Asynchronous Parallel Programming in Pei. E. Violard. Boulevard S. Brant, F Illkirch.

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1 Asynchronous Parallel Prorammin in Pei E. Violard ICPS, Universite Louis Pasteur, Strasbour Boulevard S. Brant, F Illkirch Abstract. This paper presents a transormational approach or the derivation o asynchronous parallel prorams. Transormation rules are based on a theory, called Pei. This theory includes the denitions o problems, prorams and transormation rules. It is ounded on the simple mathematical concepts o multiset and o an equivalence between their representations as data elds. Proram transormations are ounded on this equivalence and dened rom a renement relation. This paper is illustrated by the example o the shortest path problem. Keywords multiset, transormation, parallelism, asynchrony, specication Introduction Reliable parallel prorammin techniques are ounded on ormal derivations o prorams. The state o the art shows two complementary philosophies: { the denition o a stepwise renement calculus which requires a proo development, { the denition o transormation rules which apply on successive statements. I a proram F 0 satises any specication that a proram F satises, F 0 is called a renement o F [7]. This approach requires to dene a loic in order to reason on prorams. Unity [] is a undamental contribution in this domain, which allows to express specications and solutions in the same ormalism. It is associated with a non-deterministic computin model which allows to introduce sequentiality only when necessary. The second approach lies on the denition o ormal transormation rules. It supposes a convenient model to express successive statements and these rules. Ane recurrences on interal convex domains are an example o such an expression. They are mainly studied or systolic synthesis (see or example [9], [3], etc.). Alpha [6] and Crystal [2] are the main ormalisms usin this approach. These lanuaes are ounded on a deterministic computin model. In our sense, parallel proram derivation would benet by extendin the transormational approach to asynchrony, in order to reach a class o solutions as eneral as the one considered by renement techniques. This supposes to introduce an uniyin theory. This theory, called Pei (or Parallel Equations

2 Interpreter), is ounded on the simple mathematical concept o multiset, represented as a data eld. A proram is thus a unction on data elds and is denoted as a set o un-oriented equations. The two hand-sides o any equation dene two equal data elds. This expression eneralizes classical recurrence equations. Moreover transormations on these equations are ounded on a mathematical data elds equivalence reerrin to the same multiset. This equivalence induces an equivalence o prorams which is dened rom a renement relation. The paper is oranized as ollows: section 2 presents the theory Pei and its mathematical oundation. The equivalence o prorams is dened in section 3. Section 4 is devoted to the transormation rules. The concept o asynchrony is presented in section 5 and applied to the derivation o an asynchronous proram in section 6. 2 The theory Pei Pei is a notation or some semantics domains, where all names reer to mathematical objects and all operations are mathematical operations. It is ounded on the simple concepts o multiset and o equivalence between their representations as data elds. 2. Pei objects Generally speakin, we can consider a problem as a relation between input and output multisets o values. O course, prorammin may imply to put these values in a convenient oranized directory, dependin on the problem terms. In scientic computations or example, items such as arrays are unctions on indices: the index set, that is the reerence domain, is a part o some Z n. In Pei such a multiset o value items mapped on a discrete reerence domain is called a data eld. Let us consider the multiset ; 2;?3;. A possible way to map this multiset is to choose indices, or instance 0; ; 2; 3 o Z, to reer to each o the values. This mappin is shown on ure (a) and denes the data-eld called V. Obviously, this multiset miht be mapped in a dierent manner, or example onto points whose indices (i; j) in Z 2, are such that (0i; j) (ure (b)). Such a mappin thus denes another data eld, let us say M. Althouh their mappins are dierent, the two data elds represent the same multiset o values and thereore we say they are equivalent. Formally, there exists a bijection rom the rst arranement to the second one, namely (i) = (i mod 2; i div 2). This relation is expressed in Pei throuh the equation: M = alin:: V where alin(i) = (0i3) : (i mod 2; i div 2) Any Pei proram is composed o unoriented equations, each o them connectin two data eld expressions. On the example, M and (alin:: V) have the same set o value items, placed in the same ashion in the same reerence domain.

3 (a) 2 0,0,0-3 0,, (b) Fi.. Two dierent mappins o a multiset o values 2.2 Pei operations Expressions are dened by applyin operations on data elds. The operations are second-order unctions, and all into three cateories: { the operation used in the expression (alin:: V) modies the reerence domain onto which values are mapped. It is called chane o basis and is denoted by :: { another operation \moves" values in the reerence domain. It is called eometrical operation (or routin), and is denoted by C. Fiure 2(a) shows the mappin o values o the data eld W dened as W = V C shit. The unction shit shits values one place cyclically to the riht and is written in Pei as shit(i) = (0i<4) : (i?) mod 4. I the unction is not injective the operation expresses a broadcast. For instance W = V C spread, where spread(i) = (0i<4) : 0 means the value mapped at index point 0 in V is broadcasted to index points 0 to 3, to orm the data eld W as shown on ure 2(b). { the third operation computes the values o a data eld, and is called unctional operation. It is denoted by B and perorms an element-wise computation on the data eld. For example, W = inc B V, where inc(a) = a + 3, denes a data eld whose values are computed rom the V values havin the same indices (ure 3). Last, an internal operation is dened on data elds. It is called superimposition and denoted by /&/. The superimposition o several data elds results in a new data eld whose values are sequences. Each sequence is the concatenation o values mapped at the same indices. Fiure 4 shows the result o superimposin X and Y. 2.3 Pei prorams and syntactic issues A Pei statement is a set o equations which expresses the relation between input and output data elds. It is a system o equations with the input data elds

4 (a) V W (b) V W Fi. 2. Geometrical operations: (a) one-to-one relation (b) broadcast V W Fi. 3. W dened by a unctional operation applied on V bein the parameters and the output data elds bein the unknowns. A solution is a set o output data elds veriyin the system. In the Pei theory, only some statements are prorams. Denition. A Pei statement is a proram, i or any iven input data elds set, there exists at most one solution. The denition implies that we do not consider statements havin non-deterministic solutions as prorams. Let us now make precise some o the syntactic eatures o the ormalism. As typoraphic conventions, we will use X, H, W, etc. or data elds, whereas,, etc. denote unctions. The eneral orm o a Pei statement includes a header composed o the tuples I and O o input and output data elds, and an equation set S: P : I 7! O S

5 Y ;4 7;5 5;0 4 3 X Z Fi. 4. Superimposition We will also use the shorthand P S when I and O do not matter. To improve the readability, only unction names appear in the equations, their denition bein reported outside the system. Function denitions are written usin the ollowin notation: a unction o domain dom() = x j P (x) and imae im() = (x) j P (x) which associates the expression E(x) with x, is dened as (x) = P (x) : E(x) and we use # to separate alternatives in a denition by case. 2.4 Example We present an example o Pei statement or the shortest path problem. This proram is drawn rom the Unity proram presented in [], pae 04 and will be used as our runnin example: F 0 : W 7! D onirst:: (H C irst) = W H C next = min B ((H C pre) /&/ (H C shit) /&/ (H C move)) onlast:: (H C last) = D irst(i; j; k) = (0i; jn?; k=0) : (i; j; k) last(i; j; k) = (0i; jn?; k=n) : (i; j; k) onirst(i; j; k) = (0i; jn?; k=0) : (i; j) onlast(i; j; k) = (0i; jn?; k=n) : (i; j) next(i; j; k) = (0i; jn?; 0kn) : (i; j; k) pre(i; j; k) = (0i; jn?; 0kn) : (i; j; k?) shit(i; j; k) = (0i; jn?; 0kn) : (i; k; k?) move(i; j; k) = (0i; jn?; 0kn) : (k; j; k?) min(d; d2; d3) = (dd2+d3) : d # (d>d2+d3) : (d2+d3)

6 Considerin a point (i; j; k) o H, the value in that point is the weiht o the shortest path rom i to j whose indices o intermediate vertices are smaller than k. We denote it H(i; j; k) in section Semantics The previous section points out that data elds are the central concept in Pei. A data eld represents a multiset o values. It is characterized by a drawin o the multiset: a drawin associates a eometrical point on Z n with each value o a multiset. Formally, assumin the values o the multiset are in V, a drawin o the multiset is a unction v : Z n 7! V. As it has been observed in section 2., many data elds can represent the same multiset o values and a bijection links any two o them. It is the reason why, besides its drawin, a bijection characterizes also a data eld: it links the data eld with a virtual reerence domain and can be chaned by a chane o basis. In act, the bijection o a data eld is not explicit in Pei expressions and it only expresses the conormity o objects in such a way that two objects can be combined i and only i one o them conorms to the other. Formally, the bijection is denoted as, and it denes an other drawin (v? ) i dom(v) dom(). Z n v V v? Z p Denition2. A data eld is a pair, denoted as (v : ), composed o a drawin v and o a bijection such that dom(v) dom(). This denition ounds the ormal denition o operations on data elds. The superimposition combines the data elds in conormity. More precisely, we say that a data eld conorms with an other one i its bijection is a restriction o the other's bijection. The drawin o the result is the union o the drawins and the operation builds sequences o values on the intersection. As a consequence, we consider all values are sequences built rom an associative constructor \;". In the rest, we use classical notation [[:]] to associate syntax with semantics. Denition3. Let [[E ]] and [[E 2 ]] be two data elds in conormity i.e. = 2 n dom(). The superimposition denes the data eld [[E /&/ E 2 ]] as (w : 2 ), where w(z) = v (z); v 2 (z). The other operations apply a unction on a data eld [[X]] = (v : ) and orm a new data eld. We use two other notations on partial unctions: composition: the domain o a composed unction [[ o ]] is x 2 dom([[]]) j [[]](x) 2 dom([[]]) and inverse: [[inv(h)]] is the inverse o a bijection [[h]]. Denition4. Let [[]] be a partial unction rom V to W such that im(v) dom([[]]). Let [[]] be a partial unction rom dom() to dom(v). Let [[h]] be a bijection rom dom() to Z p such that dom(v) dom([[h]]).

7 { The unctional operation denes the data eld [[ B X]] as ([[]] v : ). { The eometrical operation denes the data eld [[X C ]] as (v [[]] : ). { The chane o basis denes the data eld [[h:: X]] as (v [[h]]? : [[h]]? ). Naturally, equation E = E 2 semantics where E and E 2 denotes data elds is the data elds equality [[E ]] = [[E 2 ]] (that is (v = v 2 ) ^ ( = 2 )) and the semantics o an equation set is the conjunction o the semantics o each equation. 3 Equivalence and renement o Pei statement Our denition o renement is similar to Knapp's denition, applicable to Unity prorams. Let us consider a statement P : I 7! O S and let T denote intermediate data elds in S. Accordin to the previous semantics denition, we can state what is specied by a Pei statement: we will say that P species the relation, denoted as R(P), between data elds [[I]] and [[O]]. The relation is dened by the conjunction o all equations o the statement, the intermediate data elds are existentially quantied in order to only dene the relation between [[I]] and [[O]]. Formally: R(P) = ([[I]]; [[O]]) j 9[[T]]: [[S]] Denition 5. (renement) Let P and P' be two Pei statements. We say that P is rened by P' and we note P v P 0, i R(P 0 ) R(P) The symbol denotes an inclusion relation which takes data eld equivalence into account. It is dened as ollows: Denition 6. (inclusion modulo data eld equivalence) Let P and P' be two Pei statements. R(P 0 ) R(P) i A(R(P 0 )) A(R(P)) where A is the application which returns the multisets pair represented by a iven data elds pair. Note 7. O course, i R(P 0 ) R(P), then R(P 0 ) R(P). We will speak about stron renement in that particular case. The equivalence o two statements is dened rom renement: Denition 8. (equivalence) Let P and P' be two Pei statements. We say that P and P' are equivalent and we note P P 0, i P and P 0 rene each other.

8 3. Relevance o typecheckin in Pei As seen in the previous section, Pei operations are not allowed on any data elds: this means that some phrases are orbidden accordin to some type constraints. In other words, i the constraints do not hold, then we say that no semantics is associated with such phrases. Note that this is not absolutely necessary and we could decide to associate a specic meanin with such phrases. But this addresses the ollowin crucial question: is a iven Pei specication, easible or not? It is important to be able to check or easibility at any step o the renement process [7]: in Pei, easibility checkin is just typecheckin. We presented in [] an alorithm that can iner the type o Pei expressions dened as the pair (dom(v); dom()). Our alorithm presents weak limitations. Based on this alorithm, a typechecker or Pei statements has been implemented. It uses the Omea library [5, 8] or evaluatin set expressions. This means that the alorithm is decidable i the unctions used or eometrical operations or chane o basis inside a statement can be coded into an Omea relation. 4 Transormation rules Transormation rules are partitioned in three sets: the rst rules are derived rom operation properties, the ollowin ones are derived rom equation systems and the last ones are equivalence rules. 4. Operation properties rules These rules are ounded on alebraic properties o operations. More precisely, the rened statement is obtained by replacin one occurrence o a Pei expression by an other one that can be proved equal rom some operation property or rom the mathematical structure o the data elds set. The rules derived rom operation properties all into two cateories: some are unoriented and we obtain an equivalent statement by replacin an expression by the other: ( o ') B E = B (' B E) (h o h'):: E = h:: (h':: E) (h:: E ) /&/ (h:: E 2 ) = h:: (E /&/ E 2 ) ( B E) C = B (E C ) B (h:: E) = h:: ( B E) Others are oriented because the conditions required or the expression on the riht to be well-ormed are stroner than the ones required or the expression on the let to be well-ormed. I, when substitutin the riht expression or the let one, the new statement is well-ormed, then it stronly renes the old one. Conversely, i we substitute the let expression or the riht one, then the new statement is well-ormed but it is an abstracted statement: it is equivalent only i both statements have the same type constraints.

9 E C ( o ')?! (E C ) C ' (E /&/ E 2 ) C?! (E C ) /&/ (E 2 C ) (h:: E) C?! h:: (E C (inv(h) o o h)) h:: (E C )?! (h:: E) C (h o o inv(h)) 4.2 Equations systems rules These rules are more eneral than the precedin ones. They permit to modiy not only an expression, but one or more equations o the system which denes a statement. The transormed system is a new one whose solutions set is the same: these rules maintain the equality o statements. Amon these rules let us cite classical substitution and the application o a non-sinular unction to both sides o an equation. 4.3 Equivalence rules These rules are still more eneral than the precedin ones. They consist in substitutin a data eld or an equivalent one. One o these rules is classical. It allows to chane representation o a data eld. This is its ormal denition: Theorem 9. Let PS be a Pei statement and T a data eld name in S. I PS[T=(h:: T)] is well-ormed, then it renes P. where E[E =E 2 ] denote the result o replacin all occurences o E by E 2 in E. 4.4 Operational aspects Operational aspects dene the set o computations associated with any data eld denition. This means the denition o an order on the data eld elements. This order is a partial one or parallel computations. We dene this order, denoted as `, or some iven partial order < on Z m, by considerin the bijection rom Z n to Z m which characterizes a data eld. Denition0. Let (v : ) be a data eld where is a bijection rom Z n to Z m, 8z; z 0 2 dom(v) : v(z) ` v(z 0 ), (z) < (z 0 ) The choice o the order relation < on Z m predetermines the operational deinition o a proram. In act, the aim o the transormations is to explicit or to build a "nice" bijection which introduces the "convenient" order to dene a "nice" operational behaviour o the proram. These transormations lie on the chane o basis operation. Examples

10 . Let us consider a bijection rom Z n to Z m such as (z) = (p(z); t(z)), where p is a unction rom Z n to Z m? and t a unction rom Z n to N. Note that such a denition is a classical way to dene a schedulin and a mappin o the computations on a processor set. { Let < be an order on Z m such that (z) < (z 0 ) i t(z) < t(z 0 ) on N. The induced operational denition only denes computations schedulin. { Let < be an order on Z m such that (z) < (z 0 ) i p(z) = p(z 0 ) ^ t(z) < t(z 0 ). The induced operational denition denes computations mappin and the schedulin o the processors. 2. The shortest path problem continued: an obvious synchronous operational denition o the proramm F 0 is obtained by identiyin index k with time t. In this solution, indices i; j can be identied with coordinates x; y o the computation point on the processor array. By considerin the bijection o data eld H, this solution consists in denin as the bijection rom Z 3 to Z 3 such that (z) = (p(z); t(z)), where p is the unction rom Z 3 to Z 2, dened as p(i; j; k) = (i; j) and where t is the unction rom Z 3 to N, dened as t(i; j; k) = k. 5 About asynchrony Asynchronous prorams are dened rom a unction denoted as pack or packin unction: when applied to a data eld X whose drawin domain dom(v) is a subset o Z, it denes a bijection which packs dom(v) in an interval o N, whose lenth is the cardinality o dom(v). The chane o basis applied on pack(x) and X denes a packed data eld such that dom(v) = dom(). Here is the ormal denition o the packin unction which is eneralized to be applied to any data eld: Denition. The unction pack associates with any data eld X o drawin v such that dom(v) Z n ; n>0, the bijection pack(x) dened on dom(v) which maps (i ; : : :; i n ) to (i ; : : :; i n? ; h(i n )) where h is the bijection dened on domain i n j (i ; : : :; i n ) 2 dom(v) as: { im(h) = [0::card (dom(h))[ { h strictly increases. The word \asynchronous" characterizes a non-deterministic aspect o the proram computin model. To illustrate what is asynchrony, let us consider the ollowin trivial example. Let X be a data eld whose drawin domain is a subset o Z. The values o X are ordered by their coordinates in Z. Let us dene: Sync : X 7! Y Y = X C odd odd(i) = ((i mod 2)=) : i

11 As said in section 4.4, this proram denes the instants where the values o Y are computed, or the bijection o Y is the same as the bijection o X. So, this proram can be considered as synchronous. Let us consider now the ollowin equivalent proram: Async : X 7! Y Z = X C odd pack(y):: Y = pack(z):: Z odd(i) = ((i mod 2)=) : i The packed drawins o data elds Y and Z are the same. So, the sequences o values in Y and Z are the same. But, the drawins o Y and Z are dierent. This means that the computin instants o these two sequences can be dierent. This proram can then be considered as asynchronous and the bijection pack(y) denes the computation delays in Y. 6 Prorams derivation Prorams derivation, by usin transormation rules, is illustrated here under with the shortest path problem. We present the rst steps to desin an asynchronous solution or the problem whose initial statement was previously iven. Step : distribution An asynchronous solution can be obtained by dissociatin index k rom time t, by convincin that a value in the point (i; j; k) is computed beore the computation o a value in the point (i; j; k+). In order to reach this oal, we apply a transormation which "distributes the index k on each point (i; j; k)" and leads to the ollowin new statement: F : W 7! D onirst:: (H C irst) = join 0 B W H C next = min' B ((H C pre) /&/ (H C shit) /&/ (H C move)) onlast:: (H C last) = join n B D irst(i; j; k) = (0i; jn?; k=0) : (i; j; k) last(i; j; k) = (0i; jn?; k=n) : (i; j; k) onirst(i; j; k) = (0i; jn?; k=0) : (i; j) onlast(i; j; k) = (0i; jn?; k=n) : (i; j) next(i; j; k) = (0i; jn?; 0<kn) : (i; j; k) pre(i; j; k) = (0i; jn?; 0<kn) : (i; j; k?) shit(i; j; k) = (0i; jn?; 0<kn) : (i; k; k?) move(i; j; k) = (0i; jn?; 0<kn) : (k; j; k?)

12 min 0 ((d; k); (d2; k2); (d3; k3)) = (dd2+d3) : (d; k+) # (d>d2+d3) : (d2+d3; k+) join 0 (d) = (d; 0) join n (d) = (d; n) This new proram is obtained by considerin a new intermediate data eld H' whose any point (i; j; k) is a pair o values: the value o data eld H in this point and the value k: H' = ( /&/ ) (join K B (H C select K )) K where: select K (i; j; k) = (0i; jn?; k=k) : (i; j; k) and join K (d) = (d; K) H' can be substituted or H in any equation o the proram F 0, accordin to an equation system rule which allows to apply an operation on the two hands o an equation. Let us consider, or example, the ollowin equation in F 0 : H C next = min B ((H C pre) /&/ (H C shit) /&/ (H C move)) Usin properties o the superposition, this equation can be rewritten as: ( /&/ ) K ( /&/ ) K ((H C next) C select K ) = ((min B ((H C pre) /&/ (H C shit) /&/ (H C move))) C select K ) Then, or any K: (H C next) C select K = (min B ((H C pre) /&/ (H C shit) /&/ (H C move))) C select K By usin the equation system rule, we can apply unctional operation join K on the two hands o this equation: join K B ((H C next) C select K ) = join K B ((min B ((H C pre) /&/ (H C shit) /&/ (H C move))) C select K ) Now, by usin properties o operations and by applyin an other equation system rule to substitute (H' C select K ) or (join K B (H C select K )), the ollowin equation comes: (H' C next) C select K = (min' B ((H' C pre) /&/ (H' C shit) /&/ (H' C move))) C select K where: min 0 ((d; k); (d2; k2); (d3; k3)) = (dd2+d3) : (d; k+) # (d>d2+d3) : (d2+d3; k+) For this equation is valid or any K and by renamin H' as H and min 0 as min, we deduce the correspondin equation o the proram F. Step 2 : schedulin This step allows to explicit a schedulin o the computations and leads to the ollowin new statement:

13 F 2 : W 7! D onirst':: (H C irst') = join 0 B W H C next' = min B ((H C pre') /&/ (H C shit') /&/ (H C move')) onlast':: (H C last') = join n B D irst 0 (i; j; t) = (0i; jn?; r=0) : (i; j; t) last 0 (i; j; t) = (0i; jn?; r=n) : (i; j; t) onirst 0 (i; j; t) = (0i; jn?; r=0) : (i; j) onlast 0 (i; j; t) = (0i; jn?; r=n) : (i; j) next 0 (i; j; t) = (0i; jn?; 0<rn) : (i; j; t) pre 0 (i; j; t) = (0i; jn?; 0<rn) : (i; j; k(i; j)? (r?)) shit 0 (i; j; t) = (0i; jn?; 0<rn) : (i; r; k(i; j)? (r?)) move 0 (i; j; t) = (0i; jn?; 0<rn) : (r; j; k(i; j)? (r?)) min((d; k); (d2; k2); (d3; k3)) = (dd2+d3) : (d; k+) # (d>d2+d3) : (d2+d3; k+) where r = k(i; j)(t) The transormation consists in applyin the chane o basis inv(pack(h)) on the data eld H o the previous proram. By denition o the packin unction, we can write: pack(h)(i; j; t) = (i; j; k(i; j)(t)) where k(i; j) is a bijection dened rom a subset o N to the drawin domain o H. First, we use an equation system rule to apply the chane o basis operation inv(pack(h)) on the two hands o any equation o the proram F. Then, we use operation properties in order to apply this chane o basis on H. Last, we rename inv(pack(h)):: H as H and use the equivalence rule, which allows to chane o representation in order to obtain the equivalent proram F 2. Step 3 : domain expandin The ollowin statement is obtained by considerin the unction k(i; j) dened as: k(i; j) : N! [0::n] t 7! k(i; j)(t) i t 2 dom(k(i; j)) k(i; j)(t?) else This transormation allows to expand the domain o data eld H on (i; j; t) j 0i; jn? ^ t0. It is an uniormization o the unction k(i; j).

14 F 3 : W 7! D onirst:: (H C irst) = join 0 B W H C next = min B ((H C pre) /&/ (H C shit) /&/ (H C move)) onlast:: (H C last) = join n B D irst(i; j; t) = (0i; jn?; t=0) : (i; j; t) last(i; j; t) = (0i; jn?; t=t ) : (i; j; t) onirst(i; j; t) = (0i; jn?; t=0) : (i; j) onlast(i; j; t) = (0i; jn?; t=t ) : (i; j) next(i; j; t) = (0i; jn?; t>0) : (i; j; t) pre(i; j; t) = (0i; jn?; t>0) : (i; j; t?) shit(i; j; t) = (0i; jn?; t>0) : (i; r; t?) move(i; j; t) = (0i; jn?; t>0) : (r; j; t?) min((d; k); (d2; k2); (d3; k3)) = (k2=k ^ k3=k ^ dd2+d3) : (d; k+) # (k2=k ^ k3=k ^ d>d2+d3) : (d2+d3; k+) # (k26=k _ k36=k) : (d; k) where r = k(i; j)(t) and T = k(i; j)? (n). Step 4 : abstraction From the precedin proram, we make permissible to have one process race ahead o another by usin property H(i; j; k+m)h(i; j; k), m>0. This leads to the ollowin equivalent proram that is presented in [], p. 09: F 4 : W 7! D onirst:: (H C irst) = join 0 B W H C next = min' B ((H C pre) /&/ (H C shit) /&/ (H C move)) onlast:: (H C last) = join n B D irst(i; j; t) = (0i; jn?; t=0) : (i; j; t) last(i; j; t) = (0i; jn?; t=t ) : (i; j; t) onirst(i; j; t) = (0i; jn?; t=0) : (i; j) onlast(i; j; t) = (0i; jn?; t=t ) : (i; j) next(i; j; t) = (0i; jn?; t>0) : (i; j; t) pre(i; j; t) = (0i; jn?; t>0) : (i; j; t?) shit(i; j; t) = (0i; jn?; t>0) : (i; r; t?) move(i; j; t) = (0i; jn?; t>0) : (r; j; t?) min 0 ((d; k); (d2; k2); (d3; k3)) = (k2k ^ k3k ^ dd2+d3) : (d; k+) # (k2k ^ k3k ^ d>d2+d3) : (d2+d3; k+) # (k2<k _ k3<k) : (d; k)

15 7 Conclusion The theory we have presented in this paper is ounded on the simple mathematical concepts o multiset and o an equivalence between their representations as data elds. A detailed presentation can be ound in [0, 4]. Proram transormations are ounded on this equivalence and dened rom a renement relation. Due to the uniyin aspect o this theory, solutions that can be reached by these transormations are relevant to various synchronous or asynchronous computin models. The point we have ocused in this paper concerns asynchronous computations and was illustrated by the alebraic path problem. Reerences. K.M. Chandy and J. Misra. Parallel Proram Desin : A oundation. Addison Wesley, M. Chen, Y. Choo, and J. Li. Parallel Functional Lanuaes and Compilers. Frontier Series. ACM Press, 99. chapter P. Clauss, C. Monenet, and G.-R. Perrin. Synthesis o size-optimal toroidal arrays or the alebraic path problem : A new contribution. Parallel Computin, 8:85{ 94, S. Genaud, E. Violard, and G.-R. Perrin. Transormation techniques in pei. In EUROPAR'95, Stockholm, Sweden, Auust Wayne Kelly, Vadim Maslov, William Puh, Evan Rosser, Tatiana Shpeisman, and David Wonnacott. The Omea Library - Version.00, April 996. Interace Guide. 6. C. Mauras. Alpha : un lanae equationnel pour la conception et la prorammation d'architectures paralleles synchrones. PhD thesis, Universite de Rennes, Decembre C. Moran. Prorammin rom specications. C.A.R. Hoare. Prentice Hall Ed., Endlewood Clis, N.J., William Puh. The omea test: a ast and practical inteer prorammin alorithm or dependence analysis. Communications o the ACM, Auust P. Quinton. The mappin o linear recurrence equations on reular arrays. Journal o VLSI Sinal Processin,, E. Violard. Une theorie unicatrice pour la construction de prorammes paralleles par des techniques de transormations. PhD thesis, Universite de Franche-Comte, Octobre E. Violard. Typecheckin o pei expressions. In LNCS, EUROPAR'97, volume 300, paes 52{529, Passau, 997. This article was processed usin the LaT E X macro packae with LLNCS style