Conjunctions Of Inequalities Research Articles

Maintenance of implication integrity constraints under updates to constraints

Semantic integrity constraints are used for enforcing the integrity of the database as well as for improving the efficiency of the database utilization. Although semantic integrity constraints are usually much more static as compared to the data itself, changes in the data semantics may necessitate corresponding changes in the constraint base. In this paper we address the problems related with maintaining a consistent and non-redundant set of constraints satisfied by the database in the case of updates to the constraint base. We consider implication constraints as semantic integrity constraints. The constraints are represented as conjunctions of inequalities. We present a methodology to determine whether a constraint is redundant or contradictory with respect to a set of constraints. The methodology is based on the partitioning of the constraint base which improves the efficiency of algorithms that check whether a constraint is redundant or contradictory with respect to a constraint base.

Classical numerical methods in engineering: a note on existential quantifier elimination under parametric inequality constraints

In this paper, an attempt is made to show the usefulness of computational quantifier elimination (CQE) techniques in computer algebra inside classical numerical methods in engineering for the derivation of feasibility (consistency) conditions in problems with weakly parametric linear inequality constraints (with the parameters appearing only in their right-hand sides). A simple, but non-trivial, straight edge-crack problem in fracture mechanics under linear inequality constraints both on the applied loading along the crack faces and on the value of the stress intensity factor at the crack tip (associated with the Green/weight function method, numerical approximations and classical numerical integration) is used for an elementary illustration of the proposed approach. In this application, the method used tries to imitate the theoretical principle of the linear programming methods. The manually obtained related result is expressed as a disjunction of conjunctions of inequalities (as is frequently the case in similar CQE problems), and concrete numerical results are also displayed. The related influence of various approximations and the application of the trapezoidal quadrature rule are also considered in some detail. Further possibilities could concern the application of the approach to other numerical methods in engineering (such as to the finite and the boundary element methods, to singular and hypersingular integral equation methods, etc.) combined with efficient algorithms for linear inequality constraints such as the old Fourier and the recent Weispfenning elimination methods. © 1998 John Wiley & Sons, Ltd.

On the representation and querying of sets of possible worlds

We represent a set of possible worlds using an incomplete information database. The representation techniques that we study form a hierarchy, which generalizes relations of constants. This hierarchy ranges from the very simple Codd-table, (i e , a relation of constants and distinct variables called nulls, which stand for values present but unknown), to much more complex mechanisms involving views on conditioned-tables, (i e , queries on Codd-tables together with conditions). The views we consider are the queries that have polynomial data-complexity on complete information databases. Our conditions are conjunctions of equalities and inequalities. (1) We provide matching upper and lower bounds on the data-complexity of testing containement , membership , and uniqueness for sets of possible worlds and we fully classify these problems with respect to our representation hierarchy. The most surprising result in this classification is that it is complete in Π 2 p , whether a set of possible worlds represented by a Codd-table is a subset of a set of possible worlds represented by a Codd-table with one conjuction of inequalities. (2) We investigate the data-complexity of querying incomplete information databases. We examine both asking for certain facts and for possible facts . Our approach is algebraic but our bounds also apply to logical databases. We show that asking for a certain fact is coNP-complete, even for a fixed first order query on a Codd-table. We thus strengthen a lower bound of [16], who showed that this holds for a Codd-table with a conjunction of inequalities. For each fixed positive existential query we present a polynomial algorithm solving the bounded possible fact problem of this query on conditioned-tables. We show that our approach is, in a sense, the best possible, by deriving two NP-completeness lower bounds for the bounded possible fact problem when the fixed query contains either negation or recursion.