### Generalised coinduction

FALK BARTELS
2003 Mathematical Structures in Computer Science
We introduce the λ-coiteration schema for a distributive law λ of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final F-coalgebra, generalising the basic coiteration schema as given by finality. The duals of primitive recursion and course-of-value iteration, which are known extensions of coiteration, arise as instances of our framework. One can furthermore obtain schemata justifying recursive specifications that
more » ... volve operators such as addition of power series, regular operators on languages, or parallel and sequential composition of processes. Next, the same type of distributive law λ is used to generalise coinductive proof techniques. To this end, we introduce the notion of a λ-bisimulation relation. It specialises to what could be called bisimulation up-to-equality or bisimulation upto-context for contexts built from operators of the type mentioned above. We state that every such relation is contained in some larger conventional bisimulation and demonstrate that this principle leads to simpler bisimilarity proofs using less complex relations. Open access under CC BY-NC-ND license. Bartels derived variants. The main tool for proving states equal is that of an Fbisimulation [AM89], a categorical generalisation of notions of bisimulation used for different concrete systems. But these basic principles are often too rigid to nicely cover given examples. Many functions into the carrier of a final coalgebra can be shown not to be coiterative and many statements about behavioural equivalence require bisimulations which are difficult to exhibit and check. Therefore several extensions for particular settings are studied. Examples of a more general type are the definition principles that arise as the duals of primitive recursion and course-of-value iteration. The specification of processes by systems of recursive equations forms another one. For proof purposes one often uses relations which instead of being bisimulations themselves satisfy sufficient conditions for being contained in some bisimulation. These are often called bisimulations up-to (see e.g. Milner [Mil89]). Sangiorgi [San98] has introduced a framework for deriving such principles in the context of labelled transition systems. Categorical formulations of the schemata that form the duals of primitive recursion and course-of-value iteration are known, see e.g. Vene and Uustalu [UV99]. Following common practice, we will call the first one primitive corecursion. A step towards a more general description of extended principles on this level has been made by Lenisa in the course of her comparison of set-theoretic and coalgebraic (categorical) formulations of coinduction [Len99a] . She has demonstrated that her framework captures the arrows obtainable by primitive corecursion and the same can be shown for the dual of course-of-value iteration, but in neither case does her theory directly yield the universal characterisation for these morphisms mentioned above. By modifying Lenisa's approach we arrived at a categorical description of generalised coinductive definition and proof principles based on such a universal characterisation which specialises directly to both principles from above. Simply speaking, the conventional coiteration schema assigns infinite behaviours to the states of a set X by specifying for each element a direct observation and successor states. Since these successors are taken from X again, the same specification applied to them reveals the second layer of the behaviour, and so on. A different approach is taken by the λ-coiteration schema that we introduce, which is parameterised by another functor T and a distributive law λ of T over F: it allows the successor states to be taken from TX instead of X, which may increase the expressiveness of the format in case the former can be regarded as being "richer" than the latter. For the observations to be continued with these successors, the distributive law λ lifts the specification for X to TX. In the main theorems of this paper we show for two additional assumptions that each of them is sufficient for the λ-coiteration schema to uniquely define arrows into the final F-coalgebra. Making similar use of T and λ, we define the notion of a λ-bisimulation and show that the same conditions as for the validity of the λ-coiteration schema