The term naïve evaluation refers to evaluating queries over incomplete databases as if nulls were usual data values, that is, to using the standard database query evaluation engine. Since the semantics of query answering over incomplete databases is that of certain answers, we would like to know when naïve evaluation computes them, that is, when certain answers can be found without inventing new specialized algorithms. For relational databases it is well known that unions of conjunctive queries possess this desirable property, and results on preservation of formulae under homomorphisms tell us that, within relational calculus, this class cannot be extended under the open-world assumption.Our goal here is twofold. First, we develop a general framework that allows us to determine, for a given semantics of incompleteness, classes of queries for which naïve evaluation computes certain answers. Second, we apply this approach to a variety of semantics, showing that for many classes of queries beyond unions of conjunctive queries, naïve evaluation makes perfect sense under assumptions different from open world. Our key observations are: (1) naïve evaluation is equivalent to monotonicity of queries with respect to a semanticsinduced ordering, and (2) for most reasonable semantics of incompleteness, such monotonicity is captured by preservation under various types of homomorphisms. Using these results we find classes of queries for which naïve evaluation works, for example, positive first-order formulae for the closed-world semantics. Even more, we introduce a general relation-based framework for defining semantics of incompleteness, show how it can be used to capture many known semantics and to introduce new ones, and describe classes of first-order queries for which naïve evaluation works under such semantics. ACM Reference Format:Amélie Gheerbrant, Leonid Libkin, and Cristina Sirangelo. 2014. Naïve evaluation of queries over incomplete databases.

Current methods for solving games embody a form of "procedural rationality" that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic invariant structure in games. We then generalize this to an exploration of fixed-point logics on finite trees that best fit game-theoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the forthcoming dissertation [26].Logics that describe games In recent years, many logical analyses have been given of both strategic and extensive games, through introducing formal languages that describe game structure while raising logical questions of definability and axiomatization ([17], [4], [19], [29]). A benchmark for logics in this tradition has been the definition of Backward Induction ("BI" for short), the most common method for solving finite extensive games of perfect information ([39], [40]). In this same arena, basic foundational results have been obtained in epistemic game theory, endowing bare games with epistemic assumptions about players. A pilot result was the characterization of the BI outcome in terms of assuming common knowledge, or common true belief, in rationality, meaning that players choose those actions that they believe to be best for themselves ([1]). Recently,[9] has suggested that the main focus here should be shifted: away from a static assumption of known or believed rationality to the underlying "procedural rationality" of plausible procedures that players engage in when analyzing and playing a game, and the way these result in stable limit models where rationality becomes common knowledge. 1 Thus, [4] shows how game-theoretic equilibrium fits with the computational perspective of fixed-point logics, and [11] gives several dynamic procedures that analyze BI. This paper will analyze these proposals further, and find their common mathematical background. This will then be our starting point for suggesting a more general line of investigation. Analyzing solution procedures Basics of extensive gamesWe assume some basic game theory, and we will work with finite extensive games of perfect information, i.e., finite trees with labelled nodes, where each node is either an end node, or an intermediate node that represents the turn of a unique player. 2 We will mostly think of 2player games, though much of what we say generalizes to more players. While game trees with moves are simple computational structures, the essence of rational action arises with the way players evaluate outcomes. Thus, there is also a fur...

We study static analysis, in particular the containment problem, for analogs of conjunctive queries over XML documents. The problem has been studied for queries based on arbitrary patterns, not necessarily following the tree structure of documents. However, many applications force the syntactic shape of queries to be tree-like, as they are based on proper tree patterns. This renders previous results, crucially based on having non-tree-like features, inapplicable. Thus, we investigate static analysis of queries based on proper tree patterns. We go beyond simple navigational conjunctive queries in two ways: we look at unions and Boolean combinations of such queries as well and, crucially, all our queries handle data stored in documents, i.e., we deal with containment over data trees.We start by giving a general Π p 2 upper bound on the containment of conjunctive queries and Boolean combinations for patterns that involve all types of navigation through documents. We then show matching hardness for conjunctive queries with all navigation, or their Boolean combinations with the simplest form of navigation. After that we look at cases when containment can be witnessed by homomorphisms of analogs of tableaux. These include conjunctive queries and their unions over child and next-sibling axes; however, we show that not all cases of containment can be witnessed by homomorphisms. We look at extending tree patterns used in queries in three possible ways: with wildcard, with schema information, and with data value comparisons. The first one is relatively harmless, the second one tends to increase complexity by an exponential, and the last one quickly leads to undecidability.