Boolean Arithmetic. Building a Modern Computer From First Principles.

Size: px

Start display at page:

Download "Boolean Arithmetic. Building a Modern Computer From First Principles."

Audrey Perry
6 years ago
Views:

1 Boolean Arithmetic Building a Modern Computer From First Principles Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 1

2 Counting systems quantity decimal inary 3-it register overflow overflow overflow Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 2

3 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 3 Rationale (10011) = = two i n i i n n x x x x = = )... ( (9038) = = ten

4 Binary addition Assuming a 4-it system: no overflow overflow Algorithm: exactly the same as in decimal addition Overflow (MSB carry) has to e dealt with. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 4

5 Representing negative numers (4-it system) The codes of all positive numers egin with a 0 The codes of all negative numers egin with a 1 To convert a numer: leave all trailing 0 s and first 1 intact, and flip all the remaining its Example: 2-5 = 2 (-5) = = -3 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 5

6 Building an Adder chip a it ad d e r o ut Adder: a chip designed to add two integers Proposed implementation: Half adder: designed to add 2 its Full adder: designed to add 3 its Adder: designed to add two n-it numers. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 6

7 Half adder (designed to add 2 its) a sum carry a h alf ad d er s um c arry Implementation: ased on two gates that you ve seen efore. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 7

8 Full adder (designed to add 3 its) a c sum carry a c fu ll ad d er s um c arry Implementation: can e ased on half-adder gates. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 8

9 n-it Adder (designed to add two -it numers) a it ad d e r out a out Implementation: array of full-adder gates. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 9

10 The ALU (of the Hack platform) a h alf ad d er s um c arry a c fu ll ad d er s um c arry x y -it adder out zx nx zy ny f no out(x, y, control its) = xy, x-y, y x, x y its its ALU its out 0, 1, -1, x, y, -x, -y, x!, y!, x1, y1, x-1, y-1, zr ng x&y, x y Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 10

11 ALU logic (Hack platform) Implementation: uild a logic gate architecture that executes the control it instructions : if zx==1 then set x to 0 (it-wise), etc. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 11

12 The ALU in the CPU context (a sneak preview of the Hack platform) c1,c2,,c6 D register D A register A a ALU out RAM M Mux A/M (selected register) Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 12

13 Perspective Cominational logic Our adder design is very asic: no parallelism It pays to optimize adders Our ALU is also very asic: no multiplication, no division Where is the seat of more advanced math operations? a typical hardware/software tradeoff. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 13

Historical end-note: Leinitz (46-17) The inary system may e used in place of the decimal system; express all numers y unity and

scientific literature have assuredly very often felt the want of a common scientific language, and regretted the great loss of

: a universal, formal, and decidale language of reasoning Leiniz s medallion for the Duke of Brunswick The dream s end: Turing

14 Historical end-note: Leinitz (46-17) The inary system may e used in place of the decimal system; express all numers y unity and y nothing 79: uilt a mechanical calculator (, -, *, /) CHALLENGE: All who are occupied with the reading or writing of scientific literature have assuredly very often felt the want of a common scientific language, and regretted the great loss of time and troule caused y the multiplicity of languages employed in scientific literature: SOLUTION: Characteristica Universalis : a universal, formal, and decidale language of reasoning Leiniz s medallion for the Duke of Brunswick The dream s end: Turing and Gödel in 1930 s. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 14

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

Elements of Computing Systems, Nisan & Schocken, MIT Press www.idc.ac.il/tecs Boolean Arithmetic Usage and Copyright Notice: Copyright 2005 Noam Nisan and Shimon Schocken This presentation contains lecture