# Computer Science. 19. Combinational Circuits. Computer Science. Building blocks Boolean algebra Digital circuits Adder circuit. Arithmetic/logic unit

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1 PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science 9. Combinational Circuits Computer Science 9. Combinational Circuits Building blocks Boolean algebra Digital circuits Adder circuit An Interdisciplinary Approach Section 6. Arithmetic/logic unit ROBERT SEDGEWICK K E V I N WAY N E CS.9.A.Circuits.Basics Context Wires Wires propagate on/off values ON (): connected to power. OFF (): not connected to power. Any wire connected to a wire that is ON is also ON. Drawing convention: "flow" from top, left to bottom, right. Q. What is a combinational circuit? A. A digital circuit (all signals are or ) with no feedback (no loops). analog circuit: signals vary continuously sequential circuit: loops allowed (stay tuned) Q. Why combinational circuits? A. Accurate, reliable, general purpose, fast, cheap. thick wires are ON Basic abstractions power connection On and off. Wire: propagates on/off value. Switch: controls propagation of on/off values through wires. thin wires are OFF Applications. Smartphone, tablet, game controller, antilock brakes, microprocessor,

2 Controlled Switch Controlled Switch Switches control propagation of on/off values through wires. Simplest case involves two connections: control (input) and. control OFF: ON control ON: OFF Switches control propagation of on/off values through wires. General case involves three connections: control input, data input and. control OFF: is connected to input control ON: is disconnected from input control input OFF control input OFF data input OFF control input ON control input ON OFF data input OFF control input OFF ON control input ON OFF data input ON 5 Controlled switch: example implementation ON data input ON Idealized model of pass transistors found in real integrated circuits. OFF 6 First level of abstraction A relay is a physical device that controls a switch with a magnet connections: input,, control. Magnetic force pulls on a contact that cuts electrical flow. schematic OFF control off Switches and wires model provides separation between physical world and logical world. We assume that switches operate as specified. That is the only assumption. Physical realization of switch is irrelevant to design. control on magnet (off ) contact connection broken Physical realization dictates performance Size. Speed. Power. magnet on pulls contact up New technology immediately gives new computer. spring Better switch? Better computer. Basis of Moore's law. 7 all built with "switches and wires" 8

3 Switches and wires: a first level of abstraction Switches and wires: a first level of abstraction technology technology information pneumatic air pressure switch switch VLSI = Very Large Scale Integration relay (9s) Technology Deposit materials on substrate. Key properties Lines are wires. Certain crossing lines are controlled switches. vacuum tube fluid relay (now) water pressure electric potential Amusing attempts that do not scale but prove the point Key challenge in physical world Fabricating physical circuits with billions of wires and controlled switches transistor pass transistor in integrated circuit Key challenge in abstract world Understanding behavior of circuits with billions of wires and controlled switches atom-thick transistor Bottom line. Circuit = Drawing (!) Real-world examples that prove the point 9 Circuit anatomy Image sources Need more levels of abstraction to understand circuit behavior CS.9.A.Circuits.Basics

4 PART II: ALGORITHMS, MACHINES, and THEORY Boolean algebra Developed by George Boole in 8s to study logic problems Variables represent true or false ( or for short). Basic operations are AND, OR, and NOT (see table below). Widely used in mathematics, logic and computer science. 9. Combinational Circuits Building blocks Boolean algebra Digital circuits Adder circuit Arithmetic/logic unit operation Java notation logic notation circuit design (this lecture) AND x && y x y xy OR x y x y x + y NOT! x x x' Example: (stay tuned for proof) DeMorgan's Laws (xy)' = (x' + y' ) (x + y)' = x'y' various notations in common use George Boole CS.9.B.Circuits.Algebra Relevance to circuits. Basis for next level of abstraction. Copyright, Sidney Harris Truth tables Truth table proofs A truth table is a systematic way to define a Boolean function One row for each possible set of arguments. Each row gives the function value for the specified arguments. N inputs: N rows needed. Truth tables are convenient for establishing identities in Boolean logic One row for each possibility. Identity established if columns match. x x' NOT x y xy x y x + y x y NOR x y XOR Proofs of DeMorgan's laws x y xy (xy) ' x y x' y' x' + y' x y x + y NOR (x + y)' NOR x y x' y' x'y' AND OR NOR XOR (xy)' = (x' + y' ) (x + y)' = x'y' 5 6

5 All Boolean functions of two variables Functions of three and more variables Q. How many Boolean functions of two variables? A. 6 (all possibilities for the bits in the truth table column). Q. How many Boolean functions of three variables? A. 56 (all possibilities for the 8 bits in the truth table column). Truth tables for all Boolean functions of variables x y ZERO AND x y XOR OR NOR EQ y x NAND ONE 7 AND OR NOR MAJ ODD Some Boolean functions of variables Examples AND logical AND iff any inputs is ( iff all inputs ) OR logical OR iff any input is ( iff all inputs ) NOR logical NOR iff any input is ( iff all inputs ) MAJ majority iff more inputs are than ODD odd parity iff an odd number of inputs are Q. How many Boolean functions of N variables? A. (N ) N all extend to N variables number of Boolean functions with N variables = 6 8 = 56 6 = 65,56 5 =,9,967, = 8,6,7,7,79,55,66 8 Universality of AND, OR and NOT Every Boolean function can be represented as a sum of products Form an AND term for each in Boolean function. OR all the terms together. Image sources MAJ x'yz xy'z xyz' xyz x'yz + xy'z + xyz' + xyz = MAJ Def. A set of operations is universal if every Boolean function can be expressed using just those operations. Fact. { AND, OR, NOT } is universal. Expressing MAJ as a sum of products 9 CS.9.B.Circuits.Algebra

6 PART II: ALGORITHMS, MACHINES, and THEORY A basis for digital devices Claude Shannon connected circuit design with Boolean algebra in Combinational Circuits Building blocks Boolean algebra Digital circuits Adder circuit Arithmetic/logic unit Possibly the most important, and also the most famous, master's thesis of the [th] Howard Gardner Key idea. Can use Boolean algebra to systematically analyze circuit behavior. Claude Shannon 96 CS.9.C.Circuits.Digital A second level of abstraction: logic gates Multiway OR gates boolean function NOT notation x' truth table x x' classic symbol x our symbol x' under the cover circuit (gate) x x' proof iff x is OR gates with multiple inputs. if any input is. if all inputs are. NOR (x + y)' x y NOR (x+y)' x y NOR (x+y)' x y (x+y)' iff x and y are both classic symbol u v w u+v+w+x+y+z our symbol u v w OR u+v+w+x+y+z under the cover u v w if inputs are ; if any input is u+v+w+x+y+z OR AND x + y xy x y OR x y AND x y OR x y AND x+y xy x y x y x+y xy NOR x+y = ((x + y)')' NOR xy = (x' + y')' Multiway OR gates are oriented vertically in our circuits. Learn to recognize them! examples

7 Multiway generalized AND gates Multiway generalized AND gates. for exactly set of input values. for all other sets of input values. Pop quiz on generalized AND gates Q. Give the Boolean function computed by these gates. Q. Also give the inputs for which the is. gate u v w function inputs that another set of inputs u v w AND uvwxyz u v w u v w generalized u'vwx'y'z NOR u v w u'v'w'x'y'z' same as (u + v + w + x + y + z)' Might also call these "generalized NOR gates"; we consistently use AND. 5 6 Pop quiz on generalized AND gates A useful combinational circuit: decoder Q. Give the Boolean function computed by these gates. Q. Also give the inputs for which the is. u v w uv'wxy'z x'y'z' x'y'z x'yz' Decoder n input lines (address). n s. Addressed is. All other s are. Example: -to-8 decoder = 6 u v w u'vwx'y'z x'yz xy'z' s -5 and 7 are Get the idea? If not, replay this slide, like flash cards. Note. From now on, we will not label these gates. xy'z xyz' xyz is 7 8

9 Decoder/demux Decoder/demux n address inputs. data input with value x. n pairs. Addressed pair has value (, x ). All other s are. Example: -to-8 decoder/demux = 5 x Creating a digital circuit that computes a boolean function: majority Use the truth table Identify rows where the function is. Use a generalized AND gate for each. OR the results together. Example : Majority function MAJ xyz multiway OR gate MAJ Implementation Add decoder to demux. pairs - and 6-7 are (, ) term gate pair 5 has value (, x) x'yz Application (next lecture) Access and control write of memory word [Use addr bits of instruction in IR.] MAJ = x'yz + xy'z + xyz' + xyz xy'z xyz' xyz MAJ majority circuit example MAJ is Creating a digital circuit that computes a boolean function: odd parity Combinational circuit design: Summary Use the truth table Identify rows where the function is. Use a generalized AND gate for each. OR the results together. Example : Odd parity function ODD term x'y'z x'yz' xy'z' gate xyz multiway OR gate ODD Problem: Design a circuit that computes a given boolean function. Ingredients OR gates. NOT gates. NOR gates. Wire. use to make generalized AND gates Method Step : Represent input and with Boolean variables. Step : Construct truth table to define the function. Step : Identify rows where the function is. Step : Use a generalized AND for each and OR the results. MAJ xyz MAJ ODD xyz ODD ODD = x'y'z + x'yz' + xy'z' + xyz xyz ODD odd parity circuit example ODD is 5 Bottom line (profound idea): Yields a circuit for ANY function. Caveat: Circuit might be huge (stay tuned). MAJ ODD 6

10 Pop quiz on combinational circuit design Q. Design a circuit to implement XOR(x, y). Pop quiz on combinational circuit design Q. Design a circuit to implement XOR(x, y). A. Use the truth table Identify rows where the function is. Use a generalized AND gate for each. OR the results together. XOR function circuit interface x y XOR xy term gate x'y xy' XOR = x'y + xy' XOR XOR 7 8 Encapsulation Encapsulation in hardware design mirrors familiar principles in software design Building a circuit from wires and switches is the implementation. Define a circuit by its inputs, controls, and s is the API. We control complexity by encapsulating circuits as we do with ADTs. Image sources NOT AND DECODER DEMULTIPLEXER DECODER/DEMUX ODD OR MAJ XOR 9 CS.9.C.Circuits.Digital

12 Let's make an adder circuit! Let's make an adder circuit! Goal: z = x + y for 8-bit integers. Do one bit at a time. Build truth table for carry bit. Build truth table for sum bit. carry bit xi yi ci ci+ MAJ A surprise! Carry bit is MAJ. Sum bit is ODD. sum bit xi yi ci zi c8 c7 c6 c5 c c c c x7 x6 x5 x x x x x + y7 y6 y5 y y y y y z7 z6 z5 z z z z z ODD Goal: z = x + y for -bit integers. Do one bit at a time. Carry bit is MAJ. Sum bit is ODD. Chain -bit adders to "ripple" carries. c8 c7 c6 c5 c c c c x7 x6 x5 x x x x x + y7 y6 y5 y y y y y z7 z6 z5 z z z z z x7 y7 x6 y6 x5 y5 x y x y x y x y x y c7 c6 c5 c c c c c z7 z6 z5 z z z z z 5 6 An 8-bit adder circuit Layers of abstraction Lessons for software design apply to hardware Interface describes behavior of circuit. Implementation gives details of how to build it. Exploit understanding of behavior at each level. Layers of abstraction apply with a vengeance On/off. Controlled switch. [relay, pass transistor] Gates. [NOT, OR, AND] Boolean functions. [MAJ, ODD] Adder. Arithmetic/Logic unit (next). CPU (next lecture, stay tuned). ADD NOT OR AND MAJ ODD Vastly simplifies design of complex systems and enables use of new technology at any layer 7 8

13 PART II: ALGORITHMS, MACHINES, and THEORY 9. Combinational Circuits Building blocks Boolean algebra Digital circuits Adder circuit Arithmetic/logic unit CS.9.D.Circuits.Adder CS.9.E.Circuits.ALU Next layer of abstraction: modules, busses, and control lines Arithmetic and logic unit (ALU) module Basic design of our circuits Organized as modules (functional units of TOY: ALU, memory, register, PC, and IR). Connected by busses (groups of wires that propagate information between modules). Controlled by control lines (single wires that control circuit behavior). Ex. Three functions on 8-bit words Two input busses (arguments). One bus (result). Three control lines. input busses ALU (8-bit) Conventions Bus inputs are at the top, input connections are at the left. Bus s are at the bottom, connections are at the right. Control lines are blue. input busses control lines control lines ADD XOR These conventions make circuits easy to understand. (Like style conventions in coding.) bus 5 AND bus 5

14 Arithmetic and logic unit (ALU) module A simple and useful combinational circuit: one-hot multiplexer Ex. Three functions on 8-bit words Two input busses (arguments). One bus (result). Three control lines. Left-right shifter circuits omitted (see book for details). input busses One-hot multiplexer m selection lines m data inputs. At most one selection line is. Output has value of selected input. selection inputs this is a precondition unlike other circuits we consider select input data input is all other data inputs ignored Implementation One circuit for each function. Compute all values in parallel. ADD is Q. How do we select desired? A. "One-hot muxes" (see next slide). control lines XOR AND data inputs data input is "Calculator" at the heart of your computer. bus 5 is 5 A simple and useful combinational circuit: one-hot multiplexer One-hot multiplexer m selection lines m data inputs. At most one selection line is. Output has value of selected input. Implementation AND corresponding selection and data inputs. OR all results (at most one is ). AND gate multiway OR gate select input data input is is Summary: Useful combinational circuit modules DEMUX DECODER/DEMUX ALU Applications Arithmetic-logic unit (previous slide). Main memory (next lecture). data input is Important to note. No direct connection from input to. a virtual selection switch is 55 Next: Registers, memory, connections, and control. 56

15 PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science Computer Science An Interdisciplinary Approach Section 6. ROBERT SEDGEWICK K E V I N WAY N E CS.9.D.Circuits.Adder 9. Combinational Circuits

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