4 research outputs found
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
A Coalgebraic Approach to Reducing Finitary Automata
Compact representations of automata are important for efficiency. In this
paper, we study methods to compute reduced automata, in which no two states
accept the same language. We do this for finitary automata (FA), an abstract
definition that encompasses probabilistic and weighted automata. Our procedure
makes use of Milius' locally finite fixpoint. We present a reduction algorithm
that instantiates to probabilistic and S-linear weighted automata (WA) for a
large class of semirings. Moreover, we propose a potential connection between
properness of a semiring and our provided reduction algorithm for WAs, paving
the way for future work in connecting the reduction of automata to the
properness of their associated coalgebras
A Completeness Theorem for Probabilistic Regular Expressions
We introduce Probabilistic Regular Expressions (PRE), a probabilistic
analogue of regular expressions denoting probabilistic languages in which every
word is assigned a probability of being generated. We present and prove the
completeness of an inference system for reasoning about probabilistic language
equivalence of PRE based on Salomaa's axiomatisation of Kleene Algebra