Rule Formats for Distributivity

Abstract

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Meta SOS - A Maude Based SOS Meta-Theory Framework

“…There are, naturally, many ways to improve and extend the tool. Besides checking for the commutativity format, there are many other formats to check for: determinism and idempotence [1,8], zero and unit elements [6], associativity [17], and distributivity [5]. Adapting PREG Axiomatizer and adding it as a component to Meta SOS as presented in Section 3.5 would also be of value due to its different approach to generating axiomatizations, and because it includes a GSOS format checker.…”

SOS rule formats for idempotent terms and idempotent unary operators

“…The above proposition is a typical example of a result in the meta-theory of SOS: it states that if the rules in a TSS satisfy some syntactic constraint, then some semantic result is guaranteed to hold. In the remainder of this paper, following the work presented in, e.g., [3,5,6,11,21,32], we shall present a rule format ensuring that certain unary operations are idempotent. This rule format will rely on one yielding that terms of a certain form are idempotent for some binary operator.…”

“…Note that the target of the conclusion of the rule for replication, namely x ||!x, and the above term can be proved equal using associativity of ||, the version of axiom (2) for replication and parallel composition, and the version of axiom (5) for replication, as follows:…”

“…In this approach, one gives syntactic templates for the inference rules used in defining the operational semantics for certain operators that guarantee the validity of the desired laws, thus obviating the need for an a posteriori verification. (See [3,5,6,11,21,32] for examples of rule formats for algebraic properties in the literature on SOS.) The definition of SOS rule formats is based on finding a reasonably good trade-off between generality and ease of application.…”

Algebraic Meta-Theory of Processes with Data

“…For formalisms with a Structural Operational Semantics (SOS), there exists a rich literature on metatheorems guaranteeing key algebraic properties (commutativity [32], associativity [17], zero and unit elements [4], idempotence [1], and distributivity [3]) by means of restrictions on the syntactic shape of the transition rules. At the same time, for GSOS [13], a restricted yet expressive form of SOS specifications, one can obtain a sound and ground-complete axiomatization modulo strong bisimilarity [2].…”