George Boole. IT 3123 Hardware and Software Concepts. Switching Algebra. Boolean Functions. Boolean Functions. Truth Tables

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1 George Boole IT 3123 Hrdwre nd Softwre Concepts My 28 Digitl Logic The Little Mn Computer British mthemticin nd philosopher Mny contriutions to mthemtics. Boolen lger: n lger over finite sets of discrete vlues. Notice: This session is eing recorded. Copyright 2005, 2006 y Bo Brown Switching Alger A specil cse of Boolen lger. Vriles cn only e one of two vlues, one or zero. Zero nd one cn represent flse nd true, respectively. So, n lger over sets of inry digits. Often clled Boolen lger y those involved in computing. Boolen Functions Functions in which the vriles re 0/1 nd the result is either zero or one. Consider: f(,) = (Rememer, nd cn only e 0,1) This is the AND function, or Boolen product f(,) = 1 if nd only if =1 nd =1 For ll other comintions, f(,) = 0 Truth Tles A more visul wy to represent Boolen functions. Developed from the work of Jn Łuksiewicz. Here is f(,) =. Note tht if you consider nd s digits of inry numer, the vlues re written in scending order. f (,) Boolen Functions There re only 16 Boolen functions of two vriles. Why? f (,) 0 0? 0 1? 1 0? 1 1? 1

2 Ctegories of Boolen Functions There re four ctegories of Boolen functions of two vriles: Computtionlly complete functions (defined lter) Generlly useful functions (most of our discussion is of these functions) Specil purpose functions Useless functions The AND Function f(,) is true only when oth AND re true. f (,) The AND function is lso clled the Boolen product: f(,) = The OR Function f(,) is true only when OR (or oth) re true. f (,) The OR function is lso clled the Boolen sum: f(,) = + The plus is not the sme s rithmetic ddition. The NOT Function f() is true when is flse nd flse when is true. f() The NOT function is lso clled the Boolen inverse: f() = Negtion r The EXCLUSIVE OR Function f(,) is true when is true or when is true, ut not when oth re true. f (,) The EXCLUSIVE OR (XOR) function is : f(,) = Two Useless Functions The NEVER Function f (,) The ALWAYS Function f (,) The nmes re mde up, ut the resons for them re ovious. 2

3 Electric Circuits Cn Compute Boolen Functions Circuits tht cn compute digitl logic functions re clled gtes. This circuit computes the AND function. Current cn only flow when switches A AND B re closed. Computing the OR Function This circuit computes the OR function. Current cn flow when switch A OR switch B is closed, or oth. Trnsistors s Switches Trnsistors cn function s fst electricl switches. Digitl logic gtes re mde from trnsistors. More Aout Circuits nd Functions This circuit computes the NOT function. The output is on when the input is off, nd vicevers. Astrction We use specil symols to represent the electronic circuits tht mke up digitl logic gtes. The NOT Function Inverse, or NOT function. f() f() = 3

4 The AND Function The AND function, or Boolen product f (,) f(,) = The OR Function The OR function, or Boolen sum f (,) f(,) = + + The EXCLUSIVE OR Function The EXCLUSIVE OR, or not equl, or odd function f (,) f(,) = Computtion with Digitl Logic Boolen lger dels with opertions on inry digits nd produces inry digit results. Arithmetic opertions like ddition cn e expressed s truth tles. You cn uild digitl logic circuit to compute ny function for which you cn write the truth tle. (Look up sum of products for n exmple.) Reminder: Binry Addition Binry Addition s s At the moment, we re interested only in the sum, nd not in the crry. This is the EXCLUSIVE OR function 4

5 Computing the Crry Sum nd Crry Together c o This is the AND function s c o A Hlf-Adder sum Circuit Anlysis How do we know this computes sum nd crry? sum crry crry o s c o Addition with Crry In Addition with Crry In c i s c o

6 Computtion of the Sum Computtion of the Sum c i s c o The sum is 1 when the numer of input ones is odd. s=( ) c i c i s=( ) c i V W V computes W computes V c Sum Computtion of the Crry c i s c o Crry out is true when oth nd re true or when c i nd one of nd is true. c i Computtion of the Crry c o =+( ) c i X V W Y V computes X computes Y computes ( ) c i c o Sum The Full Adder The Full Adder Two Hlf-Adders nd n OR gte c i W Sum V Y Z Crry X Z computes X + Y 6

7 Astrction Four Bit Adder Full Adder Full Adder Full Adder Full Adder Full Adder s 3 s 2 s 1 s 0 Crry Out Aout Digitl Logic Gtes Four Bit Adder The output of digitl logic gte depends only upon its inputs; chnging the inputs chnges the output. The trnsistors in digitl logic gte tke few nnoseconds (illionths of second) to switch from on to off or vice-vers. The gte dely of digitl logic gte is the time for chnge in inputs to e reflected in new output. Crry Out 3 3 Full Adder s Full Adder s Full Adder s Full Adder s 0 The NAND Function The NAND function f (,) f(,) = The NOR function The NOR Function f (,) f(,) = + + 7

8 Computtionl Completeness NAND nd NOR re computtionlly complete. Computtionlly complete: Any digitl logic function cn e computed using only one type of computtionlly complete gte. Any function cn e computed using only NAND gtes Any function cn e computed using only NOR gtes. Astrction: The NAND Gte This: Represents this: Astrction: The AND Gte This: Represents this: Or this: Circuit Equivlence Why So Mny Gtes? Question: If ny Boolen function cn e computed using only NAND or NOR, why do we other with the other gtes? Answer: Astrction! We wnt our designs to e cler, so we use the gte symols tht indicte the logicl function of gte, not its internl structure. When circuit is to e fricted, engineers use the principle of circuit equivlence to convert the logic digrm to the gtes tht re physiclly needed. Comintionl Circuits The output of digitl logic gte depends only on its inputs The output of the dder we studied lso depends only on its inputs. Such circuits re clled comintionl circuits. (Some uthors refer to them s comintoril circuits.) Becuse output depends only upon input, comintionl circuits hve no memory. Sequentil Circuits It is possile to uild circuits with memory from digitl logic S gtes. Such circuits re clled sequentil circuits. Sequentil circuits R depend on feedck for their memory. Feedck results from connecting the output of circuit ck to its input. Q Q 8

9 Another Clss Motto There is no mgic! Questions Everything tht computers do is performed y hrdwre nd softwre designed y engineers nd computer scientists. Brek IT 3123 Hrdwre nd Softwre Concepts My 28 The Little Mn Computer Notice: This session is eing recorded. Copyright 2005 y Bo Brown Complexity nd Simplicity Computers perform immensely complex tsks. However, they cn perform only few, reltively simple opertions ( few dozen to few hundred). Complexity is uilt out of these simple opertions, which cn e performed very fst. The set of opertions prticulr computer cn perform is clled its instruction set. Simple Opertions: Complex Results 9

10 One Mentl Model of Computing How the CPU processes dt, including: Instructions nd dt Min memory (RAM) Computtion Input nd output Progrm flow control (loops nd if-thenelse) The von Neumnn Architecture John von Neumnn (~1945) Stored progrm concept: Both instructions nd dt re stored in the sme memory Sequentil execution Instruction cycle Fetch instruction Decode instruction Execute instruction Binry numers Still how computers re designed in 2008! The von Neumnn Architecture CPU Control Unit Memory Arithmetic Logic Unit I-O Interfce Unit The Arithmetic Logic Unit (ALU) Performs rithmetic opertions, such s ddition, on dt Performs logicl opertions (complementtion, if-sttements, etc.) Cn e used to trnsfer dt Within the CPU From CPU to I-O The Control Unit Controls the processing of instructions Progrm counter: A smll re of memory within the CPU tht holds the ddress of the next instruction. Also clled the instruction loction counter. Controls movement of dt within the CPU. Memory Also know s: RAM (Rndom Access Memory) Min memory Primry storge Memory loctions re of fixed size Holds instructions (in mchine lnguge) nd dt for running progrms 10

11 The von Neumnn Architecture CPU LMC is von Neumnn Computer Control Unit Arithmetic Logic Unit I-O Memory Interfce Unit Copyright 2003 John Wiley nd Sons LMC is von Neumnn Computer Input nd Output Arithmetic Logic Unit Control nd Interfce Only Lels Memory The Clcultor (ALU) Three digits only Cn disply minus sign No provision for entering negtive numers No memory other thn the numer eing displyed Progrm Counter Copyright 2003 John Wiley nd Sons The Clcultor (ALU) Three opertions: Entry: new numer is typed in; it ppers in the disply remins there until chnged, i.e. it is stored. Add: A numer is typed in nd dded to the numer displyed; the sum replces the numer displyed. Sutrct: numer is typed in nd sutrcted from the numer displyed. The difference replces the numer displyed. Input nd Output An input sket An output sket Limited to three-digit numers Only one three-digit numer t time 11

12 The Counter (Progrm Counter) Two digits Cn e dvnced y one unit Cn e set to n ritrry two-digit numer Cn e reset to zero from outside Miloxes (Memory) Addresses re consecutive (00-99) Ech ox cn contin one nd only one slip of pper (dt word) Dt words re three deciml digits My e either instructions or dt The memory doesn t know which, except y context. Address Content The Little Mn Performs oth Control Unit nd Interfce functions. Cn do the following: Operte the clcultor (ALU) Red nd updte memory Get dt from the input sket Put dt in the output sket Red, dvnce, nd set the counter Knows t most ten commnds Memory Words s Instructions A memory word cn e thought of s: A three-digit numer, or A one-digit numer nd two-digit numer If the progrm counter points to prticulr memory word, it is considered to e: A one-digit opertion code, nd A two-digit ddress Together, the two prts re clled n instruction. Instruction Decoding Interpreting (understnding) n instruction is clled instruction decoding. Suppose the progrm counter contins 50 And memory looks like this: Address Content Instruction Decoding 325 is considered to e one-digit opertion code nd two-digit ddress. The op code (3) is commnd to the Little Mn, or CPU The ddress (25) is milox numer, or memory ddress. The commnd will use the specified ddress when performing the opertion Opertion code 3 25 Address 12

13 The STORE Instruction In LMC, opertion code 3 is the STORE instruction. (Becuse the designers of LMC selected 3 to e STORE, tht s why.) The STORE instruction stores the contents of the clcultor (ALU) in the specified memory ddress. So, 3 25 mens store (3) the numer displyed y the clcultor in milox 25. Another milox could hve een selected y using numer other thn 25. The Semntics of STORE The Little Mn reds the numer displyed y the clcultor, writes it on slip of pper, nd plces the pper in the milox nmed y the ddress prt of the instruction, in this cse, 25. The previous contents of milox 25 re destroyed (importnt concept!) The numer displyed y the clcultor is unchnged. The LOAD Instruction LOAD is op code 5 (Becuse the designers sid so.) Consider 5 42 The Little Mn reds the numer in the milox nmed y the ddress field of the instruction, in this cse 42. Suppose tht numer is 137. The Little Mn enters the numer (137) into the clcultor. The previous numer in the clcultor is lost. The contents of loction 42 is unchnged. The ADD Instruction ADD is op code 1 The Little Mn reds three-digit numer from the milox specified y the ddress prt of the instruction He enters the numer into the clcultor nd presses the + key to dd it to wht ws lredy there. The clcultor contins the sum The contents of the milox re unchnged The SUBTRACT Instruction SUBTRACT is op code 2 The Little Mn reds three-digit numer from the milox specified y the ddress prt of the instruction He enters the numer into the clcultor nd presses the (minus) key to sutrct it from wht ws lredy there. The clcultor contins the difference The contents of the milox re unchnged Input Opertion code 9 is used for INPUT The Little Mn gets slip of pper from the input sket. The three-digit numer from the slip is entered into the clcultor; the previous contents re lost. The slip of pper is discrded. Note: The ddress field ws not used We ritrrily ssign 01 to the ddress 13

14 Output Opertion code 9 is lso used for output. We ssign 02 to the ddress field; for opertion code 9, the Little Mn looks t the ddress field to decide whether input or output is wnted For output, the Little Mn writes the numer from the clcultor on slip of pper nd plces it in the output sket. Once in the sket, the pper is no longer ccessile. Hlt Op code 0 commnds the Little Mn to stop executing instructions (tke rek) The ddress portion is not used Instruction execution resumes only when the progrm counter is reset from outside. The LMC Instruction Set So Fr 0xx Hlt 1xx Add 2xx Sutrct 3xx Store 5xx Lod 901 Input 902 Output xx represents memory ddress. A Simple Progrm Ojective: Add two numers, output the sum Address Content Mening input numer store numer in ox input nother numer dd contents of ox output from clcultor hlt 99??? dt storge The Simple Progrm Executing the Simple Progrm The progrm counter is reset from the outside to strt the LMC The Little Mn wkes up from his coffee rek nd reds the progrm counter. It is 00 ecuse it ws just reset. The Little Mn reds the contents of milox 00. Milox 00 contins input numer 14

15 The Progrm Strts The Input Instruction The Little Mn knows (y design) tht 901 mens input. The Little Mn dvnces the progrm counter The Little Mn reds the slip of pper t the input. Assume it is 137. The Little Mn enters 137 into the clcultor. It is no longer ville t the input. The Input Instruction is Complete The Store Instruction The progrm counter now points t loction 01 The Little Mn retrieves the instruction t loction store numer in 99 Then the Little Mn dvnces the progrm counter This instruction commnds the Little Mn to store the numer in the clcultor in loction 99 The Store Instruction The Second Input Instruction The progrm counter points to loction 02, which contins n input instruction. The Little Mn reds the instruction (901) t loction 02 nd dvnces the progrm counter. The Little Mn reds slip from the input; ssume it is 42. The numer from the input is entered into the clcultor nd the input slip is discrded 15

16 The Second Input Instruction Two Things to Notice The numer 137, which ws in the clcultor efore the second input instruction, ws lost when 42 ws entered. But, 137 is still ville in loction 99 ecuse we stored it there. The Add Instruction The progrm counter points to loction 03. The Little Mn reds the instruction t loction 03 nd dvnces the progrm counter dd contents of 99 The Add Instruction The Little Mn dds the contents of loction 99 (137) to the numer in the clcultor. The sum is now 179. The Output Instruction When the Add instruction is complete, the progrm counter points to loction 04. The Little Mn reds the instruction t 04 (902: Output) nd dvnces the progrm counter. The Little Mn writes the numer from the clcultor (179) on slip of pper nd puts it in the output sket. (The clcultor still shows 179.) The Hlt Instruction The progrm counter points to loction 05, which contins 000. The Little Mn reds the 000 nd (proly) dvnces the progrm counter. Op code 0 commnds the Little Mn to stop executing instructions. The clcultor nd memory re unchnged. 16

17 Two More Things to Notice The Little Mn need not e intelligent; ll he does is: Red memory pointed y progrm counter Advnce the progrm counter Perform one of very few opertions s commnded y the opertion code Seprte equipment (like the clcultor) does the rel work The sme steps re repeted for ech instruction. The Instruction Cycle The Little Mn does three things repetedly Fetch: Get n instruction from the memory loction pointed y the progrm counter nd dvnce the progrm counter Decode: Determine wht opertion code is present, nd wht dt to use Execute: Perform the commnded opertion Progrm Flow Control Three control structures re sufficient for ll correct progrms (Böhm nd Jcopini) Sequence: one instruction fter nother Selection: if-then-else Itertion: loops tht repet The progrm counter utomticlly provides sequence. (von Neumnn s ide of sequentil execution.) The Brnch Instruction The rnch instruction (opertion code: 6) unconditionlly chnges the progrm flow. The ddress prt of the instruction is loded into the progrm counter. (The ddress digits re not used to refer to dt.) Brnch Instruction Exmple Loction Contents The instruction t loction 50 sys tht the next instruction should e fetched from loction 75 The Little Mn fetches the instruction from loction 50 The Little Mn dvnces the progrm counter (now points to 51) The Little Mn chnges the progrm counter to 75 Advncing the Progrm Counter In rel computer (nd in this description) the progrm counter is dvnced during the fetch or decode prt of the instruction cycle, i.e. efore the execute prt. This is ecuse the execute prt might chnge the progrm counter. So, the progrm counter lwys points to the next instruction to e executed. 17

18 Brnch on Zero Op code: 7 The Little Mn fetches the instruction The Little Mn Advnces the progrm counter. If the numer in the clcultor is zero, the Little Mn stores the ddress prt of the instruction in the progrm counter. Otherwise, the progrm counter lredy points to the next instruction. Brnch on Positive Op code: 8 The Little Mn fetches the instruction The Little Mn Advnces the progrm counter. If the numer in the clcultor is positive, the Little Mn stores the ddress prt of the instruction in the progrm counter. Otherwise, the progrm counter lredy points to the next instruction. Instruction Mnemonics Rememering the opertion codes is tedious. (It gets worse if there re lot of them!) The word mnemonic mens n id to memory. Instruction mnemonics re revitions tht tke the plce of numeric opertion codes. The LMC Instruction Set Mnemonic Op Code Mening HLT 0xx Hlt ADD 1xx Add SUB 2xx Sutrct STO 3xx Store LDA 5xx Lod BR 6xx Brnch BRZ 7xx Brnch Zero BRP 8xx Brnch Positive IN 901 Input OUT 902 Output Assemler Progrms An ssemler is computer progrm. It reds computer progrm written in instruction mnemonics (source progrm) Its output is the sme progrm, ut trnslted to numeric opertion codes (oject progrm) Assemlers lso provide for giving symolic nmes to memory loctions An ssemler is the simplest lnguge trnsltor A Pseudo-Instruction The LMC ssemler recognizes the DAT (for dt ) mnemonic. DAT is not n instruction to the Little Mn; it is commnd to the ssemler to reserve one loction for dt. Such ssemler commnds re clled pseudo-instructions Could lso e used to reserve nd initilize dt, e.g. DAT

19 The Positive Difference Progrm IN STO IN STO SUB BRP LDA SUB DONE OUT HLT NBR1 DAT NBR2 DAT NBR1 NBR2 NBR1 DONE NBR1 NBR2 Assemled Positive Difference IN STO NBR IN STO NBR SUB NBR BRP DONE LDA NBR SUB NBR DONE OUT HLT NBR1 DAT NBR2 DAT Only the Oject Code is Stored IN STO NBR IN STO NBR SUB NBR BRP DONE LDA NBR SUB NBR DONE OUT HLT NBR1 DAT NBR2 DAT Bootstrpping We hve skimmed over how progrms get into the computer in the first plce Computers (usully) hve smll, uilt-in progrm tht cn red nother progrm from outside, such s from disk. This process is clled ootstrpping or ooting the computer. After the computer is ootstrpped, the externlly loded progrm is responsile for loding ny other needed progrms Is LMC Relistic? The first widely successful mini-computer ws the DEC PDP-8. It hd n instruction set of eight instructions (smller thn LMC s). The PDP-8 could run complete multiuser operting system. The PDP-8 Dvid Gesswein 19

20 Something to Think Aout The PDP-8 hd n dd instruction, ut no sutrct instruction. How do you think sutrction ws performed? Hint: the mnemonic for the dd instruction ws TAD for two s complement dd. Questions 20