Basic Fuzzy Logic Research Articles

A simplified control method for a water heating load coupled to a hybrid energy system to improve inverter longevity

In this paper, the temperature characteristics of an inverter is determined through experimental validation. The experiment involves testing under two load scenarios. The first of which considers a low inverter load, which represents a baseload at approximately 18% of the maximum rated power of the inverter. The second includes an energy intensive water heater load, which increases the loading on the inverter to above 85%. The aim of this paper is to ensure that the operating limits with respect to the inverter temperature is maintained to at least 10% below the upper limit of the device. The experimental results indicated that the rate of change of the inverter temperature was 2.988 °C/min when loaded above 85% and −1.277 °C/min after returning to the 18% rated output power level. This indicated that if the inverter is subjected to a high load of above 85% for a certain amount of time and the operating temperature has reached its acceptable maximum limit, roughly twice the amount of time is necessary to cool the components down enough for another high loading event. A basic fuzzy logic control method is proposed to maintain the inverter temperature within acceptable limits to improve overall inverter longevity.

Аппаратная реализация встраиваемых нечетких систем

Principles of using analog circuitry for implementing basic fuzzy logic operations are considered. A set of developed hardware modules, where operations are executed on an analog representation of fuzzy variables is presented. Modules for converting information from digital to continuous form and vice versa are presented in order to link fuzzy system with sensors and executing circuitry, for example, in the control system of a process or an object. The components of such modules are described, along with their pros and cons. The use of circuits based on these principles seems essential for the development of fuzzy electronics.

Some Examples of BL-Algebras Using Commutative Rings

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2r+1 elements, r≥2, and BL-chains with k elements, k≥4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size (2≤n≤5).

Importation lattices

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P≠∅, we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P=[0,1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order-theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0,1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations.

Generalized backpropagation algorithm for training second-order neural networks.

The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to second-order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear quadratic operation. A single second-order neurons already have a strong nonlinear modeling ability, such as implementing basic fuzzy logic operations. In this paper, we develop a general backpropagation algorithm to train the network consisting of second-order neurons. The numerical studies are performed to verify the generalized backpropagation algorithm.

Strong standard completeness for continuous t-norms

This paper presents a proof of a strong completeness theorem for an extended axiomatic system of fuzzy logic BL with respect to all continuous t-norms. A finite strong standard completeness theorem for all continuous t-norms and their residua, the basic fuzzy logic, was proved across two papers Hájek (1998) and Cignoli et al. (2000). In Montagna (2007), the language of BL is extended by an additional connective and the axiomatic system includes an infinitary rule to achieve strong completeness result. In this paper we provide a proof of strong completeness for BL with a different infinitary inference rule but without extending the language of BL. We will also prove strong completeness for the Łukasiewicz and product t-norms using this extended axiomatic system.

Kripke semantics for fuzzy logics

Kripke frames (and models) provide a suitable semantics for sub-classical logics, for example Intuitionistic Logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the Basic Logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the Basic Fuzzy Logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the only fuzzy logics (logics containing the Basic Fuzzy Logic) which are sound and complete with respect to a class of Kripke frames/models are the extensions of the G\"odel Logic (or the super-intuitionistic logic of Dummett), indeed this logic is sound and strongly complete with respect to reflexive, transitive and connected (linear) Kripke frames (with persistent satisfaction relations). This provides a semantic characterization for the G\"odel Logic among (propositional) fuzzy logics.

Proof search and Co-NP completeness for many-valued logics

We provide a methodology to introduce proof search oriented calculi for a large class of many-valued logics, and a sufficient condition for their Co-NP completeness. Our results apply to many well known logics including Gödel, Łukasiewicz and Product Logic, as well as Hájek's Basic Fuzzy Logic.

Interpreting lattice-valued set theory in fuzzy set theory

An interpretation of lattice-valued logic, defined by Titani, in basic fuzzy logic, defined by Hájek, is presented. Moreover, Titani's axioms of lattice-valued set theory are interpreted in fuzzy set theory, as defined by the authors.

Expanding Basic Fuzzy Logic with truth constants for component delimiters

In this paper we investigate the propositional logic of standard algebras for Hájek's Basic Fuzzy Logic BL in a language expanded with propositional constants for the idempotent elements delimiting the Ł-, G-, and Π-components of the algebra. We start from a fixed standard BL-algebra; introduce new propositional constants; present a suitable set of axioms; investigate completeness results; and give some complexity results.

Representing continuous t-norms in quantum computation with mixed states

A model of quantum computation is discussed in (Aharanov et al 1997 Proc. 13th Annual ACM Symp. on Theory of Computation, STOC pp 20–30) and (Tarasov 2002 J. Phys. A: Math. Gen. 35 5207–35) in which quantum gates are represented by quantum operations acting on mixed states. It allows one to use a quantum computational model in which connectives of a four-valued logic can be realized as quantum gates. In this model, we give a representation of certain functions, known as t-norms (Menger 1942 Proc. Natl Acad. Sci. USA 37 57–60), that generalize the triangle inequality for the probability distribution-valued metrics. As a consequence an interpretation of the standard operations associated with the basic fuzzy logic (Hájek 1998 Metamathematics of Fuzzy Logic (Trends in Logic vol 4) (Dordrecht: Kluwer)) is provided in the frame of quantum computation.

Extremal states on bounded residuated $$\ell$$ -monoids with general comparability

Bounded residuated lattice ordered monoids ($$R\ell$$-monoids) are a common generalization of pseudo-$$BL$$-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. We investigate bounded $$R\ell$$-monoids satisfying the general comparability condition in connection with their states (analogues of probability measures). It is shown that if an extremal state on Boolean elements fulfils a simple condition, then it can be uniquely extended to an extremal state on the $$R\ell$$-monoid, and that if every extremal state satisfies this condition, then the $$R\ell$$-monoid is a pseudo-$$BL$$-algebra.

Classes of fuzzy filters of residuated lattice ordered monoids

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (\rl monoids) are common generalizations of $\text{\rm BL}$-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded $\text{\rm Rl}$-monoids.

On fuzzy modal logics [formula omitted

The fuzzy variant S 5 ( C ) of the well-known modal logic S5 is studied, C being a recursively axiomatized fuzzy propositional logic extending the basic fuzzy logic BL. Three kinds of Kripke models are introduced and corresponding deductive systems are found.

A Temporal Semantics for Basic Logic

In the context of truth-functional propositional many-valued logics, Hajek’s Basic Fuzzy Logic BL [14] plays a major role. The completeness theorem proved in [7] shows that BL is the logic of all continuous t-norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se. In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is Łukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for Łukasiewicz logic.

Monadic Bounded Commutative Residuated ℓ-monoids

Bounded commutative residuated l-monoids are a generalization of algebras of propositional logics such as BL-algebras, i.e. algebraic counterparts of the basic fuzzy logic (and hence consequently MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic) and Heyting algebras, i.e. algebras of the intuitionistic logic. Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. We introduce and study monadic residuated l-monoids as a generalization of monadic MV-algebras.

Interior and closure operators on bounded commutative residuated ℓ-monoids

Topological Boolean algebras are generalizations of topological spaces dened by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Lukasiewicz innite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative R‘-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the H ajek’s basic fuzzy logic) and Heyting algebras as proper subclasses.

Case-based Detection of Operating Conditions in Complex Nonlinear Systems

The case-based reasoning (CBR) system was developed for the identification of different operating situations in paper machines. Because similar break sensitivities can result from a multitude of dissimilar cases, the case base system is based on a division into categories which correspond to different levels of break sensitivity. The system is based on the linguistic equations (LE) approach and basic fuzzy logic, and it combines expert knowledge and on-line measurements from the wet-end of the paper machine. Nonlinear interactions are handled with a special scaling functions and linear equations. Each equation provides a new fact with a degree of membership, and the resulting set of facts is used in the fuzzy reasoning of process cases and break categories. The LE models are essential in compacting the system since each equation corresponds to a rule set in the fuzzy set systems. The application operates as a case retrieval and reuse application, predicting web break sensitivity in paper machine. Similarity measures based on model errors in the scaled range represent the importance of the models in activating the cases. The break category is defined by the case with the highest degree of membership. Although, the case base is fairly small the results from the on-line tests were relatively good compared to real break sensitivity. The predicted break sensitivity is an indirect measurement, which provides an early indication of process changes. The list of variables in the active cases can be used to avoid harmful operating conditions.

Non-Archimedean fuzzy and probability logic

In this paper the non-Archimedean multiple-validity is proposed for basic fuzzy logic BL∀∞ that is built as an ω-order extension of the logic BL∀. Probabilities are defined on the class of fuzzy subsets and, as a result, for the first time the non-Archimedean valued probability logic is constructed on the base of BL∀∞.

Modal operators on bounded commutative residuated ℓ-monoids

Abstract Bounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVÁ,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras.