Systems I Logic Design I Topics Topics Digital logic Logic gates Simple combinational logic circuits
A Simple C statement….. C = A + B; A + B; C = What pieces of hardware do you think you might need? pieces of hardware do you think you might need? What Storage - for values A, B, C Computation logic - to compute + A way to tell the computer to retrieve the values from storage, add them together, and put the result back in storage This could be accomplished with a single command (instruction) or with multiple of them. 2
Overview of Logic Design Fundamental Hardware Requirements Fundamental Hardware Requirements Communication How to get values from one place to another Computation Storage Bits are Our Friends Bits are Our Friends Everything expressed in terms of values 0 and 1 Communication Low or high voltage on wire Computation Compute Boolean functions Storage Store bits of information 3
Digital Signals 0 1 0 Voltage Time Use voltage thresholds to extract discrete values from continuous signal Simplest version: 1-bit signal Either high range (1) or low range (0) With guard range between them Not strongly affected by noise or low quality circuit elements Can make circuits simple, small, and fast 4
Computing with Logic Gates And Or Not a a a out out out b b out = a && b out = a || b out = ! a Logic gates constructed from transistors Outputs are Boolean functions of inputs Respond continuously to changes in inputs With some, small delay Falling Delay Rising Delay a && b b Voltage a Time 5
Truth Tables And Or Not a a a out out out b b out = a && b out = a || b out = ! a a b out a b out a out a b out a b out a out 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 6
What about this? a a a b b b c c c out out out a a a b b b c c c out out out 0 0 0 0 0 0 1 0 0 0 0 0 0 1 a 0 0 0 0 1 1 0 b 0 0 0 0 1 1 0 out c 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 out = !((a && b) || c) 1 1 0 0 1 1 0 1 1 0 0 1 1 0 out = ~(a*b + c) 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 7
Combinational Circuits Acyclic Network Primary Primary Inputs Outputs Acyclic Network of Logic Gates Acyclic Network of Logic Gates Continously responds to changes on primary inputs Primary outputs become (after some delay) Boolean functions of primary inputs 8
Bit Equality Bit equal a HCL Expression eq bool eq = (a&&b)||(!a&&!b) b Generate 1 if a and b are equal Hardware Control Language (HCL) Hardware Control Language (HCL) Very simple hardware description language Boolean operations have syntax similar to C logical operations We ʼ ll use it to describe control logic for processors 9
Hardware Control Language Very simple hardware description language Can only express limited aspects of hardware operation Parts we want to explore and modify Data Types Data Types bool : Boolean a , b , c , … int : words A , B , C , … Does not specify word size---bytes, 32-bit words, … Statements Statements bool a = bool-expr ; int A = int-expr ; 10
HCL Operations Classify by type of value returned Boolean Expressions Boolean Expressions Logic Operations a && b , a || b , !a Word Comparisons A == B , A != B , A < B , A <= B , A >= B , A > B Set Membership A in { B, C, D } » Same as A == B || A == C || A == D Word Expressions Word Expressions Case expressions [ a : A; b : B; c : C ] Evaluate test expressions a , b , c , … in sequence Return word expression A , B , C , … for first successful test 11
Word Equality Word-Level Representation b 31 eq 31 B Eq Bit equal = a 31 A b 30 eq 30 Bit equal a 30 HCL Representation bool Eq = (A == B) Eq b 1 eq 1 Bit equal 32-bit word size a 1 b 0 eq 0 HCL representation Bit equal Equality operation a 0 Generates Boolean value 12
Bit-Level Multiplexor s Bit MUX HCL Expression bool out = (s&&a)||(!s&&b) b out a Control signal s Data signals a and b Output a when s=1, b when s=0 13
Word Multiplexor Word-Level Representation s s B b 31 Out MUX out 31 A a 31 HCL Representation int Out = [ b 30 s : A; out 30 1 : B; a 30 ]; Select input word A or B depending on control signal s HCL representation Case expression Series of test : value pairs b 0 Output value for first successful out 0 test a 0 14
HCL Word-Level Examples Minimum of 3 Words Find minimum of three int Min3 = [ input words A < B && A < C : A; C HCL case expression Min3 B B < A && B < C : B; MIN3 A 1 : C; Final case guarantees ]; match How would you build this? 4-Way Multiplexor s1 s0 Select one of 4 inputs int Out4 = [ based on two control D0 !s1&&!s0: D0; bits D1 Out4 MUX4 !s1 : D1; D2 HCL case expression !s0 : D2; D3 1 : D3; Simplify tests by ]; assuming sequential matching 15
Simple computations are just combinational logic circuits One Bit Adder A B Sum = A*B + A*C in + B*C in C out = A^B^Cin C out + C in = A*B*C in + A*B*C in + A*B*C in + A*B*C in Sum How do you do subtract? Four Bit Adder How do you do multiply? A 3 B 3 A 2 B 2 A 1 B 1 A 0 B 0 0 Sum 4 + + + + Sum 3 Sum 2 Sum 1 Sum 0 16
Arithmetic Logic Unit 0 1 2 3 Y Y Y Y A A A A A A A A X + Y X - Y X & Y X ^ Y L L L L U U U U X X X X B B B B OF OF OF OF ZF ZF ZF ZF SF SF SF SF Combinational logic Continuously responding to inputs Control signal selects function computed Corresponding to 4 arithmetic/logical operations in Y86 Also computes values for condition codes OF = overflow flag, ZF = Zero Flag, SF = Sign Flag 17
Arithmetic Logic Unit <s1,s0> = 00 <s1,s0> = 01 < s 1,s0> = 10 <s1,s0> = 11 Y Y Y Y A A A A A A A A X + Y X - Y X & Y X ^ Y L L L L U U U U X X X X B B B B OF OF OF OF ZF ZF ZF ZF SF SF SF SF int Out = [ !s1&&!s0: X+Y; !s1&&s0: Y-Y; s1&&!s0: X&Y; 1 : X^Y; ]; 18
Summary Today Today Basic logic elements Combinational logic circuits Truth tables, gates Aggregating logic elements Multiplexors, ALUs, etc. Next Time Next Time Circuits that remember 19
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