We give the first two algorithms to enumerate all binary words of {0,1}^𝓁 (like Gray codes) while ensuring that the delay and the auxiliary space is independent from 𝓁, i.e., constant time for each word, and constant memory in addition to the 𝓁 bits storing the current word. Our algorithms are given in two new computational models: tape machines and deque machines. We also study more restricted models, queue machines and stack machines, and show that they cannot enumerate all binary words with constant auxiliary space, even with unrestricted delay. A tape machine is a Turing machine that stores the current binary word on a single working tape of length 𝓁 (which never increases), using no other tape. The machine has a single head and must edit its tape to reach all possible words of {0,1}^𝓁, and output them (in unit time, by entering special output states), with no duplicates. Hence a tape machine uses constant auxiliary space by definition (up to the head position). We construct a tape machine that achieves this task with constant delay between consecutive outputs, so that the machine implements a so-called skew-tolerant quasi-Gray code. We then construct a more involved tape machine that implements a Gray code. A deque machine stores the current binary word on a double-ended queue of length 𝓁, and stores a constant-size internal state. It works as a tape machine, except that it modifies the content of the deque by performing push and pop operations on the endpoints. Hence again a deque machine uses constant auxiliary space by definition. We construct deque machines that enumerate all words of {0,1}^𝓁 with constant-delay. The main technical challenge in this model is to correctly detect when enumeration has finished.

The view-based query determinacy problem asks, given a view 𝒱 and a query Q whether the information contained in 𝒱 always suffices to answer the query Q. It is a notoriously hard problem that is known to be undecidable even in very restricted settings. Here, we study it in the context of graph databases and regular path queries and views evaluated under several "walk-based semantics", that is, semantics in which queries and views return the matched walks in full as opposed to the classical "endpoint semantics". Our main finding is that view-based query determinacy is decidable for regular path queries under both trail and shortest walk semantics.

We consider the Distinct Shortest Walks problem. Given two vertices s and t of a graph database D and a regular path query, we want to enumerate all walks of minimal length from s to t that carry a label that conforms to the query. Usual theoretical solutions turn out to be inefficient when applied to graph models that are closer to real-life systems, in particular because edges may carry multiple labels. Indeed, known algorithms may repeat the same answer exponentially many times. We propose an efficient algorithm for graph databases with multiple labels. The preprocessing runs in O(DxA) and the delay between two consecutive outputs is in O(λxA), where A is a nondeterministic automaton representing the query and L is the minimal length. The algorithm can handle epsilon-transitions in A or queries given as regular expressions at no additional cost.

GQL (Graph Query Language) is being developed as a new ISO standard for graph query languages to play the same role for graph databases as SQL plays for relational. In parallel, an extension of SQL for querying property graphs, SQL/PGQ, is added to the SQL standard; it shares the graph pattern matching functionality with GQL. Both standards (not yet published) are hard-to-understand specifications of hundreds of pages. The goal of this paper is to present a digest of the language that is easy for the research community to understand, and thus to initiate research on these future standards for querying graphs. The paper concentrates on pattern matching features shared by GQL and SQL/PGQ, as well as querying facilities of GQL.

RPQs (regular path queries) are an important building block of most query languages for graph databases. They are generally evaluated under homomorphism semantics; in particular only the endpoints of the matched walks are returned. However, practical applications often need the full matched walks to compute aggregate values. In those cases, homomorphism semantics are not suitable since the number of matched walks can be infinite. Hence, graph-database engines adapt the semantics of RPQs, often neglecting theoretical red flags. For instance, the popular query language Cypher uses trail semantics, which ensures the result to be finite at the cost of making computational problems intractable. We propose a new kind of semantics for RPQs, including in particular simple-run and binding-trail semantics, as a candidate to reconcile theoretical considerations with practical aspirations. Both ensure the output to be finite in a way that is compatible with homomorphism semantics: projection on endpoints coincides with homomorphism semantics. Hence, testing the emptiness of result is tractable, and known methods readily apply. Moreover, simple-run and binding-trail semantics support bag semantics, and enumeration of the bag of results is tractable.

The development of practical query languages for graph databases runs well ahead of the underlying theory. The ISO committee in charge of database query languages is currently developing a new standard called Graph Query Language (GQL) as well as an extension of the SQL Standard for querying property graphs represented by a relational schema, called SQL/PGQ. The main component of both is the pattern matching facility, which is shared by the two standards. In many aspects, it goes well beyond RPQs, CRPQs, and similar queries on which the research community has focused for years. Our main contribution is to distill the lengthy standard specification into a simple Graph Pattern Calculus (GPC) that reflects all the key pattern matching features of GQL and SQL/PGQ, and at the same time lends itself to rigorous theoretical investigation. We describe the syntax and semantics of GPC, along with the typing rules that ensure its expressions are well-defined, and state some basic properties of the language. With this paper we provide the community a tool to embark on a study of query languages that will soon be widely adopted by industry.

As graph databases become widespread, the International Organization for Standardization (ISO) and International Electrotechnical Commission (IEC) have approved a project to create GQL, a standard property graph query language. This complements the SQL/PGQ project, which specifies how to define graph views over a SQL tabular schema, and to run read-only queries against them.

The Cypher property graph query language is an evolving language, originally designed and implemented as part of the Neo4j graph database, and it is currently used by several commercial database products and researchers. We describe Cypher 9, which is the first version of the language governed by the openCypher Implementers Group. We first introduce the language by example, and describe its uses in industry. We then provide a formal semantic definition of the core read-query features of Cypher, including its variant of the property graph data model, and its ASCII Art graph pattern matching mechanism for expressing subgraphs of interest to an application. We compare the features of Cypher to other property graph query languages, and describe extensions, at an advanced stage of development, which will form part of Cypher 10, turning the language into a compositional language which supports graph projections and multiple named graphs.

Schema mappings are a fundamental concept in data integration and exchange, and they have been thoroughly studied in different data models. For graph data, however, mappings have been studied in a restricted context that, unlike real-life graph databases, completely disregards the data they store. Our main goal is to understand query answering under graph schema mappings - in particular, in exchange and integration of graph data - for graph databases that mix graph structure with data. We show that adding data querying alters the picture in a significant way.

We consider the view determinacy problem over graph databases for queries defined as (possibly infinite) unions of path queries. These queries select pairs of nodes in a graph that are connected through a path whose length falls in a given set. A view specification is a set of such queries. We say that a view specification V determines a query Q if, for all databases D, the answers to V on D contain enough information to answer Q. Our main result states that, given a view V, there exists an explicit bound that depends on V such that we can decide the determinacy problem for all queries that ask for a path longer than this bound, and provide first-order rewritings for the queries that are determined. We call this notion asymptotic determinacy. As a corollary, we can also compute the set of almost all path queries that are determined by V.

Testing if an incomplete description of an XML document is consistent, that is, if it describes a real document conforming to the imposed schema, amounts to deciding if a given tree pattern can be matched injectively into a tree accepted by a fixed automaton. This problem can be solved in polynomial time for patterns that use the child relation and the sibling order, but do not use the descendant relation. For general patterns the problem is in NP, but no lower bound has been known so far. We show that the problem is NP-complete already for patterns using only child and descendant relations. The source of hardness turns out to be the interplay between these relations: for patterns using only descendant we give a polynomial algorithm. We also show that the algorithm can be adapted to patterns using descendant and following-sibling, but combining descendant and next-sibling leads to intractability.

We consider query answering using views on graph databases, i.e. databases structured as edge-labeled graphs. We mainly consider views and queries specified by Regular Path Queries (RPQ). These are queries selecting pairs of nodes in a graph database that are connected via a path whose sequence of edge labels belongs to some regular language. We say that a view V determines a query Q if for all graph databases D, the view image V(D) always contains enough information to answer Q on D. In other words, there is a well defined function from V(D) to Q(D). Our main result shows that when this function is monotone, there exists a rewriting of Q as a Datalog query over the view instance V(D). In particular the rewriting query can be evaluated in time polynomial in the size of V(D). Moreover this implies that it is decidable whether an RPQ query can be rewritten in Datalog using RPQ views.