Datalog programs and their persistency numbers

The relation between Datalog programs and homomorphism problems and between Datalog programs and bounded treewidth structures has been recognized for some time and given much attention recently. Additionally, the essential role of persistent variables (of program expansions) in solving several relevant problems has also started to be observed. It turns out that to understand the contribution of these persistent variables to the difficulty of some expressibility problems, we need to understand the interrelationship among different notions of persistency numbers, some of which we introduce and/or formalize in the present work.This article is a first foundational study of the various persistency numbers and their interrelationships. To prove the relations among these persistency numbers, we had to develop some nontrivial technical tools that promise to help in proving other interesting results too. More precisely, we define the adorned dependency graph of a program, a useful tool for visualizing sets of persistent variables, and we define automata that recognize persistent sets in expansions.We start by elaborating on finer definitions of expansions and queries, which capture aspects of homomorphism problems on bounded treewidth structures. The main results of this article are (a) a program transformation technique, based on automata-theoretic tools, which manipulates persistent variables (leading, in certain cases, to programs of fewer persistent variables); (b) a categorization of the different roles of persistent variables; this is done by defining four notions of persistency numbers which capture the propagation of persistent variables from a syntactical level to a semantical one; (c) decidability results concerning the syntactical notions of persistency numbers that we have defined; and (d) the exhibition of new classes of programs for which boundedness is undecidable.