Static Dataflow Graphs

Transcription

1 Static Dataflow Graphs Arvind Computer Science & Artificial Intelligence Lab Massachusetts Institute of echnology L20-1 Motivation: Dataflow Graphs A common Base Language - to serve as target representation for high-level languages; - to serve as machine language for a highly parallel machine. Jack Dennis Computation Structures Group, MI during L20-2 1

2 Dennis' Program Graphs Operators connected by arcs fork f function or predicate rue gate (alse gate) merge L20-3 Dataflow Eecution of an operation is enabled by availability of the required operand values. he completion of one operation makes the resulting values available to the elements of the program whose eecution depends on them. Dennis Eecution of an operation must not cause side-effect to preserve determinacy. he effect of an operation must be local. L20-4 2

3 iring Rules: unctional Operators y f f f(,y) L20-5 iring Rules: -Gate L20-6 3

4 he Switch Operator X X L20-7 iring Rules: Merge y y L20-8 4

5 iring Rules: Merge cont y not ready to fire L20-9 Some Conventions X1 X2 B X1 X2 B L

6 Some Conventions Cont. X1 X1 X2 X2 B X1 X1 X2 X2 B L20-11 Rules o orm Dataflow Graphs: Jutaposition Given G1 G2 G1 G2 G L

7 Rules o orm Dataflow Graphs: Iteration Given G1 G1 G L20-13 Eample: he Stream Duplicator 1-to-2 SD SD NO L

8 he Gate Operator X C X Lets X pass through only after C arrives. What happens if we don't use the gate in the Stream Duplicator? L20-15 he Stream Halver hrows away every other token. 2-to-1 SH SH L

9 Determinate Graphs Graphs whose behavior is time independent, i.e., the values of output tokens are uniquely determined by the values of input tokens. A dataflow graph formed by repeated jutaposition and iteration of deterministic dataflow operators results in a deterministic graph. Proof? L20-17 Dataflow Operators: Streams unctions add(:s,y:ys) = +(,y) : add(s,ys) -gate (:bs,:s) = : -gate(bs,s) -gate (:bs,:s) = -gate(bs,s) merge(:bs,:s,ys) = : merge(bs,s,ys) merge(:bs,s,y:ys) = y: merge(bs,s,ys) L

10 Dataflow Graphs: A Set of Recursive Equations I A B NO O O = -gate (A,I) ; A = gate (I,B) ; B = : Not (A) ; G. Kahn L20-19 Domain of Sequences Sequence: [ 1,..., n ] he least element: [ ] (aka ) he partial order ( ): prefi order on sequences [ ] [ 1 ] [ 1, 2 ]... [ 1, 2, 3... n ] [ 1, 2, 3 ] may be approimated by [ ] or [ 1 ] or[ 1, 2 ]. However, [ 1, 2 ] is a better approimation than [ 1 ] or [ ] for [ 1, 2, 3 ]. L

11 Kleene's Iterative Solution I [i1, i2, i3] [i1, i2, i3] [i1, i2, i3] A [ ] [ ] [ ] B [ ] [ ] [, ] O [ ] [ ] [ i1 ] O = -gate (A,I) ; A = gate (I,B) ; B = : Not (A) ; Is the answer unique? Yes, if all operators are monotonic and continuous! L20-21 Monotonicity y a monotonic operator on sequences can only produce more output when given more input, i.e., it can never retract a value that has been produced. Is -gate(b,x) monotonic? f() f(y) B B -gate(b,x) -gate(b,x) he proof is straightforward by the induction on the length of the sequences L

12 Continuity f (U i X i ) = U i f(x i ) A continuous operator on sequences does not suddenly produce an output after consuming an infinite amount of input. Is -gate(b,x) continuous? B 0 B 1... B n... -gate((u i B i ),X) = U i -gate(b i,x) he proof is straightforward by the induction on the length of the sequences L20-23 Kleene s ied Point heorem If D is partially ordered with one least element ( ) and is ω-complete, and f : D --> D is a monotonic and continuous function then U i f i ( ) is the least fied point solution of f. ω-complete means that every chain (U i X i ) in the domain has a least upper bound f( ) f(f( ))... he limit of this chain is denoted by U i f i ( ) L