Datalog with Negation and Monotonicity

Positive Datalog has several nice properties that are lost when the language is extended with negation. One example is that fixpoints of positive Datalog programs are robust w.r.t. the order in which facts are inserted, which facilitates efficient evaluation of such programs in distributed environments. A natural question to ask, given a (stratified) Datalog program with negation, is whether an equivalent positive Datalog program exists. In this context, it is known that positive Datalog can express only a strict subset of the monotone queries, yet the exact relationship between the positive and monotone fragments of semi-positive and stratified Datalog was previously left open. In this paper, we complete the picture by showing that monotone queries expressible in semi-positive Datalog exist which are not expressible in positive Datalog. To provide additional insight into this gap, we also characterize a large class of semi-positive Datalog programs for which the dichotomy `monotone if and only if rewritable to positive Datalog\u27 holds. Finally, we give best-effort techniques to reduce the amount of negation that is exhibited by a program, even if the program is not monotone

The umbilical cord of finite model theory

Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but didn't turn out that way. There is one application domain -- relational database management -- that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.Comment: To be published in the Logic in Computer Science column of the February 2023 issue of the Bulletin of the European Association for Theoretical Computer Scienc

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k

Degrees of Monotonicity of Spatial Transformations

. We consider spatial databases that can be defined in terms of polynomial inequalities, and we are interested in monotonic transformations of spatial databases. We investigate a hierarchy of monotonicity classes of spatial transformations that is determined by the number of degrees of freedom of the transformations. The result of a monotonic transformation with k degrees of freedom on a spatial database is completely determined by its result on subsets of cardinality at most k of the spatial database. The result of a transformation in the largest class of the hierarchy on a spatial database is determined by its result on arbitrary large subsets of the database. The latter is the class of all the monotonic spatial transformations. We give a sound and complete language for the monotonic spatial transformations that can be expressed in the relational calculus augmented with polynomial inequalities and that belong to a class with a finite number of degrees of freedom. In particular, we s..