each normed process has a unique decomposition into primes up to bisimilarity. However, the proof is non-constructive. Bisimilarity was proved to be d

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1 How to Parallelize Sequential Processes Anton n Ku era? tony@fi.muni.cz Faculty of Informatics, Masaryk University Botanick 68a, Brno Czech Republic Abstract. A process is prime if it cannot be decomposed into a parallel product of nontrivial processes. We characterize all non-prime normed BPA processes together with their decompositions by means of normal forms which are designed in this paper. Using this result we demonstrate decidability of the problem whether a given normed BPA process is prime; moreover, we show that non-prime normed BPA processes can be decomposed into primes eectively. This brings other positive decidability results. Finally, we prove that bisimilarity is decidable in a large subclass of normed PA processes. Introduction A general problem considered by many researchers is how to improve performance of sequential programs by parallelization. In this paper we study this problem within a framework of process algebras. They provide us with a pleasant formalism which allows to specify sequential as well as parallel programs. Here we adopt normed BPA processes as a simple model of sequential behaviours (they are equipped with a binary sequential operator). We examine the problem of eective decomposability of normed BPA processes into a parallel product of primes (a process is prime if it cannot be decomposed into nontrivial components). We design special normal forms for normed BPA processes which allow us to characterize all non-prime normed BPA processes together with their decompositions up to bisimilarity. As a consequence we also obtain a renement of the result achieved in [4]. Next we show that any normed BPA process can be decomposed into a parallel product of primes eectively. We also prove several related decidability results. Finally, we prove that bisimilarity is decidable in a large subclass of normed PA processes (see [2]), which consists of processes of the form k k n, where each i is a normed BPA or BPP process. In many parts of our paper we rely on results established by other researchers. The question of possible decomposition of processes into a parallel product of primes was rst addressed by Milner and Moller in [5]. A more general result was later proved by Christensen, Hirshfeld and Moller (see [8])it says that? supported by GA R, grant number 20/97/0456

2 each normed process has a unique decomposition into primes up to bisimilarity. However, the proof is non-constructive. Bisimilarity was proved to be decidable for normed BPA processes (see [,, 0]) and normed BPP processes (see [7, 9]). Blanco proved in [3] that bisimilarity is decidable even in the union of normed BPA and normed BPP processes. The same problem was independently examined by ern, K et nsk and Ku era in [5]. They demonstrated decidability of the problem whether for a given normed BPA (or BPP) process there is some unspecied normed BPP (or BPA) process 0 such that 0. If the answer is positive, then it is also possible to construct an example of such 0. Decidability of bisimilarity in the union of normed BPA and normed BPP processes is an immediate consequence. Another property of normed BPA and BPP processes which is important for us is regularity. A process is regular if it is bisimilar to a process with nitely many states. Ku era proved in [3] that regularity is decidable for normed BPA and normed BPP processes in polynomial time. 2 Preliminaries Let Act = fa; b; c; : : :g be a countably innite set of atomic actions. Let Var = fx; Y; Z; : : :g be a countably innite set of variables such that Var \ Act = ;. The class of BPA (or BPP) expressions is composed of all terms over the signature f; a; :; +g (or f; a; k; +g) where `' is a constant denoting the empty expression, `a' is a unary operator of action prexing (`a' ranges over Act), and `:', `k' and `+' are binary operators of sequential composition, parallel composition and nondeterministic choice, respectively. In the rest of this paper we do not distinguish between expressions related by structural congruence which is the smallest congruence relation over BPA and BPP expressions such that the following laws hold: associativity and `' as a unit for `:', `k', `+' operators, and commutativity for `k' and `+' operators. Moreover, we also abbreviate a as a. As usual, we restrict our attention to guarded expressions. A BPA or BPP expression E is guarded if every variable occurrence in E is within the scope of an atomic action. A guarded BPA (or BPP) process is dened by a nite family of recursive process equations = fx i = E i j i ng where X i are distinct elements of Var and E i are guarded BPA (or BPP) expressions, containing variables from fx ; : : : ; X n g. The set of variables which appear in is denoted by Var (). The variable X plays a special role (X is sometimes called the leading variable)it is a root of a labelled transition system, dened by the process and the rules of Figure. Nodes of the transition system generated by are BPA (or BPP) expressions, which are often called states of, or just states when is understood from the context. We also dene the relation!* w, where w 2 Act, as the reexive a and transitive closure of! (we often write E! F instead of E!* w F if w is irrelevant). Given two states E; F, we say that F is reachable from E, if E! F. States of which are reachable from X are said to be reachable.

3 ae a! E E! a E 0 EkF! a E 0 kf E! a E 0 E! a E 0 a F! F 0 E:F! a E 0 :F E + F! a E 0 E + F! a F 0 F! a F 0 a E! E 0 (X = E 2 ) EkF! a EkF 0 X! a E 0 Fig.. SOS rules Remark. Processes are often identied with their leading variables. Furthermore, if we assume xed processes ; 2 such that Var( ) \ Var ( 2 ) = ;, then we can view any process expression E (not necessarily guarded) whose variables are dened in ; 2 as a process tooif we denote this process by, then the leading equation of is X = E 0, where X 62 Var ( ) [ Var ( 2 ) and E 0 is a process expression which is obtained from E by substituting each variable in E with the right-hand side of its corresponding dening equation in or 2 (E 0 must be guarded now). Moreover, def. equations from ; 2 are added to. All notions originally dened for processes can be used for process expressions in this sense too. Bisimulation The equivalence between process expressions (states) we are interested in here is bisimilarity [6], dened as follows: A binary relation R over process expressions is a bisimulation if whenever (E; F ) 2 R then for each a 2 Act if E! a E 0, then F! a F 0 for some F 0 such that (E 0 ; F 0 ) 2 R if F! a F 0, then E! a E 0 for some E 0 such that (E 0 ; F 0 ) 2 R Processes and 0 are bisimilar, written 0, if their leading variables are related by some bisimulation. Normed processes An important subclass of BPA and BPP processes can be obtained by an extra restriction of normedness. A variable X 2 Var() is normed if there is w 2 Act such that X w!*. In that case we dene the norm of X, written jxj, to be the length of the shortest such w. A process is normed if all variables of Var () are normed. The norm of is then dened to be the norm of X. Note that bisimilar normed processes must have the same norm which is easily computed by the following rules: jaj =, je +F j = minfjej; jf jg, je:fj = jej+jfj, j EkF j = jej+jfj and if X i = E i and je i j = n, then jx i j = n. Greibach normal form Any BPA or BPP process can be eectively presented in a special normal form which is called 3-Greibach normal form by analogy with CF grammars (see [] and [6]). Before the denition we need to introduce the set Var () of all nite sequences of variables from Var (), and the set Var () of all nite multisets over Var (). Each multiset of Var () denotes a BPP expression by combining its elements in parallel using the `k' operator.

4 A BPA (or BPP) process is said to be in Greibach normal form (GNF) if all its equations are of the form X = P n j= a j j where n 2 N, a j 2 Act and j 2 Var() (or j 2 Var () ). We also require that each Y 2 Var () appears in some reachable state of. If length( j ) 2 (or card( j ) 2) for each j; j n, then is said to be in 3-GNF. From now on we assume that all BPA and BPP processes we are working with are presented in GNF. This justies also the assumption that all reachable states of a BPA process are elements of Var () and all reachable states of a BPP process 0 are elements of Var ( 0 ). Regular processes A process is regular if there is a process 0 with nitely many states such that 0. A regular process is said to be in normal form if all its equations are of the form P n X = j= a jx j where n 2 N, a j 2 Act and X j 2 Var (). It is easy to see that a process is regular i it can reach only nitely many states up to bisimilarity. In [4] it is shown, that regular processes can be represented in the normal form just dened. Thus a process is regular i there is a regular process 0 in normal form such that 0. A proof of the following proposition can be found in [3]. Proposition 2. Let be a normed BPA or BPP process. The problem whether is regular is decidable in polynomial time. Moreover, if is regular then a regular process 0 in normal form such that 0 can be eectively constructed. Special notation Here we summarize special notation used in this paper. nbpa and nbpp are abbreviations for normed BPA and normed BPP, respectively. if is a state of a nbpa or nbpp process such that is regular (see Remark ), then R () denotes a bisimilar regular process in normal form, which can be eectively constructed due to Proposition 2. Furthermore, we always assume that R () contains completely fresh variables which are not contained in any other process we deal with. the class of all processes for which there is a bisimilar nbpa (or nbpp) process is denoted S(nBPA) (or S(nBPP)). if ; : : : ; n are processes from nbpa[nbpp and X i is the leading variable of i for i n, then k k n denotes the process X k kx n in the sense of Remark. square brackets `[' and `]' indicate optional occurrenceif we say that some expression is of the form a[a][b], we mean that this expression is either a, aa, ab or aab. upper indexes are used heavily; they appear in two forms: i = k k {z } i.i = : : {z } i

5 Decidability of bisimilarity in nbpa[nbpp Bisimilarity is known to be decidable for nbpa (see [,, 0]) and nbpp (see [7, 9]) processes. The following result due to ern, K et nsk and Ku era (see [5]) says that bisimilarity is decidable even in the union of nbpa and nbpp processes. Proposition 3. Let be a nbpa (or nbpp) process. It is decidable, whether 2 S(nBPP) (or whether 2 S(nBPA)) and if the answer is positive, then a bisimilar nbpp (or nbpa) process can be eectively constructed. Decomposability, prime processes Let nil be a special name for the process which cannot emit any action (i.e., nil ). A nbpa or nbpp process is prime if 6 nil and whenever k 2 we have that either nil or 2 nil. Natural questions are, what processes have a decomposition into a nite parallel product of primes and whether this decomposition is unique. This problem was rst examined by Milner and Moller in [5]. They proved that each normed nite-state process has a unique decomposition up to bisimilarity. A more general result is due to Christensen, Hirshfeld and Mollerthey proved the following proposition (see [8]): Proposition 4. Each nbpp process has a unique decomposition into a parallel product of primes (up to bisimilarity). Remark 5. Proposition 4 holds for any normed process in fact (namely for nbpa). The proof is independent of a concrete syntaxit could be easily formulated in terms of normed transition systems. This proposition thus says that each normed process can be parallelized in the best way and that this way is in some sense unique. However, this nice theoretical result is non-constructive. 3 Decomposability of nbpp processes Each nbpp processes can be easily decomposed into a parallel product of primesall what has to be done is a construction of a bisimilar canonical process (see [6]). Theorem 6. Let be a nbpp process. It is decidable whether is prime and if not, its decomposition into primes can be eectively constructed. Proof. By induction on n = jj: n=: each nbpp process whose norm is is prime. Induction step: Suppose k 2. As ; 2 are reachable states of k 2, there are ; 2 2 Var () such that and 2 2, thus k 2. Furthermore, jj = j j + j 2 j. We show that there are only nitely many candidates for ; 2. First, there are only nitely many pairs [k ; k 2 ] 2 N N such that k + k 2 = jj. For each such pair [k ; k 2 ] there are

6 only nitely many pairs [ ; 2 ] such that ; 2 2 Var(), j j = k and j 2 j = k 2. It is obvious that the set M of all such pairs can be eectively constructed. For each element [ ; 2 ] of M we check whether k 2 (it can be done because bisimilarity is decidable for nbpp processes). If there is no such pair then is prime. Otherwise, we check whether ; 2 are primes (it is possible by ind. hypothesis) and construct their decompositions. If we combine these decompositions in parallel, we get a decomposition of. ut As each normed regular process in normal form can be seen as a nbpp process in GNF, Theorem 6 (and especially its constructive proof) can be also used for regular nbpa processes (see Proposition 2). In the next section we can thus concentrate on non-regular nbpa processes. 4 Decomposability of nbpa processes It this section we give an exact characterization of non-prime nbpa processes. We design special normal forms which allow us to characterize all non-prime nbpa processes together with their decompositions (up to bisimilarity). Our results bring also interesting consequenceswe obtain a renement of the result achieved in [4] (see Remark 8) and we also show that any nbpa process can be decomposed into prime processes eectively. Further positive decidability results are discussed in the end of the second subsection. Finally, we demonstrate decidability of bisimilarity in a natural subclass of normed PA processes. 4. Normal forms for non-prime nbpa processes In this subsection we design the promised normal forms for non-prime nbpa processes and for prime processes which appear in corresponding decompositions. As we already know from the previous section, the problem of possible decomposition of a nbpa process into a parallel product of primes is actually interesting only for non-regular nbpa processes, hence the main characterization theorem does not concern regular nbpa processes. The layout of this subsection is as follows: rst we present two technical lemmas (Lemma 7 and 8). Then we consider the following problem: if is a non-regular nbpa process such that k 2, where ; 2 are some (unspecied) processes, how do the processes ; ; 2 look like? It is clear that ; 2 2 S(nBPA), hence the assumption that ; 2 are nbpa processes can be used w.l.o.g. This problem is solved by Proposition and 6 with a help of several denitions. Having this, the proof of Theorem 2 is easy to complete. Lemma7. Let be a nbpa process. Let ; 2 Var() +, Q; C 2 Var () such that jqj = jcj = and kq C:. Then Q jj. Proof. We prove that for each i jj there is 2 Var() such that kq i C:. This is clearly sucient, because then kq C: Q jj+ and thus Q jj. We proceed by induction on i.

7 i=: choose =. Induction step: Let kq i C:. As jcj =, all states which are reachable from kq i in one norm-decreasing step are bisimilar. As is normed, there is 0 2 Var() such that! a 0 where jj = j 0 j +. Hence kq i? 0 kq i and by substitution we obtain kq i 0 kq i+. ut Lemma8. Let be a nbpa process, ; ; 2 Var () such that is nonregular and k. Let! Q where jqj =. Then Q jj. The proof of Lemma 8 is omitted due to the lack of space (it is rather technical). It can be found in [2]. Denition 9 (simple processes). A nbpa process is simple if Var () contains just one variable, i.e., card(var ()) =. We will often identify simple processes with their leading (and only) variables in the rest of this paper. Moreover, it is easy to see that a simple process Q is non-regular i the def. equation for Q contains a summand of the form aq.k where a 2 Act and k 2. The norm of Q is one, because Q could not be normed otherwise. Another important property of simple processes is presented in the remark below: Remark 0. Each simple nbpa process Q belongs to S(nBPP)a bisimilar nbpp process can be obtained just by replacing the `:' operator with the `k' operator in the def. equation for Q. Consequently, any process expressions built over k copies of Q using `:' and `k' operators are bisimilar (e.g., (Q:(QkQ))kQ (QkQ):(QkQ)). Proposition. Let ; 2 be non-regular nbpa processes. Then k 2 2 S(nBPA) i Q jj and 2 Q j2j for some non-regular simple process Q. Proof. ( Easysee Remark 0. ) Assume there is some nbpa process such that k 2. Then there are ; 2 2 Var () such that and 2 2. Thus k 2 and as ; 2 are non-regular, we can use Lemma 8 and conclude that there are Q ; Q 2 2 Var() such that jq j = jq 2 j =,! Q, 2! Q 2 and Q jj, 2 Q j2j 2. First we prove that Q Q for some simple process Q. To do this, it suces to prove that if a is a summand in the def. equation for Q, then Q jj (if this is the case, then also Q.jj see Remark 0). As k 2! Q k a 2! k 2, the process k 2 belongs to S(nBPA). Let! R where jrj =. Then R jj (due to Lemma 8) and as!! R, we also have R jj. Hence R Q and Q jj. To nish the proof we need to show that Q Q 2. Let m = maxfjxj; X 2 Var ()g. As is non-regular, it can reach a state of an arbitrary normlet! 0 where j 0 j = m. Then 0 kq 2 for some 2 Var () whose length

8 is at least two = A:B: 0. Clearly 0 Q j0 j (we can use the same argument as in the rst part of this proofq 2 is non-regular and 0 plays the role of ), hence Q j0 j kq 2 A:B: 0. As Q j0 j?jaj kq 2 B: 0 and Q j0 j?jaj+ B: 0, we have Q j0 j?jaj kq 2 Q j0 j?jaj+ by transitivity and thus Q Q 2. ut Proposition in fact says that if is a non-regular nbpa process such that k 2, where ; 2 are non-regular processes, then each of those three processes can be equivalently represented as a power of some non-regular simple process. This representation is very special and can be seen as normal form. If is a non-regular nbpa process such that k 2, it is also possible that is non-regular and 2 regular. Before we start to examine this sub-case, we introduce a special normal form for nbpa processes (as we shall see, and can be represented in this normal form): Denition 2 (DNF(Q)). Let be a non-regular nbpa process in GNF, Q 2 Var(). We say that is in DNF (Q) if all summands in all dening equations from are of the form a([y ]:[Q.i ]), where Y 2 Var (), i 2 N and a 2 Act. Furthermore, all summands in the def. equation for Q must be of the form a[q], where a 2 Act. Example 3. The following process is in DNF (Q): X = a(y:q:q) + bx + a(q:q:q) + c Y = bq + cx + c(y:q) + b Q = aq + bq + a + c Remark 4. Reachable states of a process in DNF (Q) are of the form [Y ]:[Q.i ] where Y 2 Var () and i 2 N [f0g. As is non-regular, the state Q.k is reachable for each k 2 N. Note that the variable Q itself is a regular simple process. The next lemma says that if is a process in DNF (Q), then the variable Q is in some sense unique: Lemma5. Let and 0 be processes in DNF (Q) and DNF (R), respectively. If 0, then Q R. Proposition 6. Let ; 2 be nbpa processes such that is non-regular and 2 is regular. Then k 2 2 S(nBPA) i there is a process 0 in DNF (Q) such that 0 and 2 Q j2j. Proof. ) Let 2! Q 0 where Q 0 2 Var( 2 ), jq 0 j =. Using the same kind of argument as in the proof of Proposition we obtain that Q 0 Q for some regular simple process Q such that 2 Q j2j. It remains to prove that there is a process 0 in DNF (Q) such that 0. We show that each summand of each dening equation from can be transformed to a form which is admitted by DNF (Q). First, let us realize two facts about summandsif a is a summand in a def. equation from, then

9 . If = :Y: where Y is a non-regular variable, then each variable P of is bisimilar to Q jp j. 2. contains at most one non-regular variable. The rst fact is a consequence of Lemma 7let be a nbpa process such that k 2. As is normed,! Y:: for some 2 Var ( ). As Y is non-regular, it can reach a state of an arbitrary lengthlet m = maxfjxj; X 2 Var ( )g and let Y!! where length(!) = m. As k 2!!::kQ 0, there is ' 2 Var () such that!::kq 0 '. Let ' = C:' 0 and let s be a norm-decreasing sequence of actions such that length(s) = jcj? and!!* s! 0. Then! 0 ::kq 0 C 0 :' 0 where jc 0 j = and due to Lemma 7 (and the fact that Q 0 Q) we have! 0 :: Q j!0 ::j, hence Q jj and P Q jp j for each variable P which appears in. The second fact is a consequence of the rst oneassume that = :Y::Z: where Y; Z are non-regular. Then Z Q jzj and as Q is regular, Q jzj is regular too. Hence Z is regular and we have a contradiction. Now we can describe the promised transformation of to 0 : if X = P n i= a i i is a def. equation in, then X = P n i= a it ( i ) is a def. equation in 0, where T is dened as follows: If i does not contain any non-regular variable, then T ( i ) = A, where A is the leading variable of R ( i ). Moreover, dening equations of R ( i ) are added to 0. If i = :Y: where Y is a non-regular variable, then T ( i ) = A, where A is the leading variable of the process 0 which is obtained by the following modication of the process R (): each summand in each def. equation of R () which is of the form b, where b 2 Act, is replaced with b(y:q.jj ) remember Q jj Q.jj. Moreover, def. equations of 0 are added to 0. The dening equation for Q is also added to 0. The resulting process is in DNF (Q) and as T preserves bisimilarity, 0. ( We show how to construct a nbpa process which is bisimilar to 0 kq j2j. Let k = j 2 j. The set of variables of looks as follows: Var () = fqg [ fy i ; Y 2 Var ( 0 ); Y 6= Q and i 2 f0; : : : ; kgg Dening equations of are constructed using the following rules: the def. equation for Q is the same as in 0 if a(y:q.j ), where j 2 N [ f0g, Y 6= Q, is a summand in the def. equation for Z 2 Var ( 0 ), then a(y i:q.j ) is a summand in the def. equation for Z i for each i 2 f0; : : : ; kg if a(q.j ) where j 2 N [ f0g is a summand in the def. equation for Z 2 Var ( 0 ), then a(q.j+i ) is a summand in the def. equation for Z i for each i 2 f0; : : : ; kg if aq is a summand in the def. equation for Q and Z 2 Var( 0 ), Z 6= Q, then az i is a summand in the def. equation for Z i for each i 2 f; : : : ; kg

10 if a is a summand in the def. equation for Q and Z 2 Var ( 0 ), Z 6= Q, then az i? is a summand in the def. equation for Z i for each i 2 f; : : : ; kg The intuition which stands behind this construction is that lower indexes of variables indicate how many copies of Q in Q j2j have not disappeared yet. The fact 0 kq j2j is easy to check. ut Example 7. If we apply the algorithm presented in the ( part of the proof of Proposition 6 to the process XkQ 2, where X; Q are variables of the process presented in Example 3, we obtain the following output: X 2 = a(y 2 :Q:Q) + bx 2 + a(q:q:q:q:q) + c(q:q) + ax 2 + bx 2 + ax + cx X = a(y :Q:Q) + bx + a(q:q:q:q) + cq + ax + bx + ax 0 + cx 0 X 0 = a(y 0 :Q:Q) + bx 0 + a(q:q:q) + c Y 2 = b(q:q:q) + cx 2 + c(y 2 :Q) + b(q:q) + ay 2 + by 2 + ay + cy Y = b(q:q) + cx + c(y :Q) + bq + ay + by + ay 0 + cy 0 Y 0 = bq + cx 0 + c(y 0 :Q) + b Q = aq + bq + a + c Remark 8. Proposition 6 can also be seen as a renement of the result presented in [4]Burkart and Steen proved that PDA processes are closed under parallel composition with nite-state processes, while BPA processes lack this property. Proposition 6 says precisely, which nbpa processes can remain nbpa if they are combined in parallel with a regular process. Moreover, it also characterizes all such regular processes. It is easy to see that the algorithm from the proof of Proposition 6 always outputs a process in DNF (Q) (see Example 7). Moreover, the structure of this process is very specic; we can observe that each variable belongs to a special level. This intuition is formally expressed by the following denition (it is a little complicatedbut it pays because we will be able to characterize all nonprime nbpa processes): Denition 9. Let be a nbpa process in DNF (Q). The level of, denoted Level (), is the maximal l 2 N such that the set Var ()? fqg can be divided into l disjoint linearly ordered subsets L ; : : : ; L l of the same cardinality k. Moreover, the following conditions must be true (the j th element of L i is denoted A i;j ): A l; is the leading variable of. Dening equations for variables of L contain only variables from L [ fqg The dening equation for A i;j, where i 2, j k, contains exactly those summands which can be derived by one of the following rules:. If aq is a summand in the dening equation for Q, then aa i;j is a summand in the dening equation for A i;j for each 2 i l, j k. 2. If a is a summand in the dening equation for Q, then aa i?;j is a summand in the dening equation for A i;j for each 2 i l, j k.

11 3. If a(a ;m :Q.n ) is a summand in the dening equation for A ;j such that A ;m 6= Q, then a(a i;m :Q.n ) is a summand in the dening equation for A i;j for each 2 i l. 4. If aq.n is a summand in the dening equation for A ;j, then aq.(n+i?) is a summand in the dening equation for A i;j, where 2 i l. Example 20. The process of Example 7 has the level 3; L = fx 0 ; Y 0 g, L 2 = fx ; Y g and L 3 = fx 2 ; Y 2 g. Now we can present the rst main theorem of this paper: Theorem 2. Let be a non-regular nbpa process and let k k n, where n 2, i is a prime process for each i n and is non-regular. Then one of the following possibilities holds: There is a non-regular simple process Q such that Q.jj and i Q for each i n. There are nbpa processes 0 ; 0 in DNF (Q) such that 0, 0, Level ( 0 ) = n, Level ( 0 ) = and i Q for each 2 i n. Proof. By a straightforward induction on nsee [2]. ut 4.2 Decidability results In this subsection we present several positive decidability results. We show that it is decidable whether a given nbpa process is prime and if the answer is negative, then its decomposition into primes can be eectively constructed. There are also other decidable properties which are summarized in Theorem 26. Finally, we demonstrate decidability of bisimilarity in a natural subclass of normed PA processes. Lemma22. Let be a nbpa process. It is decidable whether there is a nbpa process 0 in DNF (Q) such that 0. Moreover, if the answer to the previous question is positive, then the process 0 can be eectively constructed. Proof. We can assume (w.l.o.g.) that is in 3-GNF. If there is a process 0 in DNF (Q) such that 0, then there is R 2 Var() such that R Q, because Q is a reachable state of 0. As Q is a regular simple process, each summand in the def. equation for R must be of the form a[p ], where R P. As bisimilarity is decidable for nbpa processes, we can construct the set M of all variables of Var () with this property. Each variable from this set is a potential candidate for the variable which is bisimilar to Q (if the set M is empty, then cannot be bisimilar to any process in DNF (Q)). For each variable V 2 M we now modify the process slightlywe replace each summand of the form ap in the def. equation for V with av. The resulting process is denoted V (clearly V ). For each V we check whether V can be transformed to a process in DNF (V ). To do this, we rst need to realize

12 the following fact: if there is 0 V in DNF (V ) such that V 0 V and a(a:b) is a summand in a def. equation from V such that A is non-regular, then B V.jBj. It is easy to prove by the technique we already used many times in this paperas A is non-regular, it can reach a state of an arbitrary norm. Furthermore, there is a reachable state of V which is of the form A:B: where 2 Var ( V ). We choose suciently large such that A! and :B: must be bisimilar to a state of 0 V which is of the form [Y ]:V.i where i jb:j. From this we get B V.jBj. Now we can describe the promised transformation T of V to a process 0 V in DNF (V ). If this transformation fails, then there is no process in DNF (V ) bisimilar to V. T is invoked on each summand of each def. equation from V and works as follows: T (a) = a T (aa) = aa T (a(a:b)) = an if A is regular. The variable N is the leading variable of R (A), whose def. equations are also added to 0 V after the following modication: each summand in each def. equation of R (A) which is of the form b where b 2 Act is replaced with bb. T (a(a:b)) = a(a:v.jbj ) if A is non-regular and B V.jBj. If A is nonregular and B 6 V.jBj, then T fails. If there is V 2 M such that T succeeds for V, then the process 0 V is the process we are looking for. Otherwise, there is no process in DNF (Q) bisimilar to. ut Proposition 23. Let ; : : : ; n, n 2 be nbpa processes. It is decidable whether k k n 2 S(nBPA). Moreover, if the answer to the previous question is positive, then a nbpa process such that k k n can be eectively constructed. Proof. By induction on n: n=2: we distinguish three possibilities (it is decidable which one actually holdssee Proposition 2):. and 2 are regular. Then k 2 2 S(nBPA) and a bisimilar regular process in normal form can be easily constructed. 2. and 2 are non-regular. Proposition says that there is a nonregular simple process Q such that Q jj Q.jj and 2 Q j2j Q.j2j. As Q is a reachable state of Q.j2j, there is R 2 Var ( ) such that Q R. As reachable states of Q are of the form Q.i where i 2 N[f0g, each summand a in the def. equation for R has the property R.jj. As bisimilarity is decidable for nbpa processes, we can nd all variables of Var () having this propertywe obtain a set of possible candidates for R (if this set is empty, then k 2 62 S(nBPA)). Now we check whether the constructed set of candidates contains a variable R such that R.jj. If not, then k 2 62 S(nBPA). Otherwise we have R which is bisimilar to Q.

13 The same procedure is now applied to 2. If it succeeds, it outputs some S 2 Var (). Now we check whether R S. If not, then k 2 62 S(nBPA). Otherwise k 2 2 S(nBPA) and k 2 R.jj+j2j. 3. is non-regular and 2 is regular (or is regular and 2 is nonregularthis is symmetric). Due to Proposition 6 we know that there is a regular simple process Q and a nbpa process 0 in DNF (Q) such that 0 and 2 Q j2j Q.j2j. An existence of 0 can be checked eectively (see Lemma 22). If it does not exist, then k 2 62 S(nBPA). If it exists, it can be also constructed and thus the only thing which remains is to test whether 2 Q.j2j. If this test succeeds, then k 2 2 S(nBPA) and we invoke the algorithm from the proof of Proposition 6 with 0 kq j2j on inputit outputs a nbpa process which is bisimilar to k 2. Induction step: if k k n 2 S(nBPA), then also k k n? 2 S(nBPA) and this is decidable by induction hypothesisif the answer is negative, then k k n 62 S(nBPA) and if it is positive, then we can construct a nbpa process 0 such that k k n? 0. Now we check whether 0 k n 2 S(nBPA) and construct a bisimilar nbpa process. ut As an immediate consequence of Proposition 23 we get: Proposition 24. Let ; ; : : : ; n be nbpa processes. It is decidable whether k k n. Theorem 25. Let be a nbpa process. It is decidable whether is prime and if not, its decomposition into primes can be eectively constructed. Proof. The technique is the same as in the proof of Theorem 6. We can almost copy the whole proofthe crucial result which allows us to do so is Proposition 24. ut Decidability results which were proved in this subsection are summarized by the following theorem: Theorem 26. Let ; ; : : : ; n be nbpa processes. The following problems are decidable: Is prime? (If not, its decomposition can be eectively constructed) Is bisimilar to k k n? Does the process k k n belong to S(nBPA)? Is there any process 0 such that k 0 2 S(nBPA)? (if so, an example of such a process can be eectively constructed). Is there any process 0 such that k k n k 0? (if so, 0 can be eectively constructed). A structural way how to construct new processes from older ones is to combine them in parallel. If we do this with nbpa and nbpp processes, we obtain a natural subclass of normed PA processes denoted spa (simple PA processes).

14 Denition 27 (spa processes). The class of spa processes is dened as follows: spa = f k k n j n 2 N; i 2 nbpa [ nbpp for each i ng The class spa is strictly greater than the union of nbpa and nbpp processes; it suces to take a parallel composition of two normed counters specied by nbpa processes. The resulting spa process is not bisimilar to any nbpa or nbpp process. It can be easily proved with the help of pumping lemmas for CF languages and for languages generated by nbpp processessee [6]. Theorem 28. Let = ' k k' n, = k k m be spa processes. It is decidable whether. Proof. As each ' i, i n and j, j m can be eectively decomposed into a parallel product of primes, we can also construct the decompositions of and. If, then these decompositions must be the same up to bisimilarity (see Remark 5). In other words, there must be a one-to-one mapping between primes forming the two decompostions which preserves bisimilarity. An existence of such a mapping can be checked eectively, because bisimilarity is decidable in the union of nbpa and nbpp processes (see Proposition 3). ut 5 Conclusions, future work The main characterization theorem (Theorem 2) says that non-regular nbpa processes which are not prime can be divided into two groups:. processes which are bisimilar to a power of some non-regular simple process. It is obvious that each such nbpa process belongs to S(nBPP)see Remark processes which are bisimilar to some process in DNF (Q). It can be proved (with the help of results achieved in [5]) that each such process does not belong to S(nBPP). From this we can observe that our division based on normal forms corresponds to the membership to S(nBPP). We have also shown that the decomposition of non-prime nbpa processes can be eectively constructed. This algorithm can be interpreted as a construction of the most parallel version of a given sequential program. Finally, we proved that bisimilarity is decidable for spa processes. (see Denition 27). The rst possible generalization of our results could be the replacement of the `k' operator with the parallel operator of CCS which allows synchronizations on complementary actions. This should not be hard, but we can expect more complicated normal forms. Decidability results should be the same. A natural question is whether our results can be extended to the class of all (not necessarily normed) BPA processes. The answer is no, because there are quite primitive BPA processes which do not have any decomposition at alla simple example is the process X = ax.

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