From Branching to Linear Time, Coalgebraically
We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations.
D. Baelde and A. Carayol (Eds.): Fixed Points in Computer Science 2013 (FICS 2013) EPTCS 126, 2013, pp. From Branching to Linear Time, Coalgebraically–LABEL:LastPage, doi:10.4204/EPTCS.126.2 © C. Cîrstea This work is licensed under the Creative Commons Attribution License.
From Branching to Linear Time, Coalgebraically
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When analysing process behaviour, one of the early choices one has to make is between a linear and a branching view of time. In branching-time semantics, the choices a process has for proceeding from a particular state are taken into account when defining a notion of process equivalence (with bisimulation being the typical such equivalence), whereas in linear-time semantics such choices are abstracted away and the emphasis is on the individual executions that a process is able to exhibit. From a system verification perspective, one often chooses the linear-time view, as this not only leads to simpler specification logics and associated verification techniques, but also meets the practical need to verify all possible system executions.
While the theory of coalgebras has, from the outset, been able to provide a uniform account of various bisimulation-like observational equivalences (and later, of various simulation-like behavioural preorders), it has so far not been equally successful in giving a generic account of the linear-time behaviour of a state in a system whose type incorporates a notion of branching. For example, the generic trace theory of [HasuoJS07] only applies to systems modelled as coalgebras of type , with the monad specifying a branching type (e.g. non-deterministic or probabilistic), and the endofunctor defining the structure of individual transitions (e.g. labelled transitions or successful termination). The approach in loc. cit. is complemented by that of [JacobsSS12], where traces are derived using a determinisation procedure similar to the one for non-deterministic automata. The latter approach applies to systems modelled as coalgebras of type , where again a monad is used to model branching behaviour, and an endofunctor specifies the transition structure. Neither of these approaches is able to account for potentially infinite traces, as typically employed in model-based formal verification. This limitation is partly addressed in [cirstea-11], but again, this only applies to coalgebras of type , albeit with more flexibility in the underlying category (which in particular allows a measure-theoretic account of infinite traces in probabilistic systems). Finally, none of the above-mentioned approaches exploits the compositionality that is intrinsic to the coalgebraic approach. In particular, coalgebras of type (of which systems with both inputs and outputs are an example, see Example LABEL:input-output) can not be accounted for by any of the existing approaches. This paper presents an attempt to address the above limitations concerning the types of coalgebras and the nature of traces that can be accounted for, by providing a uniform and compositional treatment of (possibly infinite) linear-time behaviour in systems with branching.
In our view, one of the reasons for only a partial success in developing a fully general coalgebraic theory of traces is the long-term aspiration within the coalgebra community to obtain a uniform characterisation of trace equivalence via a finality argument, in much the same way as is done for bisimulation (in the presence of a final coalgebra). This encountered difficulties, as a suitable category for carrying out such an argument proved difficult to find in the general case. In this paper, we tackle the problem of getting a handle on the linear-time behaviour of a state in a coalgebra with branching from a different angle: we do not attempt to directly define a notion of trace equivalence between two states (e.g. via finality in some category), but focus on testing whether a state is able to exhibit a particular trace, and on measuring the extent of this ability. This ”measuring” relates to the type of branching present in the system, and instantiates to familiar concepts such as the probability of exhibiting a given trace in probabilistic systems, the minimal cost of exhibiting a given trace in weighted computations, and simply the ability to exhibit a trace in non-deterministic systems.
The technical tool for achieving this goal is a generalisation of the notions of relation and relation lifting [HermidaJ98], which lie at the heart of the definition of coalgebraic bisimulation. Specifically, we employ relations valued in a partial semiring, and a corresponding generalised version of relation lifting. Our approach applies to coalgebras whose type is obtained as the composition of several endofunctors on : one of these is a monad that accounts for the presence of branching in the system, while the remaining endofunctors, assumed here to be polynomial, jointly determine the notion of linear-time behaviour. This strictly subsumes the types of systems considered in earlier work on coalgebraic traces [HasuoJS07, cirstea-11, JacobsSS12], while also providing compositionality in the system type.
Our main contribution, presented in Section LABEL:linear-time, is a uniform and compositional account of linear-time behaviour in state-based systems with branching. A by-product of our work is an extension of the study of additive monads carried out in [Kock2011, CoumansJ2011] to what we call partially additive monads (Section LABEL:semiring). Our approach can be summarised as follows:
We move from two-valued to multi-valued relations, with the universe of truth values being induced by the choice of monad for modelling branching. This instantiates to relations valued in the interval in the case of probabilistic branching, the set in the case of weighted computations, and simply in the case of non-deterministic branching. This reflects our view that the notion of truth used to reason about the observable behaviour of a system should be dependent on the branching behaviour present in that system. Such a dependency is also expected to result in temporal logics that are more natural and more expressive, and at the same time have a conceptually simpler semantics. In deriving a suitable structure on the universe of truth values, we generalise results on additive monads [Kock2011, CoumansJ2011] to partially additive monads. This allows us to incorporate probabilistic branching under our approach. We show that for a commutative, partially additive monad on , the set carries a partial semiring structure with an induced preorder, which in turn makes an appropriate choice of universe of truth values.
We generalise and adapt the notion of relation lifting used in the definition of coalgebraic bisimulation, in order to (i) support multi-valued relations, and (ii) abstract away branching. Specifically, we make use of the partial semiring structure carried by the universe of truth values to generalise relation lifting of polynomial endofunctors to multi-valued relations, and employ a canonical extension lifting induced by the monad to capture a move from branching to linear time. The use of this extension lifting allows us to make formal the idea of testing whether, and to what extent, a state in a coalgebra with branching can exhibit a particular linear-time behaviour. Our approach resembles the idea employed by partition refinement algorithms for computing bisimulation on labelled transition systems with finite state spaces [KS90]. There, one starts from a single partition of the state space, with all states related to each other, and repeatedly refines it through stepwise unfolding of the transition structure, until a fixpoint is reached. Similarly, we start by assuming that a state in a system with branching can exhibit any linear-time behaviour, and moreover, assign the maximum possible value to each pair consisting of a state and a linear-time behaviour. We then repeatedly refine the values associated to such pairs, through stepwise unfolding of the coalgebraic structure.
The present work is closely related to our earlier work on maximal traces and path-based logics [cirstea-11], which described a game-theoretic approach to testing if a system with non-deterministic branching is able to exhibit a particular trace. Here we consider arbitrary branching types, and while we do not emphasise the game-theoretic aspect, our use of greatest fixpoints has a very similar thrust.
Several fruitful discussions with participants at the 2012 Dagstuhl Seminar on Coalgebraic Logics helped refine the ideas presented here. Our use of relation lifting was inspired by the recent work on coinductive predicates [Hasuo12], itself based on the seminal work in [HermidaJ98] on the use of predicate and relation lifting in the formalisation of induction and coinduction principles. Last but not least, the comments received from the anonymous reviewers contributed to improving the presentation of this work and to identifying new directions for future work.
2.1 Relation Lifting
The concepts of predicate lifting and relation lifting, to our knowledge first introduced in [HermidaJ98], are by now standard tools in the study of coalgebraic models, used e.g. to provide an alternative definition of the notion of bisimulation (see e.g. in [JacobsBook]), or to describe the semantics of coalgebraic modal logics [Pattinson03, Moss99]. While these concepts are very general, their use so far usually restricts this generality by viewing both predicates and relations as sub-objects in some category (possibly carrying additional structure). In this paper, we make use of the full generality of these concepts, and move from the standard view of relations as subsets to a setting where relations are valuations into a universe of truth values. This section recalls the definition of relation lifting in the standard setting where relations are given by monomorphic spans.
Throughout this section (only), denotes the category whose objects are binary relations with a monomorphic span, and whose arrows from to are given by pairs of functions s.t. factors through :