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

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1 Elements of Computing Systems, Nisan & Schocken, MIT Press Boolean Arithmetic Usage and Copyright Notice: Copyright 2005 Noam Nisan and Shimon Schocken This presentation contains lecture materials that accompany the textook The Elements of Computing Systems y Noam Nisan & Shimon Schocken, MIT Press, We provide oth PPT and PDF versions. The ook we site, features 13 such presentations, one for each ook chapter. Each presentation is designed to support aout 3 hours of classroom or self-study instruction. You are welcome to use or edit this presentation as you see fit for instructional and noncommercial purposes. If you use our materials, we will appreciate it if you will include in them a reference to the ook s we site. If you have any questions or comments, you can reach us at tecs.ta@gmail.com 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 carry sum 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 carry sum 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 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 reads each control it and does what the tale specifies: if zx=1 then set x to 0, etc. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 11

12 The ALU in the CPU context (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 / division Where is the seat of advanced math operations? a typical hardware/software tradeoff. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 13

Historical 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

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

14 Historical 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, language of reasoning Leiniz s medallion for the Duke of Brunswick The dream s end: Turing and Goedl in 1930 s. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter 2: Boolean Arithmetic slide 14

Boolean Arithmetic. Building a Modern Computer From First Principles.

Boolean Arithmetic. Building a Modern Computer From First Principles. Boolean Arithmetic Building a Modern Computer From First Principles www.nand2tetris.org Elements of Computing Systems, Nisan & Schocken, MIT Press, www.nand2tetris.org, Chapter 2: Boolean Arithmetic slide