Topics in Theoretical Computer Science: Computable Model Theory CSc 85020 – Autumn 2013 Preliminary Syllabus Class: Tuesdays 6:30 - 8:30 p.m. (tentative) at the CUNY Graduate Center, room TBA. Lecturer: Prof. Russell Miller, Queens College & CUNY Graduate Center. E-mail: [email protected]. Phone: 718-997-5853. Website: qc.edu/∼rmiller

Topics: This course is intended to introduce students to the disciplines of computable model theory and computable algebra, in which one applies the techniques of computability theory to questions from the rest of mathematics. An extremely fruitful interplay between these topics has developed over the past fifty years. It can be seen as part of computer science (since we are asking about the feasibility of writing programs to solve mathematical problems), and also as part of mathematical logic (since all computability results are in some sense proofs, and all non-computability results demonstrate the independence of the given problem from one of the basic axiom systems for mathematics). Computability theory explores the theoretical limits of algorithmic computation. We make this concept exact by defining Turing machines, and examining for which subsets of N such a machine can or cannot compute the characteristic function. Thus we define computable and noncomputable sets. We also extend the definition to encompass an oracle Turing machine, which enables us to assign levels of difficulty (the so-called Turing degrees) to the noncomputable sets and, using these degrees, to rank them according to our ability to compute one such set from another. No previous knowledge of computability or Turing machines is assumed; the first half of the course will be spent familiarizing everyone with these notions and the concepts that follow from them (as detailed on the schedule below). The second half of the schedule is more tentative: many topics are available to us, and we can go in the direction of the participants’ interests, whatever those may be. All the branches listed below (computable linear orders, computable fields, etc.) are current areas of research, but others are also available: computable differential fields, for example, or computable Boolean algebras, or computability for uncountable structures. Prof. Miller’s research includes all of these topics, and students should feel free to request particular topics. This area of mathematical logic is known by two distinct names: computability theory, and recursion theory. It is one of the four main branches of mathematical logic, along with model theory, set theory, and proof theory, and is a highly active area of current research, both by established scholars and at the doctoral level. It entered the realm of logic when the notion arose that proof-checking should be an automatic process, feasible even by machine, a notion integral to G¨odel’sIncompleteness Theorem. It is closely related to theoretical computer science, particularly to computational complexity, which uses stricter definitions of feasibility but many similar notions of reducibility. At the Graduate Center, computable model theory was the subject of Rebecca Steiner’s dissertation; she completed her degree in 2012, and now holds a postdoctoral position through 2015 at Vanderbilt University. The course is intended for doctoral students in computer science and mathematics with interest in these notions. No single course is a prerequisite. In computer science at the Graduate Center, CS 75010, Theoretical Computer Science, would give a student a leg up in understanding Turing computability and reducibility, but it is not necessary. In mathematics, Math 71200, the second half of the introductory logic sequence, often covers the topics from the first two lectures in the schedule below, but usually stops there. Feel free to contact Prof. Miller with any questions about your background. The pace of the course will depend on the background and interests of the participants, but a tentative schedule appears below. This is definitely subject to change before or during the semester. Moreover, the day of the week could yet be changed from Tuesday to another day to suit participants’ schedules.

Tentative Schedule:

Tuesday, September 3 Turing machines and computable functions Tuesday, September 10 Computably enumerable sets. Tuesday, September 17 Recursion Theorem. Tuesday, September 24 1-reducibility and Σ1-completeness. Tuesday, October 1 Turing reducibility and Turing degrees. Tuesday, October 8 Limit Lemma & Modulus Lemma. Tuesday, October 15 (no class – Monday schedule.) Tuesday, October 22 Arithmetical hierarchy & standard Σn-complete sets. Tuesday, October 29 Introduction to computable model theory. Tuesday, November 5 Computable linear orders. Tuesday, November 12 Computable algebraic fields. Tuesday, November 19 Computable categoricity. Tuesday, November 26 Computable fields with transcendentals. Tuesday, December 3 Classic undecidability results: Hilbert’s Tenth Problem and/or the Word Problem for finitely presented groups. Tuesday, December 10 Computable groups.

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