Fuzzy Modal Logics

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

We are going to study, from the logical point of view, expressions as “usually ϕ”, “ϕ is probable”, “for many x, ϕ” and similar. We shall handle them both as generalized quantifiers and as modalities. This should not look unnatural since in general modalities can be viewed as hidden quantifiers. We shall try to show how the theory of generalized quantifiers and modal logic can be applied to the above expressions (stressed by Zadeh as items of the specific agenda of fuzzy logic in the narrow sense) and that they admit “classical” logical analysis. We shall offer a logical apparatus, precise definitions, some results and various problems: here much study and development is still necessary. To be able to use and generalize the results and approaches of Boolean logic, we shall have to review and summarize several notions and facts on generalized quantifiers, some modal logics, and also on measures of uncertainty (that will be used to define various quantifiers and modalities). Thus we shall have two preliminary sections in this chapter: Section 1 on generalized quantifiers (with an appendix on uncertainty measures) and Section 2 on modal logics, both sections in the frame of Boolean (two-valued) logic. Generalized quantifiers in fuzzy logic and fuzzy modal logics are studied in Sections 3, 4.

As an extension of the traditional modal logic, the fuzzy first-order modal logic is discussed in this paper. A description of fuzzy first-order modal logic based on constant domain semantics is given, and a formal system of fuzzy reasoning based on the semantic information of models of first-order modal logic is established. It is also introduced in this paper the notion of the satisfiability of the reasoning system and some properties associated with the satisfiability are proved.

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

Modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems. In this paper, we first present a new notion of bisimulation for nondeterministic fuzzy transition systems, which is distribution based and coarser than state-based bisimulation appeared in the literature. Then, we define a distribution-based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice, which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations. We also propose an on-the-fly algorithm for computing the bisimilarity metric. Moreover, we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric. Interestingly, this characterization holds for a class of fuzzy modal logics. In addition, we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric, which makes compositional verification possible.

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 + .

The number of filaments in yarn cross-section, weave density, and weave type are considered the most important factors that affect the property of air permeability of multifilament woven fabrics. Microfilament yarns significantly affect the air permeability property of this type of fabric because of the low porosity between the filaments. This study deals with the development of a fuzzy logic model for predicting the air permeability of multifilament polyester woven fabrics produced from conventional and microfilament yarns. The polyester multifilament yarns used in this study were produced with three different microfilament fineness and two conventional filament fineness levels. The woven fabric samples used in this study were obtained in three weave types: plain, twill, and satin, and with five different weave densities. In accordance with the experimental test results, both regression analysis and fuzzy logic system were built. The air permeability results generated from the developed fuzzy model and the regression equations were compared with the experimental values. Satisfactory and accurate prediction results were obtained with the developed fuzzy logic model. The mean absolute error of the fuzzy model and regression analysis were found to be 2.32%, 12.59%, respectively. Therefore, it was confirmed that the fuzzy model was superior in predicting air permeability.

We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems.

The nontraditional hole-making process Friction Drilling (FD) receives major attention nowadays because of its operational efficiency in terms of unpolluted, chipless hole making and in fact, the holes are drilled in single step. It is a cumbersome and challenging task to predict surface finish of the work material in the final stages of operation. This difficulty arises because of nonlinear interactions between the process parameters and nonuniform nature of the heat caused by friction which occurred between the conical drill bit rotating at high speed and the workpiece. Since this process is having ambiguities and uncertainties, a model based on fuzzy logic has been developed for the prediction of surface roughness of drilled holes in the FD process. Operating parameters such as rotational speed of the spindle, feed rate, and workpiece temperature are the three membership functions chosen to propose this fuzzy model. These functions are assigned for each input of the model. This fuzzy logic model is verified by two firsthand set of parameter values. The results opine that the established fuzzy model is well in agreement with the investigational data with the maximum deviation of 3.81%. Furthermore, three-dimensional surface plots are developed using this fuzzy model to reveal the influence of individual process parameters on the surface ambiguities. The outcomes of the study attest that the three-dimensional surface plots are much useful for selecting input parameter combinations to achieve the required surface roughness.

This study aimed to use fuzzy logic model to predict various mechanical properties such as compressive strength, flexural strength and post-peak deformation of steel-fiber reinforced concrete. For this purpose, five different dosages of steel fibers (10 kg/m3, 12.5 kg/m3, 15 kg/m3, 17.5 kg/m3 and 20 kg/m3) were used in the mix design. A total of 3 specimens (cylinders and beams) were casted for each fiber dosage and experimentally tested. The experimental results were compared with the simulated results obtained from fuzzy model using fuzzy logic toolbox provided in MATLAB®. It was found that the addition of steel fibers improved the flexural strength and post-peak deformation capability of test specimens by almost 12% and 20% respectively. The compressive strength of specimens also increased by 9.15%, when the amount of steel fibers was increased from 10 to 20 kg/m3. Overall, the compressive strength reduced with the addition of fibers as compared to specimens with no fibers. Similarly, it was also observed that fuzzy model can predict the study parameters within acceptable accuracy and the percentage difference between the simulated and experimental values was below 7.5%.

This paper is concerned with a framework that combines sliding mode control and fuzzy logic systems. This controller is based on the variable structure system control theory and entails the advantages of sliding mode control and fuzzy logic control. The controlled system is a simulated two-links manipulator robot. It is found that the use of sliding mode controller gives good results yielding less coupling between freedom degrees. Moreover, the usage of fuzzy logic applied to the sliding mode controller leads a reduction of the size of the fuzzy inference system.

In the present work, compressive strength of ash-based geopolymers with different mixtures of % rice husk ash, fly ash, nano alumina and nano silica has been predicted by fuzzy logic. The alkali activating was done by a mixture of sodium hydroxide and sodium silicate solution. It was found that oven curing temperature and sodium hydroxide concentration together with SiO2 nanoparticles has the main effects on compressive strength of the specimens. The fuzzy logic model was constructed by 12 input parameters including the water curing time, the rice husk ash content, the fly ash content, the water glass content, sodium hydroxide content, the water content, the aggregate content, SiO2 nanoparticles content, Al2O3 nanoparticles content, oven curing temperature, oven curing time and test trial number. The value for the output layer was the compressive strength. According to the input parameters in fuzzy model, the input data were trained, validated and tested. The results indicate that fuzzy model is a powerful tool for predicting the effect of nanoparticles on compressive strength of the geopolymers in the considered range.

A fuzzy logic predictive model for better surface roughness of Ti–TiN coating on AL7075-T6 alloy for longer fretting fatigue life