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Bergstra, J.A. (Jan)

Klop, J.W. (Jan Willem)

Bergstra, Jan

Klop, Jan Willem

NARCIS 

Process algebra for communication and mutual exclusion

CWI's Institutional Repository

English

Coninck, H.C. de

Radboud Repository

A Matter of capabilities

AFDELING INF.ORMATICA stichting mathematisch centrum (DEPARTMENT OF COMPUTER SCIENCE) J.A. BERGSTRA & J.W. KLOP IW 218/83 PROCESS ALGEBRA FOR COMMUNICATION AND MUTUAL EXCLUSION Preprint ~ MC JANUARI kruislaan 413 1098 SJ amsterdam Printed at the Mathematical Centre, Kruislaa,1 413, Amsterdam. The Netherlands. The Mathematical Centre, founded 11th February 1946, is a non-profit institution for the promotion of pure and applied mathematics and computer science. It is sponsored by the Netherlands Government through the Netherland'i Organization for the Advancement of Pure Research (Z. W.0.). 1980 Mathematics subject classification: 68B10, 68C01, 68D25, 68F20 1982 CR. Categories: F.1.1, F.1.2, F.3.2, F.4.3 . . d 1 1 . *) Process algebra for communication an mutua exc usion by J.A. Bergstra & J.W. Klop ABSTRACT Within the context of an algebraic theory of processes we provide an equational specification of process cooperation. We consider three cases: process merging, merging with communication, and merging with mutual exclu-sion of critical sections. The term rewrite system behind the communication algebra is shown to be confluent and terminating (modulo its permutative reductions). KEY WORDS & PHRASES: nondeterministic processes, process algebra, merge, concurrency, communication, synchronisation, mutual exclusion, critical sections, term rewrite systems *) This report will be submitted for publication elsewhere. 1 0. INTRODUCTION Let A be a finite collection (alphabet) of atomic actions, ceA a dis-tinguished symbol denoting deadlock. Finite processes are generated from atomic processes in A using two operations + nondeterministic choice and sequential composition. The following equational laws will hold for finite processes. X + y = y + X X + (y + z) X + X = X (x + y) + z (X + y) . Z = X • Z + y . Z (x. y) . z x.(y.z) X + C = X c .x c Al A2 A3 A4 A5 A6 A7 The initial term algebra of these equations is (A,+, ., c). w The main source of process algebra in this style is MILNER [9]. Exactly the above processes occur as finite uniform processes in DE BAKKER & ZUCKER [1], [ 2]. After adding an extra equation: x. (y + z) = x.y + x.z, one obtains a version of trace theory as described in REM [13]. For each PE A and n ,ii 1 we have the approximation (p) of p. This is w n inductively described by (p + q)n = (p)n + (q)n (a) = a n (ax) 1 n+ a (x) • n 2 Interestingly, if A = { (p) I p EA } then (A , +, . , 15) is another model n n w n of the axioms Al, .. ,A7. Infinite processes (A00 ) can be obtained as a projective limit of the structures A. Technically this means that Am is the set of all sequences p= n (p1 ,p2 ,p3 , ... ) with piE Ai and p. = (p. 1 ) .. The operations '+' and'.' are 1 1+ 1 defined component-wise: (p + q)n = (p)n + (q)n' (p.q)n = ((p)n. (q)n)n' thus obtaining the process algebra m (A , +, • , 15 ) • m On A a metric exists: d(p,q) = {0 if p = q -n 2 with n minimal such that (p) 1 (q) if p 1 q. n n m (A ,d) is a complete metric space, in fact it is the metric completion of (A ,d). The operations+,. are continuous. w (A00 ,d) was introduced in DE BAKKER & ZUCKER [l]. MILNER[l0] uses charts modulo bisimulation (from PARK [12]) to obtain infinite processes from finite ones. Working with trace sets under the extra assumption x. (y + z) = x.y + x.z this metric occurs in NIVAT [11].In DE BAKKER et al. [31 the connections between (Am,d) and its corresponding trace space are investigated. The processes discussed so far are provided with a bare minimum of struc-ture. The crux of the algebraic method lies in algebraically defining new operators over the given process domains that will correspond to important process composition principles. We will describe operators corresponding to the following three composition principles: 1. merging two processes 2. merging with communication 3. merging processes with mutual exclusion for critical sections. 3 1. MERGING 'IWO PROCESSES The result of merging processes p and q is Pllq. For algebraic reasons (finite axiomatisability and ease of computation) an auxiliary operation ll (left merge) is used. The process pllq stands for the result of merging p and q but taking the first step from p. Both operations II and IL are specified on (A,+, ., o) by this system of equations: w xllY = xlL Y + Yll x all x = a.x ax IL Y = a(xllY> (x + y) ll z = x IL z + y ll z Ml M2 M3 M4 Here x,y,z are variables over A and a is a variable over A. Formally this w is justified as a two-sorted logic with sorts A and A where one sort is a w subset of the other one. a, The operations are extended to A as follows: (p1,P2•···> II (q1,q2,···> = ( (P1llq1)1, <P2llq2>2•···> (p1,P2,···> IL (q1,q2,···> = ( (PllJ_ql)l' <P2llq2>2,···> We omit the proof that these are indeed projective sequences (i.e. that ((pn+lllqn+l)n+l)n = (pnllqn)n ). It also follows that 11 and IL are continu-ous w.r.t. the metric d. In BERGSTRA & KLOP [4] the operation LI_ was introduced and a general existence theorem for solutions of equations of the form x= ,(x) in (A00 , +, . , 11, IL, o) is derived. 2. MERGING WITH COMMUNICATION Starting with (A , +, . , o) plus a communication function . I . : AxA + A w which describes the effect of sharing (simultaneously executing) two atomic actions, three operations II, IL and I are defined on A. Here' I' extends w 4 the given communication function. 11 and IL coincide with the operators de-fined in Section l if the effect of alb is always o (i.e. no two atomic actions communicate). For the communication function three properties are required: alb= bla (alb) le= al (blc) ola = o Then II, IL and I are specified by: xllY = xll_y + yll_x + xly all_x = a.x (ax) ll_y = a(xlly) (x + y) 11. z = x IL z + y LL z (ax) lb= (alb) .x al (bx) = (alb).x (ax) I (by) = (alb).(xllY) ex+ y> I z = xlz + ylz x I (y + z) = xly + xlz Cl C2 C3 CMl CM2 CM3 CM4 CMS CM6 CM7 CMB CM9 Let C = {a EA I 3 b EA (a I b) -'Io}. C contains all actions that must eventually communicate. So if process p runs on its own an action cEC cannot be executed and should be replaced by o. This is modeled by the operation t:,,: A + A defined by w w 6(a) = {a if a¢ C 6ifae:C 6(x + y) = 6(x) + 6(y) 6(.x.y) = 6(x).6(y) 62 63 Notice that 6(xllY> '/ 6(x)ll6(y). Thus 6 is a homomorphism on (A,+,., 6) w but not on (A , +, • , 11 , IL , I , 6) • w An important observation concerning the difference between processes 5 and trace sets is exhibited in the following example. Let A= {a,c1 ,c2 ,c,6} and let c1 1c2 = c. All other communications result in 6. Now 6(a(c1 + c 2) II c1 ) = ac, and 6((ac1 + ac2 ) II c1 ) = ac + a6 so the second process ac1 + ac2 has a deadlock possibility and the first one, a(c1 + c 2), not. As before 11 , IL and 6 can be extended to continuous operations on (A00 ,d). This formalism includes both message passing and synchronisation. In [9] and [l~]synchronisation is modeled by having alb=, whenever alb'/ 6, , denoting a silent move. In HENNESSY & PLOTKIN [6] a definition correspon-ding to the equation xllY = xll_y + yll_x + xly occurs. The heart of the system consists of Al, ... ,A7,Cl, ... ,c3,CM1, .•. ,cM9. We will call this system ACP, for algebra of communicating processes. ACP seems to provide a concise formulation of the algebraic essence of communication. In view of this we review its structure in detail here. We will show that the new operators are indeed well-defined by CM1, ... ,CM9 over Al, •.• ,A7 + Cl, ••. ,C3. We will rearrange ACP into a TRS (term rewrite system) which is shown to be confluent and strongly terminating modulo the permutative reductions Al and A2. As a consequence we find that each term built from A by +, . , 11 , IL , I can be proved equal to a unique term in A w by ACP. Finally we prove that II is associative, as well as several other interesting identities in Theorem 2.2. 6 For technical reasons we associate to each a EA a unary operator a* which acts as follows: a*x = a.x. On the term system generated by A, +, . , II, IL, I, a* (a EA) we introduce two norms 1-1 and 1- I . Here intuitively Is I computes an upper bound for the path lengths in S and 11s II computes an upper bound of the number of (nontrivial) summands in which S decomposes. lal = 1 Ja*x I = 1 + Ix I lx.yl = lxl + IYI Ix+ yl= max<jxl,IYP lxJyl = lxl + IYI - 1 Ix lL y I = Ix I + I y I Ix 11 Y I = Ix I + I Y I Ila 11 = 1 lla*x ~ = 1 llx. y II = llx II llx + Y II = llx II + IIYII llxly II= 11x11. ,1y 11 llxli_yjj = llxll llxllY II= llx II+ l!Y~ + llx~ · IIYII • Now consider the following term rewrite system RACP: X + y + y + X x + (y + z) + (x + y) + z (x + y) + z + x + (y + z) X + X + X (X + y) • Z + X. Z + y. Z a.x + a*x (a*x) .y + a*(x.y) X + 6 + X 6*x + 6 xlly + xll_ y + yll_ x + xly all x + a*x (a*x) lL y + a* (xii y) (x + y) lL z + x IL z + yLI_ z alb+ c b a, (a*x)jb + c* b.x a, alb*x + c* .x a,b (a*x)j(b*y) + c* (xllY> a,b (x + y) lz + xlz + ylz xi (y + z) + xly + xlz RAl RA2 RA2' RA3 RA4 RAS' RAS RA6 RA7 RCMl RCM2 RCM3 RCM4 RCMS' RCMS RCM6 RCM7 RCM8 RCM9 In RCM5', ••• ,RCM7 the symbol c b denotes alb EA. The axioms Cl ,C2,C3 a, translate into the commutativity and associativity of c and c = 6 for 6 ,a all a CA. Convertibility in RACP is denoted by =R. 2.1. THEOREM. For all ACP-terms without variables: ACP ACP l- s = 1- s = T# S = T R S' for some S' not containing ll , [l , I . 7 ( i) (ii) (iii) (iv) (v) ACP 1- S'= S" ~ Al, .•• ,A7 -1- S' = S" for S' ,S" not containing II, lL, I -(vi) (vii) S. (T.U) =R (S.T) .U RACP is weakly confluent (has the weak Church-Rosser property), working modulo Al and A2. RACP is strongly terminating, modulo Al and A2. RACP is confluent (has the Church-Rosser property). PROOF. we start with (vi) and we introduce the auxiliary notion of the multi-set of direct subterms DS(T) of a term T: DS(a) = 9) DS(a*x) = DS(x) DS(x + y) = DS(x) u DS(y) DS(x • y) = {x • y} u DS(x) 0 DS(y) (here ois ., II, IL, or l> Here 0 denotes the multiset union. Let [S] be the mapping from terms to wxw defined by c s 1 = c Is I , II s II> • This mapping is extended to multisets over terms, thus producing multisets over wxw [V] = {[S] I SEV}. On w x w there is the lexicographic well-ordering < which induces a well-ordering « on finite multisets over w x w • We now observe that along a re-duction path R ) 2 8 we have [ DS (T.) J » [ DS (T. l) J if R. is not RAl, RA2, RA2', and 1 1+ 1 [DS(Ti)J = [DS(Ti+l)J if Riis RAl, RA2 or RA2'. From this observation strong termination of RACP modulo Al and A2 follows. Instead of a proof of the observation we provide two characteristic examples. (1) a.x + a*x. Then: [DS(a.x)J = [a.xJ u [DS(x)J and [DS(a*x) = [DS(x)J. Now [a.xJ majorizes each element of [DS(x)J because [SJ 6 [DS (x) J • Isl~ !xi • Is I < la.x I- Hence [DS (a.x) J » [DS (a*x) J. ( 2) x 11 y • x lL y + y lL x + x I y. Then: [DS(xlly)J = [xllyJ 0 [DS(x)J 0 [DS(y)J and [DS(xll_y + yll_x + x!y)J = [xll yJ u [DS(x)J v [DS(y)J u [yll xJ v [DS (x) J 0 [DS (y) J 0 [xlyJ 0 [DS(x)J 0 [DS(y)J. Again [x!lyJmajorizes all of [xll_yJ, [yll_x], [xlyJ, [DS(x)], [DS(y)J, the first three in width and the second two in depth. An alternative proof of termination can be given by ranking all occurrences of II ,IL ,I by the I-I-norm of the term of which they are the leading operator. Using this extended set of operators a recursive path ordering can be found which is decreasing in all rewrite steps except the first three (RA1,RA2,RA2'). See DERSHOWITZ [5J. Proof of (v). RACP is weakly confluent modulo~, the congruence generated by Al and A2. (We are here working in congruence classes and reductions have the form [SJ~ + [ S' J ~ whenever S + S' . ) This is a matter of some 400 straight-forward verifications. (Of course left to the reader as an exercise.) Proof of (vii). Working modulo~ RACP is strongly terminating in view of (vi). Now combining (v) and (vi) and using Newman's Lemma· (see [BJ, Lemma 5.7. (1) or [7J where more information about reduction modulo equivalence can be found) we find that RACP is confluent modulo~ and consequently it is confluent because the reductimsgenerating ~ are symmetric. Proof of (ii). This follows immediately from (vi). Proof of (iv). First one proves the associativity of. for terms not con-taining 11 ,ll ,I , using induction on the structure of S. The result then immediately follows using (ii). 9 Proof of (i). s =RT • ACP 1- S =Tis immediate. For the other direction one uses (iv). Proof of (iii). If ACP f- S' = S" then by (i) S' = S" and by (vii) for . R some S"': S' • > S"' and S"-» S"' (here ~> is the transitive reflexive closure of •) . Now because s' and S" are free of 11 , lL , I we see that S' ~> S"' <~ S" is just a proof in Al, ... ,A7. D 2. 2. THEOREM. The following identities hold in (A , +, • , 11 , IL , I , o ). : w (1) xly = ylx (2) xllY = yllx (3) xl<ylz) = (xly>lz (4) (xlJ_y) II z = xlL (yllz) (5) x I (y LI_ z) = (x I y) lL z (6) xll<yllz) = (xllYlllz. PROOF. All proofs use induction on the structure of x,y,x written as a term over (A,+,.), which is justified by Theorem 2.1.(2). We write x = l· a.x. + I3. a 3~ l. l. l. z=t CZ +t lm mm ln c' n (1) and (2) are proved in a simultaneous induction. ylx. 10 Here we use Cl and the induction hypothesis for xi 11 y k = y k II xi. (2): xllY = xLI_y + yl1_x + xly = yLI_x + xl1_y + ylx = yllx. The proof of (3), .•. ,(6) is also done using one simultaneous induction. (3): Write x = x' + x" where x' = la.x. and x" = La' .• Likewise y = y' + y" l. l. J and z = z' + z". Then xl<ylz) = x'l(Y'lz') + x'l(Y"lz') + x'l(Y'lz") + x'l(Y"lz") + x"I (y' lz') + x"l (y"lz') + x"I (y' lz") + x"I (y"lz"). Now x'l(Y'lz') = }:(a.l(bklc ))(x.ll<Ykllz)) = l. m J. m }: ((a. I bk) I c ) ( (x. 11 yk) II z ) = J. m l. m ex, I y, > I z, . Here we used C2 and the induction hypothesis for (6). The other summands of xi (ylz) are treated similarly. Hence xi (ylz) = (xly) lz. (4) : (xll y) ILz = ((}:a.x. + }:a'.) ll_ y) LI_ z = (}:a. (x. lly) + }:a' .• y) lLz = l.l. J l. l. J }:a. ( (x. llY) llz) + }:a•. (yllz) = (induction hypothesis on (6)) l. l. J }:a.(x.ll<yllz)) + }:a~(Yllz) = l. l. J (}:a.x. + ta'.) lL (Yllz) = l. l. J x lL (YII z) • (5): Let x = x' + x" and y = y' + y" as in the proof of (3). Then xl(yll_z) = x'l(Y'ILz) + x'l(y"LI_z) + x"l(y'IL_z) + x"l(y"ll_z). Now x'l(y'LI_z) = (laixi)l(}:bkyk)llz) = (}:aixi)l(}:bk(Ykllz» = ex • I y • > u_ z . }:(ailbk) (xiii (ykllz)) = (induction hypothesis on (6)) }:(ailbk) ((xillyk) llz) = ( }:(ailbk) (xillYk)) llz = The other three summands are treated similarly. Hence x I (y LL z) = (x I y) IL. z. 11 (6) Write A (y,z) = xll_ (yllz> and B (y,z) = (ylz> ll_x. Then: X X xii (yllz> = xlJ_(yllz> + (Yllz) ll_x + xi (yllz) == A (y,z) + (yll_z) ll_x + (zll_y) ll_x + (ylz> ll_x + xi (yll_z) + xi (zll_y) + xi Eylz) = X A (y,z) + y[L(zllx) + z[L(yllx> + B (y,z) + (xly)LI_z + (xlz)IJ_y + xl(ylz) = X X A (y,z) + A (z,x) + A (y,x) + B (y,z) + B (y,x) + B (x,z) + xi (ylz>. (*) X y Z X Z y Also (xllY> llz = zll (xlly) = zll (yllx> = A (y,x) + A (x,z) + A (y,z) + B (y,x) _+ B (y,z) + B (z,x) + zl (ylx> = Z y X Z X y A (y,z) + A (x,z) + A (y,x) + B (y,z) + B (z,x) + B (y,x) + (xly) lz X y Z X y Z which equals (*) using the commutativity of the A's and B's and the induction hypothesis on (xly) lz. 12 3. MERGING WITH MUTUAL EXCLUSION OF CRITICAL REGIONS From (A , +, . , o) we derive ( (Av A) , +, • , o) by adding for each w + w atomic action a A, a new copy~- This~ stands for: do a and, if possible, immediately proceed with an action of the same process (i.e. do not allow interference of other processes until the "lowest" process containing a has terminated) . Now this algebra is enriched with operators 11 and IL and a really new one: Pt-+E• Here E stands for: execute pas a critical region (i.e. do not allow interference of other steps). The axioms for ( (Au A) , +, . , 11 , lL , , o ) are as follows: +w xllY = xll_y + yll_x MEl alJ_x = a.x ME2 ~Li_ X = a.x ME3 (ax) IJ_ y = a. (xlly) ME4 (ax) lLY = ~- (xll_y) ME5 + a = a ME6 -a = a ME7 .± a.x = a.x MEB + -a.x = a.x ME9 :t,._ + -X + y = X + X MElO 0 = + 0 MEll We omit a prooftheoretic analysis of these equations, but state without proof these useful facts: !llx = !·X + X·! · In general !l.Lx_ "I ! 11 x however. 13 Suppose one considers terms generated by A,+, ., II and • These have an intuitively clear meaning and should be given a semantics in (A,+, ., o). w Note that x frequently is denoted by <x> in the litterature on parallel programs. On the other hand, atomic actions of the form a are not common -+-and should not enter the definition of a programming system; they are here only used as a means of calculation and are eliminated in the second step below (i.e. by applying the homomorphism t). As an intermediate step a semantics in ( (Au A) , +, • , o) is found: -+- w frl* (a) = a frl*(p + q) = frl*(p) + frl*(q) frl*(p.q) = frl*(p).frl*(q) M* (£) = M* Cpl frl* <Pllq) = M* (p) IIM* (q) Then the canonical mapping (homomorphism) t: ( (Au A) , +, . , o) --+ (A , +, . , o) -+- w w generated by t(a) = t(~) = a can be used to provide a semantics for +,.,II,_ terms over At M (p) = t (M* (pl ) . Note that frl* is a homomorphism w.r.t. all program constructions +,.,II,_, as should be expected from a denotational (or rather compositional) semantics; tis a homomorphism only w.r.t. +,.,o. Hence Mis a homomorphism w.r.t +,.,o. However, M does not act as a homomorphism w.r.t. II- (I.e. frl(pllq) 'f- frl(p)II frl(q), in general. ) 14 REFERENCES [l] DE BAKKER, J.W. & J.I. ZUCKER, Denotational semantics of concurrency, Proc. 14th ACM Syrop. on Theory of Computing, pp.153-158, 1982. [2] DE BAKKER, J.W. & J.I. ZUCKER, Processes and the denotational semantics of concurrency, Department of Computer Science Technical Report IW 209/82, Mathematisch Centrum, Amsterdam 1982. [3] DE BAKKER, J.W., J.A. BERGSTRA, J.W. KLOP & J.-J.CH. MEYER, Linear time and branching time semantics for recursion with merge, Department of Computer Science Technical Report IW 211/82, Mathematisch Centrum, Amsterdam 1982. [4] BERGSTRA, J.A. & J.W. KLOP, Fixed point semantics in process algebras, Department of Computer Science Technical Report IW 206/82, Mathematisch Centrum, Amsterdam 1982. [5] DERSHOWITZ, N., Orderings for term-rewriting systems, Theoretical Computer Science 17 (1982)pp.279-301. [6] HENNESSY, M., & G. PLOTKIN, A term model for CCS, Proc. 9th MFCS, Poland (1980), LNCS 88. [ 7] HUET, G. , Confluent reduetwns: Abstroct propePties and awlwatwns to term reillitmg lY3tere, J.ACM 27 (4) (1980) pp.797-821. [8] KLOP, J.W., Combinatory Reduction Systems, Mathematical Centre Tracts 127, Mathematisch Centrum, Amsterdam 1980. [ 9] MILNER, R., A Calculus of Communicating Systems, Springer LNCS 92, 1980. [10] MILNER, R., A Complete Inference System for a class of Regular Behaviours, Preprint, Edinburgh 1982. [11] NIVAT, M., Infinite words, infinite trees, infinite computations, Foundations of Computer Science III.2 (J.W. de Bakker & J. van Leeuwen, eds.) pp.3-52, Mathematical Centre Tracts 109, Mathe-matisch Centrum, Amsterdam 1979. 15 [12] PARK, D.M.R., Concurrency and automata on finite sequences, Computer Science Department, University of Warwick (1981). [13] REM, M., Partially ordered computations, with applications to VLSI design, To appear in the proc. of the 4th Advanced Course on Foundations of Computer Science, Amsterdam, June 1982. ONTVANGEN 1 1 FEB. 1983 

A Matter of capabilities

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