Abstract

We describe a correspondence between (i) rule-based inductive definitions in the Positive GSOS format, i.e. without negative premises, and (ii) distributive laws in the spirit of Turi and Plotkin (LICS’97), that are suitably monotone. This result can be understood as an isomorphism of lattices, in which the prime elements are the individual rules. We fix a signature, i.e. a set S of operators, each op ∈ S having an arity, ar(op) ∈ N. We also fix a set L of labels. We are concerned with labelled transition relations over the terms of the signature. For an example, we recall a fragment of Milner’s CCS. Fix a set A of actions. There is a binary operation for parallel composition (written x|y), a constant for deadlock, and two unary operations for every action a ∈ A, for prefix and co-action prefix. The set of labels contains actions, co-actions, and silent steps: L = A + A + 1. x a − → x ′ y ā − → y ′ x|y τ − → x ′ |y ′ A labelled transition relation over CCS terms is defined by a transition system specification (TSS). This is a set of rules, including the one in the box on the right. Background on the Positive GSOS rule format li j − → yi j | i ≤ ar(op), j ∈ Ji} {xi lik

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