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Dissociation and propagation for approximate lifted inference with standard relational database management systems

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Abstract

Probabilistic inference over large data sets is a challenging data management problem since exact inference is generally #P-hard and is most often solved approximately with sampling-based methods today. This paper proposes an alternative approach for approximate evaluation of conjunctive queries with standard relational databases: In our approach, every query is evaluated entirely in the database engine by evaluating a fixed number of query plans, each providing an upper bound on the true probability, then taking their minimum. We provide an algorithm that takes into account important schema information to enumerate only the minimal necessary plans among all possible plans. Importantly, this algorithm is a strict generalization of all known PTIME self-join-free conjunctive queries: A query is in PTIME if and only if our algorithm returns one single plan. Furthermore, our approach is a generalization of a family of efficient ranking methods from graphs to hypergraphs. We also adapt three relational query optimization techniques to evaluate all necessary plans very fast. We give a detailed experimental evaluation of our approach and, in the process, provide a new way of thinking about the value of probabilistic methods over non-probabilistic methods for ranking query answers. We also note that the techniques developed in this paper apply immediately to lifted inference from statistical relational models since lifted inference corresponds to PTIME plans in probabilistic databases.

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Notes

  1. Also see [55] for a related discussion of fact finding algorithms, in which the approach of [78] and its use of the iterative propagation Eq. 1 is referred to as “pseudoprobabilistic”.

  2. W.l.o.g. we assume \(\varvec{\mathbf {x}}_i\) to be a tuple of only variables and don’t write the constants. Selections can always be directly pushed into the database before executing the query.

  3. Defined formally as \({\textit{ADom}}_{x_j} = \bigcup _{i: x_j \in {\texttt {Var}}(R_i)} \pi _{x_j}(R_i)\).

  4. Non-hierarchical queries can be in PTIME when considering functional dependencies or deterministic tables [12, 53] (see Sect. 5).

  5. Extensional approaches compute the probability of any formula as a function of the probabilities of its subformulas according to syntactic rules, regardless of how those were derived. Intensional approaches reason in terms of possible worlds and keep track of dependencies [56].

  6. Incidence matrices allow us to compactly reason about two types of relationships between variables and relations of sf-free CQs simultaneously: (i) in a column: a variable that is shared across relations, and (ii) in a row: relations that are joined by a variable. They thus allow us to reason about both the “query hypergraph” and the “dual query hypergraph” at the same time, which is helpful also for other types of problems involving sf-free CQs (see, e.g. [24]).

  7. A conjunctive k-chain query is a query q without self-joins in which each relation is binary, all relations are joined together, and there is no single variable common to more than two relations. Furthermore, the first and last variable are head variables and can be replaced by constants: \(q(x_1, x_{k+1}) {\,:\!\!-\,}R_1(x_1, x_2), R_2(x_2, x_3), \ldots , R_k(x_k, x_{k+1})\). The fact that relations are binary entails that the query hypergraph is actually a standard graph. Similarly, the fact that a variable is not common to more than two relations also entails the “dual hypergraph” to be a graph as well. The expression chain query derives from the observation that both its hypergraph and dual hypergraph resemble a simple chain.

  8. Notice that dissociating a table on any head variable has no implication on the probability of a query result as it does not change its lineage. We therefore only focus on dissociating existential variables.

  9. Recall that we say a query is connected if all subgoals are connected by considering only existential variables \({\texttt {EVar}}(q)\). In other words, when computing query components we remove head variables from the query: \(q - {\texttt {HVar}}(q)\). An alternative way to write this is to first substitute all head variables by constants \(q' = q[\varvec{\mathbf {a}} / \varvec{\mathbf {x}}]\) (here \(q[\varvec{\mathbf {a}} / \varvec{\mathbf {x}}]\) denotes the query obtained by substituting each head variable \(x_i \in \varvec{\mathbf {x}}\) with the constant \(a_i \in \varvec{\mathbf {a}}\)), then to let \(q_1, \ldots , q_k\) be the components of \(q'\) connected by any variable. The query is connected if \(k=1\), otherwise it is disconnected, and \(\forall i \ne j: {\texttt {Var}}(q_i) \cap {\texttt {Var}}(q_j) \subseteq {\texttt {HVar}}(q)\).

  10. This follows from the recursive definition of the unique safe plan of a query in Lemma 5: the top-most projection consists precisely of its separator variables.

  11. Note that if there are no existential variables (\(\varvec{\mathbf {z}} = \varvec{\mathbf {x}}_i\)), then there is no need for the projection operator and one could instead simplify to \(\mathscr {P} \leftarrow \{R_i(\varvec{\mathbf {z}})\}\), instead of \(\mathscr {P} \leftarrow \{\pi ^p_{\varvec{\mathbf {z}}} R_i(\varvec{\mathbf {x}}_i)\}\).

  12. A Boolean conjunctive k-star query is a query with k unary relations and one k-ary relation: \(q {\,:\!\!-\,}R_1(x_1), \ldots , R_k(x_k), U(x_1, \ldots , x_k)\). The fact that each variable appears in exactly two relations implies that the dual query hypergraph is actually a standard graph (the dual hypergraph of a query is defined by the relations as vertices and variables as the hyperedges). The expression star query derives from the observation that the query’s dual (hyper)graph resembles a star with the table U connected to all other relations.

  13. E.g., if \(\varvec{\mathbf {x}} = \{y \}\) and \(\varvec{\Gamma } = \{ x \rightarrow y, y \rightarrow z, z \rightarrow u \}\), then \({\varvec{\mathbf {x}}}^+ = \{y, z, u \}\).

  14. The time needed for the lineage query thus serves as minimum benchmark for any probabilistic approximation. The reported times for SampleSearch and MC are the sum of time for retrieving the lineage plus the actual calculations, without the time for reading and writing the input and output files for SampleSearch.

  15. Results for MC with other parameters of $2 are similar. However, the evaluation time for the experiments becomes quickly infeasible.

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Acknowledgments

This work was supported in part by NSF Grants IIS-0513877, IIS-0713576, IIS-0915054, IIS-1115188, IIS-1247469, and CAREER IIS-1553547. We like to thank Abhay Jha for help with the experiments in the workshop version of this paper, Alexandra Meliou for suggesting the name “dissociation”, and Vibhav Gogate for guidance in using his tool SampleSearch. WG would also like to thank Manfred Hauswirth for a small comment in 2007 that was crucial for the development of the ideas in this paper.

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  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Wolfgang Gatterbauer

  2. Department of Computer Science and Engineering, University of Washington, Seattle, WA, USA

    Dan Suciu

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Gatterbauer, W., Suciu, D. Dissociation and propagation for approximate lifted inference with standard relational database management systems. The VLDB Journal 26, 5–30 (2017). https://doi.org/10.1007/s00778-016-0434-5

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