From Id/pH to Dataflow Graphs Arvind Computer Science & Artificial Intelligence Lab

March 4, 2008 DF4-1http://csg.csail.mit.edu/arvind/

From Id/pH to Dataflow Graphs

ArvindComputer Science & Artificial Intelligence LabMassachusetts Institute of Technology


Parallel Language Model Global Heap ofShared Objects

Tree ofActivationFrames

h:g:

f:

loop

activethreads

asynchronous and parallelat all levels


Dataflow Graphs + I-Structures + . . .

TTDA Monsoon *T

*T-Voyager

Id Worldimplicit parallelism

Id


Id Compiler PhasesId

Kernel Id

Parallel Three Address Code, P-TAC

Dataflow Graphs/Threads

Machine Graphs von Neumann

code

R

educ

tion

desugar

select representations

F

ixed

Pro

gram

Semantics of each intermediate language is important to show the correctness of each module


Choosing Representations: Arrays

lower upper

lowerupper

bounds lower upper

lowerupper

bounds 1.

2.

3.

bounds array4.


type list *0 = nil | cons *0 (list *0)

1. nil tag=0

tag=1cons

2. nil

cons

tag

Boxed

Boxed

Unboxed

Implicit

Assumption: Pointers can be distinguished from small integers

Similar issues arise for all Algebraic Types

Choosing Representations: Lists


Case Expression and Tags

type foo .. = C1 ... | C2 ... | ... | Cn ...

The case expression for type foo

foo-case(d,e1,...,en)

is implemented using the case construct:

casek(i ,e1,...,ek)ei

where i = ifetch(d);


Top Level Functionsf = k(x1,...,xk).eDo not contain any free variables except the names of other top level functions.

In general, its arguments can be supplied in one of two ways: 1. As a data structure which is built incrementally

F = xs.{ (x1,...,xk) = destruct(xs); in e }

2. All at once - fast call Ffc = k(x1,...,xk).e


Multi-arity Functionsdef f x1 ... xk = e

can be translated as f = x1.x2. ... xk.e

However, most implementation use k-arity sf = k(x1,...,xk).e

with the rules:(k(x1,...,xk).e) a

{ x’1 = a in ((k(x1,...,xk).e) x’1)} if k > 1

((k(x1,...,xk).e) x’1 ... x’n ) a { x’n+1 = a in ((k(x1,...,xk).e) x’1 ... x’n+1)} if n < k-1

((k(x1,...,xk).e) x’1 ... x’n ) a { x’n+1 = a in e[x’1/x1]... [x’k/xk]} if n = k-1

Partial applications are values and thus substitutable


Loop Translation

TE[{while ep do S finally ef } =

{ p = k(x1,...,xk).TE[ep] ; b = k(x1,...,xk). { TS[S]

in nextx1,...,nextxk};

tp = p x ; t = Wloopk(p,b,x,tp) ; tf = TE[ef][t/x] ;in

tf }where

x = (x1,...,xk) and xi’s are the “nextified” variables of S and “next x” is replaced by “nextx” everywhere


Loop RulesWloopk(p, b, x, true)

{ t = b x; tp = p t;

t’ = Wloopk(p, b, t, tp) in t’ }

Wloopk(p, b, x, false) x


KId - The Kernel Id LanguageAn extended S-calculus

E ::= x | k(x1,...,xk).E | SE SE | { S in SE } | Casek(SE, E1,...,Ek)

| PFk(SE1,...,SEk) | CN0 | CNk(SE1,...,SEk)| allocate()| Wloop(Ep,Eb,x,bool)

S ::= x = E | S; S | S --- S| store(SE,SE,SE)

PF2 ::= ...| i-fetch | m-fetch Simple Expressions

SE ::= x | CN0 | CNk(SE1,...,SEk) | k(x1,...,xk).E | ((k(x1,...,xk).E) SE1...SEn) n < k


KId Optimizations

“Totally correct” optimizations (S rewrite rules)• Constant Propagation/Folding• Fetch Elimination• Barrier discharge• Inline Substitution/Specialization• Dead-code Elimination /Garbage collection ...

Additional transformations or rewrite rules• Algebraic Identities• Common Subexpression Elimination• Code Hoisting out of Conditionals• Loop Optimizations

• Peeling/Unrolling• Lifting loop invariants• Eliminating circulating variables and constants• Induction variable analysis


P-TAC: Three Address CodeA lower level language than KId

1. Exposes all address calculations. Requires choosing representations for all data structures

2. No nested function definitions - -abstractions exist only at the top level

3. Partial applications are represented as data structures (Closures)


Generating Dataflow Graphs

GE :: P-TAC Expression --> DFG

. . .

. . .

tg x1 xn

sg

G


P-TAC to Dataflow GraphsExpressions

apply

x1 x2

GE[c] = c

GE[x1 x2] =

GE[PFk(x1,...,xk)] = PFk

x1 ... xk

GE[x] = x


Case Schema

GE[Casek(i, e1,...,ek)] 1 k

GE[e1] GE[ek]

Switch

i x1 ... xn

...

x1 ... xn x1 ... xn

• { x1,...,xn } = FV(e1)U...U FV(ek)

• All ei’s must have the same number of outputs

• Unused xi’s in each branch have to be collected for signal generation


Graph for Function ApplicationThe Full Application case

t

((k(x1,...,xk).e) x’1 ... x’k-1 ) a { x’k = a in e[x’1/x1]... [x’k/xk]

a new copy of echange tag

a

apply

f

xt:

f

get context extract tag

change Tag0 change Tag1

a

change Tag

1:

0:

GE[e]

Apply is a conditional schema;make-closure part is not shown


Loop Schema

GE[Wloopk(p,b,(x1,...,xk),tp] =

next nextGS[Sb]

GE[ep]

x1 ... xk tp

next• {x1,...,xn} are the nextified variables

• p = k(x1...xk).ep

• b = k(x1...xk).{Sb in next x1,...,next xk }

...

...


P-TAC to Dataflow GraphsThe Block Expression

GE[{S in x}] =where{x1,...,xn} = FV(S)

GS[x = e] =where{x1,...,xn} = FV(e)

. . .tg x1 xn

sg

GE(e)

x

GS[store(x1,x2)] =

x1 x2

sg

istore

. . .

. . .

tg x1 xn

y1 ym

GS(S)

x sg


Parallel Composition''Wiring Diagrams''

{y1,...,yn} = BV(S1) U BV(S2){x1,...,xn} = FV(S1) U FV(S2) - {y1,...,yn}

GS[S1;S2] =

. . .

. . .

tg x11 x1n

y11 y1m sg

GS(S1)

. . .

. . .

tg x21 x2n

y21 y2m sg

GS(S2)

x1 ... xn

y1 ... yn

Id

Each xij is to be connected to the yij with the same name


Sequential Composition

{x1,...,xn} = FV(S1) U FV(S2){y1,...,yn} = BV(S1) U BV(S2)

. . .

. . .

tg x11 x1n

y11 y1m sg

GS(S1)

GS[S1---S2] =

. . .

. . .

tg x21 x2n

y21 y2m sg

GS(S2)

x1 ... xn

y1 ... yn

Each x2j is to be connected to the y1j with the same name

From Id/pH to Dataflow Graphs Arvind Computer Science & Artificial Intelligence Lab

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