The Hierarchical Approach to Modeling Knowledge and Common Knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the "richness" of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for "interesting" sets of events, and show that the answer depends on the definition of "interesting."

Foundations of Reasoning with Uncertainty via Real-valued Logics

Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. For the first time, we give a sound and complete axiomatization for a broad class containing all the common real-valued logics. This axiomatization allows us to derive exactly what information can be inferred about the combinations of real values of a collection of formulas given information about the combinations of real values of several other collections of formulas. We then extend the axiomatization to deal with weighted subformulas. Finally, we give a decision procedure based on linear programming for deciding, under certain natural assumptions, whether a set of our sentences logically implies another of our sentences.Comment: 9 pages (incl. references), 9 pages supplementary. In submission to NeurIPS 202

Recursive Programs for Document Spanners

A document spanner models a program for Information Extraction (IE) as a function that takes as input a text document (string over a finite alphabet) and produces a relation of spans (intervals in the document) over a predefined schema. A well-studied language for expressing spanners is that of the regular spanners: relational algebra over regex formulas, which are regular expressions with capture variables. Equivalently, the regular spanners are the ones expressible in non-recursive Datalog over regex formulas (which extract relations that constitute the extensional database). This paper explores the expressive power of recursive Datalog over regex formulas. We show that such programs can express precisely the document spanners computable in polynomial time. We compare this expressiveness to known formalisms such as the closure of regex formulas under the relational algebra and string equality. Finally, we extend our study to a recently proposed framework that generalizes both the relational model and the document spanners

A Framework for Combining Entity Resolution and Query Answering in Knowledge Bases

We propose a new framework for combining entity resolution and query answering in knowledge bases (KBs) with tuple-generating dependencies (tgds) and equality-generating dependencies (egds) as rules. We define the semantics of the KB in terms of special instances that involve equivalence classes of entities and sets of values. Intuitively, the former collect all entities denoting the same real-world object, while the latter collect all alternative values for an attribute. This approach allows us to both resolve entities and bypass possible inconsistencies in the data. We then design a chase procedure that is tailored to this new framework and has the feature that it never fails; moreover, when the chase procedure terminates, it produces a universal solution, which in turn can be used to obtain the certain answers to conjunctive queries. We finally discuss challenges arising when the chase does not terminate

Expressive Power of Entity-Linking Frameworks

We develop a unifying approach to declarative entity linking by introducing the notion of an entity linking framework and an accompanying notion of the certain links in such a framework. In an entity linking framework, logic-based constraints are used to express properties of the desired link relations in terms of source relations and, possibly, in terms of other link relations. The definition of the certain links in such a framework makes use of weighted repairs and consistent answers in inconsistent databases. We demonstrate the modeling capabilities of this approach by showing that numerous concrete entity linking scenarios can be cast as such entity linking frameworks for suitable choices of constraints and weights. By using the certain links as a measure of expressive power, we investigate the relative expressive power of several entity linking frameworks and obtain sharp comparisons. In particular, we show that we gain expressive power if we allow constraints that capture non-recursive collective entity resolution, where link relations may depend on other link relations (and not just on source relations). Moreover, we show that an increase in expressive power also takes place when we allow constraints that incorporate preferences as an additional mechanism for expressing "goodness" of links

A Declarative Framework for Linking Entities

The aim of this paper is to introduce and develop a truly declarative framework for entity linking and, in particular, for entity resolution. As in some earlier approaches, our framework is based on the systematic use of constraints. However, the constraints we adopt are link-to-source constraints, unlike in earlier approaches where source-to-link constraints were used to dictate how to generate links. Our approach makes it possible to focus entirely on the intended properties of the outcome of entity linking, thus separating the constraints from any procedure of how to achieve that outcome. The core language consists of link-to-source constraints that specify the desired properties of a link relation in terms of source relations and built-in predicates such as similarity measures. A key feature of the link-to-source constraints is that they employ disjunction, which enables the declarative listing of all the reasons as to why two entities should be linked. We also consider extensions of the core language that capture collective entity resolution, by allowing inter-dependence between links. We identify a class of "good" solutions for entity linking specifications, which we call maximum-value solutions and which capture the strength of a link by counting the reasons that justify it. We study natural algorithmic problems associated with these solutions, including the problem of enumerating the "good" solutions, and the problem of finding the certain links, which are the links that appear in every "good" solution. We show that these problems are tractable for the core language, but may become intractable once we allow inter-dependence between link relations. We also make some surprising connections between our declarative framework, which is deterministic, and probabilistic approaches such as ones based on Markov Logic Networks

Composition with Target Constraints

It is known that the composition of schema mappings, each specified by source-to-target tgds (st-tgds), can be specified by a second-order tgd (SO tgd). We consider the question of what happens when target constraints are allowed. Specifically, we consider the question of specifying the composition of standard schema mappings (those specified by st-tgds, target egds, and a weakly acyclic set of target tgds). We show that SO tgds, even with the assistance of arbitrary source constraints and target constraints, cannot specify in general the composition of two standard schema mappings. Therefore, we introduce source-to-target second-order dependencies (st-SO dependencies), which are similar to SO tgds, but allow equations in the conclusion. We show that st-SO dependencies (along with target egds and target tgds) are sufficient to express the composition of every finite sequence of standard schema mappings, and further, every st-SO dependency specifies such a composition. In addition to this expressive power, we show that st-SO dependencies enjoy other desirable properties. In particular, they have a polynomial-time chase that generates a universal solution. This universal solution can be used to find the certain answers to unions of conjunctive queries in polynomial time. It is easy to show that the composition of an arbitrary number of standard schema mappings is equivalent to the composition of only two standard schema mappings. We show that surprisingly, the analogous result holds also for schema mappings specified by just st-tgds (no target constraints). This is proven by showing that every SO tgd is equivalent to an unnested SO tgd (one where there is no nesting of function symbols). Similarly, we prove unnesting results for st-SO dependencies, with the same types of consequences.Comment: This paper is an extended version of: M. Arenas, R. Fagin, and A. Nash. Composition with Target Constraints. In 13th International Conference on Database Theory (ICDT), pages 129-142, 201

Parallel Play Saves Quantifiers

The number of quantifiers needed to express first-order properties is captured by two-player combinatorial games called multi-structural (MS) games. We play these games on linear orders and strings, and introduce a technique we call "parallel play", that dramatically reduces the number of quantifiers needed in many cases. Linear orders and strings are the most basic representatives of ordered structures -- a class of structures that has historically been notoriously difficult to analyze. Yet, in this paper, we provide upper bounds on the number of quantifiers needed to characterize different-sized subsets of these structures, and prove that they are tight up to constant factors, including, in some cases, up to a factor of $1+\varepsilon$, for arbitrarily small $\varepsilon$.Comment: 24 pages, 4 figure