Notion Of Equivalence Classes Research Articles

A new algorithm for compression of partially commutative alphabets

In this paper, we will address the problem of how to compress sequences defined in partially commutative alphabets. The use of partial order between symbols may result in far more efficient compression schemes than the most commonly used compression techniques, which use total order. We suggest a new algorithm for compressing sequences defined in alphabets with a partial order. This new algorithm is named the Lempel–Ziv with Partial Order (POLZ) and stands for a Lempel–Ziv (LZ78) generalization. We will show how to compress and decompress sequences by utilizing the POLZ. In addition, we establish the methods for determining the upper bounds for sequence complexity by using the notion of equivalence classes. We will also demonstrate that POLZ is asymptotically optimal, achieving an even greater compression rate when compared to the LZ78 when applied to sequences emanating from concurrent processes. Furthermore, we will show that the complexity obtained by using the LZ78 is also an upper bound for the POLZ complexity. We have developed examples in robotics and real-time systems to show how to use and the benefits of using our algorithm for compression within the context of concurrent systems. When the concurrency of the processes is taken into account, all experimental results show a significant compression gain.

Improving the efficiency of intrusion detection in information systems

Abstract Policy Interaction Graph Analysis is a Host-based Intrusion Detection tool that uses Linux MAC Mandatory access control policy to build the licit information flow graph and uses a detection policy defined by the administrator to extract illicit behaviour from the graph. The main limitation of this tool is the generation of a huge signature base of illicit behaviours; hence, this leads to the use of huge memory space to store it. Our primary goal in this article is to reduce this memory space while keeping the tool’s efficiency in terms of intrusion detection rate and false generated alarms. First, the interactions between the two nodes of the graph were grouped into a single interaction. The notion of equivalence class was used to classify the paths in the graph and was compressed by using a genetic algorithm. Such an approach showed its efficiency compared to the approach proposed by Pierre Clairet, by which the detection rate obtained was 99.9%, and no false-positive with a compression rate of illicit behaviour signature database reached 99.44%. Having these results is one of the critical aspects of realizing successful host-based intrusion detection systems.

Mathematical classification of regulatory logics for compound environmental changes

This paper is concerned with biological regulatory mechanisms in response to the simultaneous occurrence of a huge number of environmental changes. The restricted resources of cells strictly limit the number of their regulatory methods; hence, cells must adopt, as compensation, special mechanisms to deal with the simultaneous occurrence of environmental changes. We hypothesize that cells use various control logics to integrate information about independent environmental changes related to a cell task and represent the resulting effects of the different ways of integration by logical functions. Using the notion of equivalence classes in set theory, we describe the mathematical classification of the effects into biologically unequivalent ones realized by different control logics. Our purely mathematical and systematic classification of logical functions reveals three elementary control logics with different biological relevance. To better understand their biological significance, we consider examples of biological systems that use these elementary control logics.

Remarks on the identifiability of thurstonian ranking models: Case V, case III, or neither?

It is well-known that the representations of the Thurstonian Case III and Case V models for paired comparison data are not unique. Similarly, when analyzing ranking data, other equivalent covariance structures can substitute for those given by Thurstone in these cases. That is, we may more broadly define the family of covariance structures satisfying Case III and Case V conditions. This paper introduces the notion of equivalence classes which defines a more meaningful partition of the covariance structures of the Thurstonian ranking models. In addition, the equivalence classes of Case V and Case III are completely characterized.

Equivalence classes and generalized coherent states in quantum measurement

The notion of an equivalence class, which was first considered by Jauch, is introduced into generalized coherent-state formalism in quantum measurement. Two equivalent states in Jauch’s treatment cannot be distinguished by measurement of any observables of a physical system (the definition of an observable also being ambiguous). In our approach it is possible in principle, but becomes more difficult asN→∞,N being the number of molecules constructing a detector; we cannot distinguish them at all in the large-N limit. Two models are presented, which show that the notion of equivalence class is well suited to the generalized coherentstate approach. It is also shown that in each model a statistical operator and its mixed-state part belong to the same equivalence class.