expression simply by forming an OR of the ANDs of all input variables for which the output is


 Penelope Norman
 1 years ago
 Views:
Transcription
1 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 is true (Q = 1). However, for n ritrry truth tle such procedure could produce very lengthy nd cumersome expression which might e needlessly inecient to implement with gtes. There re severl methods for simpliction of Boolen logic expressions. The process is usully clled \logic minimiztion", nd the gol is to form result which is ecient. Two methods we will discuss re lgeric minimiztion nd Krnugh mps. For very complicted prolems the former method cn e done using specil softwre nlysis progrms. Krnugh mps re lso limited to prolems with up to 4 inry inputs. Let's strt with simple exmple. The tle elow gives n ritrry truth tle involving 2 logic inputs. Tle 1: Exmple of simple ritrry truth tle. A B Q There re two overll sttegies: 1. Write down n expression directly from the truth tle. Use Boolen lger, if desired, to simplify. 2. Use Krnugh mpping (\Kmp"). This is only pplicle if there re 4 inputs. In our exmple ove, we cnusetwo dierent wys of writin down result directly from the truth tle. We cn write down ll TRUE terms nd OR the result. This gives Q = A B + AB + AB While correct, without further simpliction this expression would involve 3 2input AND gtes, 2 inverters, nd 1 3input OR gte. Alterntively, one cn write down n expression for ll of the FALE sttes of the truth tle. This is simpler in this cse: Q = A B! Q = A B = A + B where the lst step results from Eqn. 3. Presumly, the two expressions cn e found to e equivlent with some lger. Certinly, the 2nd is simpler, nd involves only n inverter nd one 2input OR gte. 8
2 Finlly, one cn try Kmp solution. The rst step is to write out the truth tle in the form elow, with the input sttes the hedings of rows nd columns of tle, nd the corresponding outputs within, s shown elow. Tle 2: Kmp of truth tle. AnB The steps/rules re s follows: 1. Form the 2dimensionl tle s ove. Comine 2 inputs in \gry code"wy {see 2nd exmple elow. 2. Form groups of 1's nd circle them; the groups re rectngulr nd must hve sides of length 2 n 2 m, where n nd m re integers 0; 1; 2;:::. 3. The groups cn overlp. 4. Write down n expression of the inputs for ech group. 5. OR together these expressions. Tht's it. 6. Groups cn wrp cross tle edges. 7. As efore, one cn lterntively form groups of 0's to give solution for Q. 8. The igger the groups one cn form, the etter (simpler) the result. 9. There re usully mny lterntive solutions, ll equivlent, some etter thn others depending upon wht one is trying to optimize. AnB 0 1 Here is one wy of doing it: The two groupswehve drwn re A nd B. o the solution (s efore) is: Kmp Exmple 2 Q = A + B Let's use this to determine which 3it numers re prime. (This is homework prolem.) We ssume tht 0; 1; 2re not prime. We will let our input numer hve digits Here is the truth tle: Here is the corresponding Kmp nd solution. Note tht where two inputsrecomined in row or column tht their progression follows gry code, tht is only one it chnges t time. The solution shown ove is: Q = = 0 ( ) 9
3 Tle 3: 3digit prime nder. Deciml Q Tle 4: Kmp of truth tle. 2 n
4 2.4.2 Kmp Exmple 3: Full Adder In this exmple we will outline how to uild digitl full dder. It is clled \full" ecuse it will include \crryin" it nd \crryout" it. The crry its will llow succession of 1it full dders to e used to dd inry numers of ritrry length. (A hlf dder includes only one crry it.) i i i i i Figure 7: Block schemtic of full dder. (We nme our dder the \ chip"). The scheme for the full dder is outlined in Fig. 7. Imgine tht we re dding two nit inry numers. Let the inputs i nd i e the ith its of the two numers. The crry in it i represents ny crry from the sum of the neighoring less signicnt its t position i, 1. Tht is, i =1if i,1 = i,1 = 1, nd is 0 otherwise. The sum i t position i is therefore the sum of i, i,nd i. (Note tht this is n rithmetic sum, not Boolen OR.) A crry for this sum sets the crry out it, i =1,which then cn e pplied to the sum of the i + 1 its. The truth tle is given elow. i i i i i With i =0,we see tht the output sum i is just given y the XOR opertion, i i. And with i = 1, then i = i i.perhps the simplest wy to express this reltionship is the following: i = i ( i i ) To determine reltively simple expression for i,we will use Kmp: i n i i
5 This yields i = i i + i i + i i = i i + i ( i + i ) which in hrdwre would e 2 2input OR gtes nd 2 2input AND gtes. As stted ove, the crry its llow our dder to e expnded to dd ny numer of its. As n exmple, 4it dder circuit is depicted in Fig. 8. The sum cn e 5 its, where the MB is formed y the nl crry out. (ometimes this is referred to s n \overow" it.) Figure 8: Expnsion of 1it full dder to mke 4it dder Mking Multiplier from n Adder In clss we will discuss how to use our full dder (the \ chip") to mke multiplier. 2.5 Multiplexing Amultiplexer (MUX) is device which selects one of mny inputs to single output. The selection is done y using n input ddress. Hence, MUX cn tke mny dt its nd put them, one t time, on single output dt line in prticulr sequence. This is n exmple of trnsforming prllel dt to seril dt. A demultiplexer (DEMUX) performs the inverse opertion, tking one input nd sending it to one of mny possile outputs. Agin the output line is selected using n ddress. A MUXDEMUX pir cn e used to convert dt to seril form for trnsmission, thus reducing the numer of required trnsmission lines. The ddress its re shred y the MUX nd DEMUX t ech end. If n dt its re to e trnsmitted, then fter multiplexing, the numer of seprte lines required is log 2 n + 1, compred to n without the conversion to seril. Hence for lrge n the sving cn e sustntil. In L 2, you will uild such system. Multiplexers consist of two functionlly seprte components, decoder nd some switches or gtes. The decoder interprets the input ddress to select single dt it. We use the exmple of 4it MUX in the following section to illustrte how this works A 4it MUX Design We wish to design 4it multiplexer. The lock digrm is given in Fig. 9. There re 4 input dt its D 0 {D 3, 2 input ddress its A 0 nd A 1, one seril output dt it Q, nd 12
6 n (optionl) enle it E which is used for expnsion (discussed lter). First we will design the decoder. E MUX D3 D2 D1 D 0 GATE /WITCHE C 3 C C C Q A 1 A 0 DECODER Figure 9: Block digrm of 4it MUX. We need m ddress its to specify 2 m dt its. o in our exmple, we hve 2 ddress its. The truth tle for our decoder is strightforwrd: A 1 A 0 C 0 C 1 C 2 C The implementtion of the truth tle with stndrd gtes is lso strightforwrd, s given in Fig. 10. C 3 C C C A 1 A 0 Figure 10: Decoder for the 4it MUX. For the \gtes/switches" prt of the MUX, the design depends upon whether the input dt lines crry digitl or nlog signls. We will discuss the nlog possiility lter. The digitl cse is the usul nd simplest cse. Here, the dt routing cn e ccomplished 13
7 simply y forming 2input ANDs of the decoder outputs with the corresponding dt input, nd then forming n OR of these terms. Explicitly, Q = C 0 D 0 + C 1 D 1 + C 2 D 2 + C 3 D 3 Finlly, if n ENABLE line E is included, it is simply ANDed with the righthnd side of this expression. This cn e used to switch the entire MUX IC o/on, nd is useful for expnsion to more its. s we shll see. 14
Combinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So.
5/9/ Comintionl Logic ENGG05 st Semester, 0 Dr. Hyden So Representtions of Logic Functions Recll tht ny complex logic function cn e expressed in wys: Truth Tle, Boolen Expression, Schemtics Only Truth
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for contextfree grmmrs
More informationEECS 141 Due 04/19/02, 5pm, in 558 Cory
UIVERSITY OF CALIFORIA College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Lst modified on April 8, 2002 y Tufn Krlr (tufn@eecs.erkeley.edu) Jn M. Rey, Andrei Vldemirescu Homework
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 informationFast Boolean Algebra
Fst Boolen Alger ELEC 267 notes with the overurden removed A fst wy to lern enough to get the prel done honorly Printed; 3//5 Slide Modified; Jnury 3, 25 John Knight Digitl Circuits p. Fst Boolen Alger
More informationBoolean algebra.
http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing
More informationOverview of Today s Lecture:
CPS 4 Computer Orgniztion nd Progrmming Lecture : Boolen Alger & gtes. Roert Wgner CPS4 BA. RW Fll 2 Overview of Tody s Lecture: Truth tles, Boolen functions, Gtes nd Circuits Krnugh mps for simplifying
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationLecture 3. Introduction digital logic. Notes. Notes. Notes. Representations. February Bern University of Applied Sciences.
Lecture 3 Ferury 6 ern University of pplied ciences ev. f57fc 3. We hve seen tht circuit cn hve multiple (n) inputs, e.g.,, C, We hve lso seen tht circuit cn hve multiple (m) outputs, e.g. X, Y,, ; or
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
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 informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationNote 12. Introduction to Digital Control Systems
Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd . Introduction A digitl control system is one in which the
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 informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationCM10196 Topic 4: Functions and Relations
CM096 Topic 4: Functions nd Reltions Guy McCusker W. Functions nd reltions Perhps the most widely used notion in ll of mthemtics is tht of function. Informlly, function is n opertion which tkes n input
More informationDigital Control of Electric Drives
igitl Control o Electric rives Logic Circuits  Comintionl Boolen Alger, escription Form Czech Technicl University in Prgue Fculty o Electricl Engineering Ver.. J. Zdenek Logic Comintionl Circuit Logic
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationLet's start with an example:
Finite Automt Let's strt with n exmple: Here you see leled circles tht re sttes, nd leled rrows tht re trnsitions. One of the sttes is mrked "strt". One of the sttes hs doule circle; this is terminl stte
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationLecture 6. Notes. Notes. Notes. Representations Z A B and A B R. BTE Electronics Fundamentals August Bern University of Applied Sciences
Lecture 6 epresenttions epresenttions TE52  Electronics Fundmentls ugust 24 ern University of pplied ciences ev. c2d5c88 6. Integers () signndmgnitude representtion The set of integers contins the Nturl
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE2700: Digital Logic Design Fall Notes  Unit 1
INTRODUTION TO LOGI IRUITS Notes  Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl systems. It dels with the cse where vriles ssume only one o two vlues: TRUE (usully represented
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More 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 informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationpadic Egyptian Fractions
padic Egyptin Frctions Tony Mrtino My 7, 20 Theorem 9 negtiveorder Theorem 11 clss2 Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin
More informationIntroduction to Electrical & Electronic Engineering ENGG1203
Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 278 Dr. Hden KwokH So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll
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 informationFault Modeling. EE5375 ADD II Prof. MacDonald
Fult Modeling EE5375 ADD II Prof. McDonld Stuck At Fult Models l Modeling of physicl defects (fults) simplify to logicl fult l stuck high or low represents mny physicl defects esy to simulte technology
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE2700: Digital Logic Design Fall Notes  Unit 1
ELETRIL ND OMPUTER ENGINEERING DEPRTMENT, OKLND UNIVERSIT EE700: Digitl Logic Design ll 017 INTRODUTION TO LOGI IRUITS Notes  Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl systems.
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE378: Computer Hardware Design Winter Notes  Unit 1
ELETRIL ND OMPUTER ENGINEERING DEPRTMENT, OKLND UNIVERSIT EE78: omputer Hrdwre Design Winter 016 INTRODUTION TO LOGI IRUITS Notes  Unit 1 OOLEN LGER This is the oundtion or designing nd nlyzing digitl
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
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 informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationControl with binary code. William Sandqvist
Control with binry code Dec Bin He Oct 218 10 11011010 2 DA 16 332 8 E 1.1c Deciml to Binäry binry weights: 1024 512 256 128 64 32 16 8 4 2 1 71 10? 2 E 1.1c Deciml to Binäry binry weights: 1024 512 256
More informationLecture 2 : Propositions DRAFT
CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte
More information1 2 : 4 5. Why Digital Systems? Lesson 1: Introduction to Digital Logic Design. Numbering systems. Sample Problems 1 5 min. Lesson 1b: Logic Gates
Leon : Introduction to Digitl Logic Deign Computer ided Digitl Deign EE 39 meet Chvn Fll 29 Why Digitl Sytem? ccurte depending on numer of digit ued CD Muic i digitl Vinyl Record were nlog DVD Video nd
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 20181017 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More informationSpecial Numbers, Factors and Multiples
Specil s, nd Student Book  Series H + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book  Series H Contents Topics Topic  Odd, even, prime nd composite numers Topic  Divisiility tests
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 informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationEE273 Lecture 15 Asynchronous Design November 16, Today s Assignment
EE273 Lecture 15 Asynchronous Design Novemer 16, 199 Willim J. Dlly Computer Systems Lortory Stnford University illd@csl.stnford.edu 1 Tody s Assignment Term Project see project updte hndout on we checkpoint
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
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 bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationElements of Computing Systems, Nisan & Schocken, MIT Press. Boolean Logic
Elements of Computing Systems, Nisn & Schocken, MIT Press www.idc.c.il/tecs Usge nd Copyright Notice: Boolen Logic Copyright 2005 Nom Nisn nd Shimon Schocken This presenttion contins lecture mterils tht
More informationCS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p
More information6.004 Computation Structures Spring 2009
MIT OpenCourseWre http://ocw.mit.edu 6.004 Computtion Structures Spring 009 For informtion out citing these mterils or our Terms of Use, visit: http://ocw.mit.edu/terms. Cost/Performnce Trdeoffs: cse study
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationECE223. R eouven Elbaz Office room: DC3576
ECE223 R eouven Elz reouven@uwterloo.c Office room: DC3576 Outline Decoders Decoders with Enle VHDL Exmple Multiplexers Multiplexers with Enle VHDL Exmple From Decoder to Multiplexer 3stte Gtes Multiplexers
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with:  two inry opertions, commonly denoted y + nd,  unry opertion, usully denoted y ˉ or ~ or,  two elements usully clled zero nd
More informationUnit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
More informationCS103 Handout 32 Fall 2016 November 11, 2016 Problem Set 7
CS103 Hndout 32 Fll 2016 Novemer 11, 2016 Prolem Set 7 Wht cn you do with regulr expressions? Wht re the limits of regulr lnguges? On this prolem set, you'll find out! As lwys, plese feel free to drop
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
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 informationCS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata
CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges ) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Selfgrdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
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 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 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 informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationSIMPLIFICATION OF BOOLEAN ALGEBRA. Presented By: Ms. Poonam Anand
SIMPLIFITION OF OOLEN LGER Presented y: Ms. Poonam nand SIMPLIFITION USING OOLEN LGER simplified oolean expression uses the fewest gates possible to implement a given expression. ()() SIMPLIFITION USING
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationFormal Languages and Automata
Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt ChunMing Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More informationCOMPUTER SCIENCE TRIPOS
CST.2011.2.1 COMPUTER SCIENCE TRIPOS Prt IA Tuesdy 7 June 2011 1.30 to 4.30 COMPUTER SCIENCE Pper 2 Answer one question from ech of Sections A, B nd C, nd two questions from Section D. Submit the nswers
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 informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Selfgrdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationSurface maps into free groups
Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The
More informationRegular expressions, Finite Automata, transition graphs are all the same!!
CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104W11 1
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationChapter 1: Boolean Logic
Elements of Computing Systems, Nisn & Schocken, MIT Press, 2005 www.idc.c.il/tecs Chpter 1: Boolen Logic Usge nd Copyright Notice: Copyright 2005 Nom Nisn nd Shimon Schocken This presenttion contins lecture
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 informationLecture 08: Feb. 08, 2019
4CS46:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationChapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1
Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more
More 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 informationCS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University
CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted
More 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 information