Uniform Restricted Chase Termination

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The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A central problem concerning the chase procedure is uniform (a.k.a. all-instances) chase termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is, in general, undecidable, it is natural to ask whether known well-behaved classes of TGDs ensure decidability. We focus on the main paradigms that led to robust TGD-based formalisms, namely guardedness and stickiness, that have been introduced in the context of knowledge-enriched databases. Although uniform chase termination is well understood for the oblivious version of the chase (2EXPTIME-complete for guarded, and PSPACE-complete for sticky TGDs), the more subtle case of the restricted (a.k.a. the standard) chase is rather unexplored. We show that uniform restricted chase termination under guarded single-head TGDs and sticky single-head TGDs is decidable in elementary time. In the case of guardedness, we provide a reduction to the satisfiability problem of monadic second-order logic over infinite trees of bounded degree, while for stickiness we provide a reduction to the emptiness problem of deterministic Büchi automata. Those reductions build on a series of technical results of independent interest related to the notion of fairness of the restricted chase, and the existence of critical databases that characterize nontermination of the restricted chase via databases of a certain form.