Substitution in Structural Operational Semantics and value-passing process calculi

Consider a process calculus that allows agents to communicate values. The structural operational semantics involves substitution of values for variables. Existing rule formats, such as the GSOS format, do not allow this kind of explicit substitution in the semantic rules. We investigate how to derive rule formats for languages with substitution, by using categorical logic to interpret the framework of the GSOS format in different categories. The categories in question are categories of ‘substitution actions’. 1 A simple language for value-passing To set the scene, fix a set of channel names, and consider a set V of value-expressions, that includes the channel names. A simple untyped value-passing process language, V-CCS, is given in Figure 1 (c.f. [8]). The precise value expressions of V are not important, but note that since V includes the (static) channel names, V-CCS is a very primitive applied π-calculus without restriction or name generation; c.f. [1]. For the sake of illustration, consider the set Vex of value expressions determined by the following grammar: v ::= n | v + v | (v, v) | π1(v) | π2(v) | c (n is a number, c is a channel name). We will always work with value expressions up-to the evident equations (2 + 3 = 5; π1(v, w) = v; etc.), rather than explicitly evaluating or normalizing them; this is to simplify the presentation. The following transitions are derivable in Vex-CCS. (c̄〈3〉.0) | (c(v).c̄〈2 + v〉.0) τ −→ 0 | c̄〈2 + 3〉.0 c̄〈5〉 −−→ 0 |0 P ::= 0 | P |P | v(a).P | v̄〈w〉.P (v, w ∈ V) c(a).P c〈v〉 −−−→ {/a}P (input) c̄〈v〉.P c̄〈v〉 −−−→ P (output) P l −→ P ′ P |Q l −→ P ′ |Q (parallel) P c〈v〉 −−−→ P ′ Q c̄〈v〉 −−−→ Q′ P |Q τ −→ P ′ |Q′ (communication) Fig. 1. A simple value-passing language, V-CCS. We elide symmetric versions of the rules (parallel) and (communication). The set V of values is assumed to contain a set of channel names, and the (input), (output) and (communication) rules carry a side condition that c is a channel name. 2 Value-passing systems and the GSOS rule format The GSOS rule format was introduced by [2]. A transition system specification is in the positive GSOS format if it is specified by rules of the following form: (xij l − → yj | 1 ≤ j ≤ m) o(x1, . . . , xn) L −→ t where the xi’s and yj ’s are all different, and the only variables appearing in t are the xi’s and yj ’s. There are various things one can say about a language semantics if it is specified in the GSOS format; most interesting is that bisimilarity (∼) is a congruence: if x1 ∼ x1, . . . , and xn ∼ xn, then o(x1, . . . , xn) ∼ o(x1, . . . , xn). Value-passing calculi are not GSOS (well, not classically). The language V-CCS is roughly in the shape of the GSOS format, but does not fit properly into the framework. The main problems arise in the (input) rule, which includes (i) a variable a that is binding in P , and (ii) a substitution {/a}P . Neither of these features are permitted in the GSOS format. A third problem is that it is natural to consider V-CCS terms with free valuevariables, in order to define a notion of congruence that respects input contexts. For this reason we recall a more elaborate notion of bisimulation; c.f. [9]: Definition 1. A bisimulation relation R on open V-CCS terms is an open bisimulation if it is closed under substitution: if P RQ then ({/a}P ) R ({/a}Q). Open bisimilarity is the greatest open bisimulation. Value-passing calculi are GSOS, categorically. In the remainder of this note, I sketch how the specifications of value-passing calculi can be seen as GSOS specifications, by working in a category of substitution actions, rather than the category of sets and functions. We use the techniques introduced in [10] for namepassing calculi. 3 GSOS in type theory To make things slightly more general and abstract, we reformulate the structure and requirements of the positive GSOS format in more fundamental terms. We only summarize the developments here; more details are in [10], where the more general tyft/tyxt format is considered. Syntax and signatures. Traditionally, a first order signature is a set of operators, together with an arity for each operator. It is helpful to allow the arities of operators to be arbitrary sets, and not just natural numbers. In the notation of type theory: we have a set O of operators and a function O → Set assigning an arity to each operator. This is a generalization, since individual natural numbers can be thought of as sets [n] = {1 . . . n}. The usual concepts of algebra and congruence can be defined for this notion of signature, and the free algebra of terms can be built. For a signature (O,A)