, 2000, 2001, 2002, 2003,
More informationJava CUP. Java CUP Specifications. User Code Additions You may define Java code to be included within the generated parser:
Jv CUP Jv CUP is prser-genertion tool, similr to Ycc. CUP uilds Jv prser for LALR(1) grmmrs from production rules nd ssocited Jv code frgments. When prticulr production is recognized, its ssocited code
More informationGenerating In-Line Monitors For Rabin Automata
Generting In-Line Monitors For Rin Automt Hugues Chot, Rphel Khoury, nd Ndi Twi Lvl University, Deprtment of Computer Science nd Softwre Engineering, Pvillon Adrien-Pouliot, 1065, venue de l Medecine Queec
More informationBypassing Space Explosion in Regular Expression Matching for Network Intrusion Detection and Prevention Systems
Bypssing Spce Explosion in Regulr Expression Mtching for Network Intrusion Detection n Prevention Systems Jignesh Ptel, Alex Liu n Eric Torng Dept. of Computer Science n Engineering Michign Stte University
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
More informationSolution to Problem Set 1
CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let
More information! What can a computer do? ! What can a computer do with limited resources? ! Don't talk about specific machines or problems.
Introduction to Theoreticl CS ecture 18: Theory of Computtion Two fundmentl questions.! Wht cn computer do?! Wht cn computer do with limited resources? Generl pproch. Pentium IV running inux kernel.4.!
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationAutomata theory. An algorithmic approach. Lecture Notes. Javier Esparza
Automt theory An lgorithmic pproch 0 Lecture Notes Jvier Esprz My 3, 2016 2 3 Plese red this! Mny yers go I don t wnt to sy how mny, it s depressing I tught course on the utomt-theoretic pproch to model
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationFAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University
SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationPointed Regular Expressions
Pointed Regulr Expressions Andre Asperti 1, Cludio Scerdoti Coen 1, nd Enrico Tssi 2 1 Deprtment of Computer Science, University of Bologn sperti@cs.unio.it scerdot@cs.unio.it 2 INRIA-Micorsoft tssi@cs.unio.it
More information1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011 - Final Exam
1./1.1 Introduction to Computers nd Engineering Problem Solving Fll 211 - Finl Exm Nme: MIT Emil: TA: Section: You hve 3 hours to complete this exm. In ll questions, you should ssume tht ll necessry pckges
More informationModular Generic Verification of LTL Properties for Aspects
Modulr Generic Verifiction of LTL Properties for Aspects Mx Goldmn Shmuel Ktz Computer Science Deprtment Technion Isrel Institute of Technology {mgoldmn, ktz}@cs.technion.c.il ABSTRACT Aspects re seprte
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationSection 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationRegular Repair of Specifications
Regulr Repir of Specifictions Michel Benedikt Oxford University michel.enedikt@coml.ox.c.uk Griele Puppis Oxford University griele.puppis@coml.ox.c.uk Cristin Riveros Oxford University cristin.riveros@coml.ox.c.uk
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
More informationPushdown automata. Informatics 2A: Lecture 9. Alex Simpson. 3 October, 2014. School of Informatics University of Edinburgh als@inf.ed.ac.
Pushdown automata Informatics 2A: Lecture 9 Alex Simpson School of Informatics University of Edinburgh als@inf.ed.ac.uk 3 October, 2014 1 / 17 Recap of lecture 8 Context-free languages are defined by context-free
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationtrademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationVirtual Machine. Part II: Program Control. Building a Modern Computer From First Principles. www.nand2tetris.org
Virtul Mchine Prt II: Progrm Control Building Modern Computer From First Principles www.nnd2tetris.org Elements of Computing Systems, Nisn & Schocken, MIT Press, www.nnd2tetris.org, Chpter 8: Virtul Mchine,
More informationAutomated Grading of DFA Constructions
Automted Grding of DFA Constructions Rjeev Alur nd Loris D Antoni Sumit Gulwni Dileep Kini nd Mhesh Viswnthn Deprtment of Computer Science Microsoft Reserch Deprtment of Computer Science University of
More informationWhen Simulation Meets Antichains (on Checking Language Inclusion of NFAs)
When Simultion Meets Antichins (on Checking Lnguge Inclusion of NFAs) Prosh Aziz Abdull 1, Yu-Fng Chen 1, Lukáš Holík 2, Richrd Myr 3, nd Tomáš Vojnr 2 1 Uppsl University 2 Brno University of Technology
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationPushdown Automata. place the input head on the leftmost input symbol. while symbol read = b and pile contains discs advance head remove disc from pile
Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along
More informationSection 5.2, Commands for Configuring ISDN Protocols. Section 5.3, Configuring ISDN Signaling. Section 5.4, Configuring ISDN LAPD and Call Control
Chpter 5 Configurtion of ISDN Protocols This chpter provides instructions for configuring the ISDN protocols in the SP201 for signling conversion. Use the sections tht reflect the softwre you re configuring.
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
More informationStart Here. IMPORTANT: To ensure that the software is installed correctly, do not connect the USB cable until step 17. Remove tape and cardboard
Strt Here 1 IMPORTANT: To ensure tht the softwre is instlled correctly, do not connect the USB cle until step 17. Follow the steps in order. If you hve prolems during setup, see Trouleshooting in the lst
More informationSolutions for Selected Exercises from Introduction to Compiler Design
Solutions for Selected Exercises from Introduction to Compiler Design Torben Æ. Mogensen Lst updte: My 30, 2011 1 Introduction This document provides solutions for selected exercises from Introduction
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationAutomata and Computability. Solutions to Exercises
Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationAntiSpyware Enterprise Module 8.5
AntiSpywre Enterprise Module 8.5 Product Guide Aout the AntiSpywre Enterprise Module The McAfee AntiSpywre Enterprise Module 8.5 is n dd-on to the VirusScn Enterprise 8.5i product tht extends its ility
More informationOn Recognizable Timed Languages
On Recognizle Timed Lnguges Oded Mler 1 nd Amir Pnueli 2,3 1 CNRS-VERIMAG, 2 Av. de Vignte, 38610 Gières, Frnce Oded.Mler@img.fr 2 Weizmnn Institute of Science, Rehovot 76100, Isrel 3 New York University,
More informationOPTIMA QUADRANT / OFFSET QUADRANT
OPTIMA QUADRANT / OFFSET QUADRANT 71799 00 / Issue 1 / 15 Y Z DIMENSIONS Check the enclosure size in the tle elow mtches the showertry instlltion. = Widths: 800 Door = 780-805mm 900 Door = 880-905mm Y
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
More informationProtocol Analysis. 17-654/17-764 Analysis of Software Artifacts Kevin Bierhoff
Protocol Anlysis 17-654/17-764 Anlysis of Softwre Artifcts Kevin Bierhoff Tke-Awys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention
More informationQuick Reference Guide: One-time Account Update
Quick Reference Guide: One-time Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationOn the expressive power of temporal logic
On the expressive power of temporl logic Joëlle Cohen, Dominique Perrin nd Jen-Eric Pin LITP, Pris, FRANCE Astrct We study the expressive power of liner propositionl temporl logic interpreted on finite
More informationOn decidability of LTL model checking for process rewrite systems
Act Informtic (2009) 46:1 28 DOI 10.1007/s00236-008-0082-3 ORIGINAL ARTICLE On decidbility of LTL model checking for process rewrite systems Lur Bozzelli Mojmír Křetínský Vojtěch Řehák Jn Strejček Received:
More informationGeometry 7-1 Geometric Mean and the Pythagorean Theorem
Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationConcept Formation Using Graph Grammars
Concept Formtion Using Grph Grmmrs Istvn Jonyer, Lwrence B. Holder nd Dine J. Cook Deprtment of Computer Science nd Engineering University of Texs t Arlington Box 19015 (416 Ytes St.), Arlington, TX 76019-0015
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationPDF hosted at the Radboud Repository of the Radboud University Nijmegen
PDF hosted t the Rdboud Repository of the Rdboud University Nijmegen The following full text is publisher's version. For dditionl informtion bout this publiction click this link. http://hdl.hndle.net/2066/111343
More informationChapter. Contents: A Constructing decimal numbers
Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationv T R x m Version PREVIEW Practice 7 carroll (11108) 1
Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationDATABASDESIGN FÖR INGENJÖRER - 1056F
DATABASDESIGN FÖR INGENJÖRER - 06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/t-sommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse
More informationOutline of the Lecture. Software Testing. Unit & Integration Testing. Components. Lecture Notes 3 (of 4)
Outline of the Lecture Softwre Testing Lecture Notes 3 (of 4) Integrtion Testing Top-down ottom-up ig-ng Sndwich System Testing cceptnce Testing istriution of ults in lrge Industril Softwre System (ISST
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationAdvanced Baseline and Release Management. Ed Taekema
Advnced Bseline nd Relese Mngement Ed Tekem Introduction to Bselines Telelogic Synergy uses bselines to perform number of criticl configurtion mngement tsks. They record the stte of the evolving softwre
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationthe machine and check the components
Quick Setup Guide Strt Here HL-2270DW Before using this mchine for the first time, red this Quick Setup Guide to setup nd instll your mchine. To view the Quick Setup Guide in other lnguges, plese visit
More informationAPPLICATION NOTE Revision 3.0 MTD/PS-0534 August 13, 2008 KODAK IMAGE SENDORS COLOR CORRECTION FOR IMAGE SENSORS
APPLICATION NOTE Revision 3.0 MTD/PS-0534 August 13, 2008 KODAK IMAGE SENDORS COLOR CORRECTION FOR IMAGE SENSORS TABLE OF FIGURES Figure 1: Spectrl Response of CMOS Imge Sensor...3 Figure 2: Byer CFA Ptterns...4
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationUNLOCKING TECHNOLOGY IVECO
UNLOCKING TECHNOLOGY IVECO IVECO - CONTENTS PPLICTIONS PGE DS136 IVECO 3 DS177 IVECO CN 3 DIGNOSTIC SOCKETS LOCTIONS IVECO 4 GENERL OPERTION 5 6 TIPS & HINTS 15 2 Version: 2.3 July 2011 Copyright 2009
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More information1. Introduction. 1.1. Texts and their processing
Chpter 1 3 21/7/97 1. Introduction 1.1. Texts nd their processing One of the simplest nd nturl types of informtion representtion is y mens of written texts. Dt to e processed often does not decompose into
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
More informationVendor Rating for Service Desk Selection
Vendor Presented By DATE Using the scores of 0, 1, 2, or 3, plese rte the vendor's presenttion on how well they demonstrted the functionl requirements in the res below. Also consider how efficient nd functionl
More informationMultiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationEngineer-to-Engineer Note
Engineer-to-Engineer Note EE-265 Technicl notes on using Anlog Devices DSPs, processors nd development tools Contct our technicl support t dsp.support@nlog.com nd t dsptools.support@nlog.com Or visit our
More informationLec 2: Gates and Logic
Lec 2: Gtes nd Logic Kvit Bl CS 34, Fll 28 Computer Science Cornell University Announcements Clss newsgroup creted Posted on we-pge Use it for prtner finding First ssignment is to find prtners Due this
More informationSolenoid Operated Proportional Directional Control Valve (with Pressure Compensation, Multiple Valve Series)
Solenoid Operted Proportionl Directionl Control Vlve (with Pressure Compenstion, Multiple Vlve Series) Hydrulic circuit (Exmple) v Fetures hese stcking type control vlves show pressure compensted type
More informationLearning Workflow Petri Nets
Lerning Workflow Petri Nets Jvier Esprz, Mrtin Leucker, nd Mximilin Schlund Technische Universität München, Boltzmnnstr. 3, 85748 Grching, Germny {esprz,leucker,schlund}@in.tum.de Abstrct. Workflow mining
More informationHow To Network A Smll Business
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationHigh-Performance Hardware Monitors to Protect Network Processors from Data Plane Attacks
High-Perormnce Hrdwre Monitors to Protect Network Processors rom Dt Plne Attcks Hrikrishnn Chndrikkutty, Deepk Unnikrishnn, Russell Tessier nd Tilmn Wol Deprtment o Electricl nd Computer Engineering University
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More information5 a LAN 6 a gateway 7 a modem
STARTER With the help of this digrm, try to descrie the function of these components of typicl network system: 1 file server 2 ridge 3 router 4 ckone 5 LAN 6 gtewy 7 modem Another Novell LAN Router Internet
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationIaaS Configuration for Virtual Platforms
IS Configurtion for Virtul Pltforms vcloud Automtion Center 6.0 This document supports the version of ech product listed nd supports ll susequent versions until the document is replced y new edition. To
More informationEasyMP Network Projection Operation Guide
EsyMP Network Projection Opertion Guide Contents 2 About EsyMP Network Projection Functions of EsyMP Network Projection... 5 Vrious Screen Trnsfer Functions... 5 Instlling the Softwre... 6 Softwre Requirements...6
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
More information