Probabilistic Analysis of Random Mixed Horn Formulas

We present a probabilistic analysis of random mixed Horn formulas (MHF), i.e., formulas in conjunctive normal form consisting of a positive monotone part of quadratic clauses and a part of Horn clauses, with m clauses, n variables, and up to n literals per Horn clause. For MHFs parameterized by n and m with uniform distribution of instances and for large n, we derive upper bounds for the expected number of models. For the class of random negative MHFs, where only monotone negative Horn clauses are allowed to occur, we give a lower bound for the probability that formulas from this class are satisfiable. We expect that the model studied theoretically here may be of interest for the determination of hard instances, which are conjectured to be found in the transition area from satisfiability to unsatisfiability of the instances from the parameterized classes of formulas

Probabilistic Analysis of Random Mixed Horn Formulas

We present a probabilistic analysis of random mixed Horn formulas (MHF), i.e., formulas in conjunctive normal form consisting of a positive monotone part of quadratic clauses and a part of Horn clauses, with m clauses, n variables, and up to n literals per Horn clause. For MHFs parameterized by n and m with uniform distribution of instances and for large n, we derive upper bounds for the expected number of models. For the class of random negative MHFs, where only monotone negative Horn clauses are allowed to occur, we give a lower bound for the probability that formulas from this class are satisfiable. We expect that the model studied theoretically here may be of interest for the determination of hard instances, which are conjectured to be found in the transition area from satisfiability to unsatisfiability of the instances from the parameterized classes of formulas

Enhanced genetic algorithm-based fuzzy multiobjective strategy to multiproduct batch plant design

This paper addresses the problem of the optimal design of batch plants with imprecise demands in product amounts. The design of such plants necessary involves how equipment may be utilized, which means that plant scheduling and production must constitute a basic part of the design problem. Rather than resorting to a traditional probabilistic approach for modeling the imprecision on product demands, this work proposes an alternative treatment by using fuzzy concepts. The design problem is tackled by introducing a new approach based on a multiobjective genetic algorithm, combined wit the fuzzy set theory for computing the objectives as fuzzy quantities. The problem takes into account simultaneous maximization of the fuzzy net present value and of two other performance criteria, i.e. the production delay/advance and a flexibility index. The delay/advance objective is computed by comparing the fuzzy production time for the products to a given fuzzy time horizon, and the flexibility index represents the additional fuzzy production that the plant would be able to produce. The multiobjective optimization provides the Pareto's front which is a set of scenarios that are helpful for guiding the decision's maker in its final choices. About the solution procedure, a genetic algorithm was implemented since it is particularly well-suited to take into account the arithmetic of fuzzy numbers. Furthermore because a genetic algorithm is working on populations of potential solutions, this type of procedure is well adapted for multiobjective optimization

EPR Paradox,Locality and Completeness of Quantum Theory

The quantum theory (QT) and new stochastic approaches have no deterministic prediction for a single measurement or for a single time -series of events observed for a trapped ion, electron or any other individual physical system. The predictions of QT being of probabilistic character apply to the statistical distribution of the results obtained in various experiments. The probability distribution is not an attribute of a dice but it is a characteristic of a whole random experiment : '' rolling a dice''. and statistical long range correlations between two random variables X and Y are not a proof of any causal relation between these variable. Moreover any probabilistic model used to describe a random experiment is consistent only with a specific protocol telling how the random experiment has to be performed.In this sense the quantum theory is a statistical and contextual theory of phenomena. In this paper we discuss these important topics in some detail. Besides we discuss in historical perspective various prerequisites used in the proofs of Bell and CHSH inequalities concluding that the violation of these inequalities in spin polarization correlation experiments is neither a proof of the completeness of QT nor of its nonlocality. The question whether QT is predictably complete is still open and it should be answered by a careful and unconventional analysis of the experimental data. It is sufficient to analyze more in detail the existing experimental data by using various non-parametric purity tests and other specific statistical tools invented to study the fine structure of the time-series. The correct understanding of statistical and contextual character of QT has far reaching consequences for the quantum information and quantum computing.Comment: 16 pages, 59 references,the contribution to the conference QTRF-4 held in Vaxjo, Sweden, 11-16 june 2007. To be published in the Proceeding

Parameter Learning of Logic Programs for Symbolic-Statistical Modeling

We propose a logical/mathematical framework for statistical parameter learning of parameterized logic programs, i.e. definite clause programs containing probabilistic facts with a parameterized distribution. It extends the traditional least Herbrand model semantics in logic programming to distribution semantics, possible world semantics with a probability distribution which is unconditionally applicable to arbitrary logic programs including ones for HMMs, PCFGs and Bayesian networks. We also propose a new EM algorithm, the graphical EM algorithm, that runs for a class of parameterized logic programs representing sequential decision processes where each decision is exclusive and independent. It runs on a new data structure called support graphs describing the logical relationship between observations and their explanations, and learns parameters by computing inside and outside probability generalized for logic programs. The complexity analysis shows that when combined with OLDT search for all explanations for observations, the graphical EM algorithm, despite its generality, has the same time complexity as existing EM algorithms, i.e. the Baum-Welch algorithm for HMMs, the Inside-Outside algorithm for PCFGs, and the one for singly connected Bayesian networks that have been developed independently in each research field. Learning experiments with PCFGs using two corpora of moderate size indicate that the graphical EM algorithm can significantly outperform the Inside-Outside algorithm