A formal approach to the compilation of data-parallel languages

Denotational Semantics of an HPF-Like Data-Parallel Language Model

“…[9] proposed an experimental data parallel language called Booster. Some parallel features such as View concept and the separation of algorithm description and machine mapping were introduced.…”

“…M I"*l]" = Pm < ({S, a), IP mi ), IP mi = IP((i), (**2 + l)) = {(Ep],Ep*2 + l])|0 < E[t] < s'AO < E[(i*2 + 1)] < a'}, because s' = 10 and a' = 10, then IP((i), (i*2+l)) = {(i, i*2+l)|0 < i < 10A0 < i*2+l < 10} = {(0,1), (1,3),...,(4,9)}.The following mapping statement illustrates the combination from an array to a template.MIm 2 ] Pm = Pm < ((*, b), IP m2 ),IP m2 =IP((i,j),(j-2,-,i+2)) = {((t,j),E[0--2,-,i+2)]|0<E[(i,i)l<4A0< E[0"-2,-,t+2)]<a / fc ,0<fc<2} = {(*, j), (j -2, (0,..., 99), i + 2)|0, 0 < i, j < 10,10 A 0 < j -2 < 10 A 0 < i + 2 < 10} = {((t,j),0'-2,(0,...,99),t+2)|0<t<8A2< j< 10} = {((0, 2), ((0), (0,..., 99), (2))), ((0, 3), ((1), (0,..., 99), (2))),...,((7, 9), ((7),(0,...,99),(9)))}.The following mapping statement illustrates how a template is distributed onto processors.M l™3]l pm = Pm < ((S,p), SP m3 ), SP m3 = SP((i), (t/4)) = {E[(i)J, (E[(i/4)], 010 < i < 10 A 0 < t/4 < 4} = {((0), (0, /")), ((1), (0, h)), (2, (0, l 2 )), (3, (0, h)),..., ((8, (2, Jo)), ((9, (2, h))}. The following mapping statements thus determine the relationship of an array to the processors by the two-level mapping.…”