We consider evaluation strategies for database queries expressed in three functional query languages: the complex value algebra, the simply typed lambda calculus, and method schemas. Each of these query languages derives its expressive power from a different primitive: the complex value algebra from the powerset operator, the simply typed lambda calculus from list iteration, and method schemas from recursion. We show that "natural" evaluation strategies for these primitives may lead to very inefficient space usage, but that with some simple optimizations many queries can be evaluated with little or no space overhead. In particular, we show: (1) In the complex value algebra, all expressions with set nesting depth at most 2 can be evaluated in pspace, and this set of expressions is sufficient to express all queries in the polynomial hierarchy; (2) In the simply typed lambda calculus with equality and constants, all query terms of order at most 5 (where "query term" is a syntactic condition on types) can be evaluated in pspace, and this set of terms expresses exactly the pspace queries; (3) There exists a set of second-order method schemas (with no simple syntactic characterization) that can be evaluated in pspace, and this set of schemas is sufficient to express all pspace queries.

Church's simply typed -calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this thesis: (1) We re-examine the expressive power of the "pure" simply typed -calculus, but over encodings of finite relational structures, i. e., finite models or databases . In this novel framework the simply typed -calculus expresses all elementary functions from finite models to finite models. In addition, many common database query languages, e. g., relational algebra, Datalog : , and the Abiteboul/Beeri complex object algebra, can be embedded into it. The embeddings are feasible in the sense that the -terms corresponding to PTIME queries can be evaluated in polynomial time. (2) We examine fixed-order fragments of the simply typed -calculus to determine machine independent characterizations of complexity classes. For this we augment the calculus with atomic constants and equality among atomic constants. We show that over ordered structures, the order 3, 4, 5, and 6 fragments express exactly the first-order, PTIME, PSPACE, and EXPTIME queries, respectively, and we conjecture that for general k 1, order 2 k + 4 expresses exactly the k-EXPTIME queries and order 2 k + 5 expresses exactly the k-EXPSPACE queries. (3) We also re-examine other functional characterizations of PTIME and we show that method schemas with ordered objects express exactly PTIME. This is a first-order framework proposed for object-oriented databases---as opposed to the above higher-order frameworks. In summary, this research provides a link between finite model theory (and thus computational complexity), dat...

Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finite-model theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich source of questions and vitality for finite-model theory.

this technical obituary feel honored by the privilege they had in collaborating with Paris on many of these projects. In mourning his tragic death, we miss his technical facility, his broad knowledge, his insight, his commitment, and his humor. To write a research paper with Paris was also an opportunity to observe his indefatigable attention to detail, and to engage in vigorous debate with his editorial voice. To write this obituary allowed us to to feel his voice once more, to understand better what a good scientist he was, and to appreciate his uncommon decency and kindness. It is our great loss that we will not hear his voice again. 1 Deductive Databases