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We provide a wide-ranging study of the scenario where a subset of the relations in a relational vocabulary is visible to a user—that is, their complete contents are known—while the remaining relations are invisible. We also have a background theory—invariants given by logical sentences—that may relate the visible relations to invisible ones, and also may constrain both the visible and invisible relations in isolation. We want to determine whether some other information, given as a positive existential formula, can be inferred using only the visible information and the background theory. This formula whose inference we are concerned with is denoted as the query . We consider whether positive information about the query can be inferred, and also whether negative information—the sentence does not hold—can be inferred. We further consider both the instance-level version of the problem, where both the query and the visible instance are given, and the schema-level version, where we want to know whether truth or falsity of the query can be inferred in some instance of the schema.

We consider entailment problems involving powerful constraint languages such as frontier-guarded existential rules in which we impose additional semantic restrictions on a set of distinguished relations. We consider restricting a relation to be transitive, restricting a relation to be the transitive closure of another relation, and restricting a relation to be a linear order. We give some natural variants of guardedness that allow inference to be decidable in each case, and isolate the complexity of the corresponding decision problems. Finally we show that slight changes in these conditions lead to undecidability., This article was originally published at JAIR in 2018: https://www.jair.org/index.php/jair/article/view/11240 (DOI 10.1613/jair.1.11240). This version of the paper includes one modification from the publisher version: we fix an incorrect proof for one of our undecidability results (Theorem 6.2). arXiv admin note: substantial text overlap with arXiv:1607.00813

Cost monadic logic has been introduced recently as a quantitative extension to monadic second-order logic. A sentence in the logic defines a function from a set of structures to N∪{∞}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. The rich theory associated with these functions has already been studied over finite words and trees. We extend the theory to infinite trees for the weak form of the logic (where second-order quantification is interpreted over finite sets). In particular, we show weak cost monadic logic is equivalent to weak cost automata, and finite-memory strategies suffice in the infinite two-player games derived from such automata. We use these results to provide a decision procedure for the logic and to show there is a function definable in cost monadic logic which is not definable in weak cost monadic logic.

International audience; Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.