Behavioural distances provide a robust alternative to notions of equivalence such as bisimilarity in the context of probabilistic transition systems. They can be defined as least fixed points, whose universal property allows us to exhibit upper bounds on the distance between states, showing them to be at most some distance apart. In this paper, we instead consider the problem of bounding distances from below, showing states to be at least some distance apart. Contrary to upper bounds, it is possible to reason about lower bounds inductively. We exploit this by giving an inductive derivation system for lower bounds on an existing definition of behavioural distance for labelled Markov chains. This is inspired by recent work on apartness as an inductive counterpart to bisimilarity. Proofs in our system will be shown to closely match the behavioural distance by soundness and (approximate) completeness results. We further provide a constructive correspondence between our derivation system and formulas in a modal logic with quantitative semantics. This logic was used in recent work of Rady and van Breugel to construct evidence for lower bounds on behavioural distances. Our constructions provide smaller witnessing formulas in many examples.

An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has no proper subcoalgebra; and second, a coalgebra is reachable if it arises as the union of an iterative computation of successor states, starting from the initial state. In the current paper, we present corresponding universal properties and iterative constructions for trees. The universal property captures when a coalgebra is a tree, namely, when it has no proper tree unravelling. The iterative construction unravels an arbitrary coalgebra to a tree. We show that this yields the expected notion of tree for a variety of standard examples. We obtain our characterization of trees by first generalizing the previous theory of reachable coalgebras. Surprisingly, both the universal property and the iterative construction for trees arise as an instance of this generalized notion of reachability.

Abstract In the theory of coalgebras, distributive laws give a general perspective on determinisation and other automata constructions. This perspective has recently been extended to include so-called weak distributive laws, covering several constructions on state-based systems that are not captured by regular distributive laws, such as the construction of a belief-state transformer from a probabilistic automaton, and ultrafilter extensions of Kripke frames. In this paper we first observe that weak distributive laws give rise to the more general notion of what we call an invertible step: a pair of natural transformations that allows to move coalgebras along an adjunction. Our main result is that part of the construction induced by an invertible step preserves and reflects bisimilarity. This covers results that have previously been shown by hand for the instances of ultrafilter extensions and belief-state transformers.