Normal forms for fuzzy logic functions and their approximation ability

Fuzzy logic function f is unate if and only if f is either monotone increasing or monotone decreasing with regards to every variable. This paper studies some fundamental properties of fuzzy unate logic function. Based on the properties, this paper clarifies a necessary and sufficient condition of any given incompletely specified function to be unate fuzzy logic function.

Disjunctive and conjunctive normal forms of pseudo-Boolean functions

We consider the problem of efficiently indexing Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) Boolean expressions over a high-dimensional multi-valued attribute space. The goal is to rapidly find the set of Boolean expressions that evaluate to true for a given assignment of values to attributes. A solution to this problem has applications in online advertising (where a Boolean expression represents an advertiser's user targeting requirements, and an assignment of values to attributes represents the characteristics of a user visiting an online page) and in general any publish/subscribe system (where a Boolean expression represents a subscription, and an assignment of values to attributes represents an event). All existing solutions that we are aware of can only index a specialized sub-set of conjunctive and/or disjunctive expressions, and cannot efficiently handle general DNF and CNF expressions (including NOTs) over multi-valued attributes. In this paper, we present a novel solution based on the inverted list data structure that enables us to index arbitrarily complex DNF and CNF Boolean expressions over multi-valued attributes. An interesting aspect of our solution is that, by virtue of leveraging inverted lists traditionally used for ranked information retrieval, we can efficiently return the top-N matching Boolean expressions. This capability enables emerging applications such as ranked publish/subscribe systems [16], where only the top subscriptions that match an event are desired. For example, in online advertising there is a limit on the number of advertisements that can be shown on a given page and only the "best" advertisements can be displayed. We have evaluated our proposed technique based on data from an online advertising application, and the results show a dramatic performance improvement over prior techniques.

The problem is the transformation of a conjunctive normal form (CNF) into a minimized (for the inclusion operator) disjunctive normal form (DNF) and vice versa. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An algorithm built upon the classical Davis and Putnam procedure is presented for calculating, without the explicit minimization for the inclusion, this unionist product.

The paper presents a novel, transistor level, implementation of selected fuzzy set operators suitable for fuzzy control systems realized in low-power hardware. We propose a fully digital, asynchronous realization of basic fuzzy logic (FL) functions, such as the bounded sum, bounded difference, bounded product, bounded complement, fuzzy logic union (MAX) and fuzzy logic intersection (MIN). All of the proposed operators has been implemented in the CMOS TSMC 180nm Technology and verified by means of transistor level simulations in Hspice environment. The proposed structures of the FL functions can easily be scaled to any signal resolutions.

Fuzzy set theories allow us to represent our knowledge under various interpretations and axiomatic foundations from linguistic to computational representations. There are at least four levels of knowledge representation: (i) linguistic, (ii) meta-linguistic, (iii) propositional, and (iv) computational. There are three transformations, which depend on the particular interpretations put on a knowledge representation schema. There are various choices, corresponding to one's interpretation of: (a) type of set representation, (fuzzy or crisp), (b) type of propositional connectives and normal forms, and (c) type of computational connectives, i.e., weak or strong t-norms and co-norms. In the light of these selections, fuzzy disjunctive and conjunctive normal forms (FDNF, FCNF) are derived from fuzzy truth tables. It is shown that classical expressions such as excluded middle etc., when fuzzified, ought to be reinterpreted with a type II, second order, semantic uncertainty. The classical expressions should not be interpreted as to whether they are valid or not. One can only state that the well-known tautologies of classical logic are valid to many degrees specified by an interval defined by their FDNF and FCNF. FDNF and FCNF boundaries identify the nonspecificity measure associated with type II, second order, semantic uncertainty. Thus, those researchers who are not familiar or who are not concerned with type II semantic uncertainty work with a myopic understanding of fuzzy set and logic theories. >

Introduction: With regard to increasing population of country, need to high agricultural production is essential. The most suitable method for this issue is high production per area unit. Preparation much food and other environmental resources with conservation of biotic resources for futures will be possible only with optimum exploitation of soil. Among effective factors for the most production balanced addition of fertilizers increases production of crops higher than the others. With attention to this topic, determination of soil fertility degree is essential tobetter use of fertilizers and right exploitation of soils. Using fuzzy logic and Analytic Hierarchy Process (AHP) could be useful in accurate determination of soil fertility degree. Materials and Methods: The study area (at the east of Rasht city) is located between 49° 31' to 49° 45' E longitude and 37° 7' to 37° 27' N latitude in north of Guilan Province, northern Iran, in the southern coast of the Caspian sea. 117 soil samples were derived from0-30 cm depth in the study area. Air-dried soil samples were crushed and passed through a 2mm sieve. Available phosphorus, potassium and organic carbon were determined by sodium bicarbonate, normal ammonium acetate and corrected walkly-black method, respectively. In the first stage, the interpolation of data was done by kriging method in GIS context. Then S-shape membership function was defined for each parameter and prepared fuzzy map. After determination of membership function weight parameters maps were determined using AHP technique and finally soil fertility map was prepared with overlaying of weighted fuzzy maps. Relative variance and correlation coefficient criteria used tocontrol groups separation accuracy in fuzzy fertility map. Results and Discussion: With regard to minimum amounts of parameters looks some lands of study area had fertility difficulty. Therefore, soil fertility map of study area distinct these lands and present soil fertility groups for better management of soil and plant nutrition. Weight of soil parameters was0.54, 0.29 and 0.17 for organic carbon, available phosphor and potassium, respectively. Fuzzy map of study area includes five soil fertility groups as: 22.9% very high fertility, 27.7% high fertility, 35.53% medium fertility, 10.48% low fertility and 3.39% very low fertility. Consequently, a separated map for soil fertility prepared to evaluate soil fertility of study area for rice cultivation. Toinvestigatethe efficiency of fuzzy model and AHP in increasing the accuracy of soil fertility map, soil fertility map with Boolean method prepared as well. Boolean map showed 58.88% fertile and 41.12% unfertile.15 soil samples from different soil fertility groups of study area were derived fromcontrol of maps accuracy. 13 renewed samples of 15 and 9 soil samples have matched with fuzzy and Boolean map, respectively. Comparison of parameters mean in fuzzy map fertility groups showed that parameters mean amounts of very high and high fertility groups are higher than optimum level except potassium that is a few lower than optimum level in high fertility group, therefore, addition of fertilizers in these groups could not be useful to increase rice crop production. Phosphorus parameter amount is lower than the critical level in very low, low and medium fertility groups, then in these groups phosphorus fertilizer should be added to the soil toincreaserice production. The amount of potassium parameter is higher than the critical level and lower than optimum limit in very low, low, medium and high fertility groups, then in these groups addition of potassium fertilizer will results in theincrease of production. Organic carbon amount is lower than optimum level in very low and low fertility groups. With regard to the relation between organic carbon andnitrogen and phosphorus, therefore, the addition of organic carbon fertilizer could compensate deficit of nitrogen and phosphorus in these groups as well. Attention to the presented explanations and comparison of fuzzy and Boolean maps using parameters amounts in renewed sampling points for control of maps accuracy, it is distinct that fuzzy logic could influencetheoptimum using of fertilizers with increasing map efficiency and accuracy. In addition, relative variance and correlation coefficient amounts showed that fuzzy map has separatedquite wellparameters changes. Conclusion: Effective parameters in soil fertility, includingorganic carbon, phosphorus and potassium were used topreparesoil fertility map for rice cultivation. With regard to the minimum amounts of parameters looks some lands of study area had fertility difficulty. Therefore, soil fertility map of study area distinct these lands and presents soil fertility groups tobetter management of soil and plant nutrition. Fuzzy and Boolean methods were used topreparesoil fertility map. Comparison of these two approaches showed that fuzzy method with AHP caused to increase theefficiency and accuracy of fertility map for rice. Separated and distinguish soil fertility groups in fuzzy map help suitable distribution and optimum use of fertilizers for rice production.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

This paper proposes a new class of functions, i.e., fuzzy majority functions, based on the fuzzy logic which takes any value in the interval [0, 1] as the truth value of its proposition. Fuzzy majority functions are characterized by weights and thresholds, and may be represented by a logic formula involving constants. Specifically, the proposed class intersects with the well‐known fuzzy logic functions but is not a subset of them. On the other hand, the logic formula of a fuzzy majority function has a close relation with binary threshold functions. From this viewpoint, a theoretical limit in the utility of the fuzzy majority functions is discussed.

We describe the problem of mining set valued rules in large relational tables containing categorical attributes taking a finite number of values. Such rules allow for an interval of possible values to be selected for each attribute in condition instead of a single value for association rules, while conclusion contains a projection of the data restricted by the condition onto a target attribute. An example of such a rule might be “if HOUSEHOLDSIZE = {Two OR Tree} AND OCCUPATION={Professional OR Clerical} THEN PAYMENT_METHOD = {CashCheck (Max=249, Sum=4952) OR DebitCard (Max=175, Sum=3021)} WHERE Confidence=85%, Support=10%.}” We use an original conceptional and formal framework for representing multidimensional distribution induced from data by a number of so-called prime disjunctions upper bounding its surface. Each prime disjunction represents a wide multidimensional interval of impossible combinations of attribute values. This original formalism generalises the conventional boolean approach in two directions: (i) finite-valued attributes (instead of only 0 and 1), and (ii) continuous-valued semantics instead of (true and false). In addition, we describe an efficient algorithm, which carries out the generalised dual transformation from possibilistic disjunctive normal form (DNF) representing data into conjunctive normal form (CNF) representing knowledge and thus generates all the most interesting prime disjunctions. Once obtained they can be used to build different forms of rules or for other purposes (prediction, clustering etc.).

We discuss extensions to a controlled natural language allowing spatiotemporal expressions to be interpreted as fuzzy logic functions. The extensions first required new sentence templates. Next, changes to a GPSG parser modified its lexicon, and then extended its parsing and logical form rules to allow user-defined spatial and temporal constraints to be extracted. The sentence templates ground user-defined culturally-specific times and places to boundaries surrounding prototypical ideals. Query points, defined by location and time, are compared to these definitions using Gaussians centered at prototypical 'ideal' times or places. The Gaussians provide soft fall-off at the boundaries. Fuzzy logic operators allow larger expressions to be interpreted, analogous to Boolean combinations of terms. The mathematically-interpreted spatiotemporal terms act as domain features for a machine learning algorithm. They allow easy specification (compared to programming) of basis functions for an inverse reinforcement learning algorithm that detects anomalous vehicle tracks or suspicious agent behavior.

Improving feeding strategies for shrimp farming using fuzzy logic, based on water quality parameters

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

In this paper, we put forward a weak conjunctive normal form for G3, which is similar to the conjunctive normal form for classical propositional logic. We also give two algorithms to reduce a formula to weak conjunctive normal form, where one is based on charactering countermodels in model-based-formulas, and the other is based on rewriting formulas according to transformation rules. In the end, we show that there is no polynomial time algorithm to reduce an arbitrary formula to the weak conjunctive normal form.

Design of efficient control algorithms for autonomous mobile robot movement in unknown and changing environment with obstacles and walls is a difficult task. There exist different strategies to design control systems to perform the robot movement. In this article possibility to use fuzzy logic, if-then rules, and genetic algorithm for autonomous mobile robot control is presented. Control using fuzzy logic is softer and better then control using if-then rules because of absence of motors speed jumps. It is shown that the rough control system can be designed using an expert knowledge. Then genetic algorithm can be used to improve the quality of the control system.