Mathematische Logik

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Optimization and Equilibrium Problems with Equilibrium Constraints in Infinite-Dimensional Spaces

The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding sequential normal compactness (SNC) properties

Independence in CLP Languages

Studying independence of goals has proven very useful in the context of logic programming. In particular, it has provided a formal basis for powerful automatic parallelization tools, since independence ensures that two goals may be evaluated in parallel while preserving correctness and eciency. We extend the concept of independence to constraint logic programs (CLP) and prove that it also ensures the correctness and eciency of the parallel evaluation of independent goals. Independence for CLP languages is more complex than for logic programming as search space preservation is necessary but no longer sucient for ensuring correctness and eciency. Two additional issues arise. The rst is that the cost of constraint solving may depend upon the order constraints are encountered. The second is the need to handle dynamic scheduling. We clarify these issues by proposing various types of search independence and constraint solver independence, and show how they can be combined to allow dierent optimizations, from parallelism to intelligent backtracking. Sucient conditions for independence which can be evaluated \a priori" at run-time are also proposed. Our study also yields new insights into independence in logic programming languages. In particular, we show that search space preservation is not only a sucient but also a necessary condition for ensuring correctness and eciency of parallel execution

A resolution principle for clauses with constraints

We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification

Semiparametric Estimation of Long-Memory Models

This chapter reviews semiparametric methods of inference on different aspects of long memory time series. The main focus is on estimation of the memory parameter of linear models, analyzing bandwidth choice, bias reduction techniques and robustness properties of different estimates, with sorne emphasis on nonstationarity and trending behaviors. These techniques extend naturally to multivariate series, where the important issues are the estimation of the long-run relationship and testing for fractional cointegration. Specific techniques for the estimation of the degree of persistence of volatility for nonlinear time series are also considered

Logic, parallelism and semantic networks : the binary predicate execution model

This thesis develops the Binary Predicate Execution Model; a distributed, massively-parallel system for semantic networks and knowledge bases that is built on a subset of first-order predicate logic. The use of logic gives the model an easily-understood programming paradigm and a well-defined semantics of execution. When expressed in binary predicates, a simple graphical interpretation can be used. All program facts are represented in an assertion graph. Each vertex is associated with a term appearing in a fact and the edges are labeled with the predicate names. Similar graphs are also associated with each rule body and the query. Finding all possible solutions corresponds to finding all possible matches between the query graph and the assertion graph. Invoking a rule corresponds to substituting the graph of its body constrained by the dependencies between its arguments. This can be implemented in a parallel, message-passing fashion where the assertion graph vertices are active processing elements which asynchronously exchange messages identifying different parts of the query that remain to be matched and containing any binding information from previous matching required to accomplish this. The model is data-driven since every message can be immediately processed without the need for any centralized control or centralized memory. By restricting how functional terms can occur, distributed data structures and remote data look-ups for unification are eliminated. Thus, the model's performance on increasingly larger problems scales-up given increasingly larger machines in most cases. Architectural support for the model is investigated and simulation results of a relatively simple software implementation are reported. This suggests performance on the order of 10^5 logical inferences per second for 256 processing elements in an n-cube configuration. Further research directions, including that of increasing efficiency, are discussed