Kleene theorems for skew formal power series

We investigate the theory of skew (formal) power series introduced by Droste, Kuske [5, 6], if the basic semiring is a Conway semiring. This yields Kleene Theorems for skew power series, whose supports contain finite and infinite words. We then develop a theory of convergence in semirings of skew power series based on the discrete convergence. As an application this yields a Kleene Theorem proved already by Droste, Kuske [5]

Sound and complete axiomatizations of coalgebraic language equivalence

Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

Recognizable tree series with discounting

We consider weighted tree automata with discounting over commutative semirings. For their behaviors we establish a Kleene theorem and an MSO-logic characterization. We introduce also weighted Muller tree automata with discounting over the max-plus and the min-plus semirings, and we show their expressive equivalence with two fragments of weighted MSO-sentences

Wittgenstein’s Remarks on Mathematics, Turing and Computability

Typically, Wittgenstein is assumed to have been apathetic to the developments in computability theory through the 1930s. Wittgenstein’s disparaging remarks about Gödel’s incompleteness theorems, and mathematical logic in general, are well documented. It seems safe to assume the same would apply for Turing’s work. The chief aim of this thesis is to debunk this picture. I will show that: a) Wittgenstein read, understood and engaged with Turing’s proofs regarding the Entscheidungsproblem. b) Wittgenstein’s remarks on this topic are highly perceptive and have pedagogical value, shedding light on Turing’s work. c) Wittgenstein was highly supportive of Turing’s work as it manifested Wittgenstein’s prevailing approach to mathematics. d) Adopting a Wittgensteinian approach to Turing’s proofs enables us to answer several live problems in the modern literature on computability. Wittgenstein was notably resistant to Cantor’s diagonal proof regarding uncountability, being a finitist and extreme anti-platonist. He was interested, however, in the diagonal method. He made several remarks attempting to adapt the method to work in purely intensional, rule-governed terms. These are unclear and unsuccessful. Turing’s famous diagonal application realised this pursuit. Turing’s application draws conclusions from the diagonal procedure without having to posit infinite extensions. Wittgenstein saw this, and made a series of interesting remarks to that effect. He subsequently gave his own (successful) intensional diagonal proof, abstracting from Turing’s. He endorsed Turing’s proof and reframed it in terms of games to highlight certain features of rules and rule-following. I then turn to the Church-Turing thesis (CTT). I show how Wittgenstein endorsed the CTT, particularly Turing’s rendition of it. Finally, I show how adopting a family-resemblance approach to computability can answer several questions regarding the epistemological status of the CTT today

Kleene-Schützenberger and Büchi Theorems for Weighted Timed Automata

In 1994, Alur and Dill introduced timed automata as a simple mathematical model for modelling the behaviour of real-time systems. In this thesis, we extend timed automata with weights. More detailed, we equip both the states and transitions of a timed automaton with weights taken from an appropriate mathematical structure. The weight of a transition determines the weight for taking this transition, and the weight of a state determines the weight for letting time elapse in this state. Since the weight for staying in a state depends on time, this model, called weighted timed automata, has many interesting applications, for instance, in operations research and scheduling. We give characterizations for the behaviours of weighted timed automata in terms of rational expressions and logical formulas. These formalisms are useful for the specification of real-time systems with continuous resource consumption. We further investigate the relation between the behaviours of weighted timed automata and timed automata. Finally, we present important decidability results for weighted timed automata