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

Algebraic decoder specification: coupling formal-language theory and statistical machine translation: Algebraic decoder specification: coupling formal-language theory and statistical machine translation

The specification of a decoder, i.e., a program that translates sentences from one natural language into another, is an intricate process, driven by the application and lacking a canonical methodology. The practical nature of decoder development inhibits the transfer of knowledge between theory and application, which is unfortunate because many contemporary decoders are in fact related to formal-language theory. This thesis proposes an algebraic framework where a decoder is specified by an expression built from a fixed set of operations. As yet, this framework accommodates contemporary syntax-based decoders, it spans two levels of abstraction, and, primarily, it encourages mutual stimulation between the theory of weighted tree automata and the application

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton

Weighted Tree Automata -- May it be a little more?

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata

Finite Sequentiality of Unambiguous Max-Plus Tree Automata

We show the decidability of the finite sequentiality problem for unambiguous max-plus tree automata. A max-plus tree automaton is called unambiguous if there is at most one accepting run on every tree. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton