Fixed-point Characterization of Compositionality Properties of Probabilistic Processes Combinators
Bisimulation metric is a robust behavioural semantics for probabilistic processes. Given any SOS specification of probabilistic processes, we provide a method to compute for each operator of the language its respective metric compositionality property. The compositionality property of an operator is defined as its modulus of continuity which gives the relative increase of the distance between processes when they are combined by that operator. The compositionality property of an operator is computed by recursively counting how many times the combined processes are copied along their evolution. The compositionality properties allow to derive an upper bound on the distance between processes by purely inspecting the operators used to specify those processes.
Keywords: SOS, probabilistic transition systems, bisimulation metric, compositionality, continuity
Over the last decade a number of researchers have started to develop a theory of structural operational semantics for probabilistic transition systems (PTSs). Several rule formats for various PTSs were proposed that ensure compositionality of bisimilarity  and of approximate bisimilarity . We will consider specifications with rules of the probabilistic GSOS format  in order to describe nondeterministic probabilistic transition systems .
Bisimilarity is very sensitive to the exact probabilities of transitions. The slightest perturbation of the probabilities can destroy bisimilarity. Bisimulation metric  provides a robust semantics for probabilistic processes. It is the quantitative analogue to bisimulation equivalence and assigns to each pair of processes a distance which measures the proximity of their quantitative properties. The distances form a pseudometric with bisimilar processes at distance . Alternative approaches towards a robust semantics for probabilistic processes are approximate bisimulation  and bisimulation degrees . We consider bisimulation metrics as convincingly argued in e.g. .
For compositional specification and reasoning it is necessary that the considered behavioral semantics is compatible with all operators of the language. For bisimulation metric semantics this is the notion of uniform continuity. Intuitively, an operator is uniformly continuous if processes composed by that operator stay close whenever their respective subprocesses are replaced by close subprocesses.
In the 1990s, rule formats that guarantee compositionality of the specified operators have been proposed by (reasonable) argumentation for admissible rules. Prominent examples are the GSOS format  and the  format. More recently, the development of compositional proof systems for the satisfaction relation of HML-formulae  allowed to derive rule formats from the logical characterization of the behavioral relation under investigation .
We propose a new approach that allows to derive for any given specification the compositionality property of each of its specified operators. The compositionality properties are derived from an appropriate denotational model of the specified language. First, we develop for a concrete process algebra an appropriate denotational model. The denotation of an open process term describes for each resolution of the nondeterministic choices how many instances of each process variable are spawned while the process evolves. The number of spawned process replicas is weighted by the likelihood of its realization just like the bisimulation metric weights the distance between target states by their reachability. We derive from the denotation of an open process term an upper bound on the bisimulation distance between the closed instances of the denoted process. Then we generalize this method to arbitrary processes whose operational semantics is specified by probabilistic GSOS rules. In fact, the upper bound on the bisimulation distance between closed instances of is a modulus of continuity of operator if the denotation of is finitely bounded. In this case the operator is uniformly continuous and admits for compositional reasoning wrt. bisimulation metric.
This paper continues our research programme towards a theory of robust specifications for probabilistic processes. Earlier work  investigated compositional process combinators with respect to approximate bisimulation. Besides the different semantics considered in this paper, we extend substantially on the approach of  by using the newly developed denotational approach. The denotational model separates clearly between nondeterministic choice, probabilistic choice, and process replication. This answers also the open question of  how the distance of processes composed by process combinators with a nondeterministic operational semantics can be approximated.
2.1Probabilistic Transition Systems
A signature is a structure , where
gives the arity of operator . We write for . We assume an infinite set of state variables disjoint from . The set of -terms (also called state terms) over , notation , is the least set satisfying:
is the set of all closed terms and abbreviated as . is the set of open terms and abbreviated as . We may refer to operators as process combinators, to variables as process variables, and to closed terms as processes. denotes the set of all state variables in .
Probability distributions are mappings with that assign to each closed term its respective probability . By we denote the set of all probability distributions on . We let range over . The probability mass of in is defined by . Let for denote the Dirac distribution, i.e., and if and are syntactically not equal. The convex combination of a family of probability distributions with and is defined by . By we denote the distribution defined by = . We may write for .
In order to describe probabilistic behavior, we need expressions that denote probability distributions. We assume an infinite set of distribution variables . We let range over , and range over . The set of distribution terms over state variables and distribution variables , notation with denoting the signature extending by operators to describe distributions, is the least set satisfying:
A distribution variable is a variable that takes values from . An instantiable Dirac distribution is an expression that takes as value the Dirac distribution when variables in are substituted so that becomes the closed term . Case ( ?) allows to construct convex combinations of distributions. We write for . Case ( ?) lifts the structural inductive construction of state terms to distribution terms. denotes . denotes the set of all state and distribution variables in .
A substitution is a mapping such that if , and if . A substitution extends to a mapping from state terms to state terms as usual. A substitution extends to distribution terms by , and . Notice that closed instances of distribution terms are probability distributions.
Probabilistic transition systems generalize labelled transition systems (LTSs) by allowing for probabilistic choices in the transitions. We consider nondeterministic probabilistic LTSs (Segala-type systems)  with countable state spaces.
We write for , and if for some .
2.2Specification of Probabilistic Transition Systems
We specify PTSs by SOS rules of the probabilistic GSOS format  and adapt from  the language to describe distributions. We do not consider quantitative premises because they are incompatible
The expressions (resp. ) above the line are called positive (resp. negative) premises. We call in a derivative of . We denote the set of positive (resp. negative) premises of rule by (resp. ). The expression below the line is called the conclusion, notation , is called the source, notation , the are called the source variables, notation , and is called the target, notation .
A probabilistic transition system specification (PTSS) in PGSOS format is a triple , where is a signature, is a countable set of actions and is a countable set of PGSOS rules. is the set of those rules of with source . A supported model of is a PTS such that iff for some rule and some closed substitution all premises of hold, i.e. for all we have and for all we have for all , and the conclusion instantiates to and . Each PGSOS PTSS has exactly one supported model  which is moreover finitely branching.
2.3Bisimulation metric on Probabilistic Transition Systems
Behavioral pseudometrics are the quantitative analogue to behavioral equivalences and formalize the notion of behavioral distance between processes. A -bounded pseudometric is a function with
for all terms .
We define now bisimulation metrics as quantitative analogue to bisimulation equivalences. Like for bisimulation we need to lift the behavioral pseudometric on states to distributions and sets of distributions . A matching for is given if and for all . We denote the set of all matchings for by . The Kantorovich pseudometric is defined for a pseudometric by
for . The Hausdorff pseudometric is defined for a pseudometric by
for whereby and .
A bisimulation metric is a pseudometric on states such that for two states each transition from one state can be mimicked by a transition from the other state and the distance between the target distributions does not exceed the distance of the source states.
We order bisimulation metrics iff for all . The smallest bisimulation metric, notation , is called bisimilarity metric and assigns to each pair of processes the least possible distance. We call the bisimilarity metric distance also bisimulation distance. Bisimilarity equivalence  is the kernel of the bisimilarity metric , i.e. iff and are bisimilar. We say that processes and do not totally disagree if .
Bisimulation metrics can alternatively be defined as prefixed points of a monotone function. Let be the complete lattice defined by iff , for all . We define the function for and by:
We develop now a denotational model for open terms. Essentially, the denotation of an open term describes for each variable in how many copies are spawned while evolves. The denotation of allows us to formulate an upper bound on the bisimulation distance between closed instances of . In this section we consider a concrete process algebra. In the next section we generalize our method to arbitrary PGSOS specifications.
Let be the signature of the core operators of the probabilistic process algebra in  defined by the stop process , a family of -ary prefix operators with , , and , alternative composition , and parallel composition for each . We write for , and for (deterministic prefix operator). Moreover, we write for (synchronous parallel composition). The PTSS is given by the following PGSOS rules in :
We call the open terms nondeterministic probabilistic process terms. We define two important subclasses of that allow for a simpler approximation of the distance of their closed instances. Let be the set of deterministic process terms, which are those terms of that are built exclusively from the stop process , deterministic prefix , and synchronous parallel composition (no nondeterministic and no probabilistic choices). We call the open terms deterministic because all probabilistic or nondeterministic choices in the operational semantics of the closed instances , with any closed substitution, arise exclusively from the processes in . Let be the set of probabilistic process terms, which are those terms of that are built exclusively from the stop process , probabilistic prefix , and synchronous parallel composition (no nondeterministic choices). Again, all nondeterministic choices in arise exclusively from the processes in .
The denotation of a deterministic process term is a mapping that describes for each process variable how many copies of or some derivative of are spawned while evolves. We call the multiplicity of . Let be the set of all mappings . The denotation of , notation , is defined by , , if , , and .
We use notation for the multiplicity that assigns to each , and with for the multiplicity such that if and if . We write for . As it will become clear in the next sections, we need the denotation for (unbounded) recursion and replication.
We will approximate the bisimulation distance between and for closed substitutions using the denotation of and the bisimulation distances between processes and of variables . The bisimulation distance of variables is represented by a mapping . We call a process distance. Let be the set of all process distances . We henceforth assume closed substitutions with a bisimulation distance between and that is strictly less than . Practically, this is a very mild restriction because for any (non-trivial) process combinator the composition of processes that totally disagree (i.e. which are in bisimulation distance ) may lead to composed processes that again totally disagree. For any and any closed substitutions we define the associated process distance by .
To understand the functional remind that is the distance between processes and . In other words, processes and disagree by on their behavior. Hence, and agree by . Thus, copies of and copies of agree by at least , and disagree by at most .
The functional defines an upper bound on the bisimulation distance of deterministic processes.
The distance abstracts from the concrete reactive behavior of terms and . It is not hard to see that for deterministic process terms without parallel composition the approximation functional gives the exact bisimulation distance. However, the parallel composition of processes may lead to an overapproximation if the bisimulation distance of process instances arises (at least partially) from reactive behavior on which the processes cannot synchronize.
We remark that the abstraction of the closed substitutions to process distances is intentional and very much in line with common compositionality criteria that relate the distance of composed processes with the distance of the process components.
The denotation of a probabilistic process term is a distribution that describes for each multiplicity the likelihood that for each process variable exactly copies of or some derivative of are spawned while evolves. We call the probabilistic multiplicity of . Let be the set of all distributions . The denotation of , notation , is defined by with , with , , and . Notice that for all .
For important probabilistic multiplicities we use the same symbols as for multiplicities but it will always be clear from the context if we refer to probabilistic multiplicities or multiplicities. By we mean the probabilistic multiplicity that gives probability to the multiplicity . By we mean the probabilistic multiplicity that gives probability to the multiplicity .
Notice that for all . The functional defines an upper bound on the bisimulation distance of probabilistic processes.
Before we can introduce the denotation of nondeterministic probabilistic processes, we need to order the denotation of probabilistic processes. Let with be a subdistribution over multiplicities. We define the weighting of as a mapping defined if , with the size of , and if . Intuitively, the number of process copies are weighted by the probability of realization of that multiplicity. We order probabilistic multiplicities if can be decomposed into subdistributions such that each multiplicity in is above some weighted subdistribution of . The order is now defined by:
The denotation of a nondeterministic probabilistic process term is a set of probabilistic multiplicities that describes by some resolution of the nondeterministic choices in such that the process evolves as a probabilistic process described by . We construct a Hoare powerdomain over the probabilistic multiplicities and use as canonical representation for any set of probabilistic multiplicities the downward closure defined as . Let be the set of non-empty downward closed sets of probabilistic multiplicities . We use downward closed sets such that will form a complete lattice with the order defined below (esp. satisfies antisymmetry, cf. Proposition ?). The denotation of , notation , is defined by , , iff there are and such that with defined by for all , iff there are such that with defined by , and . Notice that for all . By we mean the singleton set containing the probabilistic multiplicity , and by the downward closure of the singleton set with element .
Notice that for all . Moreover, for any . The functional defines an upper bound on the bisimulation distance of nondeterministic probabilistic process terms.
Theorem ? shows that the denotation of a process term is adequate to define an upper bound on the distance of closed instances of that process term. The converse notion is full-abstraction in the sense that (no over-approximation). As demonstrated in Example ?, the approximation functionals would require for process variables besides the bisimulation distance between and also information about the reactive behavior and the branching. However, for our objective to study the distance of composed processes in relation to the distance of its components, the bisimulation distance is the right level of abstraction.
We introduce now an order on that ensures monotonicity of both the approximation functional and the functional introduced in the next section to compute the denotation of arbitrary terms of a PGSOS PTSS. The order is defined by
We order process distances by iff for all . The nondeterministic probabilistic distance approximation is monotone in both arguments.
We will see in the following section that the denotations developed for terms of are sufficient for terms of any PGSOS PTSS.
4Distance of composed processes
Now we provide a method to determine the denotation of an arbitrary term. In line with the former section this gives an upper bound on the bisimulation distance of closed instances of that term. In particular, the denotation for the term gives an upper bound on the distance of processes composed by the process combinator . This allows us in the next section to formulate a simple condition to decide if a process combinator is uniformly continuous, and hence if we can reason compositionally over processes combined by that process combinator.
4.1Operations on process denotations
We start by defining two operations on process denotations that allow us to compute the denotation of process terms by induction over the term structure. We define the operations first on and then lift them to .
The composition of two processes and which both proceed requires that their multiplicities are summed up (cf. parallel composition in the prior section). We define the summation of multiplicities by:
In order to define by structural induction the multiplicity of a term , we need an operation that composes the multiplicity denoting the operator with the multiplicity of . We define the pointed multiplication of multiplicities with respect to variable by:
Then, the multiplicity of a state term is given by:
It remains to define the multiplicity of for operators with an operational semantics defined by some rule . We define the multiplicity of in terms of the multiplicity of the target of . Let be a derivative of the source variable in rule . We use the property in order to express the multiplicity as a multiplicity of . Then, the multiplicity of is defined for any variable as the summation of the multiplicity of and its derivatives in the rule target:
Operations over lift to by
4.2Approximating the distance of composed processes
Let be any PGSOS PTSS. We compute the denotation of terms and rules as least fixed point of a monotone function. Let with and . A pair assigns to each term its denotation and to each rule its denotation . Let be a poset with ordering iff and for all and . forms a complete lattice with least element defined by for all and .
We assume that for all rules the source variable of argument is called . Let be the set of source variables for which tests the reactive behavior, i.e. iff has either some positive premise or some negative premise .
The mapping defined in Figure ? computes iteratively the nondeterministic probabilistic multiplicities for all terms and rules. As expected, the denotation of a state term is defined as the application of all rules to the denotation of the arguments. However, for distribution terms the application of the operator needs to consider two peculiarities. First, different states in the support of a distribution term may evolve according to different rules of .
Second, in the distribution term the operator may discriminate states in derivatives belonging to solely on the basis that in some rule the argument gets tested on the ability to perform or not perform some action.
To summarize Examples ? and ?: The nondeterministic probabilistic multiplicity for operator applied to some distribution term is given by (Figure ?). We explain this expression stepwise. For any rule we define by the least probabilistic multiplicity which covers all nondeterministic choices represented by the probabilistic multiplicities in . By we capture the case that premises of only test source variables in on their ability to perform an action (Example ?). By we define the least probabilistic multiplicity which covers all choices of rules (Example ?). Finally, by the downward closure we gain the nondeterministic probabilistic multiplicity that can be applied to the distribution term (Figure ?).
From Proposition ? and ? and the Knaster-Tarski fixed point theorem we derive the existence and uniqueness of the least fixed point of . We denote by the least fixed point of . We write for and for . We call the canonical denotation of . It is not hard to verify that all denotations presented in Section 3 for are canonical.
A denotation of terms is compatible with a distance function , notation , if for all and all closed substitutions . Now we can show that the functional to compute the bisimulation distance and functional to compute the denotations preserve compatibility (Proposition ?). A simple inductive argument allows then to show that the canonical denotation of terms is compatible with the bisimilarity metric (Theorem ?).
Remind that is the least fixed point of defined by and the Hausdorff metric functional (Proposition ?). Let and . Proposition ? shows that by reasoning inductively over the transitions specified by the rules. Monotonicity and upward -continuity (Proposition ?) ensures that this property is also preserved in the limit.
In order to reason compositionally over probabilistic processes it is enough if the distance of the composed processes can be related to the distance of their parts. This property is known as uniform continuity. In essence, compositional reasoning over probabilistic processes is possible whenever a small variance in the behavior of the parts leads to a bounded small variance in the behavior of the composed processes. Technically this boils down to the existence of a modulus of continuity. Uniform continuity generalizes earlier proposals of non-expansiveness  and non-extensiveness .
Intuitively, a continuous binary operator ensures that for any non-zero bisimulation distance (understood as the admissible tolerance from the operational behavior of the composed process ) there are non-zero bisimulation distances and (understood as the admissible tolerances from the operational behavior of the processes and , respectively) such that the distance between the composed processes and is at most whenever the component (resp. ) is in distance of at most from (resp. at most from ). We consider the uniform notion of continuity because we aim for universal compositionality guarantees.
The denotation of allows to derive a candidate for the modulus of continuity for operator as follows.
Trivially, we have and for all closed terms by Theorem ?. However, is continuous at only if the multiplicities in the denotation assign to each variable a finite value.
Even more, for uniformly continuous operators the function is a modulus of continuity.
In reverse, for a given modulus of continuity (as specification of some process combinator), we can derive the maximal replication of process of this operator.
To conclude, the methods provided in Section 3 and Section 4 to compute an upper bound on the distance between instances of the term can be used to derive the individual compositionality property of operator given by its the modulus of continuity . Note that depends on all those rules which define operators of processes to which an instance of may evolve to. Traditional rule formats define syntactic criteria on single rules in order to guarantee a desired compositionality property of the specified operator. In contrast, our approach derives the compositionality property of an operator from the the syntactic properties of those rules which define the operational behavior of processes composed by that operator.
6Conclusion and Future Work
We presented a method to approximate the bisimulation distance of arbitrary process terms (Theorem ? and ?). This allows to decide for any given PTSS which operators allow for compositional metric reasoning, i.e. which operators are uniformly continuous (Theorem ?). Moreover, our method allows to compute for any given PTSS a modulus of continuity of each uniformly continuous operator (Theorem ?). Additionally, for any given modulus of continuity (understood as the required compositionality property of some operator) we provide a sufficient condition to decide if an operator satisfies the modulus of continuity (Theorem ?). The condition characterizes the maximal number of times that processes combined by the operator may be replicated during their evolution in order to guarantee the modulus of continuity.
We will continue this line of research as follows. First, we will investigate the compositionality of process combinators with respect to convex bisimulation metric , discounted bisimulation metric , and generalized bisimulation metric . Second, we will explore compositionality with respect to behavioral pseudometrics based on trace semantics  and testing semantics. Finally, we will investigate how the denotational approach to decide the compositionality properties of operators relates to the logical approach to derive rule formats of . Besides this general line, we want to investigate how our structural syntactic approach to compositionality relates to the algorithmic computational approach in .
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