Relational Queries Computable in Polynomial Time

We characterize the polynomial time computable queries as those ex pressible in relational calculus plus a least xed point operator and a total ordering on the universe We also show that even without the ordering one application of xed point su ces to express any query expressible with several alternations of xed point and negation This proves that the xed point query hierarchy suggested by Chandra and Harel collapses at the rst xed point level It is also a general result showing that in nite model theory one application of xed point su ces Introduction and Summary Query languages for relational databases have received considerable attention In Codd showed that two natural languages for queries one algebraic and the other a version of rst order predicate calculus have identical powers of expressibility Cod Query languages which are as expressive as Codd s Relational Calculus are sometimes called complete This term is misleading however because many interesting queries are not expressible in complete languages In Aho and Ullman AU noted that relational calculus does not su ce to express the transitive closure property They suggested adding a least xed point operator to relational calculus in order to create a query language which can express transitive closure In Chandra and Harel CH b studied the expressive power of relational calculus with added primitives such Research supported by NSF Grant MCS and by an NSF postdoctoral fellowship A preliminary version of this paper appeared in th ACM STOC Symp as a least xed point operator They de ne a Fixed Point Hierarchy of query classes the queries in each particular class being those expressible with a certain number of applications of the least xed point operator followed by a certain number of alternations of of ordinary quanti cation and negation In this paper we show Theorem The Fixed Point Hierarchy collapses at the rst xed point level That is any query expressible with several applications of least xed point can already be expressed with one We also show Theorem Let L be a query language consisting of relational calculus plus the least xed point operator Suppose that L contains a relation symbol for a total ordering relation on the domain e g lexicographic ordering Then the queries expressible in L are exactly the queries computable in polynomial time Theorem was discovered independently by M Vardi Va It gives a syn tactic categorization of those queries which can be answered in polynomial time Of course queries requiring polynomial time in the size of the database are usu ally prohibitively expensive We also consider weaker languages for expressing less complex queries Background and Notation This section will brie y de ne and give examples of the objects under con sideration The reader is referred to Ull End AHU and Mo for excellent discussions of relational query languages rst order predicate calculus computational complexity and least xed points respectively First a relational database B hU R R ak k c csi is simply a rst order structure with nite universe U For i k Ri i is an ai ary relation on U i e Ri Ui The superscripts ai will be omitted where confusion does not arise The constants c cs are elements of U As an example we can consider a database B hU F P H Abrahami where U is a nite set of people U fAbraham Isaac Sarah Leah g F is a monadic relation true of the female elements of U