Distributive Laws and Decidable Properties of SOS Specifications

Some formats of well-behaved operational specifications, correspond to natural transformations of certain types (for example, GSOS and coGSOS laws). These transformations have a common generalization: distributive laws of monads over comonads. We prove that this elegant theoretical generalization has limited practical benefits: it does not translate to any concrete rule format that would be complete for specifications that contain both GSOS and coGSOS rules. This is shown for the case of labeled transition systems and deterministic stream systems.Comment: In Proceedings EXPRESS/SOS 2014, arXiv:1408.127

Presenting Distributive Laws

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page

Presenting Morphisms of Distributive Laws

A format for well-behaved translations between structural operational specifications is derived from a notion of distributive law morphism, previously studied by Power and Watanabe

Abstract Syntax: Substitution and Binders

AbstractWe summarise Fiore et al's paper on variable substitution and binding, then axiomatise it. Generalising their use of the category F of finite sets to model untyped cartesian contexts, we let S be an arbitrary pseudo-monad on Cat and consider (S1)op: this generality includes linear contexts, affine contexts, and contexts for the Logic of Bunched Implications. Given a pseudo-distributive law of S over the (partial) pseudo-monad Tcoc−=[(−)op,Set] for free cocompletions, one can define a canonical substitution monoidal structure on the category [(S1)op,Set], generalising Fiore et al's substitution monoidal structure for cartesian contexts: this provides a natural substitution structure for the above examples. We give a concrete description of this substitution monoidal structure in full generality. We then give an axiomatic definition of a binding signature, then state and prove an initial algebra semantics theorem for binding signatures in full generality, once again extending the definitions and theorem of Fiore et al. A delicate extension of the research includes the category Pb(Injop,Set) studied by Gabbay and Pitts in their quite different analysis of binders, which we compare and contrast with that of Fiore et al

Stream differential equations: Specification formats and solution methods

Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been developed in many papers over the past two decades. In this paper we present a survey of the many results in this area. Our focus is on the classification of different formats of stream differential equations, their solution methods, and the classes of streams they can define. Moreover, we describe in detail the connection between the so-called syntactic solution method and abstract GSOS