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F;Wa Macro Name d= d= d 靕l d= do  W܁܁Bm }蝝d = 蝝d  <W|aHeadings Table }H蝝d = H蝝d  <W}a }H蝝d = H蝝d  <W~a }H= H =!Paragraph ForPAmat }HH= HH  =WaHeading Level }H= H  =Wa Comments }H= H >W aTitle }HH= HH  >Wa }H= H  >Wa }KH = KH  ?Wa Heading1 }HKH = HKH  ?Wa }KH = KH  ?Wa }WH = WH  @Wa Heading2 }HWH = HWH  @W a }WH = WH  @W a }cH = cH  AW a }HcH = HcH  AW a }cH = cH  AW a ղT92->L(ղT92-9d=88 HH= HHL8GJ=J=d parbegin H&edp1:a := b + 1; IPdp2:c := d + 1; Jdparend Kdp3:e := a + c; L܁Ardwould be written as MMqdS ( P (p1,p2),p3) N@ndWe now prove: Od/Claim The example is not properly nested. Pd9Proof For something to be properly nested, it must be of the form  S (p i ,p j ) or  P (p i ,p j ) at the most interior level. QdClearly the example's most interior level is not  P (p i ,p j ) as there are no constructs of that form in the graph. !R$In the graph, all serially connected processes pi and pj have at least 1 more process p k  starting or finishing at the node Tn ij  between p i  and p j ; but if  S (p i ,p j ) is in the innermost level, there can be no such p k  (else a more interior  P  or  S  is DYneeded, contradiction). Hence, it's not  S (p i ,p j )) either. Ad }?H =[D #?H F;Wa Replace With }H =]D"$H F;W aHead }H =_D#%H F;W!a Comments }? =aD$&? FCW"a }?H =cD%'?H FCW#a }H =eD&(H FCW$a }H =gD')H FCW%a }d =jD(.d FDW&aCharacter Macros HH;"HH❝+G܁e HH;$3HH**靕l}?d =lD?d FDW'a }?d =nD?d FDW(a }? =pD)/? FEW)a Macro Name }?H =rD.0?H FEW*a Replace With }H =tD/1H FEW+a Comments }? =vD0B? FFW,a HUV ;.HUV ❝3G܁e HUV ;05+HUV 22靕l H$ ;1H$ 5G܁e H$ ;33H$ 44靕l HH;4HHђ:7 `Improper Nesting Example 0,` Introduction 1܁ wOne of the limits on the use of parbegin/parend, and any related constructs, is that the program involved must be propI@lerly nested. Not all programs are. For example, consider the program represented by the following graphs. 3b`The Program as Graphs 4 w h 6H`%Using  fork/join  Primitives 7܁[`GThe program equivalent to these precedence and process flow graphs is: 8J=Ρ` t6 := 2; 9o` t8 := 3; :`'S1; fork p2; fork p5; fork p7; quit; ;`!p2:S2; fork p3: fork p4; quit; <`p5:S5; join t6, p6; quit; =`p7:S7; join t8, p8; quit; >`p3:S3; join t8, p8; quit; ?`p4:S4; join t6, p6; quit; @`p6:S6; join t8, p8; quit; A`p8:S8; quit B܁&`4where S i  is the program for p i . D??`+Using  parbegin/parend  Primitives E܁Q wTo see if this is possible, we must determine if the above program is properly nested. If not, we clearly cannot repre]@rsent it using  parbegin  and  parend , which require a block structure, and hence proper nesting. aF ]Let  S ( a , b ) represent the serial execution of processes  a  and  b , and  P ( a , b ) the parallel execution of processes  a  and  b .  Then a process flow graph is properly nested if it can be described by  P ,  S , and functional composition. 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