TITLE: Asymptotic Fixed Points, Part II

ABSTRACT: Continuing the earlier seminar, I will give nonstandard proofs for

one or more (depending on time) results like the

one below (which consolidates and generalizes a number of recent results in

the area).

Suppose

$(X,d)$ is a complete metric space,

$T:Xto X$ is continuous,

$phi, phi_n:[0,infty)to[0,infty)$, and

$phi_n$ converges to $phi$ uniformly on the range of $d$,

$phi$ is semicontinuous and satisfies $phi(s)~~0$,~~

$d(T^nx,T^ny)lephi_n(d(x,y))$ for all $x,yin X$ and $ninmathbb N$.

Moreover, suppose that for any $x,y$, $phi(t)lesssim t$ on infinite

elements of

$${^*d(T^nx,T^my) : m, n text{ hyperintegers}}.$$

Then $T$ has a unique fixed point $x_infty$, and for every $xin X, limlimits_{ntoinfty}T^n(x)=x_infty$. Moreover, this convergence is

uniform on bounded subsets of $X$.

The Sporadic Logic Seminar returns this week at a new place and time

(Fridays, 2:30, K404). This week Mushfeq Khan will speak:

Title: “The Homogeneity Conjecture”

Abstract: It is often said that the theorems and methods of recursion theory

relativize. One might go as far as to say that much of its analytical power

derives from this feature. However, this power is accompanied by definite

drawbacks: There are important examples of theorems and open questions

whose statements are non-relativizing, i.e., they have been shown to be

true relative to some oracles, and false relative to others. It follows

that these questions cannot be settled purely through relativizing methods.

A famous example of such a negative result is Baker, Gill, and Solovay’s

theorem on the P vs. NP question.

The observation that techniques based on diagonalization, effective

numbering, and simulation relativize led some recursion theorists (notably

Hartley Rogers, Jr) to formulate what became known as the “Homogeneity

Conjecture”. It said that for any Turing degree d, the partial order of

degrees that are above d is isomorphic to the entire partial order of the

Turing degrees. In 1979, Richard Shore refuted it in an elegant, one-page

article which will be the subject of this talk.

Title: Some applications of logic to additive number theory

Abstract: I will review the Loeb measure construction; I will

assume some exposure to nonstandard analysis, or at least 1st order logic,

comparable to the review I gave last semester in my seminars on fixed

points. Time permitting I will give the Loeb-measure proof of Szemeredi’s

Theorem.

Logic seminar: David Ross

Title: Some applications of logic to additive number theory (cont.)

Room: Keller 404.

Abstract:

I will continue with some examples of results about sets of positive upper Banach density proved using Loeb measures.

The Logic Seminar will meet again this Friday. The speaker will be Bjørn Kjos-Hanssen.

Title:

Superposition as memory: unlocking quantum automatic complexity

Time:

Friday March 17, 2:30-3:20

Place: Keller 404 (Note: this might change)

Abstract:

Imagine a lock with two states, “locked” and “unlocked”, which may be manipulated using two operations, called 0 and 1. Moreover, the only way to (with certainty) unlock using four operations is to do them in the sequence 0011, i.e., $0^n1^n$ where $n=2$. In this scenario one might think that the lock needs to be in certain further states after each operation, so that there is some memory of what has been done so far. Here we show that this memory can be entirely encoded in superpositions of the two basic states “locked” and “unlocked”, where, as dictated by quantum mechanics, the operations are given by unitary matrices. Moreover, we show using the Jordan–Schur lemma that a similar lock is not possible for $n=60$.

The Logic Seminar will meet again this Friday, usual place: Keller Hall 404. The speaker will be Achilles Beros.

Title: Teachers, Learners and Oracles

Abstract: When identifying r.e. sets from enumeration, a teacher is a

computational aide that pre-processes the data and only passes the

“useful” examples to the learner. Access to a teacher does not affect

the learnability of a family of r.e. sets, but it can affect the speed

with which learning is accomplished. Another computational aide is the

membership oracle. We consider four different forms of polynomial

bounds on learning and compare the performance of learners equipped with

teachers and learners equipped with oracles. We find that in most cases

neither strategy is uniformly superior. In this talk I will survey the

results and show a proof that utilizes a strategy analogous to integrity

checks in TCP (Transmission Control Protocol). The paper presented is

joint work with Colin de la Higuera.