# LSEC Fellowships for Undergraduate and Graduate Students

Each year LSEC supports several Carnegie Mellon undergraduate and graduate students who wish to participate in LSEC Projects. Such participation takes place, typically, over the summer. It can be extended, however, through (parts of) the following academic year. During the summer, LSEC provides a stipend; during the academic year, LSEC Fellows can elect to receive course credit for independent research.

LSEC supports two “classes” of students; namely, rising seniors and graduate students who are in the department’s Master’s Program and are completing their first year. The expectation is that students will work *at least* half-time for *at least* two months during the summer.

A project can be an elaboration of an entry from the list below, but it can also be an independent project of the student’s choice. It may be related to an Honors Thesis (for undergraduates) or a Master’s Thesis (in the case of graduate students). Applicants must have a faculty sponsor for their project, and we highly recommend collaborating with a faculty member on the application.

We encourage undergraduates to present their research results at the *Meeting of the Minds* at the end of the following academic year. We also encourage graduate students to give conference presentations, and we will contribute to cover the costs involved. Applications for such conference support can be made at any time during the academic year.

**Due date for the application**. The submission deadline for an LSEC Fellowship is the Friday four weeks before the last day of classes of spring term. A decision will be made promptly.

Applications should be submitted to Dr. Joseph Ramsey by email or by hardcopy to his departmental address:

Dr. Joseph Ramsey

LSEC Fellowships

Department of Philosophy

161 Baker Hall

Carnegie Mellon University

Pittsburgh, PA 15213

The application should include:

- Name, Student Number
- Name of Faculty Sponsor
- Project Proposal (approximately 2 pages including a brief abstract)
- Resume
- Timeframe for Project (Summer and, possibly, Fall and Spring)

## Suggestions for Research Projects

The list below presents broader research frames only - we encourage you to contact a faculty sponsor for a project (within such a frame) you are interested in carrying out.

### Applying Causal Search to Human Data

(Contacts: Spirtes, Scheines, Glymour, Ramsey, Zhang)

Lots of science uses an old-fashioned method: the investigator has a conjecture, gathers experimental or non-experimental data, does a hypothesis test of the model on the data, and if the hypothesis passes, tries to publish a conclusion. The difficulty is that the investigator may have overlooked or been unable to think of alternative models that might better explain their data.

In recent decades several automatic methods to search for alternative explanations have been developed. These methods have been used to successfully find factors affecting national and CMU college dropout rates, to identify effects of acid rain, causes of how people have voted in past elections, and many other topics.

The summer project is to find examples where the data are publicly available and see if these newer methods can (or cannot) find models that provide explanations of the data that are statistically as good or better than published models. We are especially, but not exclusively, interested in examples from the social and behavioral sciences.

### Proof Search and Logical Reasoning

(Contacts: Wilfried Sieg or Joseph Ramsey)

*Automated Proof Search*. We have developed a very effective search method for finding natural proofs in logic. How can these techniques be extended to mathematical arguments? We are in the process of extending the underlying intercalation method to elementary set theory. This involves both interesting mathematical and computational issues.*Logic & Proofs*. The techniques of automated proof search, developed in the project above, are now being taught in a fully web-based course: Logic & Proofs. There are many areas where the logical presentation, examples, interactive learning environments and the graphical interface can be improved. However, the most important educational project is to refine an intelligent, dynamic tutor for proof construction using these techniques.*Educational experiments*. The web-based course provides an ideal setting for carrying out educational experiments; we want to investigate which methods are effective for teaching students basic notions and techniques in logic.*Mental Proofs*. Many powerful algorithms exist for finding proofs in logical systems. Some strive explicitly to use strategies that human experts are thought to employ, but little is known about how novice and experts actually search for proofs.

### Beyond Pure Logic

(Contacts: Wilfried Sieg or Joseph Ramsey)

As an extension of Logic & Proofs, we are developing a course presenting the fundamental Incompleteness and Undeidability Theorems that were first established in the 1930s by Gödel, Church and Turing. The substantive material is sketched below; however, the AProS search for proofs in set theory should also serve as the basis for intelligent tutoring in set theory.

*Elementary set theory*. Here one goal is to develop elementary set theory beginning with the axioms of Zermelo Fraenkel and ending with the famous theorems of Cantor and Cantor-Bernstein. However, apart from this purley set theoretic development, we focus on the REPRESENTATION of parts of mathematical practice in the formal framework of set theory. We do this first for elementary number theory and then for the theory of syntax. That allows us then to prove quite easily Gödel's Incompleteness theorems for set theory.*Computability theory*. The introduction to computability theory involves as the basic concept that of a Turing machine (computation). So we are aiming to implement Turing machines as a web-based application. The main undecidability result to be shown is the Halting Problem. Then we introduce a notion of EFFECTIVE REDUCIBILITY of one problem to another and show that the Halting Problem is reducible to the Decision Problem for Predicate Logic, but also to the Tiling Problem. The unsolvability of the Halting Problem implies directly the unsolvability of the other problems.

### Rational Choice

(Contacts: Teddy Seidenfeld or Kevin Zollman)

*Imprecise Probabilities and values*. The relevant information used in most real decision problems tends to be scarce, vague or even sometimes conflicting. By the same token preferences may also be incomplete. There are nevertheless well known theories of decision that can be applied to situations of this kind where both probabilities and value are imprecise. There is preliminary evidence (gathered through experiments carried by some of our faculty and students) that these theories can accommodate recalcitrant empirical evidence (like the so-called two-color Ellsberg’s paradox). Extensions of this work for non-binary choices in three color Ellsberg situations (as well as real-life versions of these situations) include the design of the experimental set-up, data-analysis, and the development of software.