Degrees of finite-state transformability

The upper semilattice of degrees of transformability by finite-state automata is defined analogously to the upper semilattice of degrees of recursive unsolvability (which arises from transformability by Turing machines). Two infinite sequences from a finite alphabet are considered equivalent if each can be transformed into the other by a finite-state automaton, perhaps after finite initial segments (not necessarily of the same length) are deleted from each. We require the output sequence to be generated at the same rate as the input, with exactly one output character for each input character. If such a transformation is possible in only one direction, an order relation holds between the equivalence classes.We show that this partially ordered set does indeed form an upper semilattice, exhibit the (unique) minimal class, and prove there is no maximal class. In the course of the proof of the last assertion, the notion of a complete sequence, a sequence in which every block of the alphabet occurs, is introduced and shown to be significant. The richness of the partial ordering is shown by two contrasting examples: We exhibit one section of it in which the partial ordering is dense, and, on the other hand, we exhibit two classes [x] > [z] having no class properly between them

An unpredictability approach to finite-state randomness

AbstractThis paper investigates the concept of randomness within a complexity theoretic framework. We consider an unpredictability approach for defining randomness in which the preditions are carried out by finite-state automata. Our model of a finite-state predicting machine (FPM) reads a binary sequence from left to right and depending on the machine's current state will generate, at each point, one of three possible values: 0, 1, or #. A response of 0 or 1 is to be taken as the FPMs prediction of the next input. A # means no prediction of the next input is made. We say that an infinite binary sequence appears random to an FPM if no more than half of the predictions made of the sequence's terms by the FPM are correct. The main result of this paper is to establish the equivalence of the sequences which appear random to all FPMs and the ∞-distributed sequences, where a binary sequence is called ∞-distributed if every string of length k occurs in the sequence with frequency 2−k, for all positive integers k. We also explicitly construct machines that exhibit success in predicting the sequences which are not ∞-distributed. Finally, we show that for any given ∞-distributed sequence, all infinite subsequences which are constructible from FPMs are also ∞-distributed

Losing Sight of the Forest for the Ψ: Beyond the Wavefunction Hegemony

Traditionally Ψ is used to stand in for both the mathematical wavefunction (the representation) and the quantum state (the thing in the world). This elision has been elevated to a metaphysical thesis by advocates of the view known as wavefunction realism. My aim in this paper is to challenge the hegemony of the wavefunction by calling attention to a little-known formulation of quantum theory that does not make use of the wavefunction in representing the quantum state. This approach, called Lagrangian quantum hydrodynamics (LQH), is not an approximation scheme, but rather a full alternative formulation of quantum theory. I argue that a careful consideration of alternative formalisms is an essential part of any realist project that attempts to read the ontology of a theory off of the mathematical formalism. In particular, I show that LQH undercuts the central presumption of wavefunction realism and falsifies the claim that one must represent the many-body quantum state as living in a 3n-dimensional configuration space. I conclude by briefly sketching three different realist approaches one could take toward LQH, and argue that both models of the quantum state should be admitted. When exploring quantum realism, regaining sight of the proverbial forest of quantum representations beyond the Ψ is just the first step

Multipartite quantum correlations: symplectic and algebraic geometry approach

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as identification of states with `the same correlations properties' i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question `what can be transformed into what \textit{via} available means?'. Exactly such an interpretations, i.e, in terms of mutual transformability can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa

An Emergent Economics of Ecosystem Management

Economics is an evolving and emerging field of study, so is the management of ecosystems. As such, this paper delineates the co-evolution of economic evaluation that reflects the various recognized ecosystem management approaches of anticipative, adaptive and capacitive ecosystem management. Each management approach is critiqued and from this theoretical analysis an emergent approach for the management of ecosystem is put forward, which accordingly suggests an alternative methodological approach for economic evaluations.Complexity, creativity, economic evaluation, ecosystem management, evolution, open systems, rationality, Resource /Energy Economics and Policy,