Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning

We introduce a simple analysis of the structural complexity of infinite-memory processes built from random samples of stationary, ergodic finite-memory component processes. Such processes are familiar from the well known multi-arm Bandit problem. We contrast our analysis with computation-theoretic and statistical inference approaches to understanding their complexity. The result is an alternative view of the relationship between predictability, complexity, and learning that highlights the distinct ways in which informational and correlational divergences arise in complex ergodic and nonergodic processes. We draw out consequences for the resource divergences that delineate the structural hierarchy of ergodic processes and for processes that are themselves hierarchical.Comment: 8 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/soi.pd

Categoricity, Open-Ended Schemas and Peano Arithmetic

One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories

Measuring Learning Complexity with Criteria Epitomizers

In prior papers, beginning with the seminal work by Freivalds et al. 1995, the notion of intrinsic complexity is used to analyze the learning complexity of sets of functions in a Gold-style learning setting. Herein are pointed out some weaknesses of this notion. Offered is an alternative based on epitomizing sets of functions -- sets, which are learnable under a given learning criterion, but not under other criteria which are not at least as powerful. To capture the idea of epitomizing sets, new reducibility notions are given based on robust learning (closure of learning under certain classes of operators). Various degrees of epitomizing sets are characterized as the sets complete with respect to corresponding reducibility notions! These characterizations also provide an easy method for showing sets to be epitomizers, and they are, then, employed to prove several sets to be epitomizing. Furthermore, a scheme is provided to generate easily very strong epitomizers for a multitude of learning criteria. These strong epitomizers are so-called self-learning sets, previously applied by Case & Koetzing, 2010. These strong epitomizers can be generated and employed in a myriad of settings to witness the strict separation in learning power between the criteria so epitomized and other not as powerful criteria

Properties grounded in identity

The topic of this dissertation are essential properties. The aim is to give an explication of the concept of essential properties in terms of the concept of metaphysical grounding. Along the way, the author proves several new results about formal theories of metaphysical grounding and develops a new hyperintensional theory of properties. Chapter 1 is the introduction of the thesis in which the author motivates the problem of explicating the concept of essential properties and gives adequacy criteria for a successful explication tracing back to Carnap. The author discusses the orthodox explication of essential properties in terms of metaphysical necessity and Fine's objections to it. He goes on to develop a new ground-theoretic explication of essential properties in an informal way, where the basic idea is that a property is essential to an object if and only if the property is metaphysically grounded in the identity or haecceity of the object. The author argues informally that the new explication provides natural solutions to the problems raised by Fine and make it the goal for the rest of the dissertation to make the account formally precise. Chapter 2 focuses on a axiomatic theories of metaphysical grounding. In particular, the author develops a new formal approach to the relation of partial ground, i.e. the relation of one truth holding partially in virtue of another. The main novelty of the chapter is the use of a grounding predicate rather than an operator to formalize statements of (partial) ground. The author develops the new theory in formal detail as a first-order theory, proves its consistency, and shows that it's a conservative extension of the well-known theory of positive truth. Moreover, the author constructs a concrete model of the theory. Then, the author extends the framework with typed truth predicates, which allow us to make statements about the grounds and truth of statements about the truth of other sentences. Also this theory the author proves consistent and a conservative extension of the ramified theory of positive truth. A model construction extending the construction of the base theory is also given. Ultimately, the author discards the theory for the purpose of the dissertation, because of technical problems that arise when we try to develop a satisfactory explication of essential properties in the framework. The author leaves further development of the framework for future work and argues that further investigating could lead to interesting and fruitful discussion between formal theorists of truth and metaphysical theorists of grounding. Chapter 3 develops a logic of iterated ground, i.e. relations of ground between statements of ground, using the operator approach. The author first discusses certain conceptual distinctions in the context of metaphysical ground in general and iterated ground in particular. The author argues that different conceptions of iterated ground lead to different logics of iterated ground. He goes on to develop the logic of iterated ground based on the idea that if one truth is grounded in others, then it's these grounds that ground the statement of ground itself. This view traces back to remarks by de Rosset and Litland. The logic is developed in formal detail, both syntactically and semantically, and its formal properties are investigated. To conclude the chapter, the author discusses certain problems that arise when we're trying to prove a completeness result for the logic and sketches a way around them. In chapter 4, the author develops a new hyperintensional theory of properties, which can distinguish in natural, semantic terms between necessarily co-extensional but intuitively distinct properties. The theory is based on the idea that we can individuate properties by means of what the author calls "exemplification criteria" of an object for a property, roughly the states of affairs that if they obtain explain why the object has the property. The author develops this theory both formally and informally and discusses in detail how it achieves a natural distinction between necessarily co-extensional but intuitively distinct properties. Chapter 5 is the conclusion, where the author brings the results of chapter 3 and 4 to bear on the informal explication of essential properties in terms of metaphysical ground from the introduction. The author argues that together the iterated logic of ground from chapter 3 and the hyperintensional property theory of chapter 4 allow us to make the explication formally precise in a way that satisfies the adequacy criteria for a successful explication laid out by Carnap. The author concludes with a brief discussion of possible ways of extending the results of the dissertation in various ways

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

The Integration of Connectionism and First-Order Knowledge Representation and Reasoning as a Challenge for Artificial Intelligence

Intelligent systems based on first-order logic on the one hand, and on artificial neural networks (also called connectionist systems) on the other, differ substantially. It would be very desirable to combine the robust neural networking machinery with symbolic knowledge representation and reasoning paradigms like logic programming in such a way that the strengths of either paradigm will be retained. Current state-of-the-art research, however, fails by far to achieve this ultimate goal. As one of the main obstacles to be overcome we perceive the question how symbolic knowledge can be encoded by means of connectionist systems: Satisfactory answers to this will naturally lead the way to knowledge extraction algorithms and to integrated neural-symbolic systems.Comment: In Proceedings of INFORMATION'2004, Tokyo, Japan, to appear. 12 page

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs