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...6) together show that ∼PTr and ∼ctPTr,dis are incomparable with each other. All the four trace equivalences above are fully backward compatible with the two trace equivalences respectively defined in =-=[5]-=- for fully nondeterministic processes – denoted by ∼Tr,fnd – and in [23] for fully probabilistic processes – denoted by ∼Tr,fpr. Moreover, they are partially backward compatible with the trace equival...

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...classes of processes. Compatibility depends on the set of admitted tests. For example, the two fully nondeterministic processes in Fig. 8 (p. 19) are identified by the original testing equivalence of =-=[11]-=-. The relation ∼PTe-unionsqu equates them if only fully nondeterministic tests are employed, but distinguishes them as soon as probabilities are admitted within tests. Dually, following the terminolog...

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...bilities. The extended model, which we have thus called NPLTS, is such that every action-labeled transition goes from a source state to a probability distribution over target states – in the style of =-=[26, 31]-=- – rather than to a single target state. The most used definition of probabilistic trace equivalence for nondeterministic and probabilistic processes is the one provided in [32]. To resolve nondetermi...

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...entative of a number of slightly different probabilistic computational models including internal nondeterminism that have appeared in the literature with names such as, e.g., concurrent Markov chains =-=[38]-=-, alternating probabilistic models [18, 39, 29], NP-systems [20], probabilistic automata in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the se...

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...o a single target state. (2) A fully probabilistic process when every state has at most one outgoing transition. (3) A reactive probabilistic process [37] – or probabilistic automaton in the sense of =-=[30]-=- – when no state has two or more outgoing transitions labeled with the same action. The NPLTS in Fig. 1(a) mixes probability and internal nondeterminism, while the one in Fig. 1(b) describes a reactiv...

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...onships. Behavioral equivalences were first of all defined for labeled transition systems (LTS – set of states related via transitions each labeled with the action that gives rise to the state change =-=[24]-=-) that were used as models of nonderministic processes. Then, they have been extended/adapted to generalizations of such models to take into account probabilistic, stochastic, or timed behaviors. Amon...

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...in which every action-labeled transition goes from a source state to a probability distribution over target states rather than to a single target state. They are essentially Markov decision processes =-=[15]-=- and are representative of a number of slightly different probabilistic computational models including internal nondeterminism that have appeared in the literature with names such as, e.g., concurrent...

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... relation ∼PTe-unionsqu equates them if only fully nondeterministic tests are employed, but distinguishes them as soon as probabilities are admitted within tests. Dually, following the terminology of =-=[37]-=-, the two generative/reactive probabilistic processes in Fig. 10 (p. 20), which are identified by the generative probabilistic testing equivalence of [9] and the reactive probabilistic testing equival...

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...zs2,o, α)) = = ∑ α∈A∗ s.t. CCmax(zs1,o,α)6=∅ prob(SCC(zs1,o, α)) = prob(SC(zs1,o)) This means that s1 ∼PTe-∀∃ s2. The fact that ∼ctPTe-∀∃ and ∼ctPTe-tbt,dis coincide immediately follows. We know from =-=[10]-=- that for fully nondeterministic processes there is a strong connection between the testing semantics of [11] and the failure semantics of [5]. Thus, for a more complete comparison of the various trac...

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...ent probabilistic computational models including internal nondeterminism that have appeared in the literature with names such as, e.g., concurrent Markov chains [38], alternating probabilistic models =-=[18, 39, 29]-=-, NP-systems [20], probabilistic automata in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the sense of [19], probabilistic transition systems i...

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...her. All the four trace equivalences above are fully backward compatible with the two trace equivalences respectively defined in [5] for fully nondeterministic processes – denoted by ∼Tr,fnd – and in =-=[23]-=- for fully probabilistic processes – denoted by ∼Tr,fpr. Moreover, they are partially backward compatible with the trace equivalence – denoted by ∼Tr,rpr – that can be defined for reactive probabilist...

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...ilistic processes) with tests admitting probabilities (resp. internal nondeterminism), because it becomes possible to make copies of intermediate states of the processes under test. As pointed out in =-=[1]-=-, this capability increases the distinguishing power of testing equivalence. In a probabilistic setting, this may lead to questionable estimations of success probabilities (see [16] and the references...

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...ns. In other words, the two resolutions must exhibit the same trace distribution. We shall denote this equivalence by ∼PTr,dis. Testing equivalence for the same class of processes has been studied in =-=[39, 21, 33, 12]-=-. It considers the probability of performing computations along which the same tests are passed, REVISITING TRACE AND TESTING EQUIVALENCES FOR NONDET. AND PROB. PROCESSES 3 called successful computati...

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...te s, we denote by CC(zs, α) the set of α-compatible computations in Cfin(zs). We now recall two variants of the probabilistic trace-distribution equivalence introduced in [32] and further studied in =-=[7, 27, 28, 6]-=-. 8 M. BERNARDO. R. DE NICOLA, AND M. LORETI s1 0.5 0.5 0.5 0.5 a a b1 b2 b3 b4 s2 0.5 0.5 0.5 0.5 a a b1 b3 b2 b4 Figure 4: NPLTS models distinguished by ∼PTr,dis/∼ctPTr,dis and identified by ∼PTr/∼c...

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...ent probabilistic computational models including internal nondeterminism that have appeared in the literature with names such as, e.g., concurrent Markov chains [38], alternating probabilistic models =-=[18, 39, 29]-=-, NP-systems [20], probabilistic automata in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the sense of [19], probabilistic transition systems i...

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...ns. In other words, the two resolutions must exhibit the same trace distribution. We shall denote this equivalence by ∼PTr,dis. Testing equivalence for the same class of processes has been studied in =-=[39, 21, 33, 12]-=-. It considers the probability of performing computations along which the same tests are passed, REVISITING TRACE AND TESTING EQUIVALENCES FOR NONDET. AND PROB. PROCESSES 3 called successful computati...

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...n tests. Dually, following the terminology of [37], the two generative/reactive probabilistic processes in Fig. 10 (p. 20), which are identified by the generative probabilistic testing equivalence of =-=[9]-=- and the reactive probabilistic testing equivalence of [25], are equated by ∼PTe-unionsqu if only generative/reactive probabilistic tests are employed, but are told apart by the same relation as soon ...

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... – in the style of [26, 31] – rather than to a single target state. The most used definition of probabilistic trace equivalence for nondeterministic and probabilistic processes is the one provided in =-=[32]-=-. To resolve nondeterminism, it resorts to the notion of scheduler (or adversary), which can be viewed as an external entity that selects the next action to perform according to the current state and ...

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...ence were defined uniformly for fully probabilistic processes [17, 23] and reactive probabilistic processes [26, 35]. In contrast, testing equivalence for fully probabilistic processes was defined in =-=[8, 9]-=- in a way that resembles ∼PTe-∀∃, while for reactive probabilistic processes it was defined in [25] in a way similar to ∼PTe-unionsqu. Our compatibility results thus show that also testing equivalence...

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...,fpr. Moreover, they are partially backward compatible with the trace equivalence – denoted by ∼Tr,rpr – that can be defined for reactive probabilistic processes by following one of the approaches in =-=[35]-=-. Theorem 3.4. Let L = (S,A,−→) be an NPLTS and s1, s2 ∈ S. (1) If L is fully nondeterministic, then: s1 ∼PTr,dis s2 ⇔ s1 ∼ctPTr,dis s2 ⇔ s1 ∼PTr s2 ⇔ s1 ∼ctPTr s2 ⇔ s1 ∼Tr,fnd s2 (2) If L is fully pr...

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...a in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the sense of [19], probabilistic transition systems in the sense of [22], and pLTS [12] (see =-=[36]-=- for an overview). We formalize them as a variant of simple probabilistic automata [31] and give them the acronym NPLTS to stress the possible simultaneous presence of nondeterminism (N) and probabili...

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...n a probabilistic setting, this may lead to questionable estimations of success probabilities (see [16] and the references therein). Indeed, taking advantage of the increased discriminating power, in =-=[12]-=- it was shown that the may-part of ∼PTe-unionsqu coincides with a simulation equivalence akin to the one in [27] and the must-part coincides with a novel failure simulation equivalence. Moreover, in [...

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...o generative/reactive probabilistic processes in Fig. 10 (p. 20), which are identified by the generative probabilistic testing equivalence of [9] and the reactive probabilistic testing equivalence of =-=[25]-=-, are equated by ∼PTe-unionsqu if only generative/reactive probabilistic tests are employed, but are told apart by the same relation as soon as internal nondeterminism is admitted within tests. Indeed...

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...tional models including internal nondeterminism that have appeared in the literature with names such as, e.g., concurrent Markov chains [38], alternating probabilistic models [18, 39, 29], NP-systems =-=[20]-=-, probabilistic automata in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the sense of [19], probabilistic transition systems in the sense of [2...

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...nating probabilistic models [18, 39, 29], NP-systems [20], probabilistic automata in the sense of [31], probabilistic processes in the sense of [21], denotational probabilistic models in the sense of =-=[19]-=-, probabilistic transition systems in the sense of [22], and pLTS [12] (see [36] for an overview). We formalize them as a variant of simple probabilistic automata [31] and give them the acronym NPLTS ...

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...te s, we denote by CC(zs, α) the set of α-compatible computations in Cfin(zs). We now recall two variants of the probabilistic trace-distribution equivalence introduced in [32] and further studied in =-=[7, 27, 28, 6]-=-. 8 M. BERNARDO. R. DE NICOLA, AND M. LORETI s1 0.5 0.5 0.5 0.5 a a b1 b2 b3 b4 s2 0.5 0.5 0.5 0.5 a a b1 b3 b2 b4 Figure 4: NPLTS models distinguished by ∼PTr,dis/∼ctPTr,dis and identified by ∼PTr/∼c...

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...edulers instead of deterministic ones and makes use of countably many success actions in place of a single one. Notably, a single success action suffices when testing finitary processes, as proved in =-=[14]-=-, and the use of different classes of schedulers does not change the discriminating power, as we now show. REVISITING TRACE AND TESTING EQUIVALENCES FOR NONDET. AND PROB. PROCESSES 15 Theorem 4.4. Let...

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...ined in the probabilistic trace-distribution equivalence of [32] and the must-part coincides with the coarsest congruence contained in a probabilistic failure-distribution equivalence. As observed in =-=[20, 13]-=-, it is easy to see that there exist fully nondeterministic NPLTS models that are identified by ∼Te,fnd but distinguished by ∼PTe-unionsqu (and also by ∼PTe-∀∃). Let us consider the two NPLTS models i...

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...te s, we denote by CC(zs, α) the set of α-compatible computations in Cfin(zs). We now recall two variants of the probabilistic trace-distribution equivalence introduced in [32] and further studied in =-=[7, 27, 28, 6]-=-. 8 M. BERNARDO. R. DE NICOLA, AND M. LORETI s1 0.5 0.5 0.5 0.5 a a b1 b2 b3 b4 s2 0.5 0.5 0.5 0.5 a a b1 b3 b2 b4 Figure 4: NPLTS models distinguished by ∼PTr,dis/∼ctPTr,dis and identified by ∼PTr/∼c...

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...hree approaches recently explored in [4] to the definition of behavioral relations for NPLTS models. We would like to mention that ∼PTr and ∼PTe-tbt did pop up when working in the framework of ULTraS =-=[3]-=-. This is a parametric model encompassing many others such as labeled transition systems, discrete-/continuous-time Markov chains, and discrete-/continuous-time Markov decision processes without/with ...

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...te s, we denote by CC(zs, α) the set of α-compatible computations in Cfin(zs). We now recall two variants of the probabilistic trace-distribution equivalence introduced in [32] and further studied in =-=[7, 27, 28, 6]-=-. 8 M. BERNARDO. R. DE NICOLA, AND M. LORETI s1 0.5 0.5 0.5 0.5 a a b1 b2 b3 b4 s2 0.5 0.5 0.5 0.5 a a b1 b3 b2 b4 Figure 4: NPLTS models distinguished by ∼PTr,dis/∼ctPTr,dis and identified by ∼PTr/∼c...

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...obabilistic choices make the latter equivalence sensitive to the moment of occurrence of internal choices, thus yielding unrealistic probability estimations. This problem has been recently tackled in =-=[8]-=- for a significantly different probabilistic model by relying on a label massaging that avoids over-/under-estimations of success probabilities in a parallel context. In this paper, by using ∼PTe,new ...

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.... As pointed out in [1], this capability increases the distinguishing power of testing equivalence. In a probabilistic setting, this may lead to questionable estimations of success probabilities (see =-=[16]-=- and the references therein). In this paper, we study new trace and testing equivalences (for nondeterministic and probabilistic processes) that, different from the old ones, do possess the above-ment...

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...ROB. PROCESSES 41 an alternative characterization of ∼PTe-unionsqu, a fact that reconciles the testing equivalence deeply investigated in the literature with the three approaches recently explored in =-=[4]-=- to the definition of behavioral relations for NPLTS models. We would like to mention that ∼PTr and ∼PTe-tbt did pop up when working in the framework of ULTraS [3]. This is a parametric model encompas...