62 research outputs found
Asymptotic analysis of Skolem's exponential functions
Skolem (1956) studied the germs at infinity of the smallest class of real
valued functions on the positive real line containing the constant , the
identity function , and such that whenever and are in the set,
and are in the set. This set of germs is well ordered and Skolem
conjectured that its order type is epsilon-zero. Van den Dries and Levitz
(1984) computed the order type of the fragment below . Here we prove
that the set of asymptotic classes within any archimedean class of Skolem
functions has order type . As a consequence we obtain, for each
positive integer , an upper bound for the fragment below . We
deduce an epsilon-zero upper bound for the fragment below , improving
the previous epsilon-omega bound by Levitz (1978). A novel feature of our
approach is the use of Conway's surreal number for asymptotic calculations
A Survey on Fixed Divisors
In this article, we compile the work done by various mathematicians on the
topic of the fixed divisor of a polynomial. This article explains most of the
results concisely and is intended to be an exhaustive survey. We present the
results on fixed divisors in various algebraic settings as well as the
applications of fixed divisors to various algebraic and number theoretic
problems. The work is presented in an orderly fashion so as to start from the
simplest case of progressively leading up to the case of Dedekind
domains. We also ask a few open questions according to their context, which may
give impetus to the reader to work further in this direction. We describe
various bounds for fixed divisors as well as the connection of fixed divisors
with different notions in the ring of integer-valued polynomials. Finally, we
suggest how the generalization of the ring of integer-valued polynomials in the
case of the ring of matrices over (or Dedekind domain) could
lead to the generalization of fixed divisors in that setting.Comment: Accepted for publication in Confluentes Mathematic
Integration on surreal numbers
The thesis concerns the (class) structure No of Conwayās surreal numbers. The main concern is the behaviour on No of some of the classical functions of real analysis, and a definition of integral for such functions. In the main texts on No, most definitions and proofs are done by transfinite recursion and induction on the complexity of elements. In the thesis I consider a general scheme of definition for functions on No, generalising those for sum, product and exponential. If a function has such a definition, and can live in a Hardy field, and satisfies some auxiliary technical conditions, one can obtain in No a substantial analogue of real analysis for that function. One example is the signchange property, and this (applied to polynomials) gives an alternative treatment of the basic fact that No is real closed. I discuss the analogue for the exponential. Using these ideas one can define a generalisation of Riemann integration (the indefinite integral falling under the recursion scheme). The new integral is linear, monotone, and satisfies integration by parts. For some classical functions (e.g. polynomials) the integral yields the traditional formulae of analysis. There are, however, anomalies for the exponential function. But one can show that the logarithm, defined as the inverse of the exponential, is the integral of 1/x as usual. Acknowledgements I wish to express my gratitude to my supervisor Angus Macintyre for his constant support and assistance. Thanks to the examiners A. Maciocia and D. Richardson for their useful suggestions. Thanks also to my colleagues and my landlord for their willingness to tolerate my company. My gratitude goes to my family for their understanding and support. I wish to thank also the Engineering and Physical Sciences Research Council an
Interactive logical analysis of planning domains
Humans exhibit a significant ability to answer a wide range of questions about previously unencountered planning domains, and leverage this ability to construct āgeneral-purpose\u27\u27 solution plans for the domain.
The long term vision of this research is to automate this ability, constructing a system that utilizes reasoning to automatically verify claims about a planning domain. The system would use this ability to automatically construct and verify a generalized plan to solve any planning problem in the domain. The goal of this thesis is to start with baseline results from the interactive verification of claims about planning domains and develop the necessary knowledge representation and reasoning methods to progressively reduce the amount of human interaction required.
To achieve this goal, a representation of planning domains in a class-based logic syntax was developed. A novel proof assistant was then used to perform semi-automatic machine analysis of two benchmark planning domains: Blocksworld and Logistics. This analysis was organized around the interactive formal verification of state invariants and specifications of the state-change effects of handwritten recursive program-like generalized plans.
The human interaction required for these verifications was metered and qualitatively characterized. This characterization motivated several algorithmic changes to the proof assistant resulting in significant savings in the interactions required. A strict limit was enforced on the time spent by the base reasoner in response to user queries; interactions taking longer were studied to direct improvements to the inference engine\u27s efficiency. A complete account of these changes is provided
Some topics in set theory
This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A+ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A + and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.<p
CUMULATIVE HIERARCHIES AND COMPUTABILITY OVER UNIVERSES OF SETS
Various metamathematical investigations, beginning with Fraenkel’s historical proof of the independence of the axiom of choice, called for suitable deļ¬nitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been veriļ¬ed with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effectively handle the Ackermann’s hereditarily ļ¬nite sets, we illustrate a particularly signiļ¬cant case among those in which the entities which form a universe of sets can be algorithmically constructed and manipulated; hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramiļ¬es into the realms of theoretical computer science and algorithmics.Various metamathematical investigations, beginning with Fraenkel’shistorical proof of the independence of the axiom of choice, called forsuitable deļ¬nitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been veriļ¬ed with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effec-tively handle the Ackermann’s hereditarily ļ¬nite sets, we illustrate a particularly signiļ¬cant case among those in which the entities which forma universe of sets can be algorithmically constructed and manipulated;hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramiļ¬es into the realms of theoretical computer science and algorithmics
Interpretations in Trees with Countably Many Branches
AbstractāWe study the expressive power of logical interpreta-tions on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by āfinitaryā set-interpretations. Index TermsāComposition method, finite-set interpretations, infinite scattered trees, monadic second order logic. I
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