Propositional Temporal Logics and Equivalences 1

2

Ursula Goltz , Ruurd Kuiper , Wo jciech Penczek

3

Abstract

We compare propositional temporal logics by comparing the equivalences that they induce on models. Linear time, branching time and partial order temporal logics are considered. The logics are interpreted on occurrence transition systems, generated by labelled prime event structures without autoconcurrency. The induced equivalences are also compared to directly de ned equivalences, e.g., history preserving bisimulation, pomset bisimulation, pomset trace equivalence, and others. It is then shown which of the induced equivalences are and which are not preserved under action re nement. Rather unexpectedly, the addition of the backward next step operator to the weakest logic considered yields a logic stronger than all others. It is shown that weak history preserving bisimulation can be obtained as the equivalence induced by a slightly constrained version of that logic.

1 Introduction Currently a lot of formalisms to describe concurrent computations exist. Even only regarding logics still leaves a large set. Our aim is to bring some structure in this set. Some comparisons between logics of course exist already. However, mostly these consider only logics interpreted over the same domain [EH86, Wo87, St87], whereas we consider logics which were originally interpreted over dierent domains. Also, these comparisons mainly address expressiveness, i.e., which sets of behaviours can be described. This is a rather strict notion, and logics quickly become incomparable ([La80]). We wish to compare logics in the large. So we need, rstly, a common framework of interpretation and, secondly, a suciently relaxed measure for comparison. Event structures ([NPW81, Wi88]) provide a very detailed model for concurrent computations. All features considered by many dierent logics are represented therein. Therefore we choose these as the natural candidate for a common framework of interpretation, satisfying our rst desire. Various equivalence notions have been de ned on event structures to obtain more abstract system representations. These equivalences themselves can then be compared, thus providing a measure for the relative precision. This we use to provide an answer to our second desire, a measure for comparison. Each logic namely induces GMD, Bonn, Germany. Supported in part by Esprit Basic Research Action 3148 (DEMON) Eindhoven University of Technology, Depart. of Computer Science, The Netherlands. Supported by Esprit-BRA project 3096: Formal Methods and Tools for the Development of Distributed and Real-Time Systems (SPEC) 3 Institute of Computer Science, Warsaw, Poland. Supported in part by the Dutch NFI/NWO project REX and also by The Wolfson Research Awards Scheme in The United Kingdom 1 2

1

an equivalence relation: the one that distinguishes just those event structures that can be distinguished by some formula. We then compare logics by comparing the induced equivalences. So the comparison is on precision rather than expressiveness. For the interpretation of temporal logics we use transition systems derived from the event structures. The equivalence relations are rede ned accordingly. We link event structures via con guration structures to occurrence transition systems. This idea has been used before, e.g., in [NRT91, Ro90]. These transition systems in turn are linked to the Kripke structures that serve as models for the logics. This idea is already present in [deNMV90], but motivated a little more extensively here. We, brie y, show how occurrence transition systems are obtained from the event structures because, as yet, we have no direct characterization of this class. As it turns out that none of the considered logics can handle autoconcurrency, we restrict the class of systems accordingly. The following temporal logics are considered: the modal logic S4 extended with Next (S4N) [HC84], Linear Time Temporal Logic (LTTL) [MP91], Computation Tree Logic (CTL) [CES83], (Quanti ed) Interleaving Set Temporal Logic ((Q)ISTL) [KP87], and Concurrent Computation Tree Logic (CCTL) [Pe90], the latter three with their *- and/or concurrently fair versions. Furthermore, the extensions of these logics with added past modalities are considered. The selection is motivated in the paper by a table presenting various combinations of features of such logics. A second aim was inspired by the fact that much research eort has been devoted already to devising useful equivalence notions directly. We compare induced and existing equivalences, to see whether or not the intuitions and motivations from the dierent elds lead to similar results. The following equivalence notions have been selected: interleaving trace equivalence (i?t), forward bisimulation (f ?b ) [Pa81, Mi80], pomset trace equivalence (p?t ), pomset bisimulation (p?b ) [BC87], history preserving bisimulation (h?b ) [TRH88], backward forward bisimulation (bf ?b ) [deNMV90]. Interleaving trace equivalence is the simplest interleaving equivalence where the branching structure (choices between alternative behaviours) is not taken into account. Pomset trace equivalence is the corresponding notion in causality based semantics, where causal dependencies in runs of systems are recorded. The selection of these equivalences is, again, motivated later by a table of features. The decisions were in uenced by intuitive expectations that the chosen equivalences would be close to induced ones. The combined result of these two eorts is a complete table, strictly comparing all equivalences considered. Two interesting facts follow from this table about the extension of S4N with past modalities, Partial Order Logic (POL) [Si90]. Rather surprisingly, the expressively quite weak language POL induces a stronger equivalence than the expressively strong (though incomparable, expressively, to POL) concurrently fair version of CCTL*. Also, a slightly modi ed form of the simple logic POL induces h?b . The only other logic, to our knowledge, that provides this correspondence can be found in [deNF90]. Constructing the table of comparisons, we learned that there is a quite close connection between induced and existing equivalence relations. Where there was no direct match, our investigations gave enough insight to enable modi cations to the logics, or, conversely, to the equivalence notions, to make them t to one another. The main dierence between equivalences induced by logics and those de ned di2

rectly turned out to be that the former use completed pomsets or sequential traces whereas the latter use simply pomsets or sequential traces. As a third, more or less independent aim, it is proven which of the equivalences are and which are not preserved under action re nement. Here our results suggest that equivalences, imposed by logics with backward modalities distinguishing branching points and concurrency, are preserved under re nement of actions, whereas for similar logics without backward modalities, preservation under re nement can not be obtained. The paper is organised as follows. Section 2 introduces the basic framework, event structures. It also contains the translation into con guration structures and the further translation of these into transition systems. In Section 3 the selected logics are presented, in Section 4 the chosen equivalence notions. In Section 5 the induced equivalences are derived. The comparison is made in Section 6. Preservation of the equivalences under action re nement is the subject of Section 7. Finally, some concluding remarks can be found in Section 8. All the proofs can be found in the full version of this paper [GKP92].

2 Representations of Concurrent Systems We are interested in comparing formalisms that describe concurrent systems performing actions from a given set Act of action names. These formalisms are interpreted over various dierent structures. As the rst, basic, structure we have chosen event structures. We use a subset of CCSP expressions to write down examples of concurrent systems in a concise and intuitive manner, assuming event structures as a semantics. An event structure semantics of CCSP can be found in, e.g., [Wi82, LG91].

2.1 Event Structures

Event structures represent a concurrent system by taking occurrences of actions as the starting point. Every occurrence of an action is modelled as a separate event; a label function indicates which action is represented. Two relations are provided that capture, respectively, the causality and con ict relationship between events. It turns out that for our purposes we only need a certain class of event structures; prime event structures with binary con ict. In the sequel we tacitly assume that event structures are taken from this class. Further details can be found in, e.g., [NPW81, Wi88]. De nition 2.1 A (prime labelled) event structure over an alphabet Act is a 4-tuple E = (E;

2

Ursula Goltz , Ruurd Kuiper , Wo jciech Penczek

3

Abstract

We compare propositional temporal logics by comparing the equivalences that they induce on models. Linear time, branching time and partial order temporal logics are considered. The logics are interpreted on occurrence transition systems, generated by labelled prime event structures without autoconcurrency. The induced equivalences are also compared to directly de ned equivalences, e.g., history preserving bisimulation, pomset bisimulation, pomset trace equivalence, and others. It is then shown which of the induced equivalences are and which are not preserved under action re nement. Rather unexpectedly, the addition of the backward next step operator to the weakest logic considered yields a logic stronger than all others. It is shown that weak history preserving bisimulation can be obtained as the equivalence induced by a slightly constrained version of that logic.

1 Introduction Currently a lot of formalisms to describe concurrent computations exist. Even only regarding logics still leaves a large set. Our aim is to bring some structure in this set. Some comparisons between logics of course exist already. However, mostly these consider only logics interpreted over the same domain [EH86, Wo87, St87], whereas we consider logics which were originally interpreted over dierent domains. Also, these comparisons mainly address expressiveness, i.e., which sets of behaviours can be described. This is a rather strict notion, and logics quickly become incomparable ([La80]). We wish to compare logics in the large. So we need, rstly, a common framework of interpretation and, secondly, a suciently relaxed measure for comparison. Event structures ([NPW81, Wi88]) provide a very detailed model for concurrent computations. All features considered by many dierent logics are represented therein. Therefore we choose these as the natural candidate for a common framework of interpretation, satisfying our rst desire. Various equivalence notions have been de ned on event structures to obtain more abstract system representations. These equivalences themselves can then be compared, thus providing a measure for the relative precision. This we use to provide an answer to our second desire, a measure for comparison. Each logic namely induces GMD, Bonn, Germany. Supported in part by Esprit Basic Research Action 3148 (DEMON) Eindhoven University of Technology, Depart. of Computer Science, The Netherlands. Supported by Esprit-BRA project 3096: Formal Methods and Tools for the Development of Distributed and Real-Time Systems (SPEC) 3 Institute of Computer Science, Warsaw, Poland. Supported in part by the Dutch NFI/NWO project REX and also by The Wolfson Research Awards Scheme in The United Kingdom 1 2

1

an equivalence relation: the one that distinguishes just those event structures that can be distinguished by some formula. We then compare logics by comparing the induced equivalences. So the comparison is on precision rather than expressiveness. For the interpretation of temporal logics we use transition systems derived from the event structures. The equivalence relations are rede ned accordingly. We link event structures via con guration structures to occurrence transition systems. This idea has been used before, e.g., in [NRT91, Ro90]. These transition systems in turn are linked to the Kripke structures that serve as models for the logics. This idea is already present in [deNMV90], but motivated a little more extensively here. We, brie y, show how occurrence transition systems are obtained from the event structures because, as yet, we have no direct characterization of this class. As it turns out that none of the considered logics can handle autoconcurrency, we restrict the class of systems accordingly. The following temporal logics are considered: the modal logic S4 extended with Next (S4N) [HC84], Linear Time Temporal Logic (LTTL) [MP91], Computation Tree Logic (CTL) [CES83], (Quanti ed) Interleaving Set Temporal Logic ((Q)ISTL) [KP87], and Concurrent Computation Tree Logic (CCTL) [Pe90], the latter three with their *- and/or concurrently fair versions. Furthermore, the extensions of these logics with added past modalities are considered. The selection is motivated in the paper by a table presenting various combinations of features of such logics. A second aim was inspired by the fact that much research eort has been devoted already to devising useful equivalence notions directly. We compare induced and existing equivalences, to see whether or not the intuitions and motivations from the dierent elds lead to similar results. The following equivalence notions have been selected: interleaving trace equivalence (i?t), forward bisimulation (f ?b ) [Pa81, Mi80], pomset trace equivalence (p?t ), pomset bisimulation (p?b ) [BC87], history preserving bisimulation (h?b ) [TRH88], backward forward bisimulation (bf ?b ) [deNMV90]. Interleaving trace equivalence is the simplest interleaving equivalence where the branching structure (choices between alternative behaviours) is not taken into account. Pomset trace equivalence is the corresponding notion in causality based semantics, where causal dependencies in runs of systems are recorded. The selection of these equivalences is, again, motivated later by a table of features. The decisions were in uenced by intuitive expectations that the chosen equivalences would be close to induced ones. The combined result of these two eorts is a complete table, strictly comparing all equivalences considered. Two interesting facts follow from this table about the extension of S4N with past modalities, Partial Order Logic (POL) [Si90]. Rather surprisingly, the expressively quite weak language POL induces a stronger equivalence than the expressively strong (though incomparable, expressively, to POL) concurrently fair version of CCTL*. Also, a slightly modi ed form of the simple logic POL induces h?b . The only other logic, to our knowledge, that provides this correspondence can be found in [deNF90]. Constructing the table of comparisons, we learned that there is a quite close connection between induced and existing equivalence relations. Where there was no direct match, our investigations gave enough insight to enable modi cations to the logics, or, conversely, to the equivalence notions, to make them t to one another. The main dierence between equivalences induced by logics and those de ned di2

rectly turned out to be that the former use completed pomsets or sequential traces whereas the latter use simply pomsets or sequential traces. As a third, more or less independent aim, it is proven which of the equivalences are and which are not preserved under action re nement. Here our results suggest that equivalences, imposed by logics with backward modalities distinguishing branching points and concurrency, are preserved under re nement of actions, whereas for similar logics without backward modalities, preservation under re nement can not be obtained. The paper is organised as follows. Section 2 introduces the basic framework, event structures. It also contains the translation into con guration structures and the further translation of these into transition systems. In Section 3 the selected logics are presented, in Section 4 the chosen equivalence notions. In Section 5 the induced equivalences are derived. The comparison is made in Section 6. Preservation of the equivalences under action re nement is the subject of Section 7. Finally, some concluding remarks can be found in Section 8. All the proofs can be found in the full version of this paper [GKP92].

2 Representations of Concurrent Systems We are interested in comparing formalisms that describe concurrent systems performing actions from a given set Act of action names. These formalisms are interpreted over various dierent structures. As the rst, basic, structure we have chosen event structures. We use a subset of CCSP expressions to write down examples of concurrent systems in a concise and intuitive manner, assuming event structures as a semantics. An event structure semantics of CCSP can be found in, e.g., [Wi82, LG91].

2.1 Event Structures

Event structures represent a concurrent system by taking occurrences of actions as the starting point. Every occurrence of an action is modelled as a separate event; a label function indicates which action is represented. Two relations are provided that capture, respectively, the causality and con ict relationship between events. It turns out that for our purposes we only need a certain class of event structures; prime event structures with binary con ict. In the sequel we tacitly assume that event structures are taken from this class. Further details can be found in, e.g., [NPW81, Wi88]. De nition 2.1 A (prime labelled) event structure over an alphabet Act is a 4-tuple E = (E;