Statement of Research Interests

My primary areas of research are complexity theory and databases. I am also interested in game theory, artificial intelligence, and logic. Everyone has data they want to store. Just as important, you have to be able to get at your data when you need it. This doesn’t just mean having access to the data, but having access to it in a timely manner. Knowing how long it will take to retrieve data from a database is especially important as the size of databases grows. In general retrieving data from a database is a hard problem. It would be nice to know if there are configurations, for either the database or the queries, that make the problem easier. In addition, we often need to transform data from one schema to another. Sometimes we need to do this because we are importing data from another source into an existing database, or because of a need to change the schema in our current database. Consistency and Games When retrieving data from a relational database, we write a query that specifies the data we want, and then evaluate that query against the database. The problem of deciding whether the answer to the query is non-empty is NP-complete in general when both the query and the database are given as input. This query evaluation problem turns out to be equivalent to another problem, known as the Constraint Satisfaction Problem. The Constraint Satisfaction Problem (CSP) has been deeply studied in the area of artificial intelligence. The idea is that you are given variables and constraints on the values those variables may take. The question is whether there is an assignment of values to variables that satisfies all of the contraints. Because it is equivalent to the query evaluation problem, CSP is also NP-complete in general. On the other hand, we know that there are subproblems that are solvable efficiently. For instance, if no constraint involves more than one variable, we can easily decide if there is a solution. There are, in fact, many large classes of subproblems for which an efficient algorithm exists to solve CSP, but because the problem is hard in general, researchers have come up with heuristics to solve problems. One of the most effective class of heuristics involve consistency properties. To talk about consistency, we must first consider the concept of a partial solution. A partial solution is an assignment of values to some of the variables in a CSP instance. The size of a partial solution is the number of variables assigned values. A CSP instance is called k-consistent, if every partial solution of size k − 1 can be extended to a partial solution of size k. An instance is strongly kconsistent if evey partial solution of size less than k can be extended to a partial solution of size k. An instance is called globally consistent if any partial solution can be extended to a solution to the instance. Dechter, in [2], showed that there are conditions where strong k-consistency implies global consistency. This gives rise to interest in algorithms to take an instance of CSP and try to change it into an equivalent instance that is strongly k-consistent. Cooper [1] has presented an algorithm that establishes strong k-consistency in exponential time. Unfortunately, this is not efficient enough to be useful on general problems. It would be nice to know if we can do better. While studying Datalog, Kolaitis and Vardi [8] found a connection to help with this issue. They introduced the existential k-pebble game, to help study Datalog and related logics. This is a game played on two relational structures A and B by two players known as Spoiler and Duplicator. They take turns placing pebbles on the structures, Spoiler playing on A and Duplicator playing on B. Duplicator wins by always making sure that the partial mapping from A to B determined by the pebbles is always a partial homomorphism. As it turns out, determining strong k-consistency for an instance of CSP is equivalent to determining a winning strategy for the Duplicator in a