Applications. Smartphone, tablet, game controller, antilock brakes, microprocessor, Wires

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1 COMPUTER SCIENCE Combinational circuits 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) Computer Science Q. Why combinational circuits? A. Accurate, reliable, general purpose, fast, cheap. 2. Combinational Circuits Computer cience S An Interdisciplinary Approach Section 6. Basic abstractions On and off. Wire: propagates on/off value. Switch: controls propagation of on/off values through wires. ROBERT SEDGEWICK K E V I N WAY N E Applications. Smartphone, tablet, game controller, antilock brakes, microprocessor, 2 COMPUTER SCIENCE 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. 2. Combinational Circuits thick wires are ON Building blocks Boolean algebra Digital circuits Adder power connection thin wires are OFF CS.2.A.Circuits.Basics 4

2 Controlled Switch Controlled Switch Switches control propagation of on/off values through wires. Simplest case involves two connections: control (input) and output. control OFF: output ON control ON: output OFF Switches control propagation of on/off values through wires. General case involves three connections: control input, data input and output. control OFF: output is connected to input control ON: output is disconnected from input control input OFF control input OFF data input OFF control input ON control input ON output OFF data input OFF control input OFF output ON control input ON output OFF data input ON 5 Controlled switch: example implementation output ON data input ON Idealized model of pass transistors found in real integrated circuits. output OFF 6 First level of abstraction A relay is a physical device that controls a switch with a magnet 3 connections: input, output, control. Magnetic force pulls on a contact that cuts electrical flow. schematic output 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 switch VLSI = Very Large Scale Integration switch Technology Deposit materials on substrate. relay pneumatic air pressure Key properties Lines are wires. Certain crossing lines are controlled switches. vacuum tube fluid relay 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 COMPUTER SCIENCE Circuit anatomy 2. Combinational Circuits Building blocks Boolean algebra Digital circuits Adder Need more levels of abstraction to understand circuit behavior CS.2.B.Circuits.Algebra

4 Boolean algebra Developed by George Boole in 84s to study logic problems Variables represent true or false ( or for short). Basic operations are AND,, and NOT (see table below). Widely used in mathematics, logic and computer science. Truth tables A truth table is a systematic way to define a Boolean function One row for each possible set of argument values. Each row gives the function value for the specified argument values. N inputs: 2 N rows needed. operation Java notation logic notation circuit design (this lecture) AND x && y x y xy x y x y x + y NOT! x x x' various notations in common use George Boole x x' x y xy x y x + y x y x y X Example: (stay tuned for proof) DeMorgan's Laws (xy)' = (x' + y' ) (x + y)' = x'y' NOT AND X Relevance to circuits. Basis for next level of abstraction. Copyright 24, Sidney Harris Truth table proofs Truth tables are convenient for establishing identities in Boolean logic One row for each possibility. Identity established if columns match. All Boolean functions of two variables Q. How many Boolean functions of two variables? A. 6 (all possibilities for the 4 bits in the truth table column). Proofs of DeMorgan's laws Truth tables for all Boolean functions of 2 variables x y ZERO AND x y X EQ y x NAND ONE x y xy (xy)' x y x' y' x' + y' x y x + y (x + y)' x y x' y' x'y' (xy)' = (x' + y' ) (x + y)' = x'y' 5 6

5 Functions of three and more variables Universality of AND, and NOT Q. How many Boolean functions of three variables? A. 256 (all possibilities for the 8 bits in the truth table column). Every Boolean function can be represented as a sum of products Form an AND term for each in Boolean function. all the terms together. x y z AND Some Boolean functions of 3 variables Examples AND logical AND iff any inputs is ( iff all inputs ) logical iff any input is ( iff all inputs ) logical iff any input is ( iff all inputs ) majority iff more inputs are than odd parity iff an odd number of inputs are Q. How many Boolean functions of N variables? A. 2 2N N all extend to N variables number of Boolean functions with N variables = = = 65, = 4,294,967, = 8,446,744,73,79,55,66 7 x y z x'yz xy'z ' Expressing as a sum of products x'yz + xy'z + ' + = Def. A set of operations is universal if every Boolean function can be expressed using just those operations. Fact. { AND,, NOT } is universal. 8 COMPUTER SCIENCE A basis for digital devices Claude Shannon connected circuit design with boolean algebra in Combinational Circuits Building blocks Boolean algebra Digital circuits Adder Possibly the most important, and also the most famous, master's thesis of the [2th] century. Howard Gardner Key idea. Can use boolean algebra to systematically analyze circuit behavior. Claude Shannon 96 2 CS.2.C.Circuits.Digital 2

6 A second level of abstraction: logic gates Gates with arbitrarily many inputs boolean function NOT notation x' truth table x x' classic symbol x our symbol x' under the cover circuit (gate) proof iff x is Multiway gates. : if any input is ; if all inputs are. : if any input is ; if all inputs are. Generalized: Negate some inputs. (x + y)' x y (x+y)' x y (x+y)' iff x and y are both multiway gate classic symbol u+v+w+x+y+z our symbol u+v+w+x+y+z under the cover if all inputs are ; if any input is example x + y x y x y x+y x+y = ((x + y)')' multiway gate (u+v+w+x+y+z)' = u'v'w'x'y'z' u'v'w'x'y'z' if all inputs are ; if any input is AND xy x y AND x y AND xy xy = (x' + y')' 2 generalized (u+v'+w'+x+y+z')' = u'vwx'y'z u'vwx'y'z iff u, x, and y are and v, w, and z are 22 Generalized gate application: Decoder Creating a digital circuit that computes a boolean function: majority A decoder uses a binary address to switch on a single output line n address inputs, 2 n outputs. Uses all 2 n different generalized gates. Addressed output line is ; all others are. x y z a a a2 a3 a4 a5 a6 a7 x'y'z' x'y'z x'yz' x'yz xy'z' xy'z ' (x+y+z)' (x+y+z')' (x+y'+z)' (x+y'+z')' (x'+y+z)' (x+y'+z)' (x'+y'+z)' (x'+y'+z')' Next. Circuits for any boolean function. = 6 example is hot 23 Use the truth table Identify rows where the function is. Use a generalized gate for each. the results together. Example : Majority function x y z = x'yz + xy'z + ' + generalized s implement AND terms in sum-of -products x'yz = (x + y' + z' )' xy'z = (x' + y + z' )' ' = (x' + y' + z)' = (x' + y' + z' )' majority circuit example is 24

7 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 gate for each. the results together. Example 2: Odd parity function x y z = x'y'z + x'yz' + xy'z' + x'y'z = (x + y + z' )' x'yz' = (x' + y' + z)' xy'z' = (x' + y + z)' = (x' + y' + z' )' odd parity circuit example is 25 Problem: Design a circuit that computes a given boolean function. Ingredients gates. NOT gates. gates. Wire. Method Step : Represent input and output with Boolean variables. Step 2: Construct truth table to define the function. Step 3: Identify rows where the function is. Step 4: Use a generalized for each and the results. Bottom line (profound idea): Yields a circuit for ANY function. Caveat (stay tuned): Circuit might be huge. x y z x y z 26 Self-assessment on combinational circuit design Encapsulation Q. Design a circuit to implement X(x, y). 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 and outputs is the API. We control complexity by encapsulating circuits as we do with ADTs. xy AND X 27 28

8 COMPUTER SCIENCE Let's make an adder circuit same ideas scale to 64-bit Goal. x + y = z for 4-bit binary integers. adder in your computer 4-bit adder: 9 inputs, 5 outputs. Each output is a boolean function of the inputs Combinational Circuits Building blocks Boolean algebra Digital circuits carry out x3 y3 x2 y2 x y x y ADD carry in + Adder carry out c4 c3 c2 c c carry in x3 x2 x x + y3 y2 y y z3 z2 z z CS.2.D.Circuits.Adder z3 z2 z z 3 Let's make an adder circuit Let's make an adder circuit Goal: x + y = z for 4-bit integers. Strawman solution: Build truth tables for each output bit. c4 c3 c2 c c x3 x2 x x + y3 y2 y y z3 z2 z z Goal: x + y = z for 4-bit integers. Do one bit at a time. Build truth table for carry bit. Build truth table for sum bit. A surprise! Carry bit is. Sum bit is. c4 c3 c2 c c x3 x2 x x + y3 y2 y y z3 z2 z z c x3 x2 x x y3 y2 y y c4 z3 z2 z z 4-bit adder truth table = 52 rows! xi yi ci ci+ xi yi ci zi carry bit sum bit Q. Why is this a bad idea? A. 28-bit adder: rows >> # electrons in universe! 3 32

9 Let's make an adder circuit Adder example (8-bit) = 23 Goal: x + y = z for 4-bit integers. x3 y3 x2 y2 x y x y = 49 Do one bit at a time. Use and circuits. Chain together -bit adders to "ripple" carries. c c4 c3 c2 c c4 c4 c3 c2 c c x3 x2 x x + y3 y2 y y z3 z2 z z z3 z2 z z 33 = Adder interface (4-bit) Adder component-level view (4-bit) A bus is a group of wires that connect components (carrying data values). x x x2 x3 y y y2 y3 input busses carry in x x x2 x3 y y y2 y3 input busses carry in ADD ADD c4 c3 c2 c c c4 c3 c2 c c x3 x2 x x x3 x2 x x + y3 y2 y y carry out + y3 y2 y y carry out z3 z2 z z z3 z2 z z z z z2 z3 output bus 35 z z z2 z3 output bus 36

10 Adder switch-level view (4-bit) Summary x x x2 x3 y y y2 y3 Lessons for software design apply to hardware! Interface describes behavior of circuit. Implementation gives details of how to build it. Boolean logic gives understanding of behavior. input busses carry in Layers of abstraction apply with a vengeance! On/off. Controlled switch. [relay, pass transistor] Gates. [NOT,,, AND] Boolean functions. [, ] Adder.... Arithmetic/Logic unit (ALU). TOY machine (stay tuned). Your computer. ADD c4 + c3 c2 c c x3 x2 x x y3 y2 y y z3 z2 z z carry out z z z2 z3 output bus COMPUTER SCIENCE Computer Science 2. Combinational Circuits Computer cience S An Interdisciplinary Approach Section 6. ROBERT SEDGEWICK K E V I N WAY N E 37 AND 38

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