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Provenance Semirings Provenance Semirings Todd J. Green Grigoris Karvounarakis Val Tannen [email protected] [email protected] [email protected] Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA ABSTRACT We observe that in all four cases, the calculations with an- We show that relational algebra calculations for incomplete notations are strikingly similar. This suggests looking for an databases, probabilistic databases, bag semantics and why- algebraic structure on annotations that captures the above provenance are particular cases of the same general algo- as particular cases. We propose using commutative semir- rithms involving semirings. This further suggests a com- ings for this purpose. In fact, we can show that the laws of prehensive provenance representation that uses semirings of commutative semirings are forced by certain expected iden- polynomials. We extend these considerations to datalog and tities in RA. Having identified commutative semirings as semirings of formal power series. We give algorithms for the right algebraic structure, we argue that a symbolic rep- datalog provenance calculation as well as datalog evaluation resentation of semiring calculations is just what is needed to for incomplete and probabilistic databases. Finally, we show record, document, and track RA querying from input to out- that for some semirings containment of conjunctive queries put for applications which require rich provenance informa- is the same as for standard set semantics. tion. It is a standard philosophy in algebra that such sym- bolic representations form the most general such structure. Categories and Subject Descriptors In the case of commutative semirings, just as for rings, the symbolic representation is that of polynomials. We therefore H.2.1 [Database Management]: Data Models propose to use polynomials to capture provenance. Next we General Terms look to extend our approach to recursive datalog queries. To achieve this we combine semirings with fixed point theory. Theory, Algorithms The contributions of the paper are as follows: Keywords • We introduce K-relations, in which tuples are anno- Data provenance, data lineage, incomplete databases, prob- tated (tagged) with elements from K. We define a gen- abilistic databases, semirings, datalog, formal power series eralized positive algebra on K-relations and argue that K must be a commutative semiring (Section 3). 1. INTRODUCTION • For provenance semirings we propose polynomials Several forms of annotated relations have appeared in var- with integer coefficients, and we show that positive ious contexts in the literature. Query answering in these algebra semantics for any commutative semirings fac- settings involves generalizing the relational algebra (RA) to tors through the provenance semantics (Section 4). perform corresponding operations on the annotations. The seminal paper in incomplete databases [19] gener- • We extend these results to datalog queries by consid- alized RA to c-tables, where relations are annotated with ering semirings with fixed points (Section 5). Boolean formulas. In probabilistic databases, [17] and [33] generalized RA to event tables, also a form of annotated • For the (possibly infinite) provenance in datalog query relations. In data warehousing, [12] and [13] compute lin- answers we propose semirings of formal power series eages for tuples in the output of queries, in effect general- that are shown to be generated by finite algebraic izing RA to computations on relations annotated with sets systems of fixed point equations (Section 6). of contributing tuples. Finally, RA on bag semantics can be • We give algorithms for deciding the finiteness of these viewed as a generalization to annotated relations, where a formal power series, for computing them when finite, tuple’s annotation is a number representing its multiplicity. and for computing the coefficient of an arbitrary mono- mial otherwise (Section 7). • We show how to specialize our algorithms for com- Permission to make digital or hard copies of all or part of this work for puting full datalog answers when K is a finite dis- personal or classroom use is granted without fee provided that copies are tributive lattice, in particular for incomplete and not made or distributed for profit or commercial advantage and that copies probabilistic databases (Section 8). bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific • We consider query containment wrt K-relation se- permission and/or a fee. PODS’07, June 11–14, 2007, Beijing, China. mantics and we show that for unions of conjunctive Copyright 2007 ACM 978-1-59593-685-1/07/0006 ...$5.00. queries and when K is a distributive lattice, query a b c a b c a c a c a b c ? a b c b1 a c (b1 ∧ b1) ∨ (b1 ∧ b1) a c b1 d b e ? d b e b2 a e b1 ∧ b2 a e b1 ∧ b2 f g e ? f g e b3 d c b1 ∧ b2 d c b1 ∧ b2 (a) (b) d e (b2 ∧ b2) ∨ (b2 ∧ b2) ∨ (b2 ∧ b3) d e b2 a c f e (b3 ∧ b3) ∨ (b3 ∧ b3) ∨ (b2 ∧ b3) f e b3 a c a e (a) (b) n a e d e a c o ∅, a c , d e , f e , , , , d c d c f e f e d e Figure 2: Result of Imielinski-Lipski computation d e f e (c) a c Figure 1: A maybe-table and a query result a b c a c 2 · 2 + 2 · 2 = 8 a b c 2 a e 2 · 5 = 10 d b e 5 d c 2 · 5 = 10 f g e 1 d e 5 · 5 + 5 · 5 + 5 · 1 = 55 containment is the same as that given by standard set f e 1 · 1 + 1 · 1 + 5 · 1 = 7 semantics. (Section 9). (a) (b) 2. QUERIES ON ANNOTATED RELATIONS Figure 3: Bag semantics example We motivate our study by considering three important ex- amples of query answering on annotated relations and high- light the similarities between them. The first example comes from the study of incomplete tions we do here for multisets, and the Boolean calculations databases, where a simple representation system is the maybe- for the c-table. table [30, 18], in which optional tuples are annotated with a A third example comes from the study of probabilistic ‘?’, as in the example of Figure 1(a). Such a table represents databases, where tuples are associated with values from [0, 1] a set of possible worlds, and the answer to a query over such which represent the probability that the tuple is present in a table is the set of instances obtained by evaluating the the database. Answering queries over probabilistic tables re- query over each possible world. Thus, given a query like quires computing the correct probabilities for tuples in the output. To do this, Fuhr and R¨ollecke [17] and Zim´anyi [33] def ` ´ q(R) = πac πabR 1 πbcR ∪ πacR 1 πbcR introduced event tables, where tuples are annotated with probabilistic events, and they gave a query answering algo- the query result is the set of possible worlds shown in Fig- rithm for computing the events associated with tuples in the ure 1(c). Unfortunately, this set of possible worlds cannot query output.1 itself be represented by a maybe-table, intuitively because Figure 4(a) shows an example of an event table with as- whenever the tuples (a, e) and (d, c) appear, then so do (a, c) sociated event probabilities (e.g., x represents the event that and (d, e), and maybe-tables cannot represent such a depen- (a, b, c) appears in the instance, and x, y, z are assumed in- dency. dependent). Considering again the same query q as above, To overcome such limitations, Imielinski and Lipski [19] the Fuhr-R¨olleke-Zim´anyi query answering algorithm pro- introduced c-tables, where tuples are annotated with Boolean duces the event table shown in Figure 4(b). Note again the formulas called conditions. A maybe-table is a simple kind similarity between this table and the example earlier with c- of c-table, where the annotations are distinct Boolean vari- tables. The probabilities of tuples in the output of the query ables, as shown in Figure 1(b). In contrast to weaker rep- can be computed from this table using the independence of resentation systems, c-tables are expressive enough to be x and y. closed under RA queries, and the main result of [19] is an algorithm for answering RA queries on c-tables, producing another c-table as a result. On our example, this algorithm 3. POSITIVE RELATIONAL ALGEBRA produces the c-table shown in Figure 2(a), which can be In this section we attempt to unify the examples above simplified to the c-table shown in Figure 2(b); this c-table by considering generalized relations in which the tuples are represents exactly the set of possible worlds shown in Fig- annotated (tagged) with information of various kinds. Then, ure 1(c). we will define a generalization of the positive relational al- Another kind of table with annotations is a multiset or bag. gebra (RA+) to such tagged-tuple relations. The examples In this case, the annotations are natural numbers which rep- in Section 2 will turn out to be particular cases. resent the multiplicity of the tuple in the multiset. (A tuple We use here the named perspective [1] of the relational not listed in the table has multiplicity 0.) Query answering model in which tuples are functions t : U → D with U a on such tables involves calculating not just the tuples in the finite set of attributes and D a domain of values. We fix output, but also their multiplicities.
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