Shape Analysis concerns the problem of determining "shape invariants"...

Views are a central component of both traditional database systems and new applications such as data warehouses. Very often the desired views (e.g. the transitive closure) cannot be defined in the standard language of the underlying database system. Fortu-nately, it is often possible to incrementally maintain these views using the standard language. For exam-ple, transitive closure of acyclic graphs, and of undi-rected graphs, can be maintained in relational cal-culus after both single edge insertions and deletions. Many such results have been published in the theoret-ical database community. The purpose of this survey is to make these useful results known to the wider database research and development community. There are many interesting issues involved in the maintenance of recursive views. A maintenance al-gorithm may be applicable to just one view, or to a class of views specified by a view definition language such as Datalog. The maintenance algorithm can be specified in a maintenance language of different ex-pressiveness, such as the conjunctive queries, the re-lational calculus or SQL. Ideally, this maintenance language should be less expensive than the view def-inition language. The maintenance algorithm may allow updates of different kinds, such as just single tu-ple insertions, just single tuple deletions, special set-based insertions and/or deletions, or combinations thereof. The view maintenance algorithms may also need to maintain auxiliary relations to help maintain the views of interest. It is of interest to know the minimal arity necessary for these auxiliary relations

In databases, as in other computational settings, one would like to efficiently update answers to queries while minor changes are being made to the data. Dynamic complexity asks what resources are required to perform such updates. In this paper our main focus will be a particular dynamic graph problem, tree isomorphism, and the efficiency of our update scheme will be measured in terms of how expressive a query language is required to express our update. Working in the framework developed by [DS93, PI97] for dynamic query evaluation, we show that dynamic tree isomorphism can be performed via first-order updates to a relational database (in [DS93] this framework is called a first-order incremental evaluation system, and in [PI97] it is called Dyn-FO). In [EI95] it was shown that tree-isomorphism can not be expressed in first-order logic augmented with a transitive closure operator and counting, (FO + TC + COUNT) (but without ordering). We thus obtain a first example of a graph problem...

this paper by reducing 3k + 1 to k + 1, or k, or even a constant. We will prove this by modifying Cai's result [3] and by modifying the reduction used in [7]. Section 2 provides a brief review the notion of "first-order incremental evaluation systems ". Section 3 establishes a necessary technical lemma, which is a variant of Cai's theorem. Section 4 gives the proof of the above theorem. 2 First-Order Incremental Evaluation Systems

It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever updates to the former occur. This leads to the concept of an incremental evaluation system, or IES. Much is already known about the theory of IES but very little has been translated into practice. The purpose of this paper is to ll in this gap by providing a gentle introduction to and an overview of some recent theoretical results on IES. The introduction is through the translation into SQL of three interesting positive maintenance results that have practical importance { the maintenance of the transitive closure of acyclic graphs, of undirected graphs, and of arbitrary directed graphs. Interestingly, these examples also allow ustoshow the relationship between power and cost in the incremental maintenance of database queries. 1

Abstract. Dynamic complexity asks for the effort needed to maintain the information about properties of a structure under operations changing the structure. This paper introduces a refined notion of dynamic problems which takes the initial structure into account. It develops the basic structural complexity notions accordingly. It also shows that the dynamic version of the LOGCFL-complete problem D2LREACH(acyclic) can be maintained with first-order updates. 1

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.

It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever updates to the former occur. This leads to the concept of an incremental evaluation system, or IES. Much is already known about the theory of IES but very little has been translated into practice. The purpose of this paper is to fill in this gap by providing a gentle introduction to and an overview of some recent theoretical results on IES. The introduction is through the translation into SQL of three interesting positive maintenance results that have practical importance -- the maintenance of the transitive closure of acyclic graphs, of undirected graphs, and of arbitrary directed graphs. Interestingly, these examples also allow us to show the relationship between power and cost in ...

. We consider IES(SQL), the incremental evaluation system over an SQL-like language with grouping, arithmetics, and aggregation. We show that every second order query is in IES(SQL) and that there are PSPACE-complete queries in IES(SQL). We further show that every PSPACE query is in IES(SQL) augmented with a deterministic transitive closure operator. Lastly, we consider ordered databases and provide a complete analysis of a hierarchy on IES(SQL) defined with respect to arity-bounded auxiliary relations. 1 Introduction There are two kinds of incremental query evaluation in general. The first kind is where a query is definable in the ambient language. In this case, incremental evaluation is possible and the main problem is to find efficient algorithms to perform it [12, 13, etc.] The second kind is where a query is not definable in the ambient language, and it is the main interest of this paper. The main questions addressed in this setting deal with conditions under which it is ...

We investigate the logical resources required to maintain knowledge about a property of a finite structure that undergoes an ongoing series of local changes such as insertion or deletion of tuples to basic relations. Our framework is closely related to the Dyn-FO-framework of Patnaik and Immerman and the FOIES-framework of Dong, Libkin, Su and Wong, and also builds on work of Weber and Schwentick. We assume that the dynamic process starts with an arbitrary, nonempty structure, but in contrast to previous work, we assume that, in general, structures are unordered. We show how to modify known dynamic algorithms for symmetric reachability, bipartiteness, k-edge connectivity and more, to work also without an order and with dynamic processes starting at an arbitrary graph. A history independent dynamic system (also called deterministic or memoryless) is one that maintains all auxiliary information independent of the update order. In 1997, Dong and Su posed the problem whether there exist history independent dynamic systems with FO-updates for symmetric reachability or bipartiteness. We give a positive answer to this question. We further show that there is a history independent system for tree isomorphism with FO+C-updates. On the other hand we show that on unordered structures first-order logic is too weak to maintain enough information to answer the equal cardinality query and the tree isomorphism query dynamically.