Normal forms and representable functions in Moisil logic

In this chapter we will give an outline of applications of logic functions and equations that are not directly related to the design of hardware. The applications in propositional logic are surely the oldest and reach back to the developments of science in ancient Greece. The names Boolean Algebra and Boolean Ring have been selected in honor of George Boole who wanted to formalize the ideas expressed by texts in natural language and to calculate the truth of complex constructions (in order to be precise and more accurate). Naturally we already had to use logic in this sense throughout the book, however, we will go back to an introductory or elementary point of view and show that the knowledge that has been prepared up to now is fully appropriate to solve these problems. There was an enormous development of logic (in a broad mathematical sense) during the last 150 years that resulted in a highly sophisticated and specialized level, however, the propositional calculus is still the foundation of all these developments, and it is still necessary and useful to have a good understanding of these concepts. Binary arithmetic is widely used in computer hardware, control systems, or other electronic devices. In this chapter we will give an outline of binary Mathematics before hardware-related problems will be discussed in more detail. Coding is a unique mapping of arbitrary objects to certain code words. Binary codes can be seen as bridges between many real world areas and logic functions. The code words can be a subset of a given finite set so that changes of one or more bits can be detected or even corrected. The introduced Specific Normal Form uniquely expresses a logic function, differs significantly from other known normal forms, and has several remarkable properties. This new normal form is used to find the most complex logic functions and to classify bent functions. Bent functions are logic functions having the largest distance to all linear function which is a useful property for cryptographic applications.

Completeness and normal form of multi-valued logical functions

In the world, research devoted to adjusting the results of heuristic methods based on forecasting, recognition, classification, and determining the absolute extremum of a multidimensional function is relevant and widely used in such fields as medicine, geology, hydrology, management, and computer technology. In this regard, it is important to construct optimal correctors of heuristic algorithms based on control materials. Therefore, checking the completeness of classes of k valued logical functions and developing methods and algorithms for minimizing functions in the class of canonical normal forms, estimating the number of monotonic functions of k valued logic, constructing minimal bases of special classes of correcting functions for correcting incorrect algorithms remains one of the important problems of computational and discrete science. mathematics. Currently, a lot of scientific research is being carried out around the world aimed at expanding the integration of science and industry, in particular the development of the theory of k-valued logical functions for correcting the results of heuristic algorithms. In this case, an important role is played by the construction of formulas in the class of canonical normal forms, the coding of elementary conjunctions and the application of the rules of gluing, absorption and idempotency for them, and checking the completeness of systems of correcting functions. Consequently, the development of effective numerical computational methods and algorithmsfor constructing correction functions based on k-valued logic to improve the accuracy of the results of heuristic methods is considered a targeted scientific research. The paper considers therepresentation of k-valued logical functions in the class of disjunctive normal forms. Various classes of monotone functions of k-valued logic are studied. Theorems are proved on the coincidence of abbreviated and shortest disjunctive normal forms of k-valued functions. For a certain class of k-valued monotone functions, we prove an estimate for the number of functions from this class. criteria for the absorption of elementary conjunctions by a first-order neighborhood of disjunctive normal forms of k-valued functions are proved

Rough set theory has seen nearly two decades of research on both foundations and on diverse applications. A substantial part of the work done on the theory has been devoted to the study of its algebraic aspects. ‘Rough algebras’ now abound, and have been shown to be instances of various algebraic structures, both well-established and relatively new, e.g., quasi-Boolean, Stone, double Stone, Nelson, Lukasiewicz algebras, on the one hand, and topological quasi-Boolean, prerough and rough algebras, on the other. More interestingly and importantly, some of these latter algebras find a new dimension (interpretation) through representations as rough structures. An attempt is made here to present the various relationships and to discuss the representation results.

Based on the direct product of Boolean algebra and Lukasiewicz algebra, six lattice-valued logic is put forward in this paper. The algebraic structure and properties of the lattice are analyzed profoundly and the tautologies of six-valued logic system L6P(X) are discussed deeply. The researches of this paper can be used in lattice-valued logic systems and can be helpful to automated reasoning systems.

In this paper, we have done new similarity measures from a continuous t-norm by implementing it in different mean measures. For the implementation, we use a Minkowsky metric based on Lukasiewicz algebra. We test these new similarities in both the generalised and normal form of Lukasiewicz algebra with weight optimisation. The mean measures examined here are arithmetic, geometric and harmonic means. We show that the magnitude order of the similarities are S/sub H//sup N/ /spl ges/S/sub G//sup N/ /spl ges/S/sub A//sup N/ . Secondly, we show that the use of different means is highly recommendable in some cases.

In this paper we study a problem of signal compression how to choose a best mother wavelet from the set S of wavelets. The approach is following: First we calculate a discrete wavelet transform of signal by using one standard wavelet. Then we form coefficients m_i for each scale i from the wavelet expansions coefficients. Coefficients m_i are used for selecting best wavelet from the set S. Selection is classification problem and we have constructed classification algorithm that uses fuzzy similarity that is based on a continuous t-norm called Lukasiewicz algebra. We are using normal and cumulative forms of generalized Lukasiewicz algebra and we have also applied a genetic algorithm into the our classifier to choose appropriate weights in our classification tasks. There are many advantages what we get by using t-norm called Lukasiewicz in classification: 1) Structure has a promising mathematical background 2) Mean of many fuzzy similarities is still a fuzzy similarity 3) Any pseudo-metric induces fuzzy similarity on a given non-empty set X with respect to the Lukasiewicz-conjunction. Algorithm is efficient especially because we have to calculate wavelet transform only once and classification is simple and fast. Algorithm is also very flexible, cause we can implement any type metrics or mean measures into it. As our results we will present a new method to select best mother wavelet from a given set S. We will also show that proposed hybrid method can be used in this kind of analytical problems. The best way to form coefficients m_i and choose metric or measure is depended of class of signals we are working with, which is still unclear.

In this chapter, we focus on a novel two-level logic representation. We define Majority Normal Form (MNF), as an alternative to the traditional Disjunctive Normal Form (DNF) and the Conjunctive Normal Form (CNF). After a brief investigation on the MNF expressive power, we study the problem of MNF-SATisfiability (MNF-SAT). We prove that MNF-SAT is NP-complete, as its CNF-SAT counterpart. However, we show practical restrictions on MNF formula whose satisfiability can be decided in polynomial time. We finally propose a simple algorithm to solve MNF-SAT, based on the intrinsic functionality of two-level majority logic. Although an automated MNF-SAT solver is still under construction, manual examples already demonstrate promising opportunities.

The article considers the implementation of the algorithm for constructing a corrector, where the features of the algorithm for constructing dead-end normal canonical forms of not everywhere defined functions of multivalued logic are studied. A criterion is proved for the representation of conjunctions of normal canonical forms of not everywhere defined functions by pairs of numbers in decimal calculus. The performance of gluing and absorption operations based on the representation of conjunctions by pairs of decimal numbers is given. A description is given of a program for constructing an optimal corrector for an arbitrary k not everywhere defined functions of many-valued logic. Descriptions of the program for implementing the invariant continuation algorithm and the program for constructing the shortest normal canonical forms of functions of k-valued logic are proposed.

Association rule mining is a very popular data mining technique. Rules in this technique are often used to identify and represent dependencies between attributes in databases. Specifically, fuzzy association rules are rules that use the concepts of fuzzy sets and can be considered as a special case of fuzzy predicates. Many quality measures have been defined for fuzzy association rules, but all consider a specific structure: antecedent and consequence. In the case of fuzzy predicates in the normal form (i.e., conjunctive or disjunctive), it is necessary to define different quality measures that do not consider the structure as an antecedent or a consequence. The only available measure for this scenario is the fuzzy predicate truth value (FPTV), which has serious limitations. The evaluation of fuzzy predicates in the normal form through appropriate quality measures has not yet been clearly defined in the literature. Thus, we propose several quality measures specifically for fuzzy predicates in the conjunctive (CNF) and disjunctive (DNF) normal forms. Experimental studies illustrate the use of the proposed measures and allow some general conclusions about each measure.

In this paper, we provide normal forms and truth tables for interval-valued fuzzy logic which are analogous to those for classical logic, i.e. analogous to the disjunctive and conjunctive normal forms and truth tables for Boolean algebras. We give an algorithm for rewriting an expression to obtain its disjunctive normal form. We also give an algorithm for obtaining the disjunctive normal form of an expression from its table of truth values.

Three types of normal forms are introduced for fuzzy logic functions: disjunctive, conjunctive and additive. Disjunctive and conjunctive normal forms are considered in two variants: infinite and finite. It is shown that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any L-valued function where L is a support set of any complete BL-algebra. The additive normal form lies in the middle of the two others. For all of them the estimation of the quality of approximation is suggested.

Three types of normal forms are introduced for fuzzy logic functions: disjunctive, conjunctive and additive. Disjunctive and conjunctive normal forms are considered in two variants: infinite and finite. It is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any L-valued function where L is a support set of any complete BL-algebra. The additive normal form “lies” in the middle of the two others. For all of them the estimation of the quality of approximation is suggested.

Normal forms lie at the heart of the theory of combinational circuits. A normal form is a general logic formula that allows us to develop a unique logic expression of a combinational circuit describe? by, say, a table of asserted events (TAE). There are a number of different normal forms, each based on one of the associative output connectives AND, OR, XOR, or EQU. In this chapter, we look at the more common normal forms whose output connectives are AND and OR, called conjunctive and disjunctive normal forms.

Chapter 7 - Equivalences in two-valued logic