Aa!Te  0 @p  ` 0P@`Pp0 ` Author Matt BishopTitleNotes for January 11, 2001Subject concurrencyKeywords2concurrency parallelism determinate noninterferingHH $ @d HHHH̀̀̀ff@  d Footnote TableFootnote**.\t.\t/ - :;,.!?4e4# eTOCHeading1Heading2   8EquationVariablesMU ;`<<=7=P=i=@E@F@;@=@?@A  <$lastpagenum><$monthname> <$daynum>, <$year>"<$monthnum>/<$daynum>/<$shortyear>J<$hour>:<$minute00> <$ampm> on <$dayname>, <$monthname> <$daynum>, <$year>"<$monthnum>/<$daynum>/<$shortyear><$monthname> <$daynum>, <$year>"<$monthnum>/<$daynum>/<$shortyear> <$fullfilename> <$filename> <$paratext[Title]> <$paratext[Heading1]> <$curpagenum> <$marker1> <$marker2> (Continued)+ (Sheet <$tblsheetnum> of <$tblsheetcount>)Heading & Page <$paratext> on page<$pagenum>Pagepage<$pagenum>See Heading & Page%See <$paratext> on page<$pagenum>. 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The question is, to what degree are these the same concepts? 1na!Formal Definitions and Notations 2*!1A system of processes  S  = (,   ) is a set of processes = {  p 1 , ,  p n  } and a precedence relation   :   . UU?The    relation is a partial ordering (we define  p      p  as true). When  p      q , process  p  must complete before proAcess  q  may begin. 3! Each process  p     has an associated set of input memory locations called  domain (p) and an associated set of *&output memory locations  range ( p )   . An interpretation  f p  of  p  associates values with each set of memory locaUUUHA3tions. The set of all inputs to  S  is abbreviated  domain ( S ), and the set of all outputs from  S  is abbreviated  range ( S ). 4UGapTwo systems of processes  S  = (,   ) and  S  = (,   ) are equivalent if 6UFa = ; :a    ; and ;a if  S  and  S  are given the same element of  domain ( S ), then they output the same element of  range ( S ). <UC!{An execution sequence  a  is any string of process initiation and termination events satisfying the precedence conUBAstraints of the system. =* UA!%V ( M i ,  a ) is the sequence of values written into memory location  M i  at the termination of processes in  a . The .*Afinal value stored in  M i  after execution sequence  a  completes is represented by  F ( M i ,  a ). ?UU>! A determinate system of processes is a system of processes  S  for which each element of  domain ( S ) produces the Jsame set  range ( S ) regardless of the order or overlapping of the elements of  S . More formally, a system  S  is deter*AIminate if, for any initial state and for all execution sequences  a  and  a  of  S ,  V ( M i ,  a ) =  V ( M i ,  a ) @UUg5!|A mutually noninterfering system of processes is a system of processes  S  in which all pairs of processes meet the 0s4Bernstein conditions. Processes  p  and  q  are noninterfering if either process is a predecessor of the other, or the A,processes satisfy the Bernstein conditions. A2a|The initiation of a process  p  is writtten  p , and the termination of  p  is written  p . B*aHRelationship of Determinate Systems and Mutually Noninterfering Systems C1aLTheorem 1 : If a system is mutually non-interfering, it is determinate. D*0!iTheorem 2 : Let  S  be a system with  domain ( p ) and  range ( p ) specified,  range ( p )   , for all  p     , and  f p  unspeciӪAdfied. Then if  S  is determinate for all  f p , it is mutually non-interfering. E*aProofs Fa The following lemma is helpful: G !=Lemma : Let  S  be a mutually noninterfering system. Let  p  be a terminal process of  S . If  a  =  b p g p d  is an execution *Aisequence of  S , then  a  =  bgd p p  is an execution sequence of  S  for which  V ( M i ,  a ) =  V ( M i ,  a ) for all  i . HUU&U$!1Proof : As  p  is a terminal process in  S , it has no successors in  S . Hence  a  satisfies the precedence constraints of  S . So 2U#A=a  is an execution sequence. We now consider two cases. I*!TM i      range ( p ). Note  p  does not write memory locations not in  range ( p ). Consider any process  p  with  p  in  d . As UUL*t=p  and  p  are mutually noninterfering,  range ( p )     domain ( p ) =   . So all such  p  find the same values in *X*sA;domain ( p ) whether the execution sequence is  a  or  a . Thus,  V ( M i ,  a ) =  V ( M i ,  a ). JfL!M i      range ( p ). Let  p  in  gd . As  p  and  p  are mutually noninterfering,  domain ( p )     range ( p ) =   . So no  p  in  gd  t!xwrites into an element of  domain ( p ). Hence for all  M j      domain ( p ),  V ( M j ,  b ) =  V ( M j ,  bgd ). By definition, for all B|M j      domain ( p ),  F ( M j ,  b ) =  F ( M j ,  bgd ). As  p  has the same input for both  a  and  a , it writes the same value into HHˆAHHˆ7H  ld@48H}?H =[ #?H ;We Replace With }H =]"$H ;W eHead }H =_#%H ;W!e Comments }? =a$&? CW"e }?H =c%'?H CW#e }H =e&(H CW$e }H =g')H CW%e }d =j(.d DW&eCharacter Macros HHˆ;"HHˆ+Ge HHˆ;$3HHˆ**l}?d =l?d DW'e }d =nd DW(e }? =p)/? EW)e Character }?H =r.0?H EW*e Replace With }H =t/1H EW+e Comments }? =v0B? FW,e HUV ;.HUV 3Ge HUV ;05+HUV 22l H$ ;1H$ 5Ge H$ ;33H$ 44l HHˆ;4HHˆ&&7  `Outline for January 11, 2001 `Greetings and felicitations! `!First part of project due Friday >`Web page up and running! Q`Process models 5]`DTheorem: If a system is mutually noninterfering, it is determinate. 7* Theorem: Let  f p  be an interpretation of process  p . Let be a system of processes, with  p     . If for all such w9p ,  domain ( p ) and  range ( p ) , but  f p  unspecified, is determinate for all  f p , then all processes in are UU@mutually noninterfering 18UH tMaximally parallel system: determinate system for which the removal of any pair from the relation    makes @5the two processes in the pair interfering processes. UF`Critical section problem UE`Mutual exclusion ` Progress ` Bounded wait UB`Classical problems UA`Producer/consumer `DReaders/writers (first: readers priority; second: writers priority)  `Dining philosophers !U>`Basic language constructs "U=` Semaphores w` Send/receive 9U;`,Evaluating higher-level language constructs EU:` Modularity ` Constraints `Expressive power ` Ease of use ` Portability `#Relationship with proram structure `H FBGseN }HTD=?H FBGteN }HTD>CH FBGue }HTD=AHFGgv% P:BodyNoInEdent }HTD@JH FGGweP }?H =x1C?H FW-e¢ }H =zB2H FW.e d=~EEd=DdFF l d=Dd& zlE zupkfa\WRMHC>v@;CFILORUX[^adgjmpsy| %).12/,)&# HUV @8!HUV :WlDLast modified at 12:35 pm on Wednesday, January 10, 2001 HHˆ@9!:HHˆII l HHˆ@:!HHˆHW` }HTDAKH FGGxeN }HTDJLH FGGyeN }HTDK;H FGGze ydAHH HHˆAFHHˆ!11HJ*Geach  M i      range ( p ) in  a  and  a . Let  v  denote the value that  p  writes into  M i  in  a . Then OV ( M i ,  a )=  V ( M i ,  b p g p d )as no process  p  in  d  writes into an element of  range ( p ) "= ( V ( M i ,  b p g ),  v )as  p  writes  v  into  M i  = ( V ( M i ,  b ),  v )as no process  p  in  g  writes into an element of  range ( p ) "= ( V ( M i , b gd ),  v )as no process  p  in  g  writes into an element of  range ( p ) r=  V ( M i , b gd p p )as  p  writes  v  into  M i A1=  V ( M i ,  a ) KUUg̴aThis proves the lemma. n Ls̳aeProof of Theorem 1 : We proceed by induction on the number  k  of processes in a system. Ma:Basis :  k  = 1. The claim is trivially true. Na Hypothesis : For  k  = 1, ,  n 1, if a system of  k  processes is mutually noninterfering, it is determinate. OabStep : Let  S  be an  n  process system of mutually noninterfering processes. P!}If  S  has exactly one execution sequence, it is determinate. So, assume that S has two distinct execution sequences Aa  and  b . QazLet  p  be a terminal process of  S , and form  a  and  b  according to the lemma. Then R*aA a  =  a  p p  V ( M i ,  a ) =  V ( M i ,  a )for all  i  such that 1  i   m SաaA b  =  b  p p  V ( M i ,  b ) =  V ( M i ,  b )for all  i  such that 1  i   m TUU!6Now form the  n 1 process system  S  = ( {  p  },   ), where    is formed by deleting from    all pairs with  p  in *wO\them. Clearly,  a  and  b  are execution sequences of  S . Further, by the induction hypothesis,  V ( M i ,  a ) =  V ( M i ,  b ) UUL?for all  i  such that 1  i   m . This means that the values in the elements of  domain ( p ) are the same in both  a  and  b ; * Lpin other words,  F ( M j ,  a ) =  F ( M j ,  b ) for all  M j      domain ( p ). As the inputs for  p  are the same in both execution !Asequences, the outputs will also be the same. It follows that  p  writes the same value  v  into  M i      range ( p ) in both  a  UUAand  b . U*0CaFHence for  M i      range ( p ): V>̕acV ( M i ,  a )=  V ( M i ,  a )by the lemma Wao=  V ( M i ,  a )as  M i      range ( p ) XaM=  V ( M i ,  b )by the induction hypothesis Yan=  V ( M i ,  b )as  M i      range ( p ) Za<=  V ( M i ,  b )by the lemma [aDand for  M i      range ( p ): \acV ( M i ,  a )=  V ( M i ,  a )by the lemma ]av= ( V ( M i ,  a ),  v ) p  writes  v  into  M i  ^a\= ( V ( M i ,  b ),  v )by the induction hypothesis _au= ( V ( M i ,  b ),  v ) p  writes  v  into  M i  `a<=  V ( M i ,  b )by the lemma aaBEither way,  V ( M i ,  a ) =  V ( M i ,  b ). Hence  S  is determinate, completing the induction step and the proof. n  bUU!Proof of Theorem 2 : We prove this theorem by contradiction. Let  S  be a determinate system. Let  p ,  p     be interlA7fering processes. Then there exist execution sequences ca0a  =  b p p p p g da1a  =  b p p p p g #e!Consider the Bernstein conditions. As  p  and  p  are interfering, at least one of those conditions does not hold. We Aexamine them separately. f*!iLet  M i      range ( p )     range ( p ). We choose the interpretation  f p  so that  p  writes the value  u  into  M i , and we :choose the interpretation  f p  so that  p  writes the value  v  into  M i , and  u   v . But then  V ( M i ,  b p p p p ) = ( V ( M i ,  b ),  u ,  v ) UUand *bL\ V ( M i ,  b p p p p ) = ( V ( M i ,  b ),  v ,  u ). UUp!AThis means  S  is not determinate, contradicting hypothesis. So  range ( p )     range ( p ) =   . Yg*|ny!Let  M i      domain ( p )     range ( p ). As  range ( p )   , take  M i      range ( p ). Choose the interpretation  f p  so that  p  HHˆAFHHˆ KGG ldAKK HHˆAIHHˆ  Kg*|reads different values in  a  and  a ; that is,  F ( M j ,  b )  F ( M j ,  b p p ) for some j such that 1  j   m . Also, choose  f p  UUso that  p  writes  u  in  a  and  v  in  a , where  u   v . But then *JV ( M i ,  b p p p p )=  V ( M i ,  b p p )as  range ( p )     range ( p ) = UM?= ( V ( M i ,  b ),  u )  V ( M i ,  b p p p p )= ( V ( M i ,  b p p ),  v )  = ( V ( M i ,  b ),  v )as  range ( p )     range ( p ) = UUIAs  u   v , this means that  S  is not determinate, contradicting hypothesis. So  domain ( p )     range ( p ) =   . [As an *cB@Raside, if  range ( p ) =   , then  M i      range ( p ) and  p  and  p  are noninterfering. Hence there is no contradiction.]  hUUq`` By symmetry, the argument for case 2 also shows that  range ( p )     domain ( p ) =   . Qi~D+`^In all three cases, the Bernstein conditions must hold. This completes the proof. n  HHˆAIHHˆHJJ l|W} ? Eu($ ? LWe }? H Ew#%? H LWe... } H Ey$ H LWe }? E{+'? MWe }?H E}&(?H MWe- }H E'#H MWe }? E.*? NWe }?H E)+?H NWe-- }H E*&H NWe }? E1-? OWe }?H E,.?H OWe° }H E-)H OWe }? E40? PWe }?H E/1?H PWe® }H E0,H PWe }? EC3? 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