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

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1 Elements of Computing Systems, Nisn & Schocken, MIT Press Usge nd Copyright Notice: Boolen Logic Copyright 2005 Nom Nisn nd Shimon Schocken This presenttion contins lecture mterils tht ccompny the textook The Elements of Computing Systems y Nom Nisn & Shimon Schocken, MIT Press, We provide oth PPT nd PDF versions. The ook we site, fetures 13 such presenttions, one for ech ook chpter. Ech presenttion is designed to support 3 hours of clssroom or self-study instruction. You re welcome to use or edit this presenttion s you see fit for instructionl nd noncommercil purposes. If you use our mterils, we will pprecite it if you will include in them reference to the ook s we site. If you hve ny questions or comments, you cn rech us t tecs.t@gmil.com Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 1

2 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,y) Nnd(x,y) Functionl expression VS truth tle expression Importnt result: Every Boolen function cn e expressed using And, Or, Not. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 2

3 All Boolen functions of 2 vriles Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 3

4 Boolen lger Given: Nnd(,), flse We cn uild: Not() = Nnd(,) true = Not(flse) And(,) = Not(Nnd(,)) George Boole, ( A Clculus of Logic ) Or(,) = Not(And(Not(),Not())) Xor(,) = Or(And(,Not()),And(Not(),))) Etc. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 4

5 Gte logic Gte logic gte rchitecture designed to implement Boolen function Elementry gtes: Composite gtes: Importnt distinction: Interfce (wht) VS implementtion (how). Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 5

6 Gte logic Interfce Xor Clude Shnnon, Implementtion Not And ( Symolic Anlysis of Rely nd Switching Circuits ) Or Not And Xor(,) = Or(And(,Not()),And(Not(),))) Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 6

7 Circuit implementtions AND gte OR gte power supply power supply c c AND (,,c) c AND AND From computer science perspective, physicl reliztions of logic gtes re irrelevnt. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 7

8 Project 1: elementry logic gtes Given: Nnd(,), flse Build: Nnd(,) Nnd(,) Not() =... true =... And(,) =... Or(,) =... Mux(,,sel) =... Etc gtes ltogether. Why these prticulr 12 gtes? Since They re commonly used gtes They provide ll the sic uilding locks needed to uild our computer. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 8

9 Multiplexer sel sel Mux sel 0 1 Proposed Implementtion: sed on Not, And, Or gtes. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 9

10 Exmple: 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; Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 10

11 Building n And gte Interfce: And(,) = 1 exctly when ==1 And And.hdl CHIP CHIP And And { IN IN,, ; ; OUT OUT ; ; // // implementtion missing missing } Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 11

12 Building n And gte Implementtion: And(,) = Not(Nnd(,)) And.hdl CHIP CHIP And And { IN IN,, ; ; OUT OUT ; ; // // implementtion missing missing } Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 12

13 Building n And gte Implementtion: And(,) = Not(Nnd(,)) Nnd x in Not And.hdl CHIP CHIP And And { IN IN,, ; ; OUT OUT ; ; // // implementtion missing missing } Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 13

14 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, = ) ) } Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 14

15 Hrdwre simultor (demonstrting Xor gte construction) HDL progrm test script Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 15

16 Hrdwre simultor HDL progrm Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 16

17 Hrdwre simultor HDL progrm put file Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 17

18 Project mterils Project 1 on the course we site And.hdl, And.tst, And.cmp files Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 18

19 Project 1 tips Red Chpter 1 of the ook Downlod the ook s softwre suite Go through the hrdwre simultor tutoril Do Project 0 (optionl) You re in usiness. Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 19

20 End notes: Cnonicl representtion Truth tle of the function s (, m, w) = ( m+ w) Cnonicl form: s (, m, w) = m w+ mw+ mw Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 20

21 End notes: Cnonicl representtion (cont.) s (, m, w) = ( m+ w) m w nd or s s (, m, w) = m w+ mw+ mw m w nd nd or s nd Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 21

22 End notes: 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) Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 22

23 Perspective 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 stndrd PLD consisting of Nnd gtes only Mss production c nd Universl uilding locks, unique topology. or f(,,c) Gtes, neurons, toms, nd Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 23

24 End note: universl uilding locks, unique topology nd c. or f(,,c) nd Elements of Computing Systems, Nisn & Schocken, MIT Press, Chpter 1: Boolen Logic slide 24