Size: px
Start display at page:

Download ""

Transcription

1 FloatingpointvericationinHOLLight:theexponentialfunction 1 Floatingpointverication theexponentialfunction inhollight: UniversityofCambridge JohnHarrison Introduction Floatingpointcorrectness Ourimplementationlanguage Thealgorithm OutlineoftheHOLproof Generalconclusions JohnHarrison UniversityofCambridge,December1997

2 FloatingpointvericationinHOLLight:theexponentialfunction 2 Introduction Floatingpointalgorithmsarefairlysmall,but Therehavebeenerrorsincommercial oftencomplicatedmathematically. Inthecaseoftranscendentalfunctionsit's systems,e.g.thepentiumfdivbugin1994. Vericationusingmodelcheckersisdicult diculteventosaywhatcorrectnessmeans. apparatus. becauseoftheneedformathematical Itcanevenbedicultusingtheoremprovers sincenotmanyofthemhavegoodtheoriesof realnumbersetc. JohnHarrison UniversityofCambridge,December1997

3 FloatingpointvericationinHOLLight:theexponentialfunction 3 Floatingpointcorrectness Wewanttospecifythecorrectnessaccordingto thefollowingdiagram: v(a) exp 6 -v(exp(a)) exp(v(a)) 6 v v a EXP -EXP(a) andexp(v(a))in`unitsinthelastplace'of Wemeasurethedierencebetweenv(EXP(a)) EXP(a). JohnHarrison UniversityofCambridge,December1997

4 FloatingpointvericationinHOLLight:theexponentialfunction 4 Ourimplementationlanguage Thisincludesthefollowingconstructs: command=variable:= command; expression if expressionthen command else command command while if expressionthen expressiondo do commandwhile expression command skip WedeneasimplerelationalsemanticsinHOL, fexpressiong correctnessrules.wethenprovetotalcorrectness andderiveweakestpreconditionsandtotal Theideaisthatthislanguagecanbeformally viavcgeneration. linkedtoc,verilog,handel,...

5 FloatingpointvericationinHOLLight:theexponentialfunction 5 Thealgorithm TanginACMTransactionsonMathematical Thealgorithmweverifyistakenfromapaperby Similartechniquesarewidelyusedforoating Software,1989. pointlibraries,and,probably,forhardware implementations. constants.tang'spapergivesactualvaluesashex Thealgorithmreliesonatableofprecomputed Thealgorithmworksinthreephases: representationsofieeenumbers. Performrangereduction Usepolynomialapproximation Reconstructanswerusingtables Thecorrectnessproofreectsthis.

6 FloatingpointvericationinHOLLight:theexponentialfunction 6 Codeforthealgorithm ifisnan(x)thene:=x elseifabs(x)>threshold_1then elseifx==plus_infinitythene:=plus_infinity elseifx==minus_infinitythene:=plus_zero elseifabs(x)<threshold_2thene:=plus_one+x elsee:=plus_zero ifx>plus_zerothene:=plus_infinity (N:=INTRND(X*Inv_L); N2:=N%Int_32; N1:=N-N2; else ifabs(n)>=int_2e9then R1:=X-Tofloat(N)*L1; R1:=(X-Tofloat(N1)*L1)-Tofloat(N2)*L1 M:=N1/Int_32; J:=N2; R2:=Tofloat(--N)*L2; R:=R1+R2; Q:=R*R*(A1+R*A2); E1:=S_Lead(J)+(S_Trail(J)+S*P); S:=S_Lead(J)+S_Trail(J); P:=R1+(R2+Q); )E:=Scalb(E1,M)

7 FloatingpointvericationinHOLLight:theexponentialfunction 7 StructureoftheHOLproof / Realnumbers / \ Programming / \ language / \ IEEEspec / Realanalysis \ / // Squarefreedecomp& // Sturm'stheorem Algorithm / /// \\\ /// \\\ FPlemmas /// \\\ /// Verification

8 FloatingpointvericationinHOLLight:theexponentialfunction 8 Floatingpointlemmas(1) roundingarealnumberxtoaoatingpoint Wedenetheerrorerror(x)resultingfrom value. Becauseoftheregularwayinwhichthe operationsaredened,alltheoperationsthen counterpartsaccordingtothesamepattern: relatetotheirabstractmathematical -Finite(a)^Finite(b)^ =)Finite(a+b)^ abs(val(a)+val(b))<threshold(float_format) error(val(a)+val(b))) (Val(a+b)=(Val(a)+Val(b))+ Thecomparisonsareevenmorestraightforward: -Finite(a)^Finite(b) =)(a<b=val(a)<val(b)) JohnHarrison UniversityofCambridge,December1997

9 FloatingpointvericationinHOLLight:theexponentialfunction 9 Floatingpointlemmas(2) Wehaveseverallemmasquantifyingtheerror,e.g. -abs(x)<threshold(float_format)^ abs(x)<(&2powj/&2pow125) Therearemanyimportantsituations,however, =)abs(error(x))<=&2powj/&2pow150 isexactlyrepresentable,e.g.subtractionof wheretheoperationsareexact,becausetheresult nearbyvalueswiththesamesign: -Finite(a)^Finite(b)^ =)Finite(a-b)^ &2*abs(Val(a)-Val(b))<=abs(Val(a)) Thisisaclassicresultinoatingpointerror (Val(a-b)=Val(a)-Val(b)) analysis. JohnHarrison UniversityofCambridge,December1997

10 FloatingpointvericationinHOLLight:theexponentialfunction 10 Informalerroranalysis HOL.Oneneedsto: Tang'serroranalysistranslatesquitedirectlyinto 1.Provethatcleverimplementationtricks ensurecertainremaindertermsarecalculated exactly.thisreliesoncancellation,andthe zeroes. factthatpre-storedconstantshavetrailing 2.Provethatthepolynomialapproximation 3.Provethattheroundingerrorswhen obeystheappropriateerrorbounds. large. reconstructingthenalanswerdonotgettoo afewlines,while3isgivenalonganddetailed InTang'spaper,1isquitebrief,2isdismissedin proof.

11 FloatingpointvericationinHOLLight:theexponentialfunction 11 HOLerroranalysis reversed! IntheHOLversion,thisorderofdicultyis 1.Therstpartisnotfundamentallydicult, butquitetrickybecauseitinvolvesalotof 2.Thesecondpartinvolvesnumerical specialcasesandlow-levelproofs. translateintoaformalproof(e.g.taylor approximation,whichneedsalotofworkto makesasmallmistakehere,thoughitdoesn't series,sturm'stheorem...).infacttang 3.Thelastpartisquiteroutine,andwecan aectthenalresult. automatically.actually,wederivebetter programholtocomposetheroundingerrors boundsthantangdoessinceweavoid thework. makingsimplifyingassumptionstocutdown

12 FloatingpointvericationinHOLLight:theexponentialfunction 12 Thenalresult conrmtang'sbottom-lineresult: Underthevarious`denitional'assumptions,we (X==Plus_infinity_ (Isnan(X)=)Isnan(E))^ Finite(X)^ (X==Minus_infinity=)E==Plus_zero)^ exp(valx)>=threshold(float_format) (Finite(X)^exp(ValX)<threshold(float_format) =)E==Plus_infinity)^ =)Isnormal(E)^ _(Isdenormal(E)_Iszero(E))^ abs(val(e)-exp(valx))<(&54/&100)*ulp(e) Infact,thisspecicationisabitmoreprecise abs(val(e)-exp(valx))<(&77/&100)*ulp(e)) overowthreshold. thantang's,e.g.weareexplicitaboutthe JohnHarrison UniversityofCambridge,December1997

13 FloatingpointvericationinHOLLight:theexponentialfunction 13 Conclusions Weconrm(andstrengthen)themainresults anduncoversubtleissues.thisclassofproofs ofthehandproof.butwedetectafewslips Theproofwasverylong(over3monthsof isagoodtargetforverication. work),butmostofthiswasdevotedto It'samistaketobelievethatonly`trivial' generalresultsthatcouldbere-used. mathematicsisneededforverication theoriesareessential. applications.hollight'smathematical Automationoflineararithmeticispractically indispensable.bettertoolsfornonlinear Theproofruntimesareverylongowingtothe reasoningareneeded. extensiveuseofarithmeticdonebyinference.

Chapter 8. Statement-Level Control Structures ISBN

Chapter 8. Statement-Level Control Structures ISBN Chapter 8 Statement-Level Control Structures ISBN 0-321-33025-0 Chapter 8 Topics Introduction Selection Statements Iterative Statements Unconditional Branching Guarded Commands Conclusions Copyright 2006

More information

Chapter 8. Statement-Level Control Structures

Chapter 8. Statement-Level Control Structures Chapter 8 Statement-Level Control Structures Chapter 8 Topics Introduction Selection Statements Iterative Statements Unconditional Branching Guarded Commands Conclusions 8-2 1 Levels of Control Flow Within

More information

Remote Monitor RC554M Website Registration and Setup Instructions.

Remote Monitor RC554M Website Registration and Setup Instructions. Remote Monitor RC554M Website Registration and Setup Instructions. -For Remote Monitor Installation Instructions See: http://www.rola-chem.com/manuals/new2016/554407-rc554m-remote-monitor-installation-manual.pdf

More information

Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions

Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a-27b. Section 8.3 on page 275 problems 1b, 8, 10a-10b, 14. Section 8.4 on page 279 problems

More information

PROPERTECHNIQUEOFSOFTWARE INSPECTIONUSING GUARDED COMMANDLANGUAGE

PROPERTECHNIQUEOFSOFTWARE INSPECTIONUSING GUARDED COMMANDLANGUAGE International Journal of Computer ScienceandCommunication Vol. 2, No. 1, January-June2011, pp. 153-157 PROPERTECHNIQUEOFSOFTWARE INSPECTIONUSING GUARDED COMMANDLANGUAGE Neeraj Kumar Singhania University,

More information

Coding and decoding with convolutional codes. The Viterbi Algor

Coding and decoding with convolutional codes. The Viterbi Algor Coding and decoding with convolutional codes. The Viterbi Algorithm. 8 Block codes: main ideas Principles st point of view: infinite length block code nd point of view: convolutions Some examples Repetition

More information

Non Depository/Mortgage Institutions Schedule D Loans Origination

Non Depository/Mortgage Institutions Schedule D Loans Origination 6/30/2015 Total Number Amount Discount Weighted (1) First mortgage FHA-15 yrs 24 $ 2,165 $0 $ 15 3.95 (2) First mortgage VA-15 yrs 12 2,428 15 13 3.61 (3) First mortgage conventional conforming 15 yrs

More information

June 2012 FORESTSAFE 4 ENTARIAN LIMITED. ForestSafe Service Configuration Adrian Owen and Jani Järvinen

June 2012 FORESTSAFE 4 ENTARIAN LIMITED. ForestSafe Service Configuration Adrian Owen and Jani Järvinen June 2012 ENTARIAN LIMITED FORESTSAFE 4 ForestSafe Adrian Owen and Jani Järvinen Table of Contents 1 INTRODUCTON...3 2 LOCAL ADMINISTRATOR GROUP...3 2.1 STEPS...3 2.2 VERIFICATION...3 3 WINDOWS DOMAIN

More information

Repetition and Loops. Additional Python constructs that allow us to effect the (1) order and (2) number of times that program statements are executed.

Repetition and Loops. Additional Python constructs that allow us to effect the (1) order and (2) number of times that program statements are executed. New Topic Repetition and Loops Additional Python constructs that allow us to effect the (1) order and (2) number of times that program statements are executed. These constructs are the 1. while loop and

More information

Algorithms are the threads that tie together most of the subfields of computer science.

Algorithms are the threads that tie together most of the subfields of computer science. Algorithms Algorithms 1 Algorithms are the threads that tie together most of the subfields of computer science. Something magically beautiful happens when a sequence of commands and decisions is able to

More information

Certificate Authorities and Public Keys. How they work and 10+ ways to hack them.

Certificate Authorities and Public Keys. How they work and 10+ ways to hack them. Certificate Authorities and Public Keys How they work and 10+ ways to hack them. -- FoxGuard Solutions Www.FoxGuardSolutions.com melkins@foxguardsolutions.com Version.05 9/2012 1 Certificate Use Overview

More information

SteganographyinaVideoConferencingSystem? AndreasWestfeld1andGrittaWolf2 2InstituteforOperatingSystems,DatabasesandComputerNetworks 1InstituteforTheoreticalComputerScience DresdenUniversityofTechnology

More information

Induction. Mathematical Induction. Induction. Induction. Induction. Induction. 29 Sept 2015

Induction. Mathematical Induction. Induction. Induction. Induction. Induction. 29 Sept 2015 Mathematical If we have a propositional function P(n), and we want to prove that P(n) is true for any natural number n, we do the following: Show that P(0) is true. (basis step) We could also start at

More information

An Example of a Numerical Calculation Class - Rational Numbers

An Example of a Numerical Calculation Class - Rational Numbers An Example of a Numerical Calculation Class - Rational Numbers Because a rational number is determined by the integer numerator and integer denominator, we defined our data structure for a rational number

More information

Section 4.2: The Division Algorithm and Greatest Common Divisors

Section 4.2: The Division Algorithm and Greatest Common Divisors Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948

More information

3. Add an Event: Alarm Alarm 0 a. Add an Action: Set Variable i. Applies to: Self ii. Variable: time_left iii. Value: +1 iv. Check the Relative box

3. Add an Event: Alarm Alarm 0 a. Add an Action: Set Variable i. Applies to: Self ii. Variable: time_left iii. Value: +1 iv. Check the Relative box Creating a Timer: You can have a timer that shows how long the player has been playing the game. 1. Create a new object and give it a name. This example is called object_timer. 2. Add an Event: Create

More information

MACM 101 Discrete Mathematics I

MACM 101 Discrete Mathematics I MACM 101 Discrete Mathematics I Exercises on Combinatorics, Probability, Languages and Integers. Due: Tuesday, November 2th (at the beginning of the class) Reminder: the work you submit must be your own.

More information

Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)

Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive

More information

SECURE EMAIL USER GUIDE OUTLOOK 2000

SECURE EMAIL USER GUIDE OUTLOOK 2000 WELLS FARGO AUTHENTICATION SERVICES DATED: MAY 2003 TABLE OF CONTENTS GENERAL INFORMATION... 1 INSTALLING THE WELLS FARGO ROOT CERTIFICATE CHAIN.. 2 INSTALLING THE CERTIFICATES INTO IE... 3 SETTING UP

More information

EKT150 Introduction to Computer Programming. Wk1-Introduction to Computer and Computer Program

EKT150 Introduction to Computer Programming. Wk1-Introduction to Computer and Computer Program EKT150 Introduction to Computer Programming Wk1-Introduction to Computer and Computer Program A Brief Look At Computer Computer is a device that receives input, stores and processes data, and provides

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 5 Proofs: Mathematical Induction 1 Outline What is a Mathematical Induction? Strong Induction Common Mistakes 2 Introduction What is the formula of the sum of the first

More information

Welcome to Meridian Mail Voice Messaging

Welcome to Meridian Mail Voice Messaging Welcome to Meridian Mail Voice Messaging Meridian Mail Voice Messaging from Nortel Networks is an advanced business messaging system that offers convenience and efficiency for managing your incoming and

More information

MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5 MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which

More information

Binary Multiplication

Binary Multiplication Binary Multiplication Q: How do we multiply two numbers? eg. 12, 345 6, 789 111105 987600 8641500 + 74070000 83, 810, 205 10111 10101 10111 00000 1011100 0000000 + 101110000 111100011 Pad, multiply and

More information

UNITED STATES PATENT AND TRADEMARK OFFICE BEFORE THE BOARD OF PATENT APPEALS AND INTERFERENCES

UNITED STATES PATENT AND TRADEMARK OFFICE BEFORE THE BOARD OF PATENT APPEALS AND INTERFERENCES UNITED STATES PATENT AND TRADEMARK OFFICE BEFORE THE BOARD OF PATENT APPEALS AND INTERFERENCES Ex parte B. REILLY BARRY, MARK A. CHODORONEK, ERIC DEROSE, CAROL Y. DEVINE, MARK N. STUDNESS, ANGELA R. JAMES,

More information

Section IV.1: Recursive Algorithms and Recursion Trees

Section IV.1: Recursive Algorithms and Recursion Trees Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller

More information

RING CYCLES ADMINISTRATIVE LINE FEATURES. AUTO DIAL Electronic Sets: 1. To program, press AUTO DIAL key 2. Dial number to be stored

RING CYCLES ADMINISTRATIVE LINE FEATURES. AUTO DIAL Electronic Sets: 1. To program, press AUTO DIAL key 2. Dial number to be stored Telephone Instructions All Washington University residence halls, fraternities and administrative offices for the Danforth Campus share the same telephone system. All lines begin with the prefix 935 or

More information

Data Representation Binary Numbers

Data Representation Binary Numbers Data Representation Binary Numbers Integer Conversion Between Decimal and Binary Bases Task accomplished by Repeated division of decimal number by 2 (integer part of decimal number) Repeated multiplication

More information

Anti Counterfeiting. GS1 Track & Trace. pointsonline

Anti Counterfeiting. GS1 Track & Trace. pointsonline Anti Counterfeiting GS1 Track & Trace Is Counterfeiting a real problem? How much is the effect on sales of NICE and OMEZ? Which are the regions that are badly effected by counterfeits? Global Overview

More information

4. PRINCIPLE OF MATHEMATICAL INDUCTION

4. PRINCIPLE OF MATHEMATICAL INDUCTION 4 PRINCIPLE OF MATHEMATICAL INDUCTION Ex Prove the following by principle of mathematical induction 1 1 + 2 + 3 + + n 2 1 2 + 2 2 + 3 2 + + n 2 3 1 3 + 2 3 + 3 3 + + n 3 + 4 (1) + (1 + 3) + (1 + 3 + 5)

More information

GarbageCollectioninObjectOriented DatabasesUsingTransactionalCyclic PrasanRoy1 ReferenceCounting S.Sudarshan1 S.Seshadri1S.Ashwin1 AviSilberschatz2 fprasan,seshadri,sudarshag@cse.iitb.ernet.in 1IndianInstituteofTechnology,

More information

Verification of Computing Policies and Other Hard Problems

Verification of Computing Policies and Other Hard Problems Verification of Computing Policies and Other Hard Problems Mohamed G Gouda University of Texas at Austin gouda@cs.utexas.edu Presentation at METIS-2014 1 Computing Policy A computing policy is a formalism

More information

How to Create a Broker Account

How to Create a Broker Account How to Create a Broker Account 1. Once you have created your NY.gov ID and set your password (see the guide, How to Create a NY.gov ID ), return to the Broker tab from the NYSOH home page and select CLICK

More information

Application Note AN-1128

Application Note AN-1128 Application Note AN-1128 IRMCK3xx One-Time Programmable Memory Considerations and Usage By Benjamin Poiesz, International Rectifier November 2006 Version 1.0 Table of Contents 1 INTRODUCTION... 2 2 BOOTLOAD

More information

Rigorous Software Engineering Hoare Logic and Design by Contracts

Rigorous Software Engineering Hoare Logic and Design by Contracts Rigorous Software Engineering Hoare Logic and Design by Contracts Simão Melo de Sousa RELEASE (UBI), LIACC (Porto) Computer Science Department University of Beira Interior, Portugal 2010-2011 S. Melo de

More information

Simulation Programming Design. PV Model, MPPT, and Simulation

Simulation Programming Design. PV Model, MPPT, and Simulation Simulation Programming Design (L3) PV Model, MPPT, and Simulation Woei-Luen Chen Simulation Programming Design PV System (L3) - 1 Outline Types of Photovoltaic Cells PV Model Simulation-1: Curve Fitting

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

Motor and Household Insurance: Pricing to Maximise Profit in a Competitive Market

Motor and Household Insurance: Pricing to Maximise Profit in a Competitive Market Motor and Household Insurance: Pricing to Maximise Profit in a Competitive Market by Tom Wright, Partner, English Wright & Brockman 1. Introduction This paper describes one way in which statistical modelling

More information

CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles)

CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) CS5314 Randomized Algorithms Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) 1 Objectives Introduce Random Graph Model used to define a probability space for all graphs with

More information

Exploring Number System

Exploring Number System Exploring Number System Mun Chou, Fong QED Education Scientific Malaysia LEVEL High schools after students have been introduced to concepts of number systems. OBJECTIVES To explore converting between binary,

More information

Telephone Instructions. Auto Dial

Telephone Instructions. Auto Dial Telephone Instructions Auto Dial The AUTO DIAL button is one that is programmed into a key. Some of you may have this button (or several buttons) programmed on the phone by the IS Dept. To store an AUTO

More information

FIRST DATA GLOBAL GATEWAY

FIRST DATA GLOBAL GATEWAY FIRST DATA GLOBAL GATEWAY Virtual Terminal Enhancements Spring 2010 Release Zero ($0) Dollar Authorizations Periodic Billing 9.15.2010 UPDATE QUICK REFERENCE MANUAL v4.2 September 14, 2010 Table of Contents

More information

Copyright 1998, American Institute of Aeronautics and Astronautics, Inc. C L 1.5 1.5 20.0 deg 22.4 0.5 0.5 Calculation Exp. Data Wing of Aeronautics and Astronautics, Inc. Figure -0.2-4 -2 0 2 4 6 8 10

More information

Instructions for Configuring Microsoft Exchange 2007/2010 Journaling

Instructions for Configuring Microsoft Exchange 2007/2010 Journaling Instructions for Configuring Microsoft Exchange 2007/2010 Journaling Versions Addressed: Microsoft Exchange 2007/2010 Document Updated: March 21, 2015 Co nfidential Copyright 2015 Smarsh, Inc. All Purpose:

More information

Work Instruction. Reverse Check Payment 11/05/07

Work Instruction. Reverse Check Payment 11/05/07 Reverse Check Payment 11/05/07 FCH8 Purpose Use this procedure to cancel a check and re-open the invoices/ credit memos in the system for future payment. The check will be voided in the system and the

More information

Documenting your research: logbooks, online reports, code archive

Documenting your research: logbooks, online reports, code archive Documenting your research: logbooks, online reports, code archive One of the most difficult things to learn, yet one of the most important for future success in physics research, is mastering the "art"

More information

R u t c o r Research R e p o r t. A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS.

R u t c o r Research R e p o r t. A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS. R u t c o r Research R e p o r t A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS Navid Hashemian a Béla Vizvári b RRR 3-2011, February 21, 2011 RUTCOR Rutgers

More information

Ordering Guide New Tech Solutions Information Technology Enterprise Solutions Hardware (ITES-3H)

Ordering Guide New Tech Solutions Information Technology Enterprise Solutions Hardware (ITES-3H) Ordering Guide New Tech Solutions Information Technology Enterprise Solutions Hardware (ITES-3H) NTS Overview: New Tech Solutions, Inc (NTS) was founded in 1997 NTS is a privately held, SLEB Certified

More information

Table: Security Services (X.800)

Table: Security Services (X.800) SECURIT SERVICES X.800 defines a security service as a service provided by a protocol layer of communicating open systems, which ensures adequate security of the systems or of data transfers. Also the

More information

Formal Verification Methods 3: Model Checking

Formal Verification Methods 3: Model Checking Formal Verification Methods 3: Model Checking John Harrison Intel Corporation Marktoberdorf 2003 Fri 1st August 2003 (11:25 12:10) This presentation was not given in Marktoberdorf since it mostly duplicates

More information

UPMC Financial Assistance Application Information

UPMC Financial Assistance Application Information UPMC Financial Assistance Application Information UPMC offers financial assistance for medical care provided by UPMC facilities and UPMC affiliated physicians to eligible individuals and families. Based

More information

Examples of decision problems:

Examples of decision problems: Examples of decision problems: The Investment Problem: a. Problem Statement: i. Give n possible shares to invest in, provide a distribution of wealth on stocks. ii. The game is repetitive each time we

More information

MATLAB Programming. Problem 1: Sequential

MATLAB Programming. Problem 1: Sequential Division of Engineering Fundamentals, Copyright 1999 by J.C. Malzahn Kampe 1 / 21 MATLAB Programming When we use the phrase computer solution, it should be understood that a computer will only follow directions;

More information

Formal models of bank cards for free

Formal models of bank cards for free Formal models of bank cards for free Fides Aarts, Joeri de Ruiter and Erik Poll Digital Security, Radboud University Nijmegen Introduction Active learning on bank cards Learn state machines of implementations

More information

An Information-Flow Type-System for Mixed Protocol Secure Computation

An Information-Flow Type-System for Mixed Protocol Secure Computation An Information-Flow Type-System for Mixed Protocol Secure Computation Florian Kerschbaum SAP Applied Research Karlsruhe, Germany florian.kerschbaum@sap.com ABSTRACT There are a number of domain-specific

More information

Divide and Conquer: Counting Inversions

Divide and Conquer: Counting Inversions Divide and Conquer: Counting Inversions Rank Analysis Collaborative filtering matches your preference (books, music, movies, restaurants) with that of others finds people with similar tastes recommends

More information

Chapter 3. Credit Cards. Ken Long New River Community College Dublin, VA 24084 http://www.nr.cc.va/fin107

Chapter 3. Credit Cards. Ken Long New River Community College Dublin, VA 24084 http://www.nr.cc.va/fin107 Chapter 3 Credit Cards Ken Long New River Community College Dublin, VA 24084 http://www.nr.cc.va/fin107 1 Should I switch to a lower rate credit card? Maybe No Not if the new card uses the two-cycle method

More information

RSA and Primality Testing

RSA and Primality Testing and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2

More information

Application Note 120 Communicating Through the 1-Wire Master

Application Note 120 Communicating Through the 1-Wire Master www.dalsemi.com Application Note 120 Communicating Through the 1-Wire Master INTRODUCTION The DS1WM 1-Wire Master was created to facilitate host CPU communication with devices over a 1-Wire bus without

More information

CHARITY CARE APPLICATION REQUIRED DOCUMENTATION CHECK LIST

CHARITY CARE APPLICATION REQUIRED DOCUMENTATION CHECK LIST CHARITY CARE APPLICATION REQUIRED DOCUMENTATION CHECK LIST Please return the items below if they apply to your situation. Theses items are required to process your application for charity care assistance.

More information

Using UMCmarket to make micro-donations to our church!

Using UMCmarket to make micro-donations to our church! Using UMCmarket to make micro-donations to our church! You can use UMCmarket from your desktop PC, laptop, smartphone or tablet. Follow these instructions to get started using UMCmarket and help our church.

More information

Enhanced Privacy ID (EPID) Ernie Brickell and Jiangtao Li Intel Corporation

Enhanced Privacy ID (EPID) Ernie Brickell and Jiangtao Li Intel Corporation Enhanced Privacy ID (EPID) Ernie Brickell and Jiangtao Li Intel Corporation 1 Agenda EPID overview EPID usages Device Authentication Government Issued ID EPID performance and standardization efforts 2

More information

REPETITION WITH PYTHON

REPETITION WITH PYTHON REPETITION WITH PYTHON José M. Garrido Department of Computer Science May 2015 College of Computing and Software Engineering Kennesaw State University c 2015, J. M. Garrido Repetition with Python 2 Repetition

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Loan Modification & Your Title Policy

Loan Modification & Your Title Policy Loan Modification & Your Title Policy Frank L. Tortora III, Esq. Senior Title Officer July 22, 2014 Title Insurance 101 Title Insurance a contract agreeing to indemnify the insured up to a specified amount

More information

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the

More information

Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

More information

Flagship Fleet Management, LLC Fuel Processing Module

Flagship Fleet Management, LLC Fuel Processing Module Flagship The Flagship fuel processing module is designed to make short work of the process of loading fuel transactions into your fleet management system. We can convert many of the industry standard transaction

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

ACCA Online Application Guidelines

ACCA Online Application Guidelines ACCA Online Application Guidelines To prevent some of the internal problems that might occur during registration process all application should be started: New Incognito window for Google Chrome It is

More information

The Feasibility and Application of using a Zero-knowledge Protocol Authentication Systems

The Feasibility and Application of using a Zero-knowledge Protocol Authentication Systems The Feasibility and Application of using a Zero-knowledge Protocol Authentication Systems Becky Cutler Rebecca.cutler@tufts.edu Mentor: Professor Chris Gregg Abstract Modern day authentication systems

More information

Factoring Algorithms

Factoring Algorithms Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

More information

Welcome to Golden Belt Bank We Appreciate Your Business

Welcome to Golden Belt Bank We Appreciate Your Business Switch Your Checking Account to Golden Belt Bank the Easy Way Welcome to Golden Belt Bank We Appreciate Your Business It has never been easier to begin enjoying the exceptional personal service you can

More information

Auditable Version Control Systems

Auditable Version Control Systems Auditable Version Control Systems Bo Chen Reza Curtmola Department of Computer Science New Jersey Institute of Technology bc47@njit.edu crix@njit.edu Abstract Version control provides the ability to track

More information

OAMulator. Online One Address Machine emulator and OAMPL compiler. http://myspiders.biz.uiowa.edu/~fil/oam/

OAMulator. Online One Address Machine emulator and OAMPL compiler. http://myspiders.biz.uiowa.edu/~fil/oam/ OAMulator Online One Address Machine emulator and OAMPL compiler http://myspiders.biz.uiowa.edu/~fil/oam/ OAMulator educational goals OAM emulator concepts Von Neumann architecture Registers, ALU, controller

More information

ALLIED PAPER : DISCRETE MATHEMATICS (for B.Sc. Computer Technology & B.Sc. Multimedia and Web Technology)

ALLIED PAPER : DISCRETE MATHEMATICS (for B.Sc. Computer Technology & B.Sc. Multimedia and Web Technology) ALLIED PAPER : DISCRETE MATHEMATICS (for B.Sc. Computer Technology & B.Sc. Multimedia and Web Technology) Subject Description: This subject deals with discrete structures like set theory, mathematical

More information

PATHWAY I: Early Learning Scholarship Application

PATHWAY I: Early Learning Scholarship Application -2014 PATHWAY I: Early Learning Scholarship Application This section to be completed by the Regional Administration Office: Application Identifier #: Region: District Number and Type: Is the Family Income

More information

NAND Flash Memories. Using Linux MTD compatible mode. on ELNEC Universal Device Programmers. (Quick Guide)

NAND Flash Memories. Using Linux MTD compatible mode. on ELNEC Universal Device Programmers. (Quick Guide) NAND Flash Memories Using Linux MTD compatible mode on ELNEC Universal Device Programmers (Quick Guide) Application Note April 2012 an_elnec_linux_mtd, version 1.04 Version 1.04/04.2012 Page 1 of 16 As

More information

Python programming: An overview Ali Dariush (Institute of Astronomy, University of Cambridge) Cambridge International School (August 2016)

Python programming: An overview Ali Dariush (Institute of Astronomy, University of Cambridge) Cambridge International School (August 2016) Python programming: An overview Ali Dariush (Institute of Astronomy, University of Cambridge) Cambridge International School (August 2016) Computer program A computer program (computer code) is a detailed

More information

ADDITIONS TO THE SDI-12 SPECIFICATION (April 18, 2016, to be integrated into the formal SDI-12 Specification document)

ADDITIONS TO THE SDI-12 SPECIFICATION (April 18, 2016, to be integrated into the formal SDI-12 Specification document) ADDITIONS TO THE SDI-12 SPECIFICATION (April 18, 2016, to be integrated into the formal SDI-12 Specification document) 10 IDENTIFY MEASUREMENT COMMANDS The identify measurement commands provide a means

More information

Outline. Conditional Statements. Logical Data in C. Logical Expressions. Relational Examples. Relational Operators

Outline. Conditional Statements. Logical Data in C. Logical Expressions. Relational Examples. Relational Operators Conditional Statements For computer to make decisions, must be able to test CONDITIONS IF it is raining THEN I will not go outside IF Count is not zero THEN the Average is Sum divided by Count Conditions

More information

UNDERGRADUATE TEACHER CERTIFICATION ENROLLMENT FORM

UNDERGRADUATE TEACHER CERTIFICATION ENROLLMENT FORM UNDERGRADUATE TEACHER CERTIFICATION ENROLLMENT FORM ELED, SPED and ECED are not available through the Teacher s Certification program. For any K 12 programs listed below, please seek advising from the

More information

61. REARRANGEMENTS 119

61. REARRANGEMENTS 119 61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0, MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

More information

Infor LN Financials User Guide for General Ledger

Infor LN Financials User Guide for General Ledger Infor LN Financials User Guide for General Ledger Publication Information Document code Release Publication date tfgldug (U9636) 10.4 (10.4) March 20, 2016 Table of Contents About this document Chapter

More information

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

More information

Introduction to Programming in C++: Algorithms, Flowcharts and Pseudocode

Introduction to Programming in C++: Algorithms, Flowcharts and Pseudocode Introduction to Programming in C++: Algorithms, Flowcharts and Pseudocode by Brent Daviduck The following material was developed by Brent Daviduck of the Computer Systems Technology program at Red Deer

More information

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

More information

Conditional and Looping Construct

Conditional and Looping Construct Chapter 3 Conditional and Looping Construct After studying this lesson, students will be able to: Understand the concept and usage of selection and iteration statements. Know various types of loops available

More information

Secure information flow: opportunities and challenges for security & forensics

Secure information flow: opportunities and challenges for security & forensics flow: and for flow: and for Michael 1 1 Department of Computing Imperial College London flow: and for Talk for ACSF1, 13 July 2006 Liverpool, United Kingdom flow: and for 1 flow: and for 2 3 4 flow: and

More information

16. Recursion. COMP 110 Prasun Dewan 1. Developing a Recursive Solution

16. Recursion. COMP 110 Prasun Dewan 1. Developing a Recursive Solution 16. Recursion COMP 110 Prasun Dewan 1 Loops are one mechanism for making a program execute a statement a variable number of times. Recursion offers an alternative mechanism, considered by many to be more

More information

IntelliPay Billpay Application Documentation

IntelliPay Billpay Application Documentation IntelliPay Billpay Application Documentation Contents Billpay Introduction First Time Payment (With username and password) First Time Payment (Without username and password) Returning Customer Introduction:

More information

What Is Recursion? Recursion. Binary search example postponed to end of lecture

What Is Recursion? Recursion. Binary search example postponed to end of lecture Recursion Binary search example postponed to end of lecture What Is Recursion? Recursive call A method call in which the method being called is the same as the one making the call Direct recursion Recursion

More information

PayBiz Bank Accounts

PayBiz Bank Accounts PayBiz Bank Accounts 30/01/2014 Document Contents Bank Accounts...2 Window Control Buttons....3 Create an Opening Balance....4 Create a Bank Transaction....5 Create the Bank Statement....5 Enter Unpresented

More information

5 Homogeneous systems

5 Homogeneous systems 5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

More information

Digital Signatures in the PDF Language

Digital Signatures in the PDF Language Developer Technical Note Introduction C ONTENTS Introduction 1 Background 1 PDF Digital Signature Basics 3 The Need for Certified Documents 8 How Field Locking is Done 9 Resources 10 Terms Used in This

More information

CIT 203-I01 Access Certification Preparation Online

CIT 203-I01 Access Certification Preparation Online CIT 203-I01 Access Certification Preparation Online 1001 Using Microsoft Access 2013 3 credits Fall 2015 August 31 December 18 Please read completely through this syllabus and print it for future reference.

More information

Multiplication with Whole Numbers

Multiplication with Whole Numbers Math 952 1.3 "Multiplication and Division with Whole Numbers" Objectives * Be able to multiply and understand the terms factor and product. * Properties of multiplication commutative, associative, distributive,

More information