Computable de Finetti measures

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor corrections. To appear in Annals of Pure and Applied Logic. Extended abstract appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23

Nuclear and Trace Ideals in Tensored *-Categories

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored *-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations which was the original motivating example. Finally, we extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. For a given symmetric monoidal category, it is not generally the case that arbitrary endomorphisms can be assigned a trace. However, we can find ideals in the category on which a trace can be defined satisfying equations analogous to those of Joyal, Street and Verity. We establish a close correspondence between nuclear ideals and trace ideals in a tensored *-category, suggested by the correspondence between Hilbert-Schmidt operators and trace operators on a Hilbert space.Comment: 43 pages, Revised versio

Łukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems

A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms

Approximate reasoning for real-time probabilistic processes

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.Comment: Preliminary version appeared in QEST 0

Probabilistic Constraint Logic Programming

This paper addresses two central problems for probabilistic processing models: parameter estimation from incomplete data and efficient retrieval of most probable analyses. These questions have been answered satisfactorily only for probabilistic regular and context-free models. We address these problems for a more expressive probabilistic constraint logic programming model. We present a log-linear probability model for probabilistic constraint logic programming. On top of this model we define an algorithm to estimate the parameters and to select the properties of log-linear models from incomplete data. This algorithm is an extension of the improved iterative scaling algorithm of Della-Pietra, Della-Pietra, and Lafferty (1995). Our algorithm applies to log-linear models in general and is accompanied with suitable approximation methods when applied to large data spaces. Furthermore, we present an approach for searching for most probable analyses of the probabilistic constraint logic programming model. This method can be applied to the ambiguity resolution problem in natural language processing applications.Comment: 35 pages, uses sfbart.cl

Continuous-time histories: observables, probabilities, phase space structure and the classical limit

In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the analogue of the Weyl group) and the implementation of a histories Wigner-Weyl transform. The latter enables us to write quantum theory solely in terms of phase space quantities. These results enable the implementation of an algorithm for identifying the classical (stochastic) limit of a general quantum system.Comment: 46 pages, latex; in this new version typographical errors have been removed and the presentation has been made cleare

Generating Interpretable Fuzzy Controllers using Particle Swarm Optimization and Genetic Programming

Autonomously training interpretable control strategies, called policies, using pre-existing plant trajectory data is of great interest in industrial applications. Fuzzy controllers have been used in industry for decades as interpretable and efficient system controllers. In this study, we introduce a fuzzy genetic programming (GP) approach called fuzzy GP reinforcement learning (FGPRL) that can select the relevant state features, determine the size of the required fuzzy rule set, and automatically adjust all the controller parameters simultaneously. Each GP individual's fitness is computed using model-based batch reinforcement learning (RL), which first trains a model using available system samples and subsequently performs Monte Carlo rollouts to predict each policy candidate's performance. We compare FGPRL to an extended version of a related method called fuzzy particle swarm reinforcement learning (FPSRL), which uses swarm intelligence to tune the fuzzy policy parameters. Experiments using an industrial benchmark show that FGPRL is able to autonomously learn interpretable fuzzy policies with high control performance.Comment: Accepted at Genetic and Evolutionary Computation Conference 2018 (GECCO '18