Arity Bounds in First-Order Incremental Evaluation and Definition of Polynomial Time Database Queries

Abstract

After inserting a tuple into or deleting a tuple from a database, a firstorder incremental evaluation system (or a``foies'') for a database query derives the new query answer by using a first-order query on the new database, the old answer, and perhaps some stored auxiliary relations. Moreover, the auxiliary relations must also be maintained similarly using first-order queries.In this paper we measure the space needed by foies in terms of the maximal arity of the auxiliary relations and present results on the existence and nonexistence of such space-restricted foies for a variety of graph queries, including counting and transitive closure. We construct space efficient foies for these queries and show that the arity bounds are tight using Ehrenfeucht Fra@ sse games. In particular, for transitive closure over undirected graphs, the minimum arity bound of its foies is shown to be exactly two; this resolves an open problem raised by Patnaik and Immerman in 1994. For the general case, we show that the arity-based hierarchy is strict for all arities. The strictness proof uses queries with input relations having arities much larger than the auxiliary relations. It is still open whether the hierarchy remains strict for arities two or greater when the input relations of queries have arities bounded by a fixed number, such as two, or by the arities of the auxiliary relations.Finally, we consider a variation of foies where the cost of storing the answer to the query is also``charged.'' We show that the arity hierarchy in this case is also strict. The positions of queries in the two arity-based hierarchies can differ; we give the positions in this hierarchy of the queries considered for the other arity hierarchy. ]

Small Dynamic Complexity Classes

“…Hesse showed that reachability on deterministic graphs can be maintained by quantifier-free update formulas (the corresponding dynamic complexity class is called DynProp), and on undirected graphs with quantifier-free update formulas that can also use auxiliary functions (the corresponding dynamic complexity class is called DynQF) [Hes03b]. Moreover other graph queries such as 2-colorability and the tree-isomorphism query have been studied, and shown to be contained in DynFO [DS98,Ete98].…”

“…Therefore several restrictions of DynFO have been studied, among them the restriction of the arity of auxiliary relations [DS98,DLW03] and syntactic restrictions of first-order update formulas [GMS12]. For a more detailed overview over previous work on lower bounds we refer to Chapter 4, which is devoted to lower bounds and methods for proving them.…”

“…Among them notions of reducibility and complete problems for dynamic complexity classes [PI97,HI02,WS07], the (in)dependence of the auxiliary relations from the actual input relations [DS97,DS98,GS12], and the relation of dynamic and static complexity classes [PI94,Ete98,WS07,GMS12]. For a detailed overview over the latter aspect we refer to Chapter 3.…”

“…Many other queries have been shown to be in DynFO as well, e.g. two-colorability of graphs [DS98] and all context-free languages are in DynFO [GMS12]. Very recently reachability for arbitrary (directed) graphs has been shown to be maintainable in DynFO [DKM + 15].…”

“…In the following we shortly discuss lower bound results for fragments obtained by restricting the arity of the auxiliary relations, by restricting the syntax of the update formulas and by restricting the power of the initialization mapping. The study of bounded arity auxiliary relations was started by Dong and Su (see, for example, [DS98]). They showed that unary auxiliary relations are not sufficient to maintain the reachability query and several other queries with first-order updates.…”