Minimal unsatisfiable sets: Classification and bounds

Proving the unsatisfiability of propositional Boolean formulas has applications in a wide range of fields. Minimal Unsatisfiable Sets (MUS) are signatures of the property of unsatisfiability in formulas and our understanding of these signatures can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this paper, we explore some combinatorial properties of MUS and use them to devise a classification scheme for MUS. We also derive bounds on the sizes of MUS in Horn, 2-SAT and 3-SAT formulas.

A general framework for measuring inconsistency through minimal inconsistent sets

Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIVC, defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller. To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets. Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief base, which is more discriminative than MIVC. Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically characterized by the corresponding revised axioms presented in this paper.

A flaw in the use of minimal defining sets for secret sharing schemes

It is shown that in some cases it is possible to reconstruct a block design D uniquely from incomplete knowledge of a minimal defining set for D. This surprising result has implications for the use of minimal defining sets in secret sharing schemes.

Uma metodologia para modelagem e avaliação da dependabilidade de redes industriais sem fio

Ensuring the dependability requirements is essential for the industrial applications since faults may cause failures whose consequences result in economic losses, environmental damage or hurting people. Therefore, faced from the relevance of topic, this thesis proposes a methodology for the dependability evaluation of industrial wireless networks (WirelessHART, ISA100.11a, WIA-PA) on early design phase. However, the proposal can be easily adapted to maintenance and expansion stages of network. The proposal uses graph theory and fault tree formalism to create automatically an analytical model from a given wireless industrial network topology, where the dependability can be evaluated. The evaluation metrics supported are the reliability, availability, MTTF (mean time to failure), importance measures of devices, redundancy aspects and common cause failures. It must be emphasized that the proposal is independent of any tool to evaluate quantitatively the target metrics. However, due to validation issues it was used a tool widely accepted on academy for this purpose (SHARPE). In addition, an algorithm to generate the minimal cut sets, originally applied on graph theory, was adapted to fault tree formalism to guarantee the scalability of methodology in wireless industrial network environments (< 100 devices). Finally, the proposed methodology was validate from typical scenarios found in industrial environments, as star, line, cluster and mesh topologies. It was also evaluated scenarios with common cause failures and best practices to guide the design of an industrial wireless network. For guarantee scalability requirements, it was analyzed the performance of methodology in different scenarios where the results shown the applicability of proposal for networks typically found in industrial environments

Reliability evaluation of distribution systems considering automatic reclosers

This paper presents a new methodology to evaluate in a predictive way the reliability of distribution systems, considering the impact of automatic recloser switches. The developed algorithm is based on state enumeration techniques with Markovian models and on the minimal cut set theory. Some computational aspects related with the implementation of the proposed algorithm in typical distribution networks are also discussed. The description of the proposed approach is carried out using a sample test system. The results obtained with a typical configuration of a Brazilian system (EDP Bandeirante Energia S.A.) are presented and discussed.

Reliability analysis for hazard and operability studies

Fault tree analysis is used as a tool within hazard and operability (Hazop) studies. The present study proposes a new methodology for obtaining the exact TOP event probability of coherent fault trees. The technique uses a top-down approach similar to that of FATRAM. This new Fault Tree Disjoint Reduction Algorithm resolves all the intermediate events in the tree except OR gates with basic event inputs so that a near minimal cut sets expression is obtained. Then Bennetts' disjoint technique is applied and remaining OR gates are resolved. The technique has been found to be appropriate as an alternative to Monte Carlo simulation methods when rare events are countered and exact results are needed. The algorithm has been developed in FORTRAN 77 on the Perq workstation as an addition to the Aston Hazop package. The Perq graphical environment enabled a friendly user interface to be created. The total package takes as its input cause and symptom equations using Lihou's form of coding and produces both drawings of fault trees and the Boolean sum of products expression into which reliability data can be substituted directly.

A genetic algorithm for the multi-source and multi-sink minimum vertex cut problem and its applications

We present a new penalty-based genetic algorithm for the multi-source and multi-sink minimum vertex cut problem, and illustrate the algorithm’s usefulness with two real-world applications. It is proved in this paper that the genetic algorithm always produces a feasible solution by exploiting some domain-specific knowledge. The genetic algorithm has been implemented on the example applications and evaluated to show how well it scales as the problem size increases.

On Solution Sets of Information Inequalities

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.

Reliability evaluations of railway power supplies by fault-tree analysis

High reliability of railway power systems is one of the essential criteria to ensure quality and cost-effectiveness of railway services. Evaluation of reliability at system level is essential for not only scheduling maintenance activities, but also identifying reliability-critical components. Various methods to compute reliability on individual components or regularly structured systems have been developed and proven to be effective. However, they are not adequate for evaluating complicated systems with numerous interconnected components, such as railway power systems, and locating the reliability critical components. Fault tree analysis (FTA) integrates the reliability of individual components into the overall system reliability through quantitative evaluation and identifies the critical components by minimum cut sets and sensitivity analysis. The paper presents the reliability evaluation of railway power systems by FTA and investigates the impact of maintenance activities on overall reliability. The applicability of the proposed methods is illustrated by case studies in AC railways.

Computational approaches to finding and measuring inconsistency in arbitrary knowledge bases

There is extensive theoretical work on measures of inconsistency for arbitrary formulae in knowledge bases. Many of these are defined in terms of the set of minimal inconsistent subsets (MISes) of the base. However, few have been implemented or experimentally evaluated to support their viability, since computing all MISes is intractable in the worst case. Fortunately, recent work on a related problem of minimal unsatisfiable sets of clauses (MUSes) offers a viable solution in many cases. In this paper, we begin by drawing connections between MISes and MUSes through algorithms based on a MUS generalization approach and a new optimized MUS transformation approach to finding MISes. We implement these algorithms, along with a selection of existing measures for flat and stratified knowledge bases, in a tool called mimus. We then carry out an extensive experimental evaluation of mimus using randomly generated arbitrary knowledge bases. We conclude that these measures are viable for many large and complex random instances. Moreover, they represent a practical and intuitive tool for inconsistency handling.

Effective atomic orbitals for fuzzy atoms

The method of extracting effective atomic orbitals and effective minimal basis sets from molecular wave function characterizing the state of an atom in a molecule is developed in the framework of the "fuzzy" atoms. In all cases studied, there were as many effective orbitals that have considerable occupation numbers as orbitals in the classical minimal basis. That is considered to be of high conceptual importance

Exact Unification and Admissibility

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification.

PC Points and their Application to Vector Optimization

AMS subject classification: 90C29, 90C48

Minimal sets of generators for the relation ideals of certain monomial curves

Let O be a monomial curve in the affine algebraic e-space over a field K and P be the relation ideal of O. If O is defined by a sequence of e positive integers some e - 1 of which form an arithmetic sequence then we construct a minimal set of generators for P and write an explicit formula for mu(P).

Chaotic planar piecewise smooth vector fields with non-trivial minimal sets

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.