Well-behaved Dataflow Graphs

Transcription

1 Well-behaved Dataflow Graphs Arvind Computer Science & Artificial Intelligence Lab Massachusetts Institute of echnology L21-1 Outline Kahnian networks and dataflow Streams with holes & agged interpretation Well behaved graphs L21-2 1

2 Kahnian Networks P Q Computing stations connected by unbounded, IO channels Each station executes a sequential program wait(ch): blocking read from a channel send(x,ch): non blocking a station either blocks for an input on a specific channel or computes L21-3 An Example h 0 Y f X Z h 1 0 g 1 X = f(y,z) 0, 1 = g(x) Y = h 0 ( 0 ) Z = h 1 ( 1 ) Process f(u,v; W) { b= true; While true do { i := if b then wait(u) else wait(v); print(i); send(i,w); b := not b} } Process g(u ; V,W) { b= true; While true do { i := wait(u); if b then send(i,v) else send(i,w) b := not b } } Process h C (U ; V) { send(c,v); While true do { i := wait(u); send(i,v) } } L21-4 2

3 Kahnian Networks & Dataflow Y not Z f h 0 g X h 1 not 0 1 Dataflow Graphs can Express any Kahnian Network and vice versa L21-5 Determinacy A computing station in Kahnian network can be viewed as a monotonic and continuous function from sequences to sequences he least fixed point solutions characterize the I/O behavior of such stations Dataflow operators can have any granularity and can be expressed in any sequential language L21-6 3

4 Missing okens x x x cannot be moved to the output because the token corresponding to is missing. How to model stream with holes? L21-7 Another Interpretation of DGs Streams with holes Streams with missing tokens {v 1, v 2,, v 4,, v 6,...} can be modeled by a set of tokens where tokens carry a tag designating their position in the stream { <1, v 1 >, <2, v 2 >, <4, v 4 >, <6, v 6 > } L21-8 4

5 agged Semantics of Operators add(xs,ys) = {<i,x+y> <i,x> xs, <i,y> ys} -gate(bs,xs) = {<k,x> <i,> bs, <i,x> xs, j i. <j,b j > bs, k = -Cnt(bs,i)} merge(bs,xs,ys) = {<i,x> <i,> bs, <k,x> xs, j i. <j,b j > bs, k = -Cnt(bs,i)} {<i,y> <i,> bs, <k,y> ys, j i. <j,b j > bs, k = -Cnt(bs,i)} D a (xs) = {<i+1,x> <i,x> xs} {<1,a>} L21-9 Ordering on Streams with Holes Stream with holes: { <i, v i >, <j, v j >, <k, v k > } he least element: { } (aka ) he partial order ( ): subset order It is easy to show that all the operators under the tagged semantics are monotonic and continuous tagged semantics are also deterministic L

6 agged versus IO Interpretation heorem: Suppose the least fixed point of a dataflow program X = { X 1,..., X n } in the IO interpretation and Y = { Y 1,..., Y n } in the tagged interpretation then X Y. Proof: Based on structural induction 1. Show it holds for each operator 2. Show it holds under juxtaposition and iteration L21-11 agged inte p etation gi es mo e defined ans e s agged versus IO Interpretation heorem: Suppose the least fixed point of a dataflow program is X = { X 1,..., X n } in the IO interpretation where X i is the stream associated with a particular arc in the program. Let Y = { Y 1,..., Y n } represent the fixed point of the same program in the tagged interpretation, then X Y. Proof: Based on structural induction 1. Show it holds for each operator 2. Show it holds under juxtaposition and iteration agged interpretation gives more defined answers and has more parallelism L

7 Well Behaved Dataflow Graphs 1. One token on each input arc produces exactly one token on each output arc. 2. he initial distribution of tokens on the arcs is restored. P P Before After L21-13 Control Operators are not well-behaved ork Primitive Ops Switch Merge + + Not well-behaved L

8 he Block Schema a b {x = a + b; y = b * 7 in (x-y) * (x+y)} Any acyclic interconnection of WBGs is a WBG *7 x y * L21-15 he Conditional Schema a b p.. + * If p then a + b else a * a L

9 Another Conditional Schema a b + * What is wrong with this schema? p If p then a + b else a * a L21-17 Merge Operator is Essential for Determinacy X 2 X 1 X 2 X 1 f g f g X Suppose g(x2) computes much faster than f(x1). okens will come out in the wrong order without the merge operator. L

10 he Loop Schema a initial x = a while p(x) do next x = f(x) ; finally x p x next x f L21-19 Well Behaved Dataflow Graphs (WBGs): Rules to form WBGs 1. Primitive operators like + and fork are WBGs (-gate, -gate and merge are not WBGs). 2. he block schema, i.e., an acyclic interconnection of graphs, is a WBG, if all its component graphs are WGBs. 3. he conditional schema is a WBG, if the graphs for the rue side and alse side are WBGs. 4. he loop schema is a WBG, if the graphs for the predicate and the body are WBGs. L

11 Unbounded Cyclic Graphs 0 1 <n f +1 + Unbounded number of tokens on an arc can only arise due to cycles. L21-21 Bounded Cyclic Graphs 0 1 <n f +1 k-bound sync + L

12 Bounded Cyclic Graphs 0 1 < n f + 1 Sync k-bound + L21-23 Well Behaved Schemas Loop Bounded Loop p f Needed for resource management L

13 New Definition of Well Behavedness P Before P After 1. One token on each input arc produces exactly one token on each output arc. 2. he initial distribution of tokens on the arcs is restored. 3. No arc can have an unbounded buildup of tokens. L21-25 Bounded Cyclic Graphs are Well Behaved Initial number of tokens at the gate input determines the maximum number of tokens on any arc. However, loop bounding can alter the "meaning" of a graph, i.e., can cause deadlock. In general, restricting the number of tokens on an arc causes deadlock. L

14 Can this program deadlock if the number of tokens per arc is restricted? 0 < L21-27 Static DGs as a Base Language Static DGs can express all recursively enumerable functions Static DGs are not sufficient as a target for compiling high level languages. Support is lacking for: procedure calls data structures Dynamic Dataflow Graphs L