Termination Problems in Chemical Kinetics

Transcription

1 Termination Problems in Cemical Kinetics Gianluigi Zavattaro and Luca Cardelli 2 Dip. Scienze dell Informazione, Università di Bologna, Italy 2 Microsoft Researc, Cambridge, UK Abstract. We consider nondeterministic and probabilistic termination problems in a process algebra tat is equivalent to basic cemistry. We sow tat te existence of a terminating computation is decidable, but tat termination wit any probability strictly greater tan zero is undecidable. Moreover, we sow tat te fairness intrinsic in stocastic computations implies tat termination of all computation pats is undecidable, wile it is decidable in a nondeterministic framework. Introduction We investigate te question of weter basic cemical kinetics (kinetics of unary and binary cemical reactions), formulated as a process algebra, is capable of general computation. In particular, we investigate nondeterministic and probabilistic termination problems in te Cemical Ground Form (CGF): a process algebra recently proposed for te compositional description of cemical systems, and proved to be bot stocastically and continuously equivalent to cemical kinetics (see [2] for te formal proof of equivalence between CGF and cemical kinetics). Te answers to tose termination problems reveal a surprisingly ric picture of wat is decidable and undecidable in basic cemistry. We consider tree variants of te termination problem: existential, universal, and probabilistic termination. By existential termination we mean te existence of a terminating computation, by universal termination we mean tat all possible computations terminate (in a probabilistic setting, by possible computation we mean tat te computation as probability > 0), by probabilistic termination we mean tat wit probability strictly greater tan a given ɛ, wit 0 < ɛ <, a terminating computation is executed. We prove tat, in te stocastic semantics of CGF, existential termination is decidable, wile bot probabilistic and universal termination are undecidable. In contrast, in a nondeterministic interpretation of te CGF tat abstracts from reaction rates, bot existential and universal termination are decidable. Tis means tat: (a) cemical kinetics is not Turing complete, (b) cemical kinetics is Turing complete up to any degree of precision, (c) existential termination is equally ard (decidable) in stocastic and nondeterministic systems, (d) universal termination is arder (undecidable) in stocastic systems tan in nondeterministic systems, (e) te fairness implicit in stocastic computations makes cecking universal termination undecidable. In recent work, Soloveicik et al. [8], prove te non-turing completeness of Stocastic Cemical Reaction Networks (wic are equivalent to te CGF [2])

2 by reduction to te decidability of cemical state coverability, wic tey call reacability. We prove more strongly tat exact cemical state reacability is also decidable, as well as tat existential termination and boundedness are decidable. (All tese argument are based on decidability results in Petri Nets.) Te same autors also prove te possibility of approximating RAM and Turing Macine computations up to an arbitrarily small error ɛ. Teir encodings allow tem to prove te undecidability of probabilistic coverability. We prove te undecidability of probabilistic termination, probabilistic reacability, probabilistic boundedness, and of universal termination. Tere are tecnical differences in our RAM encodings tat guarantee te stronger results. For example, terminating computations are still terminating in our encoding of RAMs, wile in [8] a clock process keeps running even after termination of te main computation. 2 Cemical Ground Form In te CGF eac species as an associated definition describing te possible actions for te molecules of tat species. Eac action π (r) as an associated stocastic rate r (a positive real number) wic quantifies te expected execution time for te action π. Action τ (r) indicates te possibility for a molecule to be engaged in a unary reaction. For instance, te definition A = τ (r) ; (B C) says tat one molecule of species A can be engaged in a unary reaction tat produces two molecules, one of species B and one of species C (te operator is borrowed from process algebras suc as CCS [6], were it represents parallel composition, and corresponds ere to te cemical + ). Binary reactions ave two reactants. Te two reactants perform two complementary actions?a (r) and!a (r), were a is a name used to identify te reaction; bot te name a and te rate r must matc for te reaction to be enabled. For instance, given te definitions A =?a (r) ; C and B =!a (r) ; D, we ave tat two molecules of species A and B can be engaged in a binary reaction tat produces two molecules, one of species C and one of species D. If te molecules of one species can be engaged in several reactions, ten te corresponding definition admits a coice among several actions. Te syntax of coice is as follows: A = τ (r) ; B?a (r ); C, meaning tat molecules of species A can be engaged in eiter a unary reaction, or in a binary reaction wit anoter molecule able to execute te complementary action!a (r ). Definition (Cemical Ground Form (CGF)). Consider te following denumerable sets: Species ranged over by variables X, Y,, Cannels ranged over by a, b,, Moreover, let r, s, be rates (i.e. positive real numbers). Te syntax of CGF is as follows (were te big separates syntactic alternatives wile te small denotes parallel composition): E ::= 0 X =M, E M ::= 0 π; P M Reagents Molecule P ::= 0 X P Solution π ::= τ (r)?a(r)!a(r) Internal, Input, Output prefix CGF ::= (E, P ) Reagents and initial Solution

3 Given a CGF (E, P ), we assume tat all variables in P occur also in E. Moreover, for every variable X in E, tere is exactly one definition X = M in E. In te following, trailing 0 are left implicit, and we use also as an operator over te syntax: if P and P are 0-terminated lists of variables, according to te syntax above, ten P P means appending te two lists into a single 0- terminated list. Tus, if P is a solution, ten 0 P, P 0, and P are syntactically equal. Te solution composed of k instances of X is denoted wit k X. We consider te discrete state semantics for te CGF defined in [2] in terms of Continuous Time Markov Cains (CTMCs). Te states of te CTMCs are solutions in normal form denoted wit P : for a solution P, we indicate wit P te normalized form of P were te variables are sorted in lexicograpical order (wit 0 at te end), possibly wit repetitions. Te CTMC associated to a cemical ground form is obtained in two steps: we first define te Labeled Transition Grap (LTG) of a cemical ground form, ten we sow ow to extract a CTMC from te labeled transition grap. We use te following notation. Let E.X be te molecule defined by X in E, and M.i be te i-t summand in a molecule of te form M = π ; P π n ; P n. Given a solution in normal form P, wit P.m we denote te m-t variable in P, wit P \(m,, m n ) we denote te solution obtained by removing from P te m i -t molecule for eac i {,, n}. A Labeled Transition Grap (LTG) is a set of quadruples l : S r T were te transition labels l are eiter of te form {m.x.i} or {m.x.i, n.y.j}, were m, n, i, j are positive integers, X, Y are species names, m.x.i are ordered triples and {, } are unordered pairs. Definition 2 (Labeled Transition Grap (LTG) of a Cemical Ground Form). Given te Cemical Ground Form (E, P ), we define Next(E, P ) as te set containing te following kinds of labeled transitions: {m.x.i} : P r T suc tat P.m = X and E.X.i = τ (r) ; Q and T = (P \m) Q; {m.x.i, n.y.j} : P r T suc tat P.m = X and P.n = Y and m n and E.X.i =?a (r) ; Q and E.Y.j =!a (r) ; R and T = (P \m, n) Q R. Te Labeled Transition Grap of (E, P ) is defined as follows: LT G(E, P ) = n Ψ n were Ψ 0 = Next(E, P ) and Ψ n+ = {Next(E, Q) Q is a state of Ψ n } We now define ow to extract from an LTG te corresponding CTMC. Definition 3 (Continuous Time Markov Cain associated to an LTG). If Ψ is an LTG, ten Ψ is te associated CTMC, defined as te set of te triples P r Q wit P Q, obtained by summing te rates of all te transitions in Ψ tat ave te same source and target state: Ψ = {P r Q s.t. l : P r Q Ψ wit P Q, and r = r i s.t. l i : P ri Q Ψ}.

4 It is wort noting tat two solutions Q and R are connected by a transition in LT G(E, P ) if and only if tey are connected by a transition in LT G(E, P ). In fact, te transitions of te latter are acieved by collapsing into one transition tose transitions of te former tat sare te same source and target solutions. Te rate of te new transition is te sum of te rates of te collapsed transitions. Given a CGF (E, P ), a computation is a sequence of transitions in te CTMC LT G(E, P ) starting wit a transition wit source solution P, and suc tat te target solution of one transition coincides wit te source state of te next transition. We say tat a solution Q is reacable in (E, P ) if tere exists a computation wit Q as te target solution of te last transition. A solution Q is terminated in LT G(E, P ) if Q as no outgoing transitions. Te CTMC semantics of CGF defines a probabilistic interpretation of te beavior of a CGF (E, P ): given any solution T of LT G(E, P ), if it as n outgoing transitions labeled wit r,, r n, te probability tat te j-t transition is taken is r j /( i r i). Tus, we can associate probability measures (we consider te standard probability measure for Markov cains see e.g. [5]) to computations in LT G(E, P ). We use tis tecnique to define te tree variants of te termination problem we consider in tis paper. Definition 4 (Existential, universal and probabilistic termination). Consider a CGF (E, P ) and its CTMC LT G(E, P ). Let p be te probability measure associated to te computations in LT G(E, P ) leading to a terminated solution. We say tat (E, P ) existentially terminates if p > 0, (E, P ) universally terminates if p =, (E, P ) probabilistically terminates wit probability iger tan ɛ (for 0 < ɛ < ) if p > ɛ. We will consider also probabilistic variants of oter properties. Consider a CGF (E, P ), its CTMC LT G(E, P ), and a real number ɛ suc tat 0 ɛ <. We say tat a solution Q is ɛ-reacable if te probability measure of te computations in LT G(E, P ) leading to Q is > ɛ. We say tat (E, P ) is ɛ-bound if te set of ɛ-reacable solutions is finite. We say tat (E, P ) is ɛ-terminating if te probability measure of te computations in LT G(E, P ) leading to a terminated solution is > ɛ. We say tat (E, P ) is ɛ-diverging if te probability measure of te infinite computations in LT G(E, P ) is > ɛ. It is wort noting tat, in a probabilistic setting, existential termination coincides wit 0-termination, universal termination wit te negation of 0-divergence, and probabilistic termination wit ɛ-termination for ɛ > 0. 3 Decidability Results In tis section we resort to a Place/Transition Petri net (P/T net) semantics for CGF, tat can be interpreted as a purely nondeterministic semantics of CGF tat abstracts away from te stocastic rates. In tis purely nondeterministic framework several properties are decidable. In fact, in P/T nets, properties suc as reacability (te existence of a computation leading to a given state), boundedness (te finiteness of te set of reacable states), termination (reacability of

5 a deadlocked state), and divergence (te existence of an infinite computation) are decidable (see [4] for a survey on decidable properties for Petri Nets). Definition 5 (Place/Transition Net). A P/T net is a tuple N = (S, T ) were S is te set of places, M fin (S) is te set of te finite multisets over S (eac of wic is called a marking) and T M fin (S) M fin (S) is te set of transitions. A transition (c, p) is written c p. Te marking c, represents te tokens to be consumed ; te marking p represents te tokens to be produced. A transition c p is enabled at a marking m if c m. Te execution of te transition produces te marking m = (m \ c) p (were \ and are te difference and te union operators on multisets). Tis is written as m[ m. A dead marking is a marking in wic no transition is enabled. A marked P/T net is a tuple N(m 0 ) = (S, T, m 0 ), were (S, T ) is a P/T net and m 0 is te initial marking. A computation in N(m 0 ) leading to te marking m is a sequence m 0 [ m [ m 2 m n [ m. Given a CGF (E, P ), we define a corresponding P/T net N = (S, T ) and a corresponding marked P/T net N(m 0 ). We first need to introduce an auxiliary function Mark(P ) tat associates to a solution P te multiset of its variables: Mark(P ) = { if P = 0 {X} Mark(P ) if P = X P Definition 6 (Net of a CGF). Given a CGF (E, P ), wit Net (E,P ) we denote te corresponding P/T net (S, T ) were: S = {X X occurs in E} T = { {X} Mark(X X n ) E.X.i = τ (r) ; (X X n ) } { {X, Y } Mark(X X n ) Mark(Y Y m ) E.X.i =?a (r) ; (X X n ) and E.Y.j =!a (r) ; (Y Y m ) } Te corresponding marked P/T net is Net (E,P ) (Mark(P )). Note tat te set of places S corresponds to te set of variables X defined in E, te transitions represents te possible actions, and te initial marking is te multiset of variables in te solution P. It is also wort observing tat in te net semantics we do not consider te rates (r) of te actions. We now formalize te correspondence between te beaviors of a CGF and of its corresponding P/T net. Teorem. Consider a CGF (E, P ) and te corresponding P/T net Net (E,P ) = (S, T ). We ave tat:. if l : P r Q is in Next(E, P ) (for some l and r) ten we ave also tat Mark(P )[ Mark(Q) in Net (E,P ) ; 2. if tere exists m suc tat Mark(P )[ m in Net (E,P ), ten tere exist l, r and Q suc tat l : P r Q is in Next(E, P ) and Mark(Q) = m.

6 Proof (sketc). Te proofs of te two statements are by case analysis on te possible transitions in Next(E, P ) as defined in te Definition 2 for te first statement or on te possible transitions enabled in Mark(P ) as defined in te Definition 6 for te second statement. Tis teorem allows us to conclude tat te P/T net semantics faitfully reproduces te standard CGF transitions. Te only difference is tat it abstracts away from te stocastic rates. For tis reason, we consider te P/T net semantics as a purely nondeterministic interpretation of CGF. Reacability, boundedness, termination, and divergence are decidable for P/T nets; tus we can conclude tat all tese properties are decidable also in te CGF under a purely nondeterministic interpretation. As a consequence of Teorem, existential termination is decidable. Teorem 2. Consider a CGF (E, P ). We ave tat (E, P ) existentially terminates if and only if a dead marking is reacable in te Net (E,P ) (Mark(P )). Proof. It is easy to see from te definition of LT G(E, P ) and LT G(E, P ) tat te latter contains all and only tose solutions (in normal form) reacable in (E, P ) wit a finite number of transitions, eac one aving a probability > 0 to be cosen. Tus a solution is reacable wit probability > 0 if and only if it is in LT G(E, P ). As a consequence of Teorem we ave tat a solution Q is in LT G(E, P ) if and only if Mark(Q) is reacable in Net (E,P ) (Mark(P )). Moreover, Teorem also guarantees tat Q is terminated if and only if Mark(Q) is a dead marking in Net (E,P ) (tis proves te teorem). As a corollary of Teorem we ave tat also te probabilistic variants of reacability and boundedness can be reduced to te corresponding properties in te nondeterministic setting. On te contrary, tis does not old for divergence. (Tis will be discussed in te next section.) We can summarize te results of tis section simply saying tat ɛ-termination, ɛ-reacability, and ɛ-boundedness are decidable wen ɛ = 0. 4 Undecidability Results Tis section is divided in two parts. In te first one we prove tat probabilistic termination (i.e. ɛ-termination wit ɛ > 0) is undecidable. (We also comment on ow to sow tat also ɛ-divergence, ɛ-boundedness, and ɛ-reacability are undecidable wen ɛ > 0.) In te second part we prove te undecidability of universal termination (tus also of 0-divergence). 4. Undecidability of Probabilistic Termination We prove te undecidability of probabilistic termination sowing ow to approximately model in CGF te beavior of any Random Access Macines (RAMs) [7], a well known register based Turing powerful formalism. More precisely, we reduce te termination problem for RAMs to te probabilistic termination wit probability iger tan any ɛ suc tat 0 < ɛ <. We first recall te definition of Random Access Macines.

7 Definition 7 (Random Access Macines (RAMs)). A RAM R is composed of a set of registers r,, r m tat contain non negative integer numbers and a set of indexed instructions I,, I n of two possible kinds: I i = Inc(r j ) tat increments te register r j and ten moves to te execution of te instruction wit index i + and I i = DecJump(r j, s) tat attempts to decrement te register r j ; if te register does not old 0 ten te register is actually decremented and te next instruction is te one wit index i +, oterwise te next instruction is te one wit index s. We use te following notation: (I i, r = l,, r m = l m ) represents te state of te computation of te RAM wic is going to execute te instruction I i wit registers tat contain l,, l m, respectively; (I i, r = l,, r m = l m ) (I j, r = l,, r m = l m) describes one step of computation of te RAM; (I i, r = l,, r m = l m ) denotes final states of te computation in wic I i is undefined. Witout loss of generality, we assume te existence of a special index alt suc tat all final states contain an instruction wit tat index, namely (I i, r = l,, r m = l m ) if and only if i = alt. Te basic idea tat we follow in modeling RAMs in CGF is to use one species I i for eac instruction I i, and one species R j for eac register r j. Te state (I i, r = l,, r m = l m ) of te RAM is modeled by a solution tat contains one molecule of species I i, l molecules of species R,, and l m molecules of species R m (plus a certain amount of inibitor molecules of species In, wose function will be discussed below). Te beavior of te molecules of species I i is to update te register according to te corresponding instruction I i, and to activate te execution of te next instruction I j by producing te molecule of species I j. An Inc(r j ) instruction simply produces one molecule of species R j. On te oter and, a DecJump(r j, s) instruction sould test te absence of molecules of species R j before deciding weter to execute te jump, or to consume one of te available molecules of tat species. As it is not possible to verify te absence of molecules, we admit te execution of te jump even if molecules of species R j are available. In tis case, we say tat a wrong jump is executed. In order to reduce te probability of wrong jumps, we put teir execution in competition wit alternative beaviors involving te inibitor molecules in suc a way tat te greater is te quantity of inibitor molecules in te solution, te smaller is te probability to execute a wrong jump. We are now ready to formally define our encoding of RAMs. Definition 8. Given a RAM R and one of its states (I i, r = l,, r m = l m ), let [(I i, r = l,, r m = l m )] denote te solution: I i l R l m R m In

8 were: τ; (I i+ R j ) if I i = Inc(r j ) I i =!r j ; (I i+ In) τ; Ci,s 2 if I i = DecJump(r j, s) 0 if I i = I alt Ci,s 2 =!in; Ii τ; Ci,s Ci,s =!in; Ii τ; I s R j =?r j ; 0 In =?in; In Note tat is used to denote te number of occurrences of te molecules of species In. We take all subscripts action rates equal to and we omit tem (tis coice allows us to simplify te proof of Proposition ). In te following, we use E R for te set of te above definitions of species I i, C 2 i,s, C i,s, Rj, In. Note tat before actually executing a jump, two internal τ actions must be executed in sequence (tose in te definition of te species Ci,s 2 and C i,s ), and bot of tem are in competition wit te action!in willing to perform an interaction wit one of te inibitor molecules of species In. Tus, te iger is te number of inibitor molecules, te smaller is te probability to perform tis sequence of two internal actions. We now formalize te correspondence between te beavior of a RAM and of its encoding in CGF. Proposition. Let R be a RAM. Given one of its states (I i, r = l,, r m = l m ) and [(I i, r = l,, r m = l m )], for any, we ave:. if I i = I alt ten (I i, r = l,, r m = l m ) and Next(E R, [(I i, r = l,, r m = l m )] ) as no transitions; 2. if I i = Incr j or I i = DecJump(r j, s) wit l j = 0 and (I i, r = l,, r m = l m ) (I j, r = l,, r m = l m), ten te solution [(I j, r = l,, r m = l m)] is reacable in (E R, [(I i, r = l,, r m = l m )] ) wit probability = ; 3. if I i = DecJump(r j, s) wit l j > 0 and (I i, r = l,, r m = l m ) (I j, r = l,, r m = l m), ten te solution [(I j, r = l,, r m = l m)] + is reacable in (E R, [(I i, r = l,, r m = l m )] ) wit probability >. 2 Proof. If I i = I alt or I i = Inc(r j ) te corresponding statements (te first one and te first part of te second one) are easy to prove. We detail te proof only for I i = DecJump(r j, s). If r j is empty, te probability measure for te computations in (E R, [(I i, r = l,, r m = l m )] ) passing troug [[(I j, r = l,, r m = l m)] is (see Figure ): i=0 ( ) i + ( + ) 2 = If r j is not empty, i.e. l j > 0, te standard probability measure for te computations passing troug [(I j, r = l,, r m = l m)] + is (see Figure ): i=0 ( l j l j + + ) i l j + l j + > 2

9 (a) [(I i,r = l,,r m = l m )] (b) [(I i,r = l,,r m = l m )] l j [[(I j,r = l 0,,r m = l 0 m)]] + [[(I j,r = l 0,,r m = l 0 m)]] Fig.. Fragment of te CTMC LT G(E R, [[(I i, r = l,, r m = l m)]] ) in case I i = DecJump(r j, s) wit l j = 0 (a) or l j > 0 (b). Te above proposition states te correspondence between a single RAM step and te corresponding encoding in CGF. We conclude tat a RAM terminates its computation if and only if a terminated solution is reacable wit a probability tat depends on te initial number of inibitor molecules in te encoding. Teorem 3. Let R be a RAM. We ave tat te computation of R starting from te state (I i, r = l,, r m = l m ) terminates if and only if te CGF (E R, [(I i, r = l,, r m = l m )] ) probabilistically terminates wit probability iger tan k= k. 2 Proof. In te ligt of Proposition we ave tat only decrement operations are not reproduced wit probability =, but wit probability >. Moreover, after te execution of a decrement operation, te value of inibitor 2 molecules is incremented by one. Tus, a RAM computation including d decrement operations is faitfully reproduced wit probability strictly greater tan +d ( ) k= +d k > 2 k= k. Hencefort, any terminating computation is reproduced wit probability strictly greater tan 2 k= k. 2 It is well known tat te series = is convergent (to π2 2 6 ), tus for every small value δ > 0 tere exists a corresponding initial amount of inibitor molecules suc tat k= k < δ. Hencefort, in order to reduce RAM termination to probabilistic termination wit probability iger tan any 0 < ɛ <, 2 it is sufficient to consider an initial value suc tat k= k < ( ɛ). 2

10 Te RAM encoding presented in Definition 8 reproduces also unbounded RAM computations wit any degree of precision. Tus also ɛ-divergence is undecidable wen ɛ > 0. On te contrary, suc encoding does not allow us to prove te undecidability of ɛ-boundedness and ɛ-reacability. We first sow ow to reduce te RAM divergence problem to ɛ-boundedness. Tis does not old for te encoding in te Definition 8 because tere exists divergent RAMs wit a bounded corresponding CGF. Consider, for instance, te RAM composed of only te instruction I = DecJump(r, ) tat performs an infinite loop if te register r is initially empty. It is easy to see tat te corresponding CGF is bounded. In order to guarantee tat an infinite RAM computation generates an unbounded CGF, we can simply add a new molecule of a new species A every time a jump is performed. As an infinite RAM computation executes infinitely many jump operations, an unbounded amount of molecules of species A will be generated. Te new encoding is defined as in te Definition 8 replacing te definition of te species Ci,s wit te following one: C i,s =!in; Ii τ; (I s A). We conclude tis section observing ow to reduce RAM termination to ɛ- reacability. Tis does not old for te above encodings because te solution representing te final state of a RAM computation is not known beforeand. In fact, besides te fact tat te final contents of te registers is not known, we ave tat te final solution will contain a number of inibitor molecules tat depends on te number of decrement operations executed during te computation (as eac decrement adds one molecule of species In). In order to know beforeand te final solution, we allow te molecule I alt to remove te register molecules of species R j as well as all te inibitor molecules of species In. In tis way, if te computation terminates, we ave tat te final solution surely contains only te molecule I alt. Namely, we modify in te Definition 8 te definitions of te species I alt and In as follows: I alt = m j=!r j; I alt!remove; I alt In =?in; In?remove; Undecidability of Universal Termination Te undecidability of universal termination (tus also of 0-divergence) is proved introducing an intermediary nondeterministic computational model, tat we call finitely faulting RAMs (FFRAMs). Tis model corresponds to RAMs in wic te execution of DecJump instructions is nondeterministic wen te tested register is not empty: an FFRAM can eiter decrement te register or execute a wrong jump. Te peculiarity of FFRAMs is tat in an infinite computation only finitely many wrong jumps are executed. We first sow tat it is possible to define an encoding of FFRAMs in CGF suc tat te universal termination problem for FFRAMs coincides wit te universal termination problem for te corresponding CGF. Ten we prove te undecidability of te universal termination problem for FFRAMs sowing ow to reduce te RAM termination problem to te verification of te existence of an infinite computation in

11 FFRAMs (wic corresponds to te complement of te universal termination problem). We start defining finitely faulting RAMs. Definition 9 (Finitely Faulting RAMs (FFRAMs)). Finitely Faulting RAMs are defined as traditional RAMs (see Definition 7) wit te only difference tat given an instruction I i = DecJump(r j, s) and a RAM state (I i, r = l,, r j = l j,, r m = l m ) wit l j > 0, two possible computation steps are permitted: (I i, r = l,, r j = l j,, r m = l m ) (I i+, r = l,, r j = l j,, r m = l m ) and (I i, r = l,, r j = l j,, r m = l m ) (I s, r = l,, r j = l j,, r m = l m ). Te second computation step is called wrong jump because a jump is executed even if te tested register is not empty. Te peculiar property of FFRAMs is tat in every computation (also infinite ones), finitely many wrong jumps are executed. We now sow ow to define an encoding of FFRAMs in CGF suc tat infinite computations in te FFRAMs computational model corresponds to infinite computation wit probability > 0 in te corresponding CGF. Te FFRAM encoding is defined as in Definition 8 adding a transition to a terminated state wic can be selected wit probability 2 wile executing wrong jumps. In tis way, we guarantee tat in an infinite computation infinitely many wrong jumps cannot be executed because te new transition to te terminated state cannot be avoided indefinitely. Definition 0 (FFRAM Modeling). Given a FFRAM R and one of its states (I i, r = l,, r m = l m ), [(I i, r = l,, r m = l m )] is defined as in Definition 8. Also te species I i, R j, In, and Ci,s 2 are defined as in Definition 8, wile Ci,s is defined as follows: C i,s =!in; Ii τ; C 0 s C 0 s =!r j ; I alt τ; I s In te following, we use E R for te new set of definitions of species I i, C 2 i,s, C i,s, C0 s, R j, and In. We now revisit te Proposition adapting it to te new encoding. Proposition 2. Let R be a FFRAM. Given one of its states (I i, r = l,, r m = l m ) and [(I i, r = l,, r m = l m )], for any, we ave:. (as in Proposition ); 2. (as in Proposition ); 3. if I i = DecJump(r j, s) wit l j > 0 ten wit probability one of te following states are reacable in (E R, [(I i, r = l,, r j = l j,, r m = l m )] ): [(I i+, r = l,, r j = l j,, r m = l m )] + (wit prob. > 2 ); [(I alt, r = l,, r j = l j,, r m = l m )] ; [(I s, r = l,, r j = l j,, r m = l m )] (wit probability 0 < p < 2 ).

12 [(I i,r = l,,r j = l j,,r m = l m )] l j [[(I i+,r = l,,r j = l j,,r m = l m )] + l j [[(I alt,r = l,,r j = l j,,r m = l m )]] [[(I s,r = l,,r j = l j,,r m = l m )]] Fig. 2. Fragment of te CTMC LT G(E R, [[(I i, r = l,, r m = l m)]] ) in case I i = DecJump(r j, s) wit l j > 0. Proof. Te first two statements are proved as in Proposition. We sketc te proof for te tird statement. Te probability measure for te computations in (E R, [(I i, r = l,, r j = l j,, r m = l m )] ) passing troug [(I i+, r = l,, r j = l j,, r m = l m )] + is computed as in Proposition. Te probability measure p for te computations passing troug [[(I s, r = l,, r j = l j,, r m = l m )] is (see Figure 2): i=0 ( l j l j + + ) i ( ) 2 ( ) 2 + l j + + It is easy to see tat as l j > 0, ten p < 2. Finally, we observe tat te probability measure of te computations leading to [(I alt, r = l,, r j = l j,, r m = l m )] is equal to minus te probability measure of te computations passing troug eiter [[(I i+, r = l,, r j = l j,, r m = l m )] + or [(I s, r = l,, r j = l j,, r m = l m )]. Te above proposition states te correspondence between a single computation step of a FFRAM and tat of te corresponding CGF. We conclude tat a FFRAM as an infinite computation if and only if tere exists an infinite computation wit probability > 0 in te corresponding CGF. Teorem 4. Let R be a FFRAM. We ave tat R as an infinite computation starting from te state (I i, r = l,, r m = l m ) if and only if te CGF

13 (E R, [(I i, r = l,, r m = l m )] ) as an infinite computation for some initial amount of inibitor molecules. ( k 2 ) w k= p k were w Proof. We first consider te only if part. Assume te existence of an infinite computation of R starting from te state (I i, r = l,, r m = l m ). Tis computation will execute infinitely many DecJump instructions, but only finitely many wrong jumps. We now consider te CGF (E R, [(I i, r = l,, r m = l m )] ) for a generic. According to te Proposition 2, it can reproduce te same infinite computation wit probability strictly greater tan k= is te number of wrong jumps, and p k is te probability for te k-t wrong jump computed as in Proposition 2. Let p min be te minimum among p,, p w. We ave tat te above probability is strictly greater tan ( ) ( k= k ) w. 2 p min We ave already discussed, after Teorem 3, tat te series = is conver- 2 gent, tus tere exists suc tat k= k <. If we consider tis particular 2 value, te overall probability for te infinite computation is > 0. We now consider te if part. Assume te existence of an infinite computation wit probability > 0 in te CGF (E R, [(I i, r = l,, r m = l m )] ) for some. Tis computation corresponds to an infinite computation of R for te two following reasons. We first observe tat te infinite computation reproduces infinitely many correct computation steps (I i, r = l,, r m = l m ) (I j, r = l,, r m = l m) of R. In fact, te unique wrong computation step could be te one described in te second item of te tird statement of Proposition 2. Tis computation step leads to te encoding of te terminated state (I alt, r = l,, r j = l j,, r m = l m ), but in tis case te computation cannot be infinite. Ten, we observe tat te number of wrong jumps is finite. In fact, if we assume (by contradiction) tat te computation contains infinitely many wrong jumps, we ave tat (in te ligt of te tird item of te tird statement of Proposition 2) te probability of te infinite computation is smaller tan i= 2, tus it cannot be > 0. We now prove tat te existence of an infinite computation in FFRAMs is undecidable by defining an encoding tat reduces te termination problem for RAMs to te divergence problem for FFRAMs. As it is not restrictive, we consider only RAMs starting wit all registers empty. Our tecnique as been inspired by a similar one used in [3]. We initially assume tat an arbitrary number k of wrong jumps occurs and, as a consequence, te number k is introduced in a special register. Ten we let te FFRAM repeat indefinitely te simulation of te beavior of te corresponding RAM, but if tis simulation requires more tan k steps, te encoding blocks (tis is ensured by decrementing te special register before simulating every computational step). In tis way, if a RAM terminates, ten te corresponding FFRAM (wit k greater tan te lengt of te RAM computation) can diverge. On te oter and, if an infinite computation of te FFRAM exists, tis as an infinite suffix tat does not contain wrong jumps. In tis correct part of te computation, te encoding faitfully simulates te RAM computation infinitely often; tis is possible only if te RAM terminates.

14 Teorem 5. Given a RAM R, tere exists a corresponding FFRAM [R] suc tat te computation of R (starting wit all registers empty) terminates if and only if [R] as an infinite computation (starting wit all registers empty). Proof. Given a RAM R wit instructions I,, I n (assuming I n = I alt ) and registers r 0,, r m, wit [R] we denote te FFRAM composed of te registers r 0, r,, r m, r m+, r m+2, r m+3 and of te following instructions: J = Inc(r m+ ) J 2 = DecJump(r m+, ) J 3i = DecJump(r m+, alt) (for i < n) J 3i+ = Inc(r { m+2 ) (for i < n) Inc(rj ) if I J 3i+2 = i = Inc(r j ) DecJump(r j, 3s) if I i = DecJump(r j, s) J 3n+2j = DecJump(r j, 3n + 2j + 2) (for i m) J 3n+2j+ = DecJump(r m+3, 3n + 2j) (for i m) J 3n+2m+2 = DecJump(r m+2, 3) J 3n+2m+3 = Inc(r m+ ) J 3n+2m+4 = DecJump(r m+3, 3n + 2m + 2) (for i < n) We prove tat te computation of R starting from te state (I, r 0 = 0,, r m = 0) terminates if and only if [R] as an infinite computation starting from te state (J, r 0 = 0,, r m+3 = 0). We first consider te only-if part. We assume tat te RAM R terminates after te execution of k steps. Te corresponding FFRAM [R] as te following infinite computation wic contains exactly k wrong jumps. Te wrong jumps are all executed at te beginning of te compuation in order to introduce in r m+ te value k. Ten te computation proceeds simulating infinitely many times te computation of R. Note tat at te end of eac simulation, all te registers r,, r m are emptied, and te value k (wic is introduced in r m+2 during te computation, is moved back in r m+ ). Note also tat te register r m+3 is always empty, and tat it is simply tested for zero by instructions tat must always perform a jump. We now consider te if part. Assume tat te FFRAM [R] as an infinite computation. Tis computation starts wit k executions of te instructions J and J 2. Te loop between tese two instructions cannot proceed indefinitely as it contains a wrong jump. At te end of tis first pase, te register r m+ contains k. Ten te computation continues by simulating te beavior of te RAM R, and before executing every instruction te register r m+ is decremented and te register r m+2 is incremented. If (by contradiction) te register r m+ becomes empty before completing te simulation of R, te computation sould block. Tus, te simulation completes before simulating k steps. After, all te registers r 0,, r m are emptied, te value k is reintroduced in r m+, and a new simulation is started. Tis part of te computation, i.e. simulation of R and subsequent reset of te registers, surely terminates because te simulation of R includes at most k steps, and te subsequent reset of te registers cannot proceed indefinitely. We can conclude tat an infinite computation includes infinitely many simulations

15 of te computation of R and, as an FFRAM can perform only finitely many wrong jumps, infinitely many of tese simulations are correct. Tis implies tat te RAM R terminates witin k computation steps. 5 Conclusion In tis paper we ave investigated te decidability of termination problems in CGF, a process algebra proposed in [2] for te compositional description of cemical systems. In particular, we ave proved tat existential termination is decidable, probabilistic termination is undecidable, and universal termination is decidable under a purely nondeterministic interpretation of CGF wile it turns to be undecidable under te stocastic semantics. It is wort saying tat similar results old also for lossy cannels: universal termination is decidable in lossy cannels wile it turns out to be undecidable in teir probabilistic variant []. Neverteless, te result on lossy cannels is not comparable wit ours. In fact, in CGF process communication is syncronous (in lossy cannels syncronous communication is not admitted) wile in te lossy cannel model it is asyncronous troug unbounded FIFO buffers (tat cannot be directly encoded in CGF). Acknowledgement. We would like to acknowledge M. Bravetti, D. Soloveicik, H. Wiklicky, E. Winfree, and te anonymous referees for teir insigtful comments on previous versions of tis paper. References. P. Abdulla, C. Baier, P. Iyer, and B. Jonsson. Reasoning about Probabilistic Lossy Cannel Systems. In Proc. of t International Conference on Concurrency Teory (Concur), volume 877 of LNCS, pages , L. Cardelli. On Process Rate Semantics. Teoretical Computer Science, in press, Available at ttp://dx.doi.org/0.06/j.tcs H. Carstensen. Decidability Questions for Fairness in Petri Nets. In Proc. of 4t Annual Symposium on Teoretical Aspects of Computer Science (STACS), volume 247 of LNCS, pages , J. Esparza and M. Nielsen. Decidability Issues for Petri Nets, 994. Tecnical report BRICS RS J.G. Kemeny, J.L. Snell, and A.W. Knapp. Denumerable Markov Cains. Springer Verlag, R. Milner. Communication and Concurrency. Prentice-Hall, M. L. Minsky. Computation: finite and infinite macines. Prentice-Hall, D. Soloveicik, M. Cook, E. Winfree, and J. Bruck. Computation wit Finite Stocastic Cemical Reaction Networks. Natural Computing, in press, Available at ttp://dx.doi.org/0.007/s y.