BRICS RS-03-18 Klin & Soboci
´
nski: Syntactic Formats for Free: An Abstract Approach to Process Equivalence
BRICS
Basic Research in Computer Science
Syntactic Formats for Free:
An Abstract Approach to
Process Equivalence
Bartek Klin
Paweł Soboci
´
nski
BRICS Report Series RS-03-18
ISSN 0909-0878 April 2003
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c
2003, Bartek Klin & Paweł Soboci
´
nski.
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This document in subdirectory RS/03/18/
Syntactic Formats for Free:
An Abstract Approach to Process
Equivalence
Bartek Klin Pawe l Soboci´nski
BRICS
∗
University of Aarhus, Denmark
April, 2003
Abstract
A framework of Plotkin and Turi’s, originally aimed at provid-
ing an abstract notion of bisimulation, is modified to cover other
operational equivalences and preorders. Combined with bialge-
braic methods, it yields a technique for the derivation of syntactic
formats for transition system specifications which guarantee that
various operational preorders are precongruences. The technique
is applied to the trace preorder, the completed trace preorder and
the failures preorder. In the latter two cases, new syntactic for-
mats guaranteeing precongruence properties are introduced.
1 Introduction
Structural operational semantics [21, 2] is one of the most fundamental
frameworks for providing a precise interpretation of programming and
specification languages. Due to its flexibility and generality, it has gained
much popularity in the theory of concurrent processes. It is usually pre-
sented as a labelled transition system (LTS), in which states (sometimes
∗
Basic Research in Computer Science (www.brics.dk),
funded by the Danish National Research Foundation.
1
called processes) are closed terms over some syntactic signature, and
transitions are labelled with elements of some fixed set of actions. The
transition relation is in turn defined by a transition system specification,
i.e., a set of derivation rules.
Many operational equivalences and preorders have been defined on pro-
cesses. Among these are: bisimulation equivalence [19], simulation pre-
order, trace preorder, completed trace preorder, failures preorder [13, 23]
and many others (for a comprehensive list see [10]). In the case of pro-
cesses without internal actions, all of the above have been given modal
characterisations [10], obtained by considering appropriate subsets of the
Hennessy-Milner logic [12].
Reasoning about operational equivalences and preorders is significantly
easier when they are congruences (resp. precongruences). This facili-
tates compositional reasoning and full substitutivity. In general, opera-
tional equivalences are not necessarily congruences on processes defined
by operational rules. Similarly, operational preorders are not necessarily
precongruences. Proofs of such congruence results for given transition
system specifications can be quite demanding. This has been an acute
problem in the process calculus community and has led to new reasoning
approaches, for instance, the notion of barbed congruence [18].
One way to ensure congruential properties is to impose syntactic restric-
tions (called syntactic formats) on operational rules. Many such formats
have been developed. For bisimulation equivalence, the examples are: de
Simone format [27], GSOS [8], and ntyft/ntyxt [11], each of these gener-
alising the previous one. For trace equivalence, examples include [31, 5],
while several versions of decorated trace preorders have been provided
with formats in [6]. For an overview of the subject see [2]. Another
approach which generates LTS on which bisimulation is a congruence is
the smallest-contexts-as-labels approach [26, 17, 25].
The search for an abstract theory of processes, bisimulation and ’well-
behaved’ operational semantics has led to development of final coalge-
bra semantics [24], and — later — of bialgebraic semantics [29, 30] of
processes. In these frameworks, the notion of a transition system is
parametrised by a notion of behaviour. Bisimulation is modelled ab-
stractly as a span of coalgebra morphisms. The abstract notion spe-
cialises to the classical one, indeed, the class of finitely branching la-
belled transition systems is in 1-1 correspondence with the coalgebras of
the functor P
f
(A ×−), and to give a (classical) bisimulation relation is
2
to give a span of coalgebra morphisms for this functor [3, 24]. Another
abstract approach to bisimulation is via open maps [15].
In [29, 30] it was shown how to define operational rules on an abstract
level. For abstract transition system specifications defined in this way,
bisimulation equivalence (defined abstractly, using spans of coalgebra
morphisms) is guaranteed to be a congruence.
At the core of this so-called abstract GSOS is the modelling of a transition
system specification as a natural transformation
λ :Σ(id×B)→BT
where Σ is the syntactic endofunctor, T is the monad freely generated
from Σ, and B is some behaviour endofunctor. In the special case of the
behaviour endofunctor P
f
(A×−), the abstract operational rules specialise
to GSOS rules.
The abstract framework which defines bisimulation as a span of coalgebra
morphisms is not sufficient for certain purposes [22] and in particular one
runs into problems when working with complete partial orders. Recently,
another abstract notion of bisimulation, based on topologies (or complete
boolean algebras) of tests, has been proposed [20, 28]. Again, for the
familiar process behaviour the novel abstract notion is equivalent to the
classical one.
In this paper we show that the latter abstract definition of bisimulation
can in fact be modified in a structured manner, to yield other known
operational equivalences and preorders. We illustrate this approach on
trace preorder, completed trace preorder and failures preorder (and re-
spective equivalences). This constitutes another systematic approach to
various operational preorders and equivalences, such as those based on
testing scenarios and modal logics [10], as well as quantales [1].
Although the framework is general, in this paper we shall concentrate on
the category of sets and functions, Set. We define the test-suite fibration
with total category Set
∗
having as objects pairs consisting of a set X
and a test suite (a subset of PX)overX. We define a way of lifting the
abstract-GSOS framework to Set
∗
by describing how to lift the syntax
functor Σ and the behaviour functor B. By changing how B lifts to Set
∗
we alter the specialisation preorder of certain test suites in Set
∗
.Inpar-
ticular, taking particular liftings which strongly resemble fragments of
the Hennessy-Milner logic [12] causes the specialisation preorder to vary
between known operational preorders. The abstract framework guaran-
3