Rational streams coalgebraically

We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams: by finite dimensional linear systems; by finite stream circuits; by finite weighted stream automata; and by finite dimensional subsystems of the set of streams

QStream: A Suite of Streams

We present a simple tool in Haskell, QStream, implementing the technique of coinductive counting by making use of Haskell's built-in coinduction capabilities. We furthermore provide a number of useful tools for stream exploration, including a number of pretty print functions and integration with the Online Encyclopedia of Integer Sequences

Newton series, coinductively

We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shue, (iii) the inltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a nal coalgebra, we use coinduction to prove that a classical operator from dierence calculus in mathematics: the Newton transform, generalises (from innite sequences) to weighted lan- guages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into an inltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them

Context-free Coalgebras

In this article, we provide a coalgebraic account of parts of the mathematical theory underlying context-free languages. We characterize context-free languages, and power series and streams generalizing or corresponding to the context-free languages, by means of systems of behavioural differential equations; and prove a number of results, some of which are new, and some of which are new proofs of existing theorems, using the techniques of bisimulation and bisimulation up to linear combinations. Furthermore, we establish a link between automatic sequences and these systems of equations, allowing us to, given an automaton generating an automatic sequence, easily construct a system of behavioural differential equations yielding this sequence as a context-free stream

Newton series, coinductively: a comparative study of composition

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Proper Functors and Fixed Points for Finite Behaviour

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata