Computer Architecture: A Constructive Approach Combinational ALU Arvind Computer Science & Artificial Intelligence Lab. Massachusetts Institute of Technology.

Computer Architecture: A Constructive Approach

Combinational ALU

ArvindComputer Science & Artificial Intelligence Lab.Massachusetts Institute of Technology

January 9, 2012 L1-1http://csg.csail.mit.edu/SNU

Computing Devices Then…EDSAC, University of Cambridge, UK, 1949

January 9, 2012

L01-2http://csg.csail.mit.edu/SNU

Computing Devices Now

Dramatic progress in terms of size, speed, cost, reliability

January 9, 2012

L01-3http://csg.csail.mit.edu/SNU

How does a program execute on hardware

A C program to add two arrays:

The hardware must know, for example, How to add and compare two numbers Must have a place to keep the program and data Must know how to fetch instructions and data Must know how to sequence instructions:

Fetch a[i], Fetch b[i], add, store results in c[i], increment i, …

January 9, 2012 L1-4http://csg.csail.mit.edu/SNU

void vvadd( int n, int a[], int b[], int c[] )

{ int i;  for ( i = 0; i < n; i++ )    c[i] = a[i] + b[i];}

Computer Architecture is learning about how programs execute and designing hardware to execute them efficiently

Instruction Set Architecture (ISA)

Computer architecture is the discipline of designing and implementing interfaces through which hardware and software interact This interface is often referred to as the Instruction Set Architecture (ISA)

Examples: Intel’s IA-32, ARM, ARM-Thumb, PowerPC In this class we will use SMIPS, a subset of MIPS ISA

Implementations are deeply affected by the technology issues; we will assume a simple and abstract model of technology based on Silicon

January 9, 2012 L1-5http://csg.csail.mit.edu/SNU

Goal of the Intensive Course

Learn a constructive approach to studying computer architectureLearn a new method of describing architectures where there is less emphasis on figures/diagrams and more emphasis on executable descriptions

Each architecture and each part of it would be defined as executable code in Bluespec

By this Friday you will have designed several different computers on which you will be able to execute your C programs

January 9, 2012 L1-6http://csg.csail.mit.edu/SNU

In this lecture we will study how to implement simple arithmetic-logic operations

we will first look at circuits for component functions such as Add, Shift and then put them together to form an Arithmetic-Logic Unit (ALU)

January 9, 2012 L1-7http://csg.csail.mit.edu/SNU

Arithmetic-Logic Unit (ALU)Computers have many important interconnected parts or subsystems:

Central Processing Unit Memory system including caches I/O systems

ALU resides inside the CPU and carried out all the arithmetic and logical functions

Op - Add, Sub, ... - And, Or, Xor, Not, ... - GT, LT, EQ, Zero, ... Result

Comp?

A

BALU

January 9, 2012 L1-8http://csg.csail.mit.edu/SNU

Full Adder: A one-bit adderfunction fa(a, b, c_in); s = (a ^ b)^ c_in; c_out = (a & b) | (c_in & (a ^ b)); return {c_out,s}; endfunction

Not quite correct –needs type annotations

January 9, 2012 L1-9http://csg.csail.mit.edu/SNU

Full Adder: A one-bit addercorrectedfunction Bit#(2) fa(Bit#(1) a, Bit#(1) b, Bit#(1) c_in); Bit#(1) s = (a ^ b)^ c_in; Bit#(1) c_out = (a & b) | (c_in & (a ^ b)); return {c_out,s}; endfunction

Bit#(1) a declaration says that a is one bit wide

{c_out,s} represents bit concatenation

How big is {c_out,s}?

January 9, 2012 L1-10http://csg.csail.mit.edu/SNU

TypesEvery expression and variable in a Bluespec program has a type; sometimes it is specified explicitly and sometimes it is deduced by the compilerA type is a grouping of values:

Integer: 1, 2, 3, … Bool: True, False Bit: 0,1 A pair of Integers: Tuple2#(Integer, Integer) A function fname from Integers to Integers: function Integer fname (Integer arg)

Thus we say an expression has a type, belongs to a type

January 9, 2012 L1-11http://csg.csail.mit.edu/SNU

The type of each expression is unique

Type declaration versus deduction

User writes down types of some expressions in a program and the compiler deduces the types of the rest of expressionsIf the type deduction cannot be performed or the type declarations are inconsistent then the compiler complains

function Bit#(2) fa(Bit#(1) a, Bit#(1) b, Bit#(1) c_in); Bit#(1) s = (a ^ b)^ c_in; Bit#(2) c_out = (a & b) | (c_in & (a ^ b)); return {c_out,s}; endfunction

type error

January 9, 2012 L1-12http://csg.csail.mit.edu/SNU

Type checking prevents lots of silly mistakes

2-bit Ripple-Carry Adderfunction Bit#(3) add(Bit#(2) x, Bit#(2) y, Bit#(1) c0); Bit#(2) s = 0; Bit#(3) c; c[0] = c0; let cs0 = fa(x[0], y[0], c[0]); c[1] = cs0[1]; s[0] = cs0[0]; let cs1 = fa(x[1], y[1], c[1]); c[2] = cs1[1]; s[1] = cs1[0];

return {c[2],s}; endfunction

fa fa

x[0] y[0]

c[0]

s[0]

x[1] y[1]

c[1]

s[1]

c[2]

January 9, 2012 L1-13http://csg.csail.mit.edu/SNU

“let” syntax

January 9, 2012 L1-14http://csg.csail.mit.edu/SNU

The “let” syntax: avoids having to write down types explicitly

let cs0 = fa(x[0], y[0], c[0]); Bits#(2) cs0 = fa(x[0], y[0], c[0]);

The same

Parameterized types: #A type declaration itself can be parameterized – the parameters are indicated by using the syntax ‘#’ For example Bit#(n) represents n bits and

can be instantiated by specifying a value of n Bit#(1), Bit#(32), Bit#(8), …

January 9, 2012 L1-15http://csg.csail.mit.edu/SNU

An w-bit Ripple-Carry Adderfunction Bit#(w+1) addN(Bit#(w) x, Bit#(w) y,

Bit#(1) c0); Bit#(w) s; Bit#(w+1) c; c[0] = c0; for(Integer i=0; i<w; i=i+1) begin let cs = fa(x[i],y[i],c[i]); c[i+1] = cs[1]; s[i] = cs[0]; endreturn {c[w],s}; endfunction

Not quite correct

January 9, 2012 L1-16http://csg.csail.mit.edu/SNU

// concrete instances of addN!function Bit#(33) add32(Bit#(32) x, Bit#(32) y,

Bit#(1) c0) = addN(x,y,c0);function Bit#(4) add3(Bit#(3) x, Bit#(3) y, Bit#(1) c0) = addN(x,y,c0);

valueOf(w) versus w Each expression has a type and a value and these come from two entirely disjoint worldsn in Bit#(n) resides in the types worldSometimes we need to use values from the types world into actual computation. The function valueOf allows us to do that

Thus i<w is not type correct i<valueOf(w)is type correct

January 9, 2012 L1-17http://csg.csail.mit.edu/SNU

Tadd#(w,1) versus w+1 Sometimes we need to perform operations in the types world that are very similar to the operations in the value world

Examples: Add, Mul, Log

We define a few special operators in the types space for such operations

Examples: Tadd#(m,n), Tmul#(m,n), …

January 9, 2012 L1-18http://csg.csail.mit.edu/SNU

A w-bit Ripple-Carry Addercorrected

function Bit#(Tadd#(w,1)) addN(Bit#(w) x, Bit#(w) y,Bit#(1) c0);

Bit#(w) s; Bit#(Tadd#(w,1)) c; c[0] = c0; let valw = valueOf(w); for(Integer i=0; i<valw; i=i+1) begin let cs = fa(x[i],y[i],c[i]); c[i+1] = cs[1]; s[i] = cs[0]; endreturn {c[valw],s}; endfunction

January 9, 2012 L1-19http://csg.csail.mit.edu/SNU

Integer versus Int#(32)In mathematics integers are unbounded but in computer systems integers always have a fixed sizeBluespec allows us to express both types of integers, though unbounded integers are used only as a programming convenience

January 9, 2012 L1-20http://csg.csail.mit.edu/SNU

for(Integer i=0; i<valw; i=i+1) begin let cs = fa(x[i],y[i],c[i]); c[i+1] = cs[1]; s[i] = cs[0]; end

Static Elaboration phaseWhen Bluespec program are compiled, first type checking is done and then the compiler gets rid of many different constructs which have no direct hardware meaning, like Integers, loops

cs0 = fa(x[0], y[0], c[0]); c[1]=cs0[1]; s[0]=cs0[0]; cs1 = fa(x[1], y[1], c[1]); c[2]=cs1[1]; s[1]=cs1[0]; …csw = fa(x[w-1], y[w-1], c[w-1]); c[w] = csw[1]; s[w-1] = csw[0];

January 9, 2012 L1-21http://csg.csail.mit.edu/SNU

for(Integer i=0; i<valw; i=i+1) begin let cs = fa(x[i],y[i],c[i]); c[i+1] = cs[1]; s[i] = cs[0];end

0 0

Logical right shift by 2

Fixed size shift operation is cheap in hardware – just wire the circuit appropriatelyRotate, sign-extended shifts – all are equally easy

January 9, 2012 L1-22http://csg.csail.mit.edu/SNU

a b c d

0 0 a b

Conditional operation: shift versus no-shift

We need a Mux to select the appropriate wires: if s is one the Mux will select the wires on the left otherwise it would select wires on the right

s

(s==0)?{a,b,c,d}:{0,0,a,b};

January 9, 2012 L1-23http://csg.csail.mit.edu/SNU

0 0

Gate-level implementation of a multiplexer

January 9, 2012 L1-24http://csg.csail.mit.edu/SNU

(s==0)?A:B

case {s1,s0} matches 0: return A; 1: return B; 2: return C; 3: return D;endcase

AND

AND

OR

S

A

B

S

S0

S0

S1A 2-way multiplexer

A

B

C

D

A

B

A 4-way multiplexer

Logical right shift by nShift n can be broken down in several steps of fixed-length shifts of 1, 2, 4, …Thus a shift of size 3 can be performed by first doing a shift of length 2 and then a shift of length 1We need a Mux to select whether we need to shift at all

The whole structure can be expressed an a bunch of nested conditional expressions

You will write a Blusepec program to produce a variable size shifter in Lab 1 this afternoon

January 9, 2012 L1-25http://csg.csail.mit.edu/SNU

00

0

s0

s1

Combinational ALUfunction Bit#(width) combALU(Bit#(width) a, Bit#(width) b, OP op); case op matches NOT: return ~a; AND: return a&b; OR: return a|b; SHL: return a<<b; SHR: return a>>b; ADD: return addN(a,b); SUB: return a-b; MUL: return combMulN(a,b); endcaseendfunction

Once we have implemented the primitive operations like addN, >>, the ALU can be implemented simply by introducing a Mux controlled by op to select the appropriate circuit

January 9, 2012 L1-26http://csg.csail.mit.edu/SNU

Other operations may be needed

slightly stylized code

Conditional operationsGenerate one-bit result which is used for branching

Eq (a,b) : (a == b); Neq(a,b) : (a != b); Le(a) : signedLE(a, 0); Lt(a) : signedLT(a, 0); Ge(a) : signedGE(a, 0); Gt(a) : signedGT(a, 0);

Straightforward to implement; can be combined with the ALU with an extra bit of output

January 9, 2012 L1-27http://csg.csail.mit.edu/SNU

ALU with Conditionals

mux

add

combMulN

a b

op

January 9, 2012 L1-28http://csg.csail.mit.edu/SNU

…

Next - Sequential Circuits

mux

0EQ