Regular Sets and Expressions


 Barnard Sharp
 4 years ago
 Views:
Transcription
1 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 utomt re circuits!) Computer scientists dore them ecuse they dpt very nicely to lgorithm design, for exmple the lexicl nlysis portion of compiling nd trnsltion. Mthemticins re intrigued y them too due to the fct tht there re severl nifty mthemticl chrcteriztions of the sets they ccept. This is prtilly wht this section is out. We shll uild expressions from the symol, 1, +, nd & using the opertions of union, conctention, nd Kleene closure. Severl intuitive exmples of our nottion re: ) 01 mens zero followed y one (conctention) ) 0+1 mens either zero or one (union) c) 0 * mens ^ (Kleene closure) With prentheses we cn uild lrger expressions. And we cn ssocite menings with our expressions. Here's how: Expression Set Represented (0+1) * ll strings over {0,1}. 0 * 10 * 10 * strings contining exctly two ones. (0+1) * 11 strings which end with two ones. Tht is the intuitive pproch to these new expressions or formuls. Now for precise, forml view. Severl definitions should do the jo. Definition. 0, 1, ε, nd re regulr expressions. Definition. If α nd β re regulr expressions, then so re (αβ), (α + β), nd (α) *. OK, fine. Regulr expressions re strings put together with zeros, ones, epsilons, strs, plusses, nd mtched prentheses in certin wys. But why did we do it? And wht do they men? We shll nswer this with list of wht vrious generl regulr expressions represent. First, let us define wht some specific regulr expressions represent.
2 Regulr Sets 2 ) 0 represents the set {0} ) 1 represents the set {1} c) ε represents the set {ε} (the empty string) d) represents the empty set Now for some generl cses. If α nd β re regulr expressions representing the sets A nd B, then: ) (αβ) represents the set AB ) (α + β) represents the set A B c) (α) * represents the set A * The sets which cn e represented y regulr expressions re clled regulr sets. When writing down regulr expressions to represent regulr sets we shll often drop prentheses round conctentions. Some exmples re 11(0 + 1) * (the set of strings eginning with two ones), 0 * 1 * (ll strings which contin possily empty sequence of zeros followed y possily null string of ones), nd the exmples mentioned erlier. We lso should note tht {0,1} is not the only lphet for regulr sets. Any finite lphet my e used. From our precise definitions of the regulr expressions nd the sets they represent we cn derive the following nice chrcteriztion of the regulr sets. Then, very quickly we shll relte them to finite utomt. Theorem 1. The clss of regulr sets is the smllest clss contining the sets {0}, {1}, {ε}, nd which is closed under union, conctention, nd Kleene closure. See why the ove chrcteriztion theorem is true? And why we left out the proof? Anywy, tht is ll rther net ut, wht exctly does it hve to do with finite utomt? Theorem 2. Every regulr set cn e ccepted y finite utomton. Proof. The singleton sets {0}, {1}, {ε}, nd cn ll e ccepted y finite utomt. The fct tht the clss of sets ccepted y finite utomt is closed under union, conctention, nd Kleene closure completes the proof. Just from closure properties we know tht we cn uild finite utomt to ccept ll of the regulr sets. And this is indeed done using the constructions
3 Regulr Sets 3 from the theorems. For exmple, to uild mchine ccepting ( + ) *, we design: M which ccepts {}, M which ccepts {}, M + which ccepts {, } (from M nd M ), M * which ccepts *, nd so forth until the desired mchine hs een uilt. This is esily done utomticlly, nd is not too d fter the finl mchine is reduced. But it would e nice though to hve some lgorithm for converting regulr expressions directly to utomt. The following lgorithm for this will e presented in intuitive terms in lnguge reminiscent of lnguge prsing nd trnsltion. Initilly, we shll tke regulr expression nd rek it into suexpressions. For exmple, the regulr expression ( + ) * () * cn e roken into the three suexpressions: ( + ) *,, nd () *. (These cn e roken down lter on in the sme mnner if necessry.) Then we numer the symols in the expression so tht we cn distinguish etween them lter. Our three suexpressions now re: ( ) *, 3 2, nd ( 3 4 ) *. Symols which led n expression re importnt s re those which end the expression. We group these in sets nmed FIRST nd LAST. These sets for our suexpressions re: Expression FIRST LAST ( ) * 1, 1 2, ( 3 4 ) * 3 4 Note tht since the FIRST suexpression contined union there were two symols in its FIRST set. The FIRST set for the entire expression is: { 1, 3, 1 }. The reson tht 3 ws in this set is tht since the first suexpression ws strred, it could e skipped nd thus the first symol of the next suexpression could e the first symol for the entire expression. For similr resons, the LAST set for the whole expression is { 2, 4 }. Forml, precise rules do govern the construction of the FIRST nd LAST sets. We know tht FIRST() = {} nd tht we lwys uild FIRST nd LAST sets from the ottom up. Here re the remining rules for FIRST sets.
4 Regulr Sets 4 Definition. If α nd β re regulr expressions then: ) FIRST(α + β) = FIRST(α) FIRST(β) ) FIRST(α*) = FIRST(α) {ε} FIRST(α) if ε FIRST(α) c) FIRST(αβ) = FIRST(α) FIRST(β) otherwise Exmining these rules with cre revels tht the ove chrt ws not quite wht the rules cll for since empty strings were omitted. The correct, complete chrt is: Expression FIRST LAST ( ) * 1, 1, ε 2, 1, ε ( 3 4 ) * 3, ε 4, ε Rules for the LAST sets re much the sme in spirit nd their formultion will e left s n exercise. One more notion is needed, the set of symols which might follow ech symol in ny strings generted from the expression. We shll first provide n exmple nd explin in moment. Symol FOLLOW 2 1, 3, 2 1, 3, Now, how did we do this? It is lmost ovious if given little thought. The FOLLOW set for symol is ll of the symols which could come next. The lgorithm goes s follows. To find FOLLOW(), we keep reking the expression into suexpressions until the symol is in the LAST set of suexpression. Then FOLLOW() is the FIRST set of the next suexpression. Here is n exmple. Suppose tht we hve αβ s our expression nd know tht LAST(α). Then FOLLOW() = FIRST(β). In most cses, this is the wy it we compute FOLLOW sets.
5 Regulr Sets 5 But, there re three exceptions tht must e noted. 1) If n expression of the form γ* is in α then we must lso include the FIRST set of this strred suexpression γ. 2) If α is of the form β* then FOLLOW() lso contins α's FIRST set. 3) If the suexpression to the right of α hs n ε in its FIRST set, then we keep on to the right unioning FIRST sets until we no longer find n ε in one. Another exmple. Let's find the FOLLOW set for 1 in the regulr expression ( * ) * 2 * (3 + 3 ). First we rek it down into suexpressions until 1 is in LAST set. These re: ( * ) * 2 * ( ) Their FIRST nd LAST sets re: Expression FIRST LAST ( * 2 ) * 1, 1, ε 1, 1, 2, ε 2 * 2, ε 2, ε ( ) 3, 3 3, 3 Since 1 is in the LAST set of suexpression which is strred then we plce tht suexpression's FIRST set { 1, 1 } into FOLLOW( 1 ). Since * 2 cme fter 1 nd ws strred we must include 2 lso. We lso plce the FIRST set of the next suexpression ( * 2 ) in the FOLLOW set. Since tht set contined n ε, we must put the next FIRST set in lso. Thus in this exmple, ll of the FIRST sets re comined nd we hve: FOLLOW( 1 ) = { 1, 1, 2, 2, 3, 3 } Severl other FOLLOW sets re: FOLLOW( 1 ) = { 1, 1, 2, 3, 3 } FOLLOW( 2 ) = { 2, 3, 3 } After computing ll of these sets it is not hrd to set up finite utomton for ny regulr expression. Begin with stte nmed. Connect it to sttes
6 Regulr Sets 6 denoting the FIRST sets of the expression. (By sets we men: split the FIRST set into two prts, one for ech type of symol.) Our first exmple ( ) * 3 2 ( 3 4 ) * provides: 1,3 1 Next, connect the sttes just generted to sttes denoting the FOLLOW sets of ll their symols. Agin, we hve: 1, Continue on until everything is connected. Any edges missing t this point should e connected to rejecting stte nmed s r. The sttes contining symols in the expression's LAST set re the ccepting sttes. The complete construction for our exmple ( + ) * () * is: 1, s r 4,
7 Regulr Sets 7 This construction did indeed produce n equivlent finite utomton, nd in not too inefficient mnner. Though if we note tht 2 nd 4 re siclly the sme, nd tht 1 nd 2 re similr, we cn esily stremline the utomton to: 1,3 2,4 2 1 s r 3, Our construction method provides: s 0 1, for our finl exmple. There is very simple equivlent mchine. Try to find it! We now close this section with the equivlence theorem concerning finite utomt nd regulr sets. Hlf of it ws proven erlier in the section, ut the trnsltion of finite utomt into regulr expressions remins. This is not included for two resons. First, tht it is very tedious, nd secondly tht noody ever ctully does tht trnsltion for ny prcticl reson! (It is n interesting demonstrtion of correctness proof which involves severl levels of itertion nd should e looked up y the interested reder.) Theorem 3. The regulr sets re exctly those sets ccepted y finite utomt.
Homework 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 informationOne Minute To Learn Programming: Finite Automata
Gret Theoreticl Ides In Computer Science Steven Rudich CS 15251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge
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 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 informationSection 54 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 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 informationPentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simplelooking set of objects through which some powerful
Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this
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 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 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 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 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 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 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 welldefined
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 pointdirection nd twopoint
More informationUnambiguous Recognizable Twodimensional Languages
Unmbiguous Recognizble Twodimensionl 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 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 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 informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
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 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 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 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 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 informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 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 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 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 prsergenertion 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 information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
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 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 ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
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 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 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 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 information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
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 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 informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
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 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 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 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 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 relvlued
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationQuick Reference Guide: Onetime Account Update
Quick Reference Guide: Onetime 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 informationLearning Outcomes. Computer Systems  Architecture Lecture 4  Boolean Logic. What is Logic? Boolean Logic 10/28/2010
/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems  Architecture Lecture 4  Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationProtocol Analysis. 17654/17764 Analysis of Software Artifacts Kevin Bierhoff
Protocol Anlysis 17654/17764 Anlysis of Softwre Artifcts Kevin Bierhoff TkeAwys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention
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 informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
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 informationGenerating InLine Monitors For Rabin Automata
Generting InLine Monitors For Rin Automt Hugues Chot, Rphel Khoury, nd Ndi Twi Lvl University, Deprtment of Computer Science nd Softwre Engineering, Pvillon AdrienPouliot, 1065, venue de l Medecine Queec
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 informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
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 informationDATABASDESIGN FÖR INGENJÖRER  1056F
DATABASDESIGN FÖR INGENJÖRER  06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/tsommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
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 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 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 INRIAMicorsoft tssi@cs.unio.it
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 6483.
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationYour duty, however, does not require disclosure of matter:
Your Duty of Disclosure Before you enter into contrct of generl insurnce with n insurer, you hve duty, under the Insurnce Contrcts Act 1984 (Cth), to disclose to the insurer every mtter tht you know, or
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 informationOn the expressive power of temporal logic
On the expressive power of temporl logic Joëlle Cohen, Dominique Perrin nd JenEric Pin LITP, Pris, FRANCE Astrct We study the expressive power of liner propositionl temporl logic interpreted on finite
More informationObject Semantics. 6.170 Lecture 2
Object Semntics 6.170 Lecture 2 The objectives of this lecture re to: to help you become fmilir with the bsic runtime mechnism common to ll objectoriented lnguges (but with prticulr focus on Jv): vribles,
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 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 business. Introducing technology
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 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 informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
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 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 business. Introducing technology
More informationSmall Businesses Decisions to Offer Health Insurance to Employees
Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employersponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
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 wepge Use it for prtner finding First ssignment is to find prtners Due this
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 informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationBrillouin Zones. Physics 3P41 Chris Wiebe
Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction
More information1 Fractions from an advanced point of view
1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning
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 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 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 760190015
More informationINTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović
THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn
More informationSecondDegree Equations as Object of Learning
Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct SecondDegree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,
More informationLinear Equations in Two Variables
Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then
More information