Knowledge Graph Foundation Models (KGFMs) are at the frontier for deep learning on knowledge graphs (KGs), as they can generalize to completely novel knowledge graphs with different relational vocabularies. Despite their empirical success, our theoretical understanding of KGFMs remains very limited. In this paper, we conduct a rigorous study of the expressive power of KGFMs. Specifically, we show that the expressive power of KGFMs directly depends on the *motifs* that are used to learn the relation representations. We then observe that the most typical motifs used in the existing literature are *binary*, as the representations are learned based on how pairs of relations interact, which limits the model's expressiveness. As part of our study, we design more expressive KGFMs using richer motifs, which necessitate learning relation representations based on, e.g., how triples of relations interact with each other. Finally, we empirically validate our theoretical findings, showing that the use of richer motifs results in better performance on a wide range of datasets drawn from different domains.

Spatial data play an important role in many applications built over knowledge graphs, and are frequently referenced in queries posed to public query services, such as that of Wikidata.Querying for spatial data presents a significant challenge, as topological relations such as adjacent or contains imply inferred information, such as through the transitivity of the containment relation.However, despite all the recent advances in querying knowledge graphs, we still lack techniques specifically tailored for topological information.Applications looking to incorporate topological relations must either materialize the inferred relations, incurring high space and maintenance overheads, or query them with less efficient recursive algorithms, incurring high runtime overheads.In this paper we address the problem of leveraging topological information in knowledge graphs by designing efficient algorithms to process these queries.Our solution involves building a specific index that stores the topological information in a convenient compact form, and includes specialized algorithms that infer every possible relation from the basic topological facts in the graph.We show that, while using essentially the same space required to solve standard graph pattern queries, we can incorporate topological predicates, accounting for all the inferred information, all within worst-caseoptimal time.We implement our scheme and show experimentally that it outperforms baseline solutions by a notable margin.

We extend the concept of worst-case optimal equijoins in graph databases to the case where some nodes are required to be within the k-nearest neighbors (kNN) of others under some similarity function. We model the problem by superimposing the database graph with the kNN graph and show that a variant of Leapfrog TrieJoin (LTJ) implemented over a compact data structure called the Ring can be seamlessly extended to integrate similarity clauses with the equijoins in the LTJ query process, retaining worst-case optimality in many relevant cases. Our experiments on a benchmark that combines Wikidata and IMGpedia show that our enhanced LTJ algorithm outperforms by a considerable margin a baseline that first applies classic LTJ and then completes the query by applying the similarity predicates. The difference is more pronounced on queries where the similarity clauses are more densely connected to the query, becoming of an order of magnitude in some cases.

We present an indexing scheme for triple-based graphs that supports join queries in worst-case optimal (wco) time within compact space. This scheme, called a ring , regards each triple as a cyclic string of length 3. Each rotation of the triples is lexicographically sorted and the values of the last attribute are stored as a column, so we obtain the order of the next column by stably re-sorting the triples by its attribute. We show that, by representing the columns with a compact data structure called a wavelet tree, this ordering enables forward and backward navigation between columns without needing pointers. These wavelet trees further support wco join algorithms and cardinality estimations for query planning. While traditional data structures such as B-Trees, tries, and so on, require 6 index orders to support all possible wco joins over triples, we can use one ring to index them all. This ring replaces the graph and uses only sublinear extra space, thus supporting wco joins in almost no space beyond storing the graph itself. Experiments querying a large graph (Wikidata) in memory show that the ring offers nearly the best overall query times while using only a small fraction of the space required by several state-of-the-art approaches. We then turn our attention to some theoretical results for indexing tables of arity d higher than 3 in such a way that supports wco joins. While a single ring of length d no longer suffices to cover all d ! orders, we need much fewer rings to index them all: O (2 d ) rings with a small constant. For example, we need 5 rings instead of 120 orders for d =5. We show that our rings become a particular case of what we dub order graphs , whose nodes are attribute orders and where stably sorting by some attribute leads us from an order to another, thereby inducing an edge labeled by the attribute. The index is then the set of columns associated with the edges, and a set of rings is just one possible graph shape. We show that other shapes, like for example a single ring instead of several ones of length d , can lead us to even smaller indexes, and that other more general shapes are also possible. For example, we handle d =5 attributes within space equivalent to 4 rings.

Conjunctive regular path queries (CRPQs) are one of the core classes of queries over graph databases. They are join intensive, inheriting their structure from the relational setting, but they also allow arbitrary length paths to connect points that are to be joined. However, despite their popularity, little is known about what are the best algorithms for processing CRPQs. We focus on worst-case optimal algorithms, which are algorithms that run in time bounded by the worst-case output size of queries, and have been recently deployed for simpler graph queries with very promising results. We show that the famous bound on the number of query results by Atserias, Grohe and Marx can be extended to CRPQs, but to obtain tight bounds one needs to work with slightly stronger cardinality profiles. We also discuss what algorithms follow from our analysis. If one pays the cost for fully materializing graph queries, then the techniques developed for conjunctive queries can be reused. If, on the other hand, one imposes constraint on the working memory of algorithms, then worst-case optimal algorithms must be adapted with care: the order of variables in which queries are processed can have striking implications on the running time of queries.

Worst-case optimal join algorithms have gained a lot of attention in the database literature. We now count several algorithms that are optimal in the worst case, and many of them have been implemented and validated in practice. However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality one either needs to build completely new indexes or must populate the database with several instantiations of indexes such as B \( + \) -trees. Either way, this means spending an extra amount of storage space that is typically one or two orders of magnitude more than what is required to store the raw data. We show that worst-case optimal algorithms can be obtained directly from a representation that regards the relations as point sets in variable-dimensional grids, without the need of any significant extra storage. Our representation is a compressed quadtreefor the static indexes and a quadtreebuilt on the fly that shares subtrees (which we dub a qdag) for intermediate results. We develop a compositional algorithm to process full join queries under this representation, which simulates navigation of the quadtreeof the output, and show that the running time of this algorithm is worst-case optimal in data complexity. We implement our index and compare it experimentally with state-of-the-art alternatives. Our experiments show that our index uses even less space than what is needed to store the data in raw form (and replaces it) and one or two orders of magnitude less space than the other indexes. At the same time, our query algorithm is competitive in time, even sharply outperforming other indexes in various cases. Finally, we extend our framework to evaluate more expressive queries from relational algebra, including not only joins and intersections but also unions and negations. To obtain optimality on those more complex formulas, we introduce a lazy version of qdagswe dub lqdags, which allow us navigate over the quadtreerepresenting the output of a formula while only evaluating what is needed from its components. We show that the running time of our query algorithms on this extended set of operations is worst-case optimal under some constraints. Moving to full relational algebra, we also show that lqdagscan handle selections and projections. While worst-case optimality is no longer guaranteed, we introduce a partial materialization scheme that extends results from Deep and Koutris regarding compressed representation of query results.

Various recent proposals increase the distinguishing power of Graph Neural Networks (GNNs) by propagating features between k-tuples of vertices. The distinguishing power of these “higher-order” GNNs is known to be bounded by the k-dimensional Weisfeiler-Leman (WL) test, yet their O(n^k) memory requirements limit their applicability. Other proposals infuse GNNs with local higher-order graph structural information from the start, hereby inheriting the desirable O(n) memory requirement from GNNs at the cost of a one-time, possibly non-linear, preprocessing step. We propose local graph parameter enabled GNNs as a framework for studying the latter kind of approaches and precisely characterize their distinguishing power, in terms of a variant of the WL test, and in terms of the graph structural properties that they can take into account. Local graph parameters can be added to any GNN architecture, and are cheap to compute. In terms of expressive power, our proposal lies in the middle of GNNs and their higher-order counterparts. Further, we propose several techniques to aide in choosing the right local graph parameters. Our results connect GNNs with deep results in finite model theory and finite variable logics. Our experimental evaluation shows that adding local graph parameters often has a positive effect for a variety of GNNs, datasets and graph learning tasks.

We present an indexing scheme that supports worst-case optimal (wco) joins over graphs within compact space. Supporting all possible wco joins using conventional data structures - based on B(+)-Trees, tries, etc. - requires 6 index orders in the case of graphs represented as triples. We rather propose a form of index, which we call a ring, that indexes each triple as a set of cyclic bidirectional strings of length 3. Rather than maintaining 6 orderings, we can use one ring to index them all. This ring replaces the graph and uses only sublinear extra space on top of the graph; in order words, the ring supports worst-case optimal graph joins in almost no space beyond storing the graph itself. We perform experiments using our representation to index a large graph (Wikidata) in memory, over which wco join algorithms are implemented. Our experiments show that the ring offers the best overall performance for query times while using only a small fraction of the space when compared with several state-of-the-art approaches.

Worst-case optimal join algorithms have gained a lot of attention in the database literature. We now count with several algorithms that are optimal in the worst case, and many of them have been implemented and validated in practice. However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality we either need to build completely new indexes, or we must populate the database with several instantiations of indexes such as B+-trees. Either way, this means spending an extra amount of storage space that may be non-negligible. We show that optimal algorithms can be obtained directly from a representation that regards the relations as point sets in variable-dimensional grids, without the need of extra storage. Our representation is a compact quadtree for the static indexes, and a dynamic quadtree sharing subtrees (which we dub a qdag) for intermediate results. We develop a compositional algorithm to process full join queries under this representation, and show that the running time of this algorithm is worst-case optimal in data complexity. Remarkably, we can extend our framework to evaluate more expressive queries from relational algebra by introducing a lazy version of qdags (lqdags). Once again, we can show that the running time of our algorithms is worst-case optimal.

As graph databases grow in popularity, decades of work in graph query languages and models are materialising in industry standards and in the construction of new graph database systems. However, this surge in graph systems has in turn opened up a series of new, interesting research problems related to graph databases. Our first set of problems has to do with more efficient ways of computing the answers of graph queries, specifically graph patterns, path queries, and combinations between them. Traditionally, researchers in graph databases have pointed out that relational systems are ill-equipped to process these types of queries, and if one looks at the performance of native graph database systems, there is clearly a lot of room for improvement. The talk focuses on two possible directions for improving the state of the art in graph query processing. The first is implementing worst-case optimal algorithms for processing graph patterns that traduce in relational queries with several joins. Some advances are already in development (see e.g. Nguyen, Dung, et al. "Join processing for graph patterns: An old dog with new tricks." GRADES'15. or Hogan, Aidan, et al. "A Worst-Case Optimal Join Algorithm for SPARQL." ISWC’19.), but we are still far from a full fledged solution: most algorithms require complex data structures, or need further support in terms of heuristics to select an order in which joins are processed. Second, we need to understand what is the best way of evaluating path queries (that is, finding all pairs of nodes connected by a path), in such a way that these results can be further integrated with other query results in a graph system pipeline. We already have complexity results regarding path computation and enumeration for different semantics of path queries (see e.g. Martens, Wim, and Tina Trautner. "Evaluation and enumeration problems for regular path queries." ICDT'18. or Bagan, Guillaume, Angela Bonifati, and Benoit Groz. "A trichotomy for regular simple path queries on graphs." PODS'13.), but still very little is known in terms of optimal processing of path queries when inside a tractable fragment. Our second set of problems is related to graph analytics, one of the current selling points of graph databases. Systems should be able to run more complex analytical queries involving tasks such as more complex path finding, centrality or clustering. It is also important to be able to run these algorithms not over native graphs, but perhaps over a certain set of nodes or edges previously selected by a graph query, and one may also want to pose further queries over the result of the analytics task. Finally, all of this should be done in an efficient way, specially in the prospect that graph databases may contain a huge amount of nodes. In this talk I will discuss possible approaches to perform these operations, covering aspects from the design of languages for graph analytics to efficient ways of processing them, and also comparing the expressive power of graph analytics solutions with other forms of graph computation.

A widespread security claim of the Bitcoin system, presented in the original Bitcoin white-paper, states that the security of the system is guaranteed as long as there is no attacker in possession of half or more of the total computational power used to maintain the system. This claim, however, is proved based on theoretically flawed assumptions. In the paper we analyze two kinds of attacks based on two theoretical flaws: the Block Discarding Attack and the Difficulty Raising Attack. We argue that the current theoretical limit of attacker's fraction of total computational power essential for the security of the system is in a sense not $\frac{1}{2}$ but a bit less than $\frac{1}{4}$, and outline proposals for protocol change that can raise this limit to be as close to $\frac{1}{2}$ as we want. The basic idea of the Block Discarding Attack has been noted as early as 2010, and lately was independently though-of and analyzed by both author of this paper and authors of a most recently pre-print published paper. We thus focus on the major differences of our analysis, and try to explain the unfortunate surprising coincidence. To the best of our knowledge, the second attack is presented here for the first time.

SHACL (SHape Constraint Language) is a W3C recommendation for validating graph-based data against a set of constraints (called shapes). Importantly, SHACL allows to define recursive shapes, i.e. a shape may refer to itself, directly of indirectly. The recommendation left open the semantics of recursive shapes, but proposals have emerged recently to extend the official semantics to support recursion. These proposals are based on the principle of possibility (or non-contradiction): a graph is considered valid against a schema if one can assign shapes to nodes in such a way that all constraints are satisfied. This semantics is not constructive, as it does not provide guidelines about how to obtain such an assignment, and it may lead to unfounded assignments, where the only reason to assign a shape to a node is that it allows validating the graph.

Navigational queries over RDF data are viewed as one of the main applications of graph query languages, and yet the standard model of graph databases—essentially labeled graphs—is different from the triples-based model of RDF. While encodings of RDF databases into graph data exist, we show that even the most natural ones are bound to lose some functionality when used in conjunction with graph query languages. The solution is to work directly with triples, but then many properties taken for granted in the graph database context (e.g., reachability) lose their natural meaning. Our goal is to introduce languages that work directly over triples and are closed, i.e., they produce sets of triples, rather than graphs. Our basic language is called TriAL, or Triple Algebra: it guarantees closure properties by replacing the product with a family of join operations. We extend TriAL with recursion and explain why such an extension is more intricate for triples than for graphs. We present a declarative language, namely a fragment of datalog, capturing the recursive algebra. For both languages, the combined complexity of query evaluation is given by low-degree polynomials. We compare our language with previously studied graph query languages such as adaptations of XPath, regular path queries, and nested regular expressions; many of these languages are subsumed by the recursive triple algebra. We also provide an implementation of recursive TriAL on top of a relational query engine, and we show its usefulness by running a wide array of navigational queries over real-world RDF data, while at the same time testing how our implementation compares to existing RDF systems.

Despite the fact that JSON is currently one of the most popular formats for exchanging data on the Web, there are very few studies on this topic and there is no agreement upon a theoretical framework for dealing with JSON. Therefore in this paper we propose a formal data model for JSON documents and, based on the common features present in available systems using JSON, we define a lightweight query language allowing us to navigate through JSON documents. We also introduce a logic capturing the schema proposal for JSON and study the complexity of basic computational tasks associated with these two formalisms.

JSON -- the most popular data format for sending API requests and responses -- is still lacking a standardized schema or meta-data definition that allows the developers to specify the structure of JSON documents. JSON Schema is an attempt to provide a general purpose schema language for JSON, but it is still work in progress, and the formal specification has not yet been agreed upon. Why this could be a problem becomes evident when examining the behaviour of numerous tools for validating JSON documents against this initial schema proposal: although they agree on most general cases, when presented with the greyer areas of the specification they tend to differ significantly. In this paper we provide the first formal definition of syntax and semantics for JSON Schema and use it to show that implementing this layer on top of JSON is feasible in practice. This is done both by analysing the theoretical aspects of the validation problem and by showing how to set up and validate a JSON Schema for Wikidata, the central storage for Wikimedia.

Applications of description logics (DLs) such as ontology-based data access (OBDA) require understanding of how to pose database queries over DL knowledge bases. While there have been many studies regarding traditional relational query formalisms such as conjunctive queries and their extensions, little attention has been paid to graph database queries, despite the fact that graph databases have essentially the same structure as knowledge bases. In particular, not much is known about the interplay between DLs and XPath. The latter is a powerful formalism for querying semistructured data: it is in the core of most practical query languages for XML trees, and it is also gaining popularity in theory and practice of graph databases. In this paper we make a step towards coupling knowledge bases and graph databases by studying how to answer powerful XPath-style queries over simple DLs like DL-Lite and EL. We start with adapting the definition of XPath to the DL context, and then proceed to study the complexity of evaluating XPath queries over knowledge bases. Results show that, while query answering is undecidable for the full XPath, by carefully tuning the shape of negation allowed in the queries we can arrive at XPath fragments that have a potential to be used in practice.

SPARQL has become the most popular language for querying RDF datasets, the standard data model for representing information in the Web. This query language has received a good deal of attention in the last few years: two versions of W3C standards have been issued, several SPARQL query engines have been deployed, and important theoretical foundations have been laid. However, many fundamental aspects of SPARQL queries are not yet fully understood. To this end, it is crucial to understand the correspondence between SPARQL and well-developed frameworks like relational algebra or first order logic. But one of the main obstacles on the way to such understanding is the fact that the well-studied fragments of SPARQL do not produce RDF as output. In this paper we embark on the study of SPARQL CONSTRUCT queries, that is, queries which output RDF graphs. This class of queries takes rightful place in the standards and implementations, but contrary to SELECT queries, it has not yet attracted a worth-while theoretical research. Under this framework we are able to establish a strong connection between SPARQL and well-known logical and database formalisms. In particular, the fragment which does not allow for blank nodes in output templates corresponds to first order queries, its well-designed sub-fragment corresponds to positive first order queries, and the general language can be re-stated as a data exchange setting. These correspondences allow us to conclude that the general language is not composable, but the aforementioned blank-free fragments are. Finally, we enrich SPARQL with a recursion operator and establish fundamental properties of this extension.

Graph databases are currently one of the most popular paradigms for storing data. One of the key conceptual differences between graph and relational databases is the focus on navigational queries that ask whether some nodes are connected by paths satisfying certain restrictions. This focus has driven the definition of several different query languages and the subsequent study of their fundamental properties. We define the graph query language of Regular Queries, which is a natural extension of unions of conjunctive 2-way regular path queries (UC2RPQs) and unions of conjunctive nested 2-way regular path queries (UCN2RPQs). Regular queries allow expressing complex regular patterns between nodes. We formalize regular queries as nonrecursive Datalog programs with transitive closure rules. This language has been previously considered, but its algorithmic properties are not well understood. Our main contribution is to show elementary tight bounds for the containment problem for regular queries. Specifically, we show that this problem is 2EXPSPACE-complete. For all extensions of regular queries known to date, the containment problem turns out to be non-elementary. Together with the fact that evaluating regular queries is not harder than evaluating UCN2RPQs, our results show that regular queries achieve a good balance between expressiveness and complexity, and constitute a well-behaved class that deserves further investigation.

The graph database model is currently one of the most pop-ular paradigms for storing data, used in applications such as social networks, biological databases and the Semantic Web. Despite the popularity of this model, the develop-ment of graph database management systems is still in its infancy, and there are several fundamental issues regarding graph databases that are not fully understood. Indeed, while graph query languages that concentrate on topological prop-erties are now well developed, not much is known about lan-guages that can query both the topology of graphs and their underlying data. Our goal is to conduct a detailed study of static analysis problems for such languages. In this paper we consider the containment problem for several recently proposed classes of queries that manipulate both topology and data: regu-lar queries with memory, regular queries with data tests, and graph XPath. Our results show that the problem is in general undecidable for all of these classes. However, we find natural fragments that enjoy much better static analy-sis properties: the containment problem is decidable, and its computational complexity ranges from PSPACE-complete to EXPSPACE-complete. We also propose several exten-sions of the classes and study containment for them. 1.

Graph data appears in a variety of application domains, and many uses of it, such as querying, matching, and transforming data, naturally result in incompletely specified graph data, that is, graph patterns. While queries need to be posed against such data, techniques for querying patterns are generally lacking, and properties of such queries are not well understood. Our goal is to study the basics of querying graph patterns. The key features of patterns we consider here are node and label variables and edges specified by regular expressions. We provide a classification of patterns, and study standard graph queries on graph patterns. We give precise characterizations of both data and combined complexity for each class of patterns. If complexity is high, we do further analysis of features that lead to intractability, as well as lower-complexity restrictions. Since our patterns are based on regular expressions, query answering for them can be captured by a new automata model. These automata have two modes of acceptance: one captures queries returning nodes, and the other queries returning paths. We study properties of such automata, and the key computational tasks associated with them. Finally, we provide additional restrictions for tractability, and show that some intractable cases can be naturally cast as instances of constraint satisfaction problems.

We study the problem of query containment of conjunctive queries over annotated databases. Annotations are typically attached to tuples and represent metadata, such as probability, multiplicity, comments, or provenance. It is usually assumed that annotations are drawn from a commutative semiring. Such databases pose new challenges in query optimization, since many related fundamental tasks, such as query containment, have to be reconsidered in the presence of propagation of annotations. We axiomatize several classes of semirings for each of which containment of conjunctive queries is equivalent to existence of a particular type of homomorphism. For each of these types, we also specify all semirings for which existence of a corresponding homomorphism is a sufficient (or necessary) condition for the containment. We develop new decision procedures for containment for some semirings which are not in any of these classes. This generalizes and systematizes previous approaches.

One of the main applications of description logics is the ontology-based data access model, which requires algorithms for query answering over ontologies. In fact, some description logics, like those in the DL-Lite family, are designed so that simple queries, such as conjunctive queries, are efficiently computable. In this paper we study counting aggregate queries over ontologies, i.e. queries which use aggregate functions COUNT and COUNT DISTINCT. We propose an intuitive semantics for certain answers for these queries, which conforms to the open world assumption. We compare our semantics with other approaches that have been proposed in different contexts. We establish data and combined computational complexity for the problems of answering counting aggregate queries over ontologies for several variants of DL-Lite.

Data exchange and schema mapping management have received little attention so far in the graph database scenario, and tools developed in this context for relational databases have significant drawbacks in the context of graph-structured data. In this paper we embark on the study of interoperability issues for graph databases, including schema mappings, data exchange and certain answers computation.

Querying RDF data is viewed as one of the main applications of graph query languages, and yet the standard model of graph databases -- essentially labeled graphs -- is different from the triples-based model of RDF. While encodings of RDF databases into graph data exist, we show that even the most natural ones are bound to lose some functionality when used in conjunction with graph query languages. The solution is to work directly with triples, but then many properties taken for granted in the graph database context (e.g., reachability) lose their natural meaning.

In the traditional data exchange setting, source instances are restricted to be complete in the sense that every fact is either true or false in these instances. Although natural for a typical database translation scenario, this restriction is gradually becoming an impediment to the development of a wide range of applications that need to exchange objects that admit several interpretations. In particular, we are motivated by two specific applications that go beyond the usual data exchange scenario: exchanging incomplete information and exchanging knowledge bases . In this article, we propose a general framework for data exchange that can deal with these two applications. More specifically, we address the problem of exchanging information given by representation systems , which are essentially finite descriptions of (possibly infinite) sets of complete instances. We make use of the classical semantics of mappings specified by sets of logical sentences to give a meaningful semantics to the notion of exchanging representatives, from which the standard notions of solution, space of solutions, and universal solution naturally arise. We also introduce the notion of strong representation system for a class of mappings , that resembles the concept of strong representation system for a query language. We show the robustness of our proposal by applying it to the two applications mentioned above: exchanging incomplete information and exchanging knowledge bases, which are both instantiations of the exchanging problem for representation systems. We study these two applications in detail, presenting results regarding expressiveness, query answering and complexity of computing solutions, and also algorithms to materialize solutions.

We study the problem of query containment of (unions of) conjunctive queries over annotated databases. Annotations are typically attached to tuples and represent metadata such as probability, multiplicity, comments, or provenance. It is usually assumed that annotations are drawn from a commutative semiring. Such databases pose new challenges in query optimization, since many related fundamental tasks, such as query containment, have to be reconsidered in the presence of propagation of annotations.

We consider data exchange for XML documents: given source and target schemas, a mapping between them, and a document conforming to the source schema, construct a target document and answer target queries in a way that is consistent with source information. The problem has primarily been studied in the relational context, in which data-exchange systems have also been built. Since many XML documents are stored in relations, it is natural to consider using a relational system for XML data exchange. However, there is a complexity mismatch between query answering in relational and XML data exchange, which indicates that restrictions have to be imposed on XML schemas and mappings, and on XML shredding schemes, to make the use of relational systems possible. We isolate a set of five requirements that must be fulfilled in order to have a faithful representation of the XML data-exchange problem by a relational translation. We then demonstrate that these requirements naturally suggest the inlining technique for dataexchange tasks. Our key contribution is to provide shredding algorithms for schemas, documents, mappings and queries, and demonstrate that they enable us to correctly perform XML data-exchange tasks using a relational system.

We study regular expressions that use variables, or parameters, which are interpreted as alphabet letters. We consider two classes of languages denoted by such expressions: under the possibility semantics, a word belongs to the language if it is denoted by some regular expression obtained by replacing variables with letters; under the certainty semantics, the word must be denoted by every such expression. Such languages are regular, and we show that they naturally arise in several applications such as querying graph databases and program analysis. As the main contribution of the paper, we provide a complete characterization of the complexity of the main computational problems related to such languages: nonemptiness, universality, containment, membership, as well as the problem of constructing NFAs capturing such languages. We also look at the extension when domains of variables could be arbitrary regular languages, and show that under the certainty semantics, languages remain regular and the complexity of the main computational problems does not change.

In the traditional data exchange setting, source instances are restricted to be complete in the sense that every fact is either true or false in these instances. Although natural for a typical database translation scenario, this restriction is gradually becoming an impediment to the development of a wide range of applications that need to exchange objects that admit several interpretations. In particular, we are motivated by two specific applications that go beyond the usual data exchange scenario: exchanging incomplete information and exchanging knowledge bases.

Graph data appears in a variety of application domains, and many uses of it, such as querying, matching, and transforming data, naturally result in incompletely specified graph data, i.e., graph patterns. While queries need to be posed against such data, techniques for querying patterns are generally lacking, and properties of such queries are not well understood.

In the last few years, a lot of attention has been paid to the specification and subsequent manipulation of schema mappings, a problem which is of fundamental importance in metadata management. There have been many achievements in this area, and semantics have been defined for operators on schema mappings such as composition and inverse. However, little research has been pursued towards providing formal tools to compare schema mappings, in terms of their ability to transfer data and avoid storing redundant information, which has hampered the development of foundations for more complex operators as many of them involve these notions.

The class of unions of conjunctive queries (UCQ) has been shown to be particularly well-behaved for data exchange; its certain answers can be computed in polynomial time (in terms of data complexity). However, this is not the only class with this property; the certain answers to any Datalog program can also can be computed in polynomial time. The problem is that both UCQ and Datalog do not allow negated atoms, as adding an unrestricted form of negation to these languages yields to intractability.

The inversion of schema mappings has been identified as one of the fundamental operators for the development of a general framework for metadata management. In fact, during the last years three alternative notions of inversion for schema mappings have been proposed (Fagin-inverse [10], quasi-inverse [14] and maximum recovery [2]). However, the procedures that have been developed for computing these operators have some features that limit their practical applicability. First, these algorithms work in exponential time and produce inverse mappings of exponential size. Second, these algorithms express inverses in some mappings languages which include features that are difficult to use in practice. A typical example is the use of disjunction in the conclusion of the mapping rules, which makes the process of exchanging data much more complicated. In this paper, we propose solutions for the two problems mentioned above. First, we provide a polynomial time algorithm that computes the three inverse operators mentioned above given a mapping specified by a set of tuple-generating dependencies (tgds). This algorithm uses an output mapping language that can express these three operators in a compact way and, in fact, can compute inverses for a much larger class of mappings. Unfortunately, it has already been proved that this type of mapping languages has to include some features that are difficult to use in practice and, hence, this is also the case for our output mapping language. Thus, as our second contribution, we propose a new and natural notion of inversion that overcomes this limitation. In particular, every mapping specified by a set of tgds admits an inverse under this new notion that can be expressed in a mapping language that slightly extends tgds, and that has the same good properties for data exchange as tgds. Finally, as our last contribution, we provide an algorithm for computing such inverses.