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...bility values. This relation preserves the fairness introduced by the probability distributions. Finally, we study the bisimulation equivalence resulting from considering Park & Milner’s bisimulation =-=[22]-=- on the usual stripped probabilistic transition system (translated into a labeled transition system). Here we call it probability obliterated bisimulation. This is the usual way to abstract probabilit...

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... the probability values. In addition, we compare these bisimulation equivalences and provide a logic characterization for each of them. 1 Introduction Structural operational semantics (SOS for short) =-=[24]-=- is a powerful tool to provide semantics to programming languages. In SOS, process behavior is described using transition systems and the behavior of a composite process is given in terms of the behav...

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...ongruences for all operator definable in the respective format. The first relation we focus on is Segala’s variant of bisimulation that considers combined transitions, here called convex bisimulation =-=[25]-=-. The second relation we explore originates here and we call it probability abstracted bisimulation. Like bisimulation and unlike convex bisimulation, it preserves the structure of the distributions o...

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...arget of the transition is a probability distribution on states) we manage to obtained the most general format for bisimulation equivalence, which we called ntµfθ/ntµxθ, following the nomenclature of =-=[14, 15]-=-. Starting from the ntµfθ/ntµxθ format, in this paper we define formats to guarantee that three coarser versions of bisimulation equivalence are congruences for all operator definable in the respectiv...

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...arget of the transition is a probability distribution on states) we manage to obtained the most general format for bisimulation equivalence, which we called ntµfθ/ntµxθ, following the nomenclature of =-=[14, 15]-=-. Starting from the ntµfθ/ntµxθ format, in this paper we define formats to guarantee that three coarser versions of bisimulation equivalence are congruences for all operator definable in the respectiv...

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... the least 3-valued stable model for P. PTSSs with least 3-valued stable model that are also a 2-valued model are particularly interesting, since this model is actually the only 3-valued stable model =-=[7, 13]-=-. A PTSS P is said to be complete if its least 3-valued stable model 〈CT,PT〉 satisfies that CT = PT (i.e., the model is also 2-valued). We associate a probabilistic transition system to each complete ...

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...uction of probabilistic process algebras motivated the need for a theory of structural operational semantics to define probabilistic transition systems. Few earlier results appeared in this direction =-=[4, 5, 17, 18]-=- presenting congruence theorems for Larsen & Skou bisimulation equivalence [19]. Most of these formats have complicated restrictions that extend to sets of rules due to the fact that they considered t...

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...als such that P ` Nψ and PT |= N (b) ψ ∈ PT iff there is a set N of closed negative literals such that P ` Nψ and CT |= N. The least 3-valued stable model of a PTSS can be constructed using induction =-=[8, 11, 12]-=-. Lemma 1. Let P be a PTSS. For each ordinal α, define the pair 〈CTα,PTα〉 as follows: • CT0 = ∅ and PT0 = T(Σs)×A×T(Σd). • For every non-limit ordinal α > 0, define: CTα = { t a−→ θ | for some set N o...

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...uction of probabilistic process algebras motivated the need for a theory of structural operational semantics to define probabilistic transition systems. Few earlier results appeared in this direction =-=[4, 5, 17, 18]-=- presenting congruence theorems for Larsen & Skou bisimulation equivalence [19]. Most of these formats have complicated restrictions that extend to sets of rules due to the fact that they considered t...

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...als such that P ` Nψ and PT |= N (b) ψ ∈ PT iff there is a set N of closed negative literals such that P ` Nψ and CT |= N. The least 3-valued stable model of a PTSS can be constructed using induction =-=[8, 11, 12]-=-. Lemma 1. Let P be a PTSS. For each ordinal α, define the pair 〈CTα,PTα〉 as follows: • CT0 = ∅ and PT0 = T(Σs)×A×T(Σd). • For every non-limit ordinal α > 0, define: CTα = { t a−→ θ | for some set N o...

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...riables in Yl for all l ∈ L, hence condition 6. The precise understanding of conditions 5 and 6 requires a rather lengthy explanation that is beyond the scope of this paper. The reader is referred to =-=[8, 9]-=- for details. All congruence theorems in this article apply only to PTSSs whose rules are well-founded. A rule r is well-founded if there is no infinite backward chain in the dependency graph Gr = (V,...

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... the least 3-valued stable model for P. PTSSs with least 3-valued stable model that are also a 2-valued model are particularly interesting, since this model is actually the only 3-valued stable model =-=[7, 13]-=-. A PTSS P is said to be complete if its least 3-valued stable model 〈CT,PT〉 satisfies that CT = PT (i.e., the model is also 2-valued). We associate a probabilistic transition system to each complete ...

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...erational semantics to define probabilistic transition systems. Few earlier results appeared in this direction [4, 5, 17, 18] presenting congruence theorems for Larsen & Skou bisimulation equivalence =-=[19]-=-. Most of these formats have complicated restrictions that extend to sets of rules due to the fact that they considered transitions labeled both with an action and a probability value. By using a more...

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...i for (t,a,pi) ∈ −→. Transition relations are usually defined by means of structured operational semantics in Plotkin’s style [24]. For PTS, algebraic characterizations of this style were provided in =-=[8, 9, 21]-=- where the term probabilistic transition system specification was used and which we adopt in our paper. Definition 3 (PTSS). A probabilistic transition system specification (PTSS) is a triple P = (Σ,A...

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...mat [7, 14]. For completeness we mention two other similar results on PTSSs that fall out of Larsen & Skou’s bisimulation. They are [20], that presents a format for rooted branching bisimulation, and =-=[26]-=-, that presents a format for non-expansiveness of -bisimulations. Prior to the congruence theorems, we presented the different bisimulation equivalences, compare them, and, in particular, we gave a l...

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... each term of the language. A particular focus of these formalizations was to provide a meta-theory that ensures a diversity of semantic properties by simple inspection on the form of the rules. (See =-=[1, 2, 23]-=- for overviews.) One of such kind of properties is to ensure that a given equivalence relation is a congruence for all operators whose semantics is defined in a TSS whose rules complies to a particula...

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...i for (t,a,pi) ∈ −→. Transition relations are usually defined by means of structured operational semantics in Plotkin’s style [24]. For PTS, algebraic characterizations of this style were provided in =-=[8, 9, 21]-=- where the term probabilistic transition system specification was used and which we adopt in our paper. Definition 3 (PTSS). A probabilistic transition system specification (PTSS) is a triple P = (Σ,A...

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...so that all stated inclusions are proper. In the rest of the section we present logical characterizations for the different bisimulation equivalences. This work has already been done for bisimulation =-=[10, 16]-=- and convex bisimulation [16]. We adopt here the two-level logic style of [10]. We define the logic Lb as the set of all formulas with the following syntax: φ := > | 〈a〉ψ | 〈a〉cψ | ∧i∈I φi | ¬φ ψ := [...

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...uction of probabilistic process algebras motivated the need for a theory of structural operational semantics to define probabilistic transition systems. Few earlier results appeared in this direction =-=[4, 5, 17, 18]-=- presenting congruence theorems for Larsen & Skou bisimulation equivalence [19]. Most of these formats have complicated restrictions that extend to sets of rules due to the fact that they considered t...

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...uction of probabilistic process algebras motivated the need for a theory of structural operational semantics to define probabilistic transition systems. Few earlier results appeared in this direction =-=[4, 5, 17, 18]-=- presenting congruence theorems for Larsen & Skou bisimulation equivalence [19]. Most of these formats have complicated restrictions that extend to sets of rules due to the fact that they considered t...

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... be seen as generalizations of the non-probabilistic ntyft/ntyxt format [7, 14]. For completeness we mention two other similar results on PTSSs that fall out of Larsen & Skou’s bisimulation. They are =-=[20]-=-, that presents a format for rooted branching bisimulation, and [26], that presents a format for non-expansiveness of -bisimulations. Prior to the congruence theorems, we presented the different bisi...

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... each term of the language. A particular focus of these formalizations was to provide a meta-theory that ensures a diversity of semantic properties by simple inspection on the form of the rules. (See =-=[1, 2, 23]-=- for overviews.) One of such kind of properties is to ensure that a given equivalence relation is a congruence for all operators whose semantics is defined in a TSS whose rules complies to a particula...