Spring 2008
Q520: Mathematics and Logic for Cognitive Science

The course web page is here.

This graduate class introduces a wide variety of mathematical topics pertinent to cognitive science, artificial intelligence, and related fields. Those topics are primarily taken from probability and linear algebra, and secondarily from logic, algorithms and complexity theory, and optimization. The intended applications will be to models for uncertain reasoning, such as Bayesian nets and other graphical models, hidden Markov models, Markov decision processes, and some types of neural nets. In addition, the class will also present the basics of logic, both in its classical form and some systems for areas like default reasoning. It will also introduce concepts like entropy and game-theoretic equilibrium. So the class will see many application areas. But it will not be a class in using or even building application tools. And although the applications include some of the main tools in cognitive science and artificial intelligence, the overall point is to introduce a large body of related mathematical ideas in a friendly way, so that you will be stimulated to continue learning.

Fall 2007
M781: Coalgebra

Coalgebra is a field of study which aims to study the general properties of a great variety of state-based dynamical systems, like transition systems, automata, process calculi, Markov chains, etc. These all can be captured uniformly as coalgebras, and coalgebra aims to be the mathematics of computational dynamics. Coalgebra is developing into a field of its own interest presenting a deep mathematical foundation, a growing field of applications and interactions with various other fields such as formal system specification, modal logic, dynamical systems, control systems, category theory, set theory, algebra, analysis, enumerative combinatorics, recursive program schemes, circularity, etc. The field as a whole is fairly new, and it is currently very active.

Fall 2007
COLL-S 105: Freshman Seminar in Computability and Logic

The theory of computation is one of the most important intellectual developments of the first half of the twentieth century. From very slender roots, a tree blossomed in the 1930's whose fruit is the development of computers as we know them. But the same tree contains thorns, as it were; these are the 'negative results' which talk about computer programs that we can never write, and true sentences which we can never prove. These results are often taken to imply fundamental limitations on what human beings can know. They are on a cultural par with other developments that came at roughly the same time: the uncertainty principle in physics, and even with Freud's notion of an unconscious which cannot know itself.

The course will be an entry point to both the mathematical theory of computation, and also to discussions of the place that the theory occupies in broader intellectual discourse. This is not a course only for mathematics or computer science majors.

For more on the course, see the course web page.

Spring 2006
Q520: Mathematics and Logic for Cognitive Science

Please see my description of it above, in the section on Spring 2008 classes.

Spring 2006
M584: Recursion Theory

Recursion theory is the mathematical study of computability. It is connected to areas of theoretical computer science, and in a sense it is a topic at the heart of that subject. But recursion theory is usually taken to be a branch of mathematical logic. Our course will be a standard introduction to the subject, using Robert Soare's revised textbook. We will mainly cover part 1 of the book. One certainly can study recursion theory without ever seeing logic, but our course will go into detail on one of the main applications of recursion theory in logic, to the Incompleteness Theorems.

The course also developed text register machines. These are machines whose registers hold words over the same alphabet that the instructions are written in. The purpose is to obtain explicit proofs of results like the Recursion Theorem and self-replicating computer programs. You can find out more about this project from this link.

Fall 2005
M781: Coalgebra

Coalgebra is a field of study which aims to study the general properties of a great variety of state-based dynamical systems, like transition systems, automata, process calculi, Markov chains, etc. These all can be captured uniformly as coalgebras, and coalgebra aims to be the mathematics of computational dynamics. Coalgebra is developing into a field of its own interest presenting a deep mathematical foundation, a growing field of applications and interactions with various other fields such as formal system specification, modal logic, dynamical systems, control systems, category theory, set theory, algebra, analysis, enumerative combinatorics, recursive program schemes, circularity, etc. The field as a whole is fairly new, and it is currently very active.

This course will be an introduction to coalgebra. The main intended audience is students of logic or theoretical computer science, but students in other areas of math are of course welcome. The central results in coalgebra are best formulated with the help of category theory. So the course would serve as an introduction to that subject, even for people whose main interest is another field of mathematics. It would be good to have seen logic and recursive definitions at some level, but this is probably something someone can pick up during the course.

The text will be a draft textbook on the subject by Bart Jacobs entitled Introduction to Coalgebra: Towards Mathematics of States and Observations. In addition, we'll read several other papers.

Spring 2005
L546: Semantics

An introduction to semantics for linguistics graduate students and others with similar backgrounds. My current goal is to do enough formal semantics of extensional topics so that people are conversant with the ideas, then to do a unit on intensional phenomena and also a unit on computational semantics. The textbook is Henriette de Swart's book Introduction to Natural Language Semantics. The class will meet TR 1:00-2:15.

Fall 2004
M682: Model Theory

This is really a course in logical systems: syllogistic, propositional, intuitionistic, equational, first-order and second-order; at the end I reach the traditional topics of model theory. The text will be some notes that I'm working on. If you write to me, I'll send you the version of the notes from the last time I taught this material.

Fall 2004
P550/x384: Modal Logic

An introduction to modal logic, emphasizing applications in epistemic logic and also conditionals. The course could be of interest to people in AI, Informatics, and Linguistics as well as Philosophy. The course may be taken either for undergraduate math credit or (with appropriate extra work) graduate credit as a Philosophy course. The text is some notes that I'm writing up. If you'd like to see them, please write to me.

Spring 2004
M583: Set Theory

Our graduate course in axiomatic set theory. The text was Yiannis Moschovakis' book Set Theory Notes. We covered most of the book, and then turned to the main easy consistency results concerning set theory.

Spring 2004
Q520: Mathematics and Logic for Cognitive Science.

This graduate class introduces a wide variety of mathematical topics pertinent to cognitive science, artificial intelligence, and related fields. Those topics are primarily taken from probability and linear algebra, and secondarily from logic, algorithms and complexity theory, and optimization. The intended applications will be to models for uncertain reasoning, such as Bayesian nets and other graphical models, hidden Markov models, and some types of neural nets. In addition, the class will also present the basics of logic, both in its classical form and some systems for areas like default reasoning. It will also introduce concepts like entropy and game-theoretic equilibrium. So the class will see many application areas. But it will not be a class in using or even building application tools. And although the applications include some of the main tools in cognitive science and artificial intelligence, the overall point is to introduce a large body of related mathematical ideas in a friendly way, so that you will be stimulated to continue learning.

The recommended text is Dana H. Ballard's book Introduction to Natural Computation. The book will be useful as something to read, but I hope to write up notes, distribute other readings, and point people to articles on the web. Probably the book will be useful for half the semester, so it is a good idea to purchase it.

Fall 2003
M385: Mathematics from Language

An undergraduate course in mathematics for linguistics students. The text is a book I am writing with Edward L. Keenan for use both here and at UCLA.

Spring 2003
P550/X384: Modal Logic

This was a graduate/undergraduate course in modal logic emphasizing applications in many areas, especially epistemic logic. It ended with conditional logic.

Spring 2003
I201: Mathematical Foundations of Informatics

The School of Informatics' required course for majors, covering topics in finite mathematics that build on the material of M118.

Fall 2002
M781: Coalgebra

This might have been one of the first graduate seminars on coalgebra in the world.

Fall 2002
M680: Logic and Decidability

The course studied automata on infinite objects. The main goal was the decidability of S2S. The textbook was "Automata Theory and Its Applications" by Bakhadyr Khoussainov and Anil Nerode.

Spring 2002
Q520: Mathematics and Logic for Cognitive Science

This was be a new graduate course, designed for our new Cognitive Science PhD program. It also was designed to be suitable for students in other fields. The textbook was Mathematical Methods in Artificial Intelligence by Edward A. Bender and published by the IEEE Computer Society.

Fall 2001
M682: Model Theory

A graduate class covering some standard material on first-order logic and also finite model theory.

I organized a graduate student reading/working group on Logic, Games, and Decisions. We read recent papers on connections of logic with game theory, rationality, and related areas.