Abstract. Distributive laws of a monad T over a functor F are categor-ical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specifica-tion 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 trans-formations. 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. 1

Language equivalence and inclusion can be checked coinductively by establishing a (bi)simulation on suitable deterministic automata. In this paper we present an enhancement of this technique called (bi)simulation-up-to. We give general conditions on language operations for which bisimulation-up-to is sound. These results are illustrated by a large number of examples, giving new proofs of clas-sical results such as Arden’s rule, and involving the regular operations of union, concatenation and Kleene star as well as language equations with complement and intersection, and shuffle (closure).

Abstract. We show that finite λ-bisimulations (closely related to bisim-ulations up to context) are sound and complete for finitely generated λ-bialgebras for distributive laws λ of a monad T on Set over an end-ofunctor F on Set, such that F preserves weak pullbacks and finitely generated T-algebras are closed under taking kernel pairs. This result is used to infer the decidability of weighted language equivalence when the underlying semiring is a subsemiring of an effectively presentable Noetherian semiring. These results are closely connected to [ÉM10] and [BMS13], concerned with respectively the decidability and axiomatiza-tion of weighted language equivalence w.r.t. Noetherian semirings. 1

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking prop-erties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinduc-tive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for simi-larity, and a new categorical treatment of weak bisimilarity. Categories and Subject Descriptors F.3 [Logics and meanings of programs]; F.4 [Mathematical logic and formal languages]