Nested Semantics over Finite Trees are Equationally Hard

This paper studies nested simulation and nested trace semantics over the language BCCSP, a basic formalism to express finite process behaviour. It is shown that none of these semantics affords finite (in)equational axiomatizations over BCCSP. In particular, for each of the nested semantics studied in this paper, the collection of sound, closed (in)equations over a singleton action set is not finitely based

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)

2-Nested Simulation is not Finitely Equationally Axiomatizable

2-nested simulation was introduced by Groote and Vaandrager [10] as the coarsest equivalence included in completed trace equivalence for which the tyft/tyxt format is a congruence format. In the lineartime-branching time spectrum of van Glabbeek [8], 2-nested simulationis one of the few equivalences for which no finite equational axiomatization is presented. In this paper we prove that such an axiomatizationdoes not exist for 2-nested simulation.Keywords: Concurrency, process algebra, basic CCS, 2-nested simulation, equational logic, complete axiomatizations

On the Axiomatisability of Parallel Composition

This paper studies the existence of finite equational axiomatisations of the interleaving parallel composition operator modulo the behavioural equivalences in van Glabbeek's linear time-branching time spectrum. In the setting of the process algebra BCCSP over a finite set of actions, we provide finite, ground-complete axiomatisations for various simulation and (decorated) trace semantics. We also show that no congruence over BCCSP that includes bisimilarity and is included in possible futures equivalence has a finite, ground-complete axiomatisation; this negative result applies to all the nested trace and nested simulation semantics

A Linear-Time Branching-Time Spectrum for Behavioral Specification Theories

We propose behavioral specification theories for most equivalences in the linear-time--branching-time spectrum. Almost all previous work on specification theories focuses on bisimilarity, but there is a clear interest in specification theories for other preorders and equivalences. We show that specification theories for preorders cannot exist and develop a general scheme which allows us to define behavioral specification theories, based on disjunctive modal transition systems, for most equivalences in the linear-time--branching-time spectrum

A Fast Compiler for NetKAT

High-level programming languages play a key role in a growing number of networking platforms, streamlining application development and enabling precise formal reasoning about network behavior. Unfortunately, current compilers only handle "local" programs that specify behavior in terms of hop-by-hop forwarding behavior, or modest extensions such as simple paths. To encode richer "global" behaviors, programmers must add extra state -- something that is tricky to get right and makes programs harder to write and maintain. Making matters worse, existing compilers can take tens of minutes to generate the forwarding state for the network, even on relatively small inputs. This forces programmers to waste time working around performance issues or even revert to using hardware-level APIs. This paper presents a new compiler for the NetKAT language that handles rich features including regular paths and virtual networks, and yet is several orders of magnitude faster than previous compilers. The compiler uses symbolic automata to calculate the extra state needed to implement "global" programs, and an intermediate representation based on binary decision diagrams to dramatically improve performance. We describe the design and implementation of three essential compiler stages: from virtual programs (which specify behavior in terms of virtual topologies) to global programs (which specify network-wide behavior in terms of physical topologies), from global programs to local programs (which specify behavior in terms of single-switch behavior), and from local programs to hardware-level forwarding tables. We present results from experiments on real-world benchmarks that quantify performance in terms of compilation time and forwarding table size