Rick Statman: Backus FP is Turing complete

Mathematical logic seminar – Feb 28 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Rick Statman
Department of Mathematical Sciences
CMU

Title:     Backus FP is Turing complete

Abstract:

Cartesian monoids are rather simple algebraic structures of which you know many examples. They also travel under many assumed names such as Cantor algebras, Jonsson-Tarski algebras, and Freyd-Heller monoids. John Backus’s FP is just the theory of Cartesian monoids together with fixed points for all Cartesian monoid polynomials.

In his 1977 Turing Award address, John Backus introduced the model of functional programming called “FP”. FP is a descendant of the Herbrand-Godel notion of recursive definablity and the ancestor of the programming language Haskell. One reason that FP is attractive is that it provides “an algebra of functional programs”. However, Backus did not believe that basic FP was powerful enough;

“FP systems have a number of limitations….. If the set of primitive functions and functional forms is weak, it may not be able to express every computable function”. John Backus, 1977 ACM Turing award lecture.

and he moved on to stronger systems. It turns out that, in this respect, Backus was mistaken. Here we shall show that FP can compute every partial recursive function.

Deirdre Haskell: Using model theory to find upper bounds on VC density

Mathematical logic seminar – Feb 21 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Deirdre Haskell
Department of Mathematics and Statistics
McMaster University

Title:     Using model theory to find upper bounds on VC density

Abstract:

The VC dimension of a collection of sets is a concept used in probability and learning theory. It is closely related to the model-theoretic concept of the independence property. In this talk, I will illustrate these concepts in various examples, and show how the model-theoretic approach can give some bounds on VC density.

Sergio Garcia-Balan: On star selection principles

Place: Fields Institute (Room 210)

Date: February 17th, 2017 (13:30-15:00)

Speaker: Segio Garcia-Balan

Title: On star selection principles

Abstract: In the theory of selection principles, an important result (due to L. Aurichi), states that every Menger space is a D-space. Motivated by this result, we will discuss the star versions of the Menger property and some other selection principles in specific topological spaces. We will also talk about the game version of some of these principles. This is joint work with Javier Casas de la Rosa and Paul Szeptycki.

Sam Dworetzky: The classification problem for models of Peano Arithmetic

Tuesday, February 14 from 3 to 4pm
Room: MB 124
Speaker: Sam Dworetzky (BSU)
Title: The classification problem for models of Peano Arithmetic

Abstract: It is well known that the countable models of Peano Arithmetic (PA) are complicated to classify. In this talk we will make this precise using the language of Borel complexity theory. We will use a construction of Gaifman to show there is a Borel reduction from the isomorphism relation on linear orders to the isomorphism relation on models of PA.

Monika Seisenberger: Programs from constructive and classical proofs

Tuesday, February 28, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Monika Seisenberger (Swansea University)

Title: Programs from constructive and classical proofs

Abstract:

Program extraction from formal proofs is a powerful proof theoretic technique based on realisability to obtain provably correct programs. In this talk we give a short overview on the state-of-the-art, give several examples for program extraction from constructive proofs in different areas (Well-quasiorders, Sat Solving, Parsing), and also show how classical proofs which are often more concise and elegant compared to a constructive counterpart can lead to interesting programs. The case studies are carried out in the interactive theorem prover Minlog.

Philipp Lücke: Sigma_1-partition properties

Tuesday, February 14, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Philipp Lücke (Hausdorff Centre, University of Bonn)

Title: Sigma_1-partition properties

Abstract:

We consider colourings of the set of pairs of countable ordinals with two colours that are definable by Sigma_1-formulas that only use the first uncountable cardinal omega_1 and real numbers as parameters. We present results showing that the existence of a measurable cardinal above a Woodin cardinal implies that uncountable homogeneous sets exist for all such colourings. In contrast, a failure of this partition property is compatible with the existence of a single Woodin cardinal. Finally, we show that similar definable partition properties can hold for large cardinals that are not weakly compact; e.g. stationary limits of omega_1-iterable cardinals.

Ioannis Souldatos: L_{omega_1,omega}-sentences with maximal models in two cardinalities, part II

Thursday, February 16, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Ioannis Souldatos (University of Detroit Mercy)

Title: L_{omega_1,omega}-sentences with maximal models in two cardinalities, part II

Abstract:

This will be part II of the talk on complete L_{omega_1,omega}-sentences with maximal models
in (at least) two cardinalities. The talk will be self-contained.

Sample theorems

Theorem: If kappa is homogeneously characterizable and mu is the least such that 2^mu>=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities
2^lambda, for all mu<=lambdaaleph_0 is the least such that mu^omega>=kappa, then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities kappa^omega and kappa.

Theorem (Baldwin-Shelah) If mu is the first measurable cardinal and phi belongs to L_{omega_1,omega}, then no model of phi of size greater or equal to mu is maximal with respect to the L_{omega_1,omega}-elementary substructure relation.

André Nies: Structure within the class of K-trivial sets

SEMINAR ANNOUNCEMENT

Title: Structure within the class of K-trivial sets

Prof. André Nies
Department of Computer Science
University of Auckland
Date:    6 February 2017 (Monday)
Time:      4.00pm – 5.00pm
Venue:     MAS Executive Classroom 1 #03-06,
School of Physical and Mathematical Sciences  and quantum settings

The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Since 2002, many  alternative characterisations of this class have been found: properties such as low for K, low for Martin-Löf (ML) randomness, and basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially the class looked quite amorphous. More recently, internal structure has been found. Bienvenu et al. (JEMS 2016) showed that there is a “smart” K-trivial set, in the sense that any random oracle computing it computes all K-trivials. Greenberg, Miller and Nies(submitted)  showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches using cost functions. ML-reducibility (A is below B if every random oracle computing B also computes A) appears to a good way to compare the complexity of K-trivials, but the vexing question remains whether this reducibility is arithmetical.

Ioannis Souldatos: L_{omega_1,omega}-sentences with maximal models in two cardinalities

Thursday, February 9, 2017, from 4 to 5:30pm
East Hall, room 2866

Speaker: Ioannis Souldatos (University of Detroit Mercy)

Title: L_{omega_1,omega}-sentences with maximal models in two cardinalities

Abstract:

In this talk, we will present some examples on complete L_{omega_1,omega}-sentences with maximal models in (at least) two cardinalities.

Sample theorems:

Theorem: There is a complete L_{omega_1,omega}-sentence that characterizes aleph_2 and has maximal models in aleph_1 and aleph_2.

Theorem: Assume 2^{aleph_0}>aleph_n. Then there is a complete L_{omega_1,omega}-sentence with maximal models in cardinalities 2^{aleph_0}, 2^{aleph_1},…,2^{aleph_n}.

The main construction behind these theorems is a refinement of a construction of J. Knight. This is recent work of J. Baldwin and the speaker.

Yinhe Peng: Product of countable Frechet spaces

Place: Fields Institute (Room 210)

Date: February 3rd, 2017 (13:30-15:00)

Speaker: Yinhe Peng, University of Toronto

Title: Product of countable Frechet spaces

Abstract: I will discuss the preservation of Frechet property in the
product, mainly in the class of countable spaces. Some result in the
higher powers will also be mentioned.