CSCI 1010 Models of Computa3on. Lecture 02 Func3ons and Circuits

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1 CSCI 1010 Models of Computa3on Lecture 02 Func3ons and Circuits

2 Overview Func3ons and languages Designing circuits from func3ons Minterms and the DNF Maxterms and CNF Circuit complexity Algebra of Boolean expressions Illustra3on using the Full Adder Dual rail logic John E. Savage CSCI 1010 Lect 02 2

3 Mathema3cal Preliminaries The Cartesian product A B of two sets is the set of pairs, the first from A, the second from B. If A = {0,1}, B = {2,3}, A B = {02, 03, 12, 13} The n-fold Cartesian product of a set A with itself is denoted A n. E.g. A 3 = {0,1} 3 = {000,001,010,011, } If a, b are in A, a b denotes the string ab in A B. The opera3on is called concatena3on. The empty string ϵ sa3sfies a ϵ = a, ϵ a = a. John E. Savage CSCI 1010 Lect 02 3

4 Mathema3cal Preliminaries The Kleene star A * = {ϵ} U A U A 2 U A 3 U, that is, the set of strings of all lengths including the empty string. {0,1} * = {ϵ, 0, 1, 00, 01, 10, 11, 000, 001, } John E. Savage CSCI 1010 Lect 02 4

5 Languages A language is a set of strings over an alphabet. A language L over the alphabet A sa3sfies L A *, that is, L is a subset of A *. Examples of languages The set of 10-bit binary strings (in {0,1}) with an odd number of 1s is a subset of {0,1} 10. The strings that form acceptable Pascal programs. John E. Savage CSCI 1010 Lect 02 5

6 Boolean Func3ons A Boolean func3on f : {0,1} k {0,1} f has k inputs and one output. f(x 1, x 2,, x k ) is the value of f on inputs x 1, x 2,, x k Standard Boolean func3ons: AND on two binary inputs, f AND (x,y) = x ᴧ y OR on two binary inputs, f OR (x,y) = x v y EXOR on two binary inputs, f EXOR (x,y) = x y NOT on one binary input, f NOT (x) = x or f NOT (x) = x John E. Savage CSCI 1010 Lect 02 6

7 Languages and Func3ons Let f : {0,1} k {0,1} The k-tuples x for which f(x) = 1 is a language They form a subset of the strings in {0,1} k. Given a language L A *, there is a natural characteris3c func3on f L : A * {0,1} associated with the language L. If x is in L, f L (x) = 1. If x is not in L, f L (x) = 0. John E. Savage CSCI 1010 Lect 02 7

8 Logic Circuits and Func3ons A logic circuit is a directed acyclic graph (DAG) in which node labels are Boolean func3ons. g 7 := g 5 v g 6 ; g 5 := g 1 ᴧ g 4 ; g 6 := g 3 ᴧ g 2 ; g 4 := g 2 ; g 3 := g 1 ; g 1 := x; g 2 = y The output func3on is g 7 = x y How would you show this? John E. Savage CSCI 1010 Lect 02 8

9 Straight-Line Programs A straight-line program (SLP) is a program that has no branching or looping. An SLP can use arithme3c or Boolean opera3ons A Boolean SLP is an SLP in which the opera3ons are Boolean. The graph of a Boolean SLP is a circuit. The graph is directed and acyclic (a dag) John E. Savage CSCI 1010 Lect 02 9

10 Boolean Straight-Line Program (1 READ x) (2 READ y) (3 NOT 1) (4 NOT 2) (5 AND 1 4) (6 AND 3 2) (7 OR 5 6) Nota3on: (r OP s t) g r := g s OP g t This SLP computes the Exclusive OR of x and y: g 7 = x y = (x y) (x y) John E. Savage CSCI 1010 Lect 02 10

11 Compu3ng Mul3-Output Func3ons Computes f : {0,1} 3 {0,1} 2, f(x,y,z) = (c,s) This Full Adder uses AND, OR, XOR carry sum X Y Z C S is Exclusive OR John E. Savage CSCI 1010 Lect 02 11

12 Designing Circuits From Func3ons Consider binary func3on g : {0,1} n {0,1} m with m outputs. It can be viewed as m func3ons, that is, g = (f 1,f 2,, f m ) where f j : {0,1} n {0,1} are Boolean func3ons. We can realize each f j separately by a circuit or find common sub-circuits to reduce number of gates (opera3ons). John E. Savage CSCI 1010 Lect 02 12

13 Designing Circuits From Func3ons Start by realizing one-output func3ons f(x 1,,x n ) A literal is a variable (x j ) or its complement (the NOT of the variable) (x j ) A minterm is the AND of one literal for each variable of a func3on. E.g. x 1 x 2 x n A minterm = 1 exactly when each literal = 1. E.g. x 1 = 0, x 2 = 1,, x n = 0. It has value 0 on 2 n -1 inputs. John E. Savage CSCI 1010 Lect 02 13

14 Designing Circuits From Func3ons Construct circuit for f : {0,1} n {0,1} by forming the OR of those minterms in the inputs to f for which f has value 1. E.g. c = xyz xyz xyz xyz X Y Z C Minterm expansions can oqen be simplified E.g. c = xy xz yz Tables offer clear (but long) way to show func3onal equivalence output John E. Savage CSCI 1010 Lect 02 14

15 Disjunc3ve Normal Form The minterm expansion is also called the disjunc3ve normal form (DNF). Disjunct is an old word for OR. Defini3on: The DNF is the OR of minterms corresponding to the inputs for which a Boolean func3on has value 1. Why does a formula compute the func3on? John E. Savage CSCI 1010 Lect 02 15

16 Maxterms The maxterm of a Boolean func3on is the OR of one literal for each variable of the func3on. E.g. x y z A maxterm is 1 except when all literals are 0. E.g. The above maxterm is 0 when x = 1, y = 1, and z = 0. It has value 1 on 7 of the 8 inputs! X Y Z C John E. Savage CSCI 1010 Lect 02 16

17 Conjunc3ve Normal Form The conjunc3ve normal form (CNF) of func3on f is the AND of the maxterms for which f = 0. Conjunct is an old word for AND. John E. Savage CSCI 1010 Lect 02 17

18 CNF Example The CNF of f is 1 except where maxterms = 0. E.g. c = (x y z ) (x y z) (x y z) (x y z) The CNF can oqen be simplified. E.g. c = (x y) (x z) (y z) = 0 if 2 or more inputs are 0 X Y Z C John E. Savage CSCI 1010 Lect 02 18

19 Circuit Complexity Given a binary func3on f : {0,1} n {0,1} m its circuit size, denoted C Ω (f), is the size of the smallest circuit (fewest gates) drawn from the basis Ω. A basis Ω is complete if every binary func3on can be realized by a circuit with gates from Ω. Ω = {AND, OR, NOT} is complete. Ω = {AND, EXOR} and Ω = {NAND} are complete. John E. Savage CSCI 1010 Lect 02 19

20 Compu3ng Circuit Size For complete bases the circuit size changes by a mul3plica3ve factor when the basis changes. Why is this statement true? It is hard to compute C Ω (f) for arbitrary func3on f. Thus, we compute upper and lower bounds. Best approach: enumerate circuits contain k gates, star3ng at k = 0, un3l we find a circuit compu3ng f. Courses are taught on circuit complexity! John E. Savage CSCI 1010 Lect 02 20

21 Monotone Func3ons A func3on f : {0,1} n {0,1} m is monotone if its none of its outputs decreases when an input is increased from 0 to 1. Not all func3ons are monotone. NOT is not monotone Monotone func3ons can be realized over the basis {AND, OR}, which is not complete. Adding NOT this basis can greatly reduce circuit size of a monotone func3on. John E. Savage CSCI 1010 Lect 02 21

22 The Algebra of Boolean Expressions Commuta3vity Absorp3on Rules x y = y x, x x = x, x x = 1 x y = y x Distribu3vity x x = x, x x = 0 1. x (y z) = (x y) (x z) (AND distributes over OR) 2. x (y z) = (x y) (x z) (OR distributes over AND) E.g. simplify (a b c) (a b c) Let x = (a b) and apply rule 1. (a b c) (a b c) = (x c) (x c) = x (c c) = x = (a b) John E. Savage CSCI 1010 Lect 02 22

23 The Algebra of Boolean Expressions DeMorgan s Rules x y = x y x y = x y Exclusive OR defini3on x y = (x y) (x y) Exclusive OR expansion (= 1 if 1 or 3 inputs = 1) x y z = x (y z)= (x (y z)) (x (y z)) = (x ((y z) (y z))) (x ((y z) (y z))) ((y z) (y z)) = (y z) (y z) = (y z) (y z) = (y y) (y z) (z y) (z z)=(y z) (z y) x y z = (x y z) (x y z) (x y z) (x y z) John E. Savage CSCI 1010 Lect 02 23

24 Full Adder s = x y z c = = xy xz yz X Y Z C S The formulas use 7 binary opera3ons. The above circuit uses only 5! John E. Savage CSCI 1010 Lect 02 24

25 Implica3on Boolean Operator Implica3on is wriyen x y. It is interpreted, if x is True, then y is cannot be False. The truth table for z = x y is shown below. Formula for implica3on over {AND, OR, NOT}? X Y Z John E. Savage CSCI 1010 Lect 02 25

26 Dual Rail Logic Instead of represen3ng a truth value by a literal x, let s represent it by (x,x). Thus, True and False are represented by (0,1) and (1,0). How is AND represented in this new system? How about OR and NOT? What does a circuit look like? John E. Savage CSCI 1010 Lect 02 26

27 Dual Rail Logic AND: Inputs (x,x) and (y,y). Output (z,z) where z = x y and z = x y. OR: Inputs (x,x) and (y,y). Output (z,z) where z = x y and z = x y. NOT: Input (x,x), output (z,z) where z = x, z = x. Thus, flip the inputs. AND, OR, NOT circuit can be converted to dual rail by doubling AND, OR gates, dropping NOTs John E. Savage CSCI 1010 Lect 02 27

28 Review Func3ons and languages Designing circuits from func3ons Minterms and the DNF Maxterms and CNF Circuit complexity Algebra of Boolean expressions Illustra3on using the Full Adder Dual rail logic John E. Savage CSCI 1010 Lect 02 28

CHAPTER1: Digital Logic Circuits Combination Circuits

CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.