Nested Semantics over Finite Trees are Equationally Hard Invited Talk

AbstractOne of the criteria that has been put forward for studying the mathematical tractability of the behavioural equivalences in van Glabbeek's linear time- banching time spectrum is that they afford elegant, finite equational axiomatizations over fragments of process algebraic languages. A review of existing complete equational axiomatizations for many of the behavioural semantics in van Glabbeek's spectrum is offered in his encyclopedic chapter in the [Handbook of Process Algebra, Jan A. Bergstra, Alban Ponse, and Scott A. Smolka, Editors. North-Holland, Amsterdam, 2001]. The equational axiomatizations offered ibidem are over the language BCCSP, a common fragment of Milner's CCS and Hoare's CSP suitable for describing finite synchronization trees, and characterize the differences between behavioural semantics in terms of a few revealing axioms.The main omissions in this menagerie of equational axiomatizations for the behavioural semantics in van Glabbeek's spectrum are axiomatizations for 2-nested simulation semantics and possible futures semantics. The relation of 2-nested simulation was introduced by Groote and Vaandrager as the coarsest equivalence included in completed trace equivalence for which the tyft/tyxt format is a congruence format. Possible futures semantics, on the other hand, was proposed by Brookes and Rounds as far back as 1981, and is one of the nested trace equivalences introduced by Hennessy and Milner.In this talk, I shall present, amongst other results, a mathematical answer to the following natural question:Why have these semantics not been given complete axiomatizations yet, even for the language of finite synchronization trees?More precisely, I shall argue that the 2-nested simulation and the possible futures preorder and equivalence do not admit a finite (in)equational axiomatization over the language BCCSP. I then generalize these negative results to show that, for n⩾2, none of the n-nested simulation or trace preorders and equivalences defined by Groote and Vaandrager, and Hennessy and Milner, respectively, afford finite equational axiomatizations over the language BCCSP.Interestingly, the intersection of all of the n-nested simulation or trace equivalences or preorders over image-finite labelled transition systems, and therefore over the language BCCSP, is bisimulation equivalence. Hennessy and Milner proved that bisimulation equivalence is axiomatized over the language BCCSP by means of four classic equations. It follows that this fundamental behavioural equivalence, albeit finitely based over BCCSP, is the limit of sequences of relations that do not afford finite equational axiomatizations themselves. Thus “finite axiomatizability” is an example from process theory of a “discontinuous” property of a behavioural equivalence—i.e., of a property that “appears at the limit”, but is not afforded by its finite approximations. This talk is based upon joint work with Wan Fokkink (CWI), Rob van Glabbeek (INRIA, Sophia Antipolis) and Anna Ingólfsdóttir (BRICS, Aalborg University)

CPO Models for GSOS Languages - Part I: Compact GSOS Languages

In this paper, we present a general way of giving denotational semantics to a class of languages equipped with an operational semantics that fits the GSOS format of Bloom, Istrail and Meyer. The canonical model used for this purpose will be Abramsky's domain of synchronization trees, and the denotational semantics automatically generated by our methods will be guaranteed to be fully abstract with respect to the finitely observable part of the bisimulation preorder. In the process of establishing the full abstraction result, we also obtain several general results on the bisimulation preorder (including a complete axiomatization for it), and give a novel operational interpretation of GSOS languages

A Complete Axiomatization of Timed Bisimulation for a Class of Timed Regular Behaviours (Revised Version)

One of the most satisfactory results in process theory is Milner's axiomatization of strong bisimulation for regular CCS. This result holds for open terms with finite-state recursion. Wang has shown that timed bisimulation can also be axiomatized, but only for closed terms without recursion. In this paper, we provide an axiomatization for timed bisimulation of open terms with finite-state recursion

Rule Formats for Nominal Process Calculi

The nominal transition systems (NTSs) of Parrow et al. describe the operational semantics of nominal process calculi. We study NTSs in terms of the nominal residual transition systems (NRTSs) that we introduce. We provide rule formats for the specifications of NRTSs that ensure that the associated NRTS is an NTS and apply them to the operational specification of the early pi-calculus. Our study stems from the recent Nominal SOS of Cimini et al. and from earlier works in nominal sets and nominal logic by Gabbay, Pitts and their collaborators

Meta SOS - A Maude Based SOS Meta-Theory Framework

Meta SOS is a software framework designed to integrate the results from the meta-theory of structural operational semantics (SOS). These results include deriving semantic properties of language constructs just by syntactically analyzing their rule-based definition, as well as automatically deriving sound and ground-complete axiomatizations for languages, when considering a notion of behavioural equivalence. This paper describes the Meta SOS framework by blending aspects from the meta-theory of SOS, details on their implementation in Maude, and running examples.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690

Resumptions, Weak Bisimilarity and Big-Step Semantics for While with Interactive I/O: An Exercise in Mixed Induction-Coinduction

We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive trace-based semantics for While, we define several big-step semantics for While with interactive I/O, based on resumptions and termination-sensitive weak bisimilarity. These require nesting inductive definitions in coinductive definitions, which is interesting both mathematically and from the point-of-view of implementation in a proof assistant. After first defining a basic semantics of statements in terms of resumptions with explicit internal actions (delays), we introduce a semantics in terms of delay-free resumptions that essentially removes finite sequences of delays on the fly from those resumptions that are responsive. Finally, we also look at a semantics in terms of delay-free resumptions supplemented with a silent divergence option. This semantics hinges on decisions between convergence and divergence and is only equivalent to the basic one classically. We have fully formalized our development in Coq.Comment: In Proceedings SOS 2010, arXiv:1008.190

Testing Hennessy-Milner Logic with Recursion

This study offers a characterization of the collection of propertiesexpressible in Hennessy-Milner Logic (HML) with recursion that can be testedusing finite LTSs. In addition to actions used to probe the behaviour of thetested system, the LTSs that we use as tests will be able to perform a distinguished action nok to signal their dissatisfaction during the interaction with the tested process. A process s passes the test T iff T does not perform the action nok when it interacts with s. A test T tests for a property phi in HML with recursion iff it is passed by exactly the states that satisfy phi. The paper gives an expressive completeness result offering a characterization of the collection of properties in HML with recursion that are testable in the above sense