Conservative Extension in Structural Operational Semantics
Structural operational semantics (SOS)  provides a framework to give
an operational semantics to programming and specification languages. In
particular, because of its intuitive appeal and flexibility, SOS has found considerable application in the study of the semantics of concurrent processes. SOS generates a labelled transition system, whose states are the closed terms over an algebraic signature, and whose transitions are supplied with labels. The transitions between states are obtained inductively from a transition system specification (TSS), which consists of so-called transition rules of the form premises / conclusion . A typical example of a transition rule is:
stipulating that if t -> t' holds for closed terms t and t', then so does t||u -> t'||u for each closed term u. In general, validity (or invalidity) of the positive (or negative) premises of a transition rule, under a certain substitution implies validity of the conclusion of this rule under the same substitution. This column is an excerpt from , which gives an overview of recent results in the field of SOS, with an emphasis on existing formats for TSSs. Each of these formats comes equipped with a rich body of results that are guaranteed to hold for any process calculus whose TSS is within that format. Over and over again, process calculi such as CCS , CSP , and ACP  have been extended with new features, and the TSSs that provide the operational semantics for these process algebras were extended with transition rules to describe these features; see, e.g.  for a systematic approach. A question that arises naturally is whether or not the original and the extended TSS induce the same transitions t -> t' for closed terms t in the original domain. Usually it is desirable that an extension is operationally conservative, meaning that the provable transitions for an original term are the same both in the original and in the extended TSS. Groote and Vaandrager [34, Thm. 7.6] proposed syntactic restrictions on a TSS, which automatically yield that an extension of this TSS with transition
rules that contain fresh function symbols in their sources is operationally
conservative. Bol and Groote [18, 33] supplied this conservative extension
format with negative premises. Verhoef  showed that, under certain conditions,
a transition rule in the extension can be allowed to have an original
term as its source. D'Argenio and Verhoef [22, 23] formulated a generalization
in the context of inequational specifications. Fokkink and Verhoef
 relaxed the syntactic restrictions on the original TSS, and lifted the
operational conservative extension result to higher-order languages. This
column contains an exposition on existing conservative extension formats
for SOS, and their applications with respect to term rewriting systems and
completeness of axiomatizations.
Predicates in SOS semantics can be coded as binary relations . Moreover, negative premises can often be expressed positively using predicates . However, in the literature SOS definitions are often decorated with predicates and/or negative premises. For example, predicates are used to express matters like (un)successful termination, convergence, divergence , enabledness , maximal delay, and side conditions . Negative premises
are used to describe, e.g., deadlock detection , sequencing , priorities [7, 21], probabilistic behaviour , urgency , and various real  and discrete time [6, 35] settings. Since predicates and negative premises are so pervasive, and often lead to cleaner semantic descriptions for many features and constructs of interest, we deal explicitly with these notions. The organization of this column is as follows. Sect. 2 gives an overview of the basics of SOS. Sect. 3 presents syntactic constraints to ensure that an extension of a TSS is operationally conservative. Sect. 4 and 5 contain applications of conservative extension in equational specification and term rewriting. Sect. 6 nishes with some conclusions.
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