The Dichotomy of Evaluating Homomorphism-Closed Queries on Probabilistic Graphs

We study the problem of probabilistic query evaluation on probabilistic graphs, namely, tuple-independent probabilistic databases on signatures of arity two. Our focus is the class of queries that is closed under homomorphisms, or equivalently, the infinite unions of conjunctive queries. Our main result states that all unbounded queries from this class are #P-hard for probabilistic query evaluation. As bounded queries from this class are equivalent to a union of conjunctive queries, they are already classified by the dichotomy of Dalvi and Suciu (2012). Hence, our result and theirs imply a complete data complexity dichotomy, between polynomial time and #P-hardness, for evaluating infinite unions of conjunctive queries over probabilistic graphs. This dichotomy covers in particular all fragments of infinite unions of conjunctive queries such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries on arity-two signatures. Our result is shown by reducing from counting the valuations of positive partitioned 2-DNF formulae for some queries, or from the source-to-target reliability problem in an undirected graph for other queries, depending on properties of minimal models. The presented dichotomy result applies to even a special case of probabilistic query evaluation called generalized model counting, where fact probabilities must be 0, 0.5, or 1.Comment: 30 pages. Journal version of the ICDT'20 paper https://drops.dagstuhl.de/opus/volltexte/2020/11939/. Submitted to LMCS. The previous version (version 2) was the same as the ICDT'20 paper with some minor formatting tweaks and 7 extra pages of technical appendi

A Dichotomy for Homomorphism-Closed Queries on Probabilistic Graphs

We study the problem of probabilistic query evaluation (PQE) over probabilistic graphs, namely, tuple-independent probabilistic databases (TIDs) on signatures of arity two. Our focus is the class of queries that is closed under homomorphisms, or equivalently, the infinite unions of conjunctive queries, denoted UCQ?. Our main result states that all unbounded queries in UCQ? are #P-hard for PQE. As bounded queries in UCQ? are already classified by the dichotomy of Dalvi and Suciu [Dalvi and Suciu, 2012], our results and theirs imply a complete dichotomy on PQE for UCQ? queries over probabilistic graphs. This dichotomy covers in particular all fragments in UCQ? such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries on arity-two signatures. Our result is shown by reducing from counting the valuations of positive partitioned 2-DNF formulae (#PP2DNF) for some queries, or from the source-to-target reliability problem in an undirected graph (#U-ST-CON) for other queries, depending on properties of minimal models

Tuple-Independent Representations of Infinite Probabilistic Databases

Probabilistic databases (PDBs) are probability spaces over database instances. They provide a framework for handling uncertainty in databases, as occurs due to data integration, noisy data, data from unreliable sources or randomized processes. Most of the existing theory literature investigated finite, tuple-independent PDBs (TI-PDBs) where the occurrences of tuples are independent events. Only recently, Grohe and Lindner (PODS '19) introduced independence assumptions for PDBs beyond the finite domain assumption. In the finite, a major argument for discussing the theoretical properties of TI-PDBs is that they can be used to represent any finite PDB via views. This is no longer the case once the number of tuples is countably infinite. In this paper, we systematically study the representability of infinite PDBs in terms of TI-PDBs and the related block-independent disjoint PDBs. The central question is which infinite PDBs are representable as first-order views over tuple-independent PDBs. We give a necessary condition for the representability of PDBs and provide a sufficient criterion for representability in terms of the probability distribution of a PDB. With various examples, we explore the limits of our criteria. We show that conditioning on first order properties yields no additional power in terms of expressivity. Finally, we discuss the relation between purely logical and arithmetic reasons for (non-)representability

A dichotomy for homomorphism−closed queries on probabilistic graphs

We study the problem of probabilistic query evaluation (PQE) over probabilistic graphs, namely, tuple-independent probabilistic databases (TIDs) on signatures of arity two. Our focus is the class of queries that is closed under homomorphisms, or equivalently, the infinite unions of conjunctive queries, denoted UCQ∞. Our main result states that all unbounded queries in UCQ∞ are #P-hard for PQE. As bounded queries in UCQ∞ are already classified by the dichotomy of Dalvi and Suciu [Dalvi and Suciu, 2012], our results and theirs imply a complete dichotomy on PQE for UCQ∞ queries over probabilistic graphs. This dichotomy covers in particular all fragments in UCQ∞ such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries on arity-two signatures. Our result is shown by reducing from counting the valuations of positive partitioned 2-DNF formulae (#PP2DNF) for some queries, or from the source-to-target reliability problem in an undirected graph (#U-ST-CON) for other queries, depending on properties of minimal models