47 research outputs found
Some natural zero one laws for ordinals below Δ0
We are going to prove that every ordinal α with Δ_0â>âαââ„âÏ^Ï satisfies a natural zero one law in the following sense. For αâ<âΔ_0 let Nα be the number of occurences of Ï in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with Ï Ï ââ€âαâ<âΔ_0 and any sentence Ï in the language of linear orders the asymptotic density of Ï along α is an element of â{0,1}. We further show that for any such sentence Ï the asymptotic density along Δ_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below Ï^Ï
Phase transitions related to the pigeonhole principle
Since Paris introduced them in the late seventies (Paris1978), densities turned out to be useful for studying independence results. Motivated by their simplicity and surprising strength we investigate the combinatorial complexity of two such densities which are strongly related to the pigeonhole principle. The aim is to miniaturise Ramsey's Theorem for -tuples. The first principle uses an unlimited amount of colours, whereas the second has a fixed number of two colours. We show that these principles give rise to Ackermannian growth. After parameterising these statements with respect to a function f:N->N, we investigate for which functions f Ackermannian growth is still preserved
A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets
We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings
Complexity Bounds for Ordinal-Based Termination
`What more than its truth do we know if we have a proof of a theorem in a
given formal system?' We examine Kreisel's question in the particular context
of program termination proofs, with an eye to deriving complexity bounds on
program running times.
Our main tool for this are length function theorems, which provide complexity
bounds on the use of well quasi orders. We illustrate how to prove such
theorems in the simple yet until now untreated case of ordinals. We show how to
apply this new theorem to derive complexity bounds on programs when they are
proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability
Problems (RP 2014, 22-24 September 2014, Oxford
An order-theoretic characterization of the Howard-Bachmann-hierarchy
In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Î âÂč-comprehension
Ackermann and Goodstein go functorial
We present variants of Goodsteinâs theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits
Boundedness Theorems for Flowers and Sharps
Abstract. We show that the Sigma11
- and Sigma12
-boundedness theorems extend to
the category of continuous dilators. We then apply these results to conclude
the corresponding theorems for the category of sharps of real numbers, thus
establishing another connection between Proof Theory and Set Theory, and
extending work of Girard-Normann and Kechris-Woodin
Well-partial-orderings and the big Veblen number
In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with the classical tree-embeddability, but in this article we are interested in more general situations (transfinite addresses and well-partially ordered addresses). We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ÏΩΩ . Somewhat surprisingly, changing a little bit the well-partially ordered addresses (going from multisets to finite sequences), the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ÏΩΩΩ . Our results contribute to the research program (originally initiated by Diana Schmidt) on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings
AGI and the Knight-Darwin Law: why idealized AGI reproduction requires collaboration
Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: âNot without outside helpâ. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and so on forever: that any sequence of organisms (each one a child of the previous) must contain occasional multi-parent organisms, or must terminate. By proving that a certain measure (arguably an intelligence measure) decreases when an idealized parent AGI single-handedly creates a child AGI, we argue that a similar Law holds for AGIs