Course Overview. Shimon Schocken. Spring Course Overview 1 Elements of Computing Systems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Course Overview. Shimon Schocken. Spring Course Overview 1 Elements of Computing Systems"

Transcription

1 IDC Herzliy Shimon Schocken Course Overview Shimon Schocken Spring 2005 Course Overview 1 Elements of Computing Systems Course gols Course ojectives: Understnd how HW+SW systems re uilt, nd how they work Lern how to rek complex prolems into simpler ones Lern how lrge scle development projects re plnned nd executed Hve fun. Methodology: Build / experiment with trnsprent computer tht we cn fully understnd. Course Overview 2 Elements of Computing Systems

2 Course theme nd structure Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics (Astrction implementtion prdigm) Course Overview 3 Elements of Computing Systems Resources nd rules Book Lectures Exercises Tools Course site Exm Projects Individul work policy Course Overview 4 Elements of Computing Systems

3 Appliction level: Pong Bll strction Bt strction Course Overview 5 Elements of Computing Systems The ig picture Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics Course Overview 6 Elements of Computing Systems

4 High-level progrmming /** A Grphic Bt for Pong Gme */ clss Bt { field int x, y; // screen loction of the t's top-left corner field int width, height; // t's width & height // The clss constructor nd most of the clss methods re omitted } /** Drws (color=true) or erses (color=flse) the t */ method void drw(oolen color) { do Screen.setColor(color); do Screen.drwRectngle(x,y,x+width,y+height); return; } /** Moves the t one step (4 pixels) to the right. */ method void mover() { do drw(flse); // erse the t t the current loction let x = x + 4; // chnge the t's X-loction // ut don't go eyond the screen's right order if ((x + width) > 511) { let x = width; } do drw(true); // re-drw the t in the new loction return; } A typicl cll to n operting system method Bll strction Bt strction Course Overview 7 Elements of Computing Systems Operting system level /** An OS-level screen driver tht strcts the computer's physicl screen */ clss Screen { sttic oolen currentcolor; // the current color // The Screen clss is collection of methods, ech implementing one // strct screen-oriented opertion. Most of this code is omitted. } /** Drws rectngle in the current color. */ // the rectngle's top left corner is nchored t screen loction (x0,y0) // nd its width nd length re x1 nd y1, respectively. function void drwrectngle(int x0, int y0, int x1, int y1) { vr int x, y; let x = x0; while (x < x1) { let y = y0; while(y < y1) { do Screen.drwPixel(x,y); let y = y+1; } let x = x+1; } } Course Overview 8 Elements of Computing Systems

5 The ig picture Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics Course Overview 9 Elements of Computing Systems The complete compiltion model Some... lnguge Some Other lnguge... Jck lnguge Proj. 9: uilding n pp. Proj. 12: uilding the OS Some compiler Some Other compiler Jck compiler Projects VM lnguge VM implementtion over CISC pltforms VM imp. over RISC pltforms VM emultor VM imp. over the Hck pltform Projects 7-8 CISC mchine lnguge RISC mchine lnguge... written in high-level lnguge Hck mchine lnguge Projects 1-6 CISC mchine RISC mchine other digitl pltforms, ech equipped with its VM implementtion Any computer Hck computer Course Overview 10 Elements of Computing Systems

6 Compiltion > Intermedite code Source code (x+width)>511 prsing x width code genertion push x push width dd push 511 gt Astrction Syntx Anlysis Prse Tree Semntic Synthesis Implementtion Modulrity The implementtion is lso n strction. Course Overview 11 Elements of Computing Systems The Virtul Mchine (VM) if ((x+width)>511) { let x=511-width; } // VM implementtion push x // s1 push width // s2 dd // s3 memory (efore)... width x sp s s4 s5 s9 push 511 gt if-goto L1 goto L2 // s4 // s5 // s6 // s7 sp sp 1 sp L1: push 511 push width su pop x L2:... // s8 // s9 // s10 // s11 sp s10 61 memory (fter)... width x Course Overview 12 Elements of Computing Systems

7 The ig picture Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics Course Overview 13 Elements of Computing Systems Low-level progrmming Virtul mchine progrm... push x push width dd push 511 gt if-goto L1 goto L2 L1: push 511 push width su pop x L2: VM trnsltor Assemly progrm // push D=A // A=M M=D M=M+1 // SP++ Assemler Executle Course Overview 14 Elements of Computing Systems

8 The ig picture Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics Course Overview 15 Elements of Computing Systems Mchine lnguge semntics Code semntics, s interpreted y the Hck hrdwre pltform Instruction code (0= ddress inst.) Address Code syntx M=M Instruction code (1= compute inst.) ALU opertion code Destintion Code Jump Code (M-1) (M) (no jump) We need HW rchitecture tht will relize this semntics The HW pltform should e designed to: Prse instructions, nd Execute them Course Overview 16 Elements of Computing Systems

9 Computer rchitecture instruction Instruction Memory D A M ALU dt Dt Memory (M) Progrm Counter ddress of next instruction dt in RAM(A) A typicl Von Neumnn mchine Course Overview 17 Elements of Computing Systems The ig picture Humn Thought Astrct design Chpters 9, 12 H.L. Lnguge & Operting Sys. Compiler Chpters Virtul Mchine Softwre hierrchy VM Trnsltor Chpters 7-8 Assemly Lnguge Assemler Chpter 6 Mchine Lnguge Computer Architecture Chpters 4-5 Hrdwre hierrchy Hrdwre Pltform Gte Logic Chpters 1-3 Chips & Logic Gtes Electricl Engineering Physics Course Overview 18 Elements of Computing Systems

10 Gte logic HW pltform = inter-connected set of chips Chips re mde of simpler chips, ll the wy down to logic gtes Logic gte = HW element tht implements certin Boolen function Every chip nd gte hs n interfce, specifying WHAT it is doing, nd n implementtion, specifying HOW it is doing it. Interfce Implementtion Xor Not And Or Not And Course Overview 19 Elements of Computing Systems Hrdwre Description Lnguge (HDL) Not And Or Not And CHIP Xor { IN,; OUT ; PARTS: Not(in=,=Not); Not(in=,=Not); And(=,=Not,=w1); And(=Not,=,=w2); Or(=w1,=w2,=); } Course Overview 20 Elements of Computing Systems

11 The tour ends Interfce Implementtion (CMOS) Nnd Course Overview 21 Elements of Computing Systems On the power of strctions Astrction: n ttempt to cpture in thought the essence of something Cognitive Mth Sciences Computer science Top down / ottom up Astrction Implementtion ( ) the sic theme Built using / Reduced into: Astrction Course Overview 22 Elements of Computing Systems

12 Fmous strction Course Overview 23 Elements of Computing Systems Finl note We delierte not ends, ut mens. For doctor does not delierte whether he shll hel, nor n ortor whether he shll persude... They ssume the end nd consider how nd y wht mens it is ttined, nd if it seems esily nd est produced therey; And if it is chieved y some mens, they consider how it will e chieved, nd y wht mens this will e chieved, until they come to the first cuse. And wht is lst in the order of nlysis seems to e first in the order of ecoming. Course Overview 24 Elements of Computing Systems

13 IDC Herzliy Shimon Schocken Boolen Logic Shimon Schocken Course Overview 25 Elements of Computing Systems Boolen lger Some elementry Boolen opertors: Not(x) And(x,y) Or(x,y) Nnd(x,y) Boolen functions: x y z f ( x, y, z) = ( x + y) z x Not(x) Not(x) x y Or(x,y) Or(x,y) x y And(x,y) And(x,y) x y Nnd(x,x) Nnd(x,x) Functionl expression VS truth tle expression Importnt result: Every Boolen function cn e expressed using And, Or, Not Course Overview 26 Elements of Computing Systems

14 All Boolen functions of 2 vriles Course Overview 27 Elements of Computing Systems Boolen lger Given: Nnd(,), flse Not() = Nnd(,) true = Not(flse) And(,) =... George Boole, ( A Clculus of Logic ) Or(,) =... Xor(,) =... Etc. Course Overview 28 Elements of Computing Systems

15 Gte logic Gte logic gte rchitecture designed to implement oolen function Elementry gtes: Composite gtes: Interfce VS implementtion. Course Overview 29 Elements of Computing Systems Circuit implementtions AND gte OR gte power supply power supply c c AND (,,c) c AND AND Physicl reliztions of logic gtes re irrelevnt to computer science. Course Overview 30 Elements of Computing Systems

16 Gte Logic Interfce Xor Clude Shnnon, Implementtion Not And ( Symolic Anlysis of Rely nd Switching Circuits ) Or Not And Xor(,)=Or(,Not(),Not(),)) Course Overview 31 Elements of Computing Systems Project 1: elementry logic gtes Given: Nnd(,), flse Build: Nnd(,) Nnd(,) Not() =... true =... And(,) =... Or(,) =... Mux(s,,) =... Etc gtes ltogether. Course Overview 32 Elements of Computing Systems

17 Building n And gte And And.cmp Contrct: When running your.hdl on our.tst, your. should e the sme s our.cmp. And.hdl And.tst CHIP CHIP And And { IN IN,, ; ; OUT OUT ; ; // // implementtion missing missing } lod lod And.hdl, And.hdl, put-file And., And., compre-to And.cmp, And.cmp, put-list ; ; set set 0,set 0,set 0,evl,put; set set 0,set 0,set 1,evl,put; set set 1,set 1,set 0,evl,put; set set 1, 1, set set 1, 1, evl, evl, put; put; Course Overview 33 Elements of Computing Systems Building n And gte Implementtion: And(,) = Not(Nnd(,)) NAND x in NOT And.hdl CHIP CHIP And And { IN IN,, ; ; OUT OUT ; ; Nnd( Nnd( =,, =,, = x); x); Not(in Not(in = x, x, = ) ) } Course Overview 34 Elements of Computing Systems

18 Hrdwre simultor HDL progrm test script put file Course Overview 35 Elements of Computing Systems Multiplexor (n interesting chip) sel sel Mux sel 0 1 Implementtion: sed on Not, And, Or gtes. Course Overview 36 Elements of Computing Systems

19 Cnonicl representtion Suspect function (-l-leinitz): Ech suspect my or my not hve n lii (), motivtion to commit the crime (m), nd reltionship to the wepon found in the scene of the crime (w). The police decides to focus ttention only on suspects for whom the proposition Not() And (m Or w) is true. Truth tle of the "suspect" function s (, m, w) = ( m + w) Cnonicl form: s (, m, w) = m w + mw + mw Course Overview 37 Elements of Computing Systems Two possile implementtions s (, m, w) = ( m + w) m w nd or s s (, m, w) = m w + mw + mw m w nd nd or s nd Course Overview 38 Elements of Computing Systems

20 Progrmmle Logic Device for 3-wy functions c nd legend: ctive fuse lown fuse 8 nd terms connected to the sme 3 inputs. or f(,,c) nd single or term connected to the puts of 8 nd terms _ PLD implementtion of f(,,c)= c + c (the on/off sttes of the fuses determine which gtes prticipte in the computtion) Course Overview 39 Elements of Computing Systems Some oservtions Ech Boolen function hs cnonicl representtion The cnonicl representtion is expressed in terms of And, Not, Or And, Not, Or cn e expressed in terms of Nnd lone Ergo, every Boolen function cn e relized y strndrd PLD consisting of Nnd gtes only Mss production c nd Universl uilding locks, unique connectivity (neurons).. or f(,,c) nd Course Overview 40 Elements of Computing Systems

21 IDC Herzliy Shimon Schocken Boolen Arithmetic Shimon Schocken Course Overview 41 Elements of Computing Systems Counting systems quntity deciml inry 3-it register overflow overflow overflow Course Overview 42 Elements of Computing Systems

22 Rtionle (9038) ten = = 9038 (10011) two = = 19 ( x n x n 1... x0 ) n = x i= 0 i i Course Overview 43 Elements of Computing Systems Binry ddition Assuming 4-it system: no overflow overflow Algorithm: exctly the sme s in deciml ddition Overflow (MSB crry) hs to e delt with. Course Overview 44 Elements of Computing Systems

23 Representing negtive numers (4-it system) The codes of ll positive numers egin with 0 The codes of ll negtive numers egin with 1 To convert numer: leve ll triling 0 s nd first 1 intct, nd flip ll the remining its Exmple: 2-5 = 2 + (-5) = = -3 Course Overview 45 Elements of Computing Systems Building n Adder chip it dder 16 Adder: chip designed to dd two integers The construction hierrchy: Hlf dder: designed to dd 2 its Full dder: designed to dd 3 its Adder: designed to dd two n-it numers Course Overview 46 Elements of Computing Systems

24 Hlf dder (designed to dd 2 its) crry sum hlf dder sum crry Implementtion: sed on two gtes tht you ve seen efore. Course Overview 47 Elements of Computing Systems Full dder (designed to dd 3 its) c crry sum c full dder sum crry Implementtion: cn e sed on hlf-dder gtes. Course Overview 48 Elements of Computing Systems

25 n-it Adder it dder Implementtion: rry of full-dder gtes Course Overview 49 Elements of Computing Systems The ALU (of the Hck pltform) hlf dder sum crry c full dder sum crry x y it dder 16 zx nx zy ny f no (x, y, control its) = x+y, x-y, y x, x y 16 its 16 its ALU 16 its 0, 1, -1, x, y, -x, -y, x!, y!, x+1, y+1, x-1, y-1, zr ng x&y, x y Course Overview 50 Elements of Computing Systems

26 ALU logic Course Overview 51 Elements of Computing Systems The ALU in the CPU context c1,c2,,c6 D D A A M Mux A/M ALU RAM Course Overview 52 Elements of Computing Systems

27 End note: Leinitz Descried inry clculus nd 4-it dder in 1694 The inry system my e used in plce of the deciml system; express ll numers y unity nd y nothing Prctice: uilt one of the first mechnicl clcultors Theory: dremed univerl, forml, lnguge of thought -- the Chrcteristic Universlis The drem s end: Turing nd Goedl in 1930 s. Course Overview 53 Elements of Computing Systems

Elements of Computing Systems, Nisan & Schocken, MIT Press. Boolean Logic

Elements 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 information

Boolean Logic. Building a Modern Computer From First Principles.

Boolean Logic. Building a Modern Computer From First Principles. Boolen Logic Building Modern Computer From First Principles www.nnd2tetris.org Elements of Computing Systems, Nisn & Schocken, MIT Press, www.nnd2tetris.org, Chpter 1: Boolen Logic slide 1 Usge nd Copyright

More information

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

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: 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 information

Introduction to Electrical & Electronic Engineering ENGG1203

Introduction to Electrical & Electronic Engineering ENGG1203 Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll

More information

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

More information

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 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 information

Boolean Algebra. Boolean Algebra

Boolean 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 information

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1 CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

More information

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch. Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

More information

CS 330 Formal Methods and Models

CS 330 Formal Methods and Models CS 330 Forml Methods nd Models Dn Richrds, section 003, George Mson University, Fll 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 7 1. Prove (p q) (p q), () (5pts) using truth tles. p q

More information

CS375: Logic and Theory of Computing

CS375: Logic and Theory of Computing CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper 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

Chapter 2 Finite Automata

Chapter 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 information

Good Review book ( ) ( ) ( )

Good Review book ( ) ( ) ( ) 7/31/2011 34 Boolen (Switching) Algebr Review Good Review book BeBop to the Boolen Boogie: An Unconventionl Guide to Electronics, 2 nd ed. by Clive Mxwell Hightext Publictions Inc. from Amzon.com for pprox.

More information

How 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?

How 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 information

Homework 3 Solutions

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 information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing 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 information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1

Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1 Chpter 1, Prt 1 Regulr Lnguges CSC527, Chpter 1, Prt 1 c 2012 Mitsunori Ogihr 1 Finite Automt A finite utomton is system for processing ny finite sequence of symols, where the symols re chosen from finite

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Normal Forms for Context-free Grammars

Normal Forms for Context-free Grammars Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S

More information

CSCI FOUNDATIONS OF COMPUTER SCIENCE

CSCI FOUNDATIONS OF COMPUTER SCIENCE 1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not

More information

Addition and Subtraction

Addition and Subtraction ddition and Subtraction Philipp Koehn 9 February 2018 1 addition 1-it dder 2 Let s start simple: dding two 1-it numbers Truth table + 0 0 0 0 1 1 1 0 1 1 1 10 Really 2 Operations 3 Truth table for "position

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, 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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb. CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

Lexical Analysis Finite Automate

Lexical Analysis Finite Automate Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

More information

1 Nondeterministic Finite Automata

1 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 information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 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 information

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016 CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

More information

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap)

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap) . Boolen Alger Fchgeiet Rechnersysteme. Boolen Alger Veriiction Technology Content. Boolen lger sics (recp).2 Resoning out Boolen expressions . Boolen Alger 2 The prolem o logic veriiction: Show tht two

More information

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck. Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

Lecture 2 : Propositions DRAFT

Lecture 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 information

Recursively Enumerable and Recursive. Languages

Recursively Enumerable and Recursive. Languages Recursively Enumerble nd Recursive nguges 1 Recll Definition (clss 19.pdf) Definition 10.4, inz, 6 th, pge 279 et S be set of strings. An enumertion procedure for Turing Mchine tht genertes ll strings

More information

Lecture 9: LTL and Büchi Automata

Lecture 9: LTL and Büchi Automata Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

More information

3 Regular expressions

3 Regular expressions 3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 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 information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

Designing Information Devices and Systems I Spring 2018 Homework 8

Designing 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. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission

More information

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection,

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

input tape head moves current state

input tape head moves current state CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter

More information

Semantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4

Semantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4 Semnti nlysis SI 16 Priniples of Progrmming Lnguges Fulty of omputer Siene Dlhousie University Winter 2012 Reding: hpter 4 Motivtion Soure progrm (hrter strem) Snner (lexil nlysis) Front end Prse tree

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

Introduction to Computer Engineering EECS 203 dickrp/eecs203/ Brief course overview. What s your major?

Introduction to Computer Engineering EECS 203  dickrp/eecs203/ Brief course overview. What s your major? http://ziyng.eecs.northwestern.edu/ dickrp/eecs203/ Brief course overview Instructor: Roert Dick Office: L477 Tech Emil: dickrp@northwestern.edu Phone: 847 467 2298 TA: Emil: TT: Emil: Nel Oz neloz@u.northwestern.edu

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

MTH 505: Number Theory Spring 2017

MTH 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 information

Lecture 7 notes Nodal Analysis

Lecture 7 notes Nodal Analysis Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey 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 information

Exam 2 Solutions ECE 221 Electric Circuits

Exam 2 Solutions ECE 221 Electric Circuits Nme: PSU Student ID Numer: Exm 2 Solutions ECE 221 Electric Circuits Novemer 12, 2008 Dr. Jmes McNmes Keep your exm flt during the entire exm If you hve to leve the exm temporrily, close the exm nd leve

More information

State Minimization for DFAs

State Minimization for DFAs Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chapter Five - Eigenvalues, Eigenfunctions, and All That Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

Section 3.1: Exponent Properties

Section 3.1: Exponent Properties Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted

More information

Section: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.

Section: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language. Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 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 information

Where did dynamic programming come from?

Where did dynamic programming come from? Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/1526-5463-2002-50-01-0048.pdf

More information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 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 information

Software Engineering using Formal Methods

Software Engineering using Formal Methods Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-* Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion

More information

EE273 Lecture 15 Asynchronous Design November 16, Today s Assignment

EE273 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 information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

1 From NFA to regular expression

1 From NFA to regular expression Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 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 information

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.

More information

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C. Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

More information

UNIT 8A Computer Circuitry: Layers of Abstraction. Boolean Logic & Truth Tables

UNIT 8A Computer Circuitry: Layers of Abstraction. Boolean Logic & Truth Tables UNIT 8 Computer Circuitry: Layers of bstraction 1 oolean Logic & Truth Tables Computer circuitry works based on oolean logic: operations on true (1) and false (0) values. ( ND ) (Ruby: && ) 0 0 0 0 0 1

More information

Quadratic reciprocity

Quadratic reciprocity Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

Operations Algorithms on Quantum Computer

Operations Algorithms on Quantum Computer IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 85 Opertions Algorithms on Quntum Computer Moyd A. Fhdil, Ali Foud Al-Azwi, nd Smmer Sid Informtion Technology Fculty,

More information

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic

More information

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages 5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

More information

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and. Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

More information

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

More information

Refined interfaces for compositional verification

Refined interfaces for compositional verification Refined interfces for compositionl verifiction Frédéric Lng INRI Rhône-lpes http://www.inrilpes.fr/vsy Motivtion Enumertive verifiction of concurrent systems Prllel composition of synchronous processes

More information

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart. MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of

More information

On Determinisation of History-Deterministic Automata.

On Determinisation of History-Deterministic Automata. On Deterministion of History-Deterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YR-ICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil

More information

Preview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms

Preview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

Module 2. Basic Digital Building Blocks. Binary Arithmetic & Arithmetic Circuits Comparators, Decoders, Encoders, Multiplexors Flip-Flops

Module 2. Basic Digital Building Blocks. Binary Arithmetic & Arithmetic Circuits Comparators, Decoders, Encoders, Multiplexors Flip-Flops Module 2 asic Digital uilding locks Lecturer: Dr. Yongsheng Gao Office: Tech 3.25 Email: Web: Structure: Textbook: yongsheng.gao@griffith.edu.au maxwell.me.gu.edu.au 6 lecturers 1 tutorial 1 laboratory

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Plane Surveying Levelling

Plane Surveying Levelling Deprtment of Civil Engineering, UC Introduction: Levelling is mens y which surveyors cn determine the elevtion of points, using other known points s references. Levelling is perhps the most sic of surveying

More information

5.1 Estimating with Finite Sums Calculus

5.1 Estimating with Finite Sums Calculus 5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information