# Fuzzy Logic

### Provably Efficient $Q$-learning with Function Approximation via Distribution Shift Error Checking Oracle

$Q$-learning with function approximation is one of the most popular methods in reinforcement learning. Though the idea of using function approximation was proposed at least $60$ years ago, even in the simplest setup, i.e, approximating $Q$-functions with linear functions, it is still an open problem how to design a provably efficient algorithm that learns a near-optimal policy. The key challenges are how to efficiently explore the state space and how to decide when to stop exploring in conjunction with the function approximation scheme. The current paper presents a provably efficient algorithm for $Q$-learning with linear function approximation. Under certain regularity assumptions, our algorithm, Difference Maximization $Q$-learning (DMQ), combined with linear function approximation, returns a near optimal policy using polynomial number of trajectories. Our algorithm introduces a new notion, the Distribution Shift Error Checking (DSEC) oracle. This oracle tests whether there exists a function in the function class that predicts well on a distribution $\mathcal{D}_1$, but predicts poorly on another distribution $\mathcal{D}_2$, where $\mathcal{D}_1$ and $\mathcal{D}_2$ are distributions over states induced by two different exploration policies. For the linear function class, this oracle is equivalent to solving a top eigenvalue problem. We believe our algorithmic insights, especially the DSEC oracle, are also useful in designing and analyzing reinforcement learning algorithms with general function approximation.

### Extension of Rough Set Based on Positive Transitive Relation

The application of rough set theory in incomplete information systems is a key problem in practice since missing values almost always occur in knowledge acquisition due to the error of data measuring, the limitation of data collection, or the limitation of data comprehension, etc. An incomplete information system is mainly processed by compressing the indiscernibility relation. The existing rough set extension models based on tolerance or symmetric similarity relations typically discard one relation among the reflexive, symmetric and transitive relations, especially the transitive relation. In order to overcome the limitations of the current rough set extension models, we define a new relation called the positive transitive relation and then propose a novel rough set extension model built upon which. The new model holds the merit of the existing rough set extension models while avoids their limitations of discarding transitivity or symmetry. In comparison to the existing extension models, the proposed model has a better performance in processing the incomplete information systems while substantially reducing the computational complexity, taking into account the relation of tolerance and similarity of positive transitivity, and supplementing the related theories in accordance to the intuitive classification of incomplete information. In summary, the positive transitive relation can improve current theoretical analysis of incomplete information systems and the newly proposed extension model is more suitable for processing incomplete information systems and has a broad application prospect.

### Anticipation in collaborative music performance using fuzzy systems: a case study

The creation and performance of music has inspired AI researchers since the very early times of artificial intelligence [8, 13, 10], and there is today a rich literature of computational approaches to music [11], including AI systems for music composition [3] and improvisation [2]. As pointed out by Thom [15], however, these systems rarely focus on the spontanous interaction between the human and the artificial musicians. We claim that such interaction demands a combination of reactivity and anticipation, that is, the ability to act now based on a predictive model of the companion player [12]. This paper reports our initial steps in the generation of collaborative human-machine music performance, as a special case of the more general problem of anticipation and creative processes in mixed human-robot, or anthrobotic systems [4]. We consider a simple case study of a duo consisting of a human pianist accompained by an off-the-shelf virtual drummer, and we design an AI system to control the key perfomance parameters of the virtual drummer (patterns, intensity, complexity, fills, and so on) as a function of what the human pianist is playing. The AI system is knowledge-based: it relies on an internal model represented by means of fuzzy logic.

### Function approximation by deep networks

We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thus, the blessing of compositionality mitigates the curse of dimensionality. On the other hand, a theorem called good propagation of errors allows to `lift' theorems about shallow networks to those about deep networks with an appropriate choice of norms, smoothness, etc. We illustrate this in three contexts where each channel in the deep network calculates a spherical polynomial, a non-smooth ReLU network, or another zonal function network related closely with the ReLU network.

### Deep Fuzzy Systems

ABSTRACT-An investigation of deep fuzzy systems is presented in this paper. A deep fuzzy system is represented by recursive fuzzy systems from an input terminal to output terminal. Recursive fuzzy systems are sequences of fuzzy grade memberships obtained using fuzzy transmition functions and recursive calls to fuzzy systems. A recursive fuzzy system which calls a fuzzy system times includes fuzzy chains to evaluate the final grade membership of this recursive system. A connection matrix which includes recursive calls are used to represent recursive fuzzy systems.

### Descriptive evaluation of students using fuzzy approximate reasoning

In recent years, descriptive evaluation has been introduced as a new model for educational evaluation of Iranian students. The current descriptive evaluation method is based on four-valued logic. Assessing all students with only four values is led to a lack of relative justice and the creation of unrealistic equality. Also, the complexity of the evaluation process in the current method increases teacher errors likelihood. As a suitable solution, in this paper, a fuzzy descriptive evaluation system has been proposed. The proposed method is based on fuzzy logic, which is an infinite-valued logic and it can perform approximate reasoning on natural language propositions. By the proposed fuzzy system, student assessment is performed over the school year with infinite values instead of four values. But to eliminate the diversity of assigned values to students, at the end of the school year, the calculated values for each student will be rounded to the nearest value of the four standard values of the current descriptive evaluation system. It can be implemented easily in an appropriate smartphone app, which makes it much easier for the teachers to evaluate the evaluation process. In this paper, the evaluation process of the elementary third-grade mathematics course in Iran during the period from the beginning of the MEHR (The Seventh month of Iran) to the end of BAHMAN (The Eleventh Month of Iran) is examined by the proposed system. To evaluate the validity of this system, the proposed method has been simulated in MATLAB software.

### A knowledge-based intelligence system for control of dirt recognition process in the smart washing machines

In this paper, we propose an intelligence approach based on fuzzy logic to modeling human intelligence in washing clothes. At first, an intelligent feedback loop is designed for perception-based sensing of dirt inspired by human color understanding. Then, when color stains leak out of some colored clothes the human probabilistic decision making is computationally modeled to detect this stain leakage and thus the problem of recognizing dirt from stain can be considered in the washing process. Finally, we discuss the fuzzy control of washing clothes and design and simulate a smart controller based on the fuzzy intelligence feedback loop.

### A Fuzzy Inference System for the Identification

Odor identification is an important area in a wide range of industries like cosmetics, food, beverages and medical diagnosis among others. Odor detection could be done through an array of gas sensors conformed as an electronic nose where a data acquisition module converts sensor signals to a standard output to be analyzed. To facilitate odors detection a system is required for the identification. This paper presents the results of an automated odor identification process implemented by a fuzzy system and an electronic nose. First, an electronic nose prototype is manufactured to detect organic compounds vapor using an array of five tin dioxide gas sensors, an arduino uno board is used as a data acquisition section. Second, an intelligent module with a fuzzy system is considered for the identification of the signals received by the electronic nose. This solution proposes a system to identify odors by using a personal computer. Results show an acceptable precision.

### Fuzzy Rule Interpolation Methods and Fri Toolbox

FRI methods are less popular in the practical application domain. One possible reason is the missing common framework. There are many FRI methods developed independently, having different interpolation concepts and features. One trial for setting up a common FRI framework was the MATLAB FRI Toolbox, developed by Johany\'ak et. al. in 2006. The goals of this paper are divided as follows: firstly, to present a brief introduction of the FRI methods. Secondly, to introduce a brief description of the refreshed and extended version of the original FRI Toolbox. And thirdly, to use different unified numerical benchmark examples to evaluate and analyze the Fuzzy Rule Interpolation Techniques (FRI) (KH, KH Stabilized, MACI, IMUL, CRF, VKK, GM, FRIPOC, LESFRI, and SCALEMOVE), that will be classified and compared based on different features by following the abnormality and linearity conditions [15].

### Resilient Supplier Selection in Logistic 4.0: An integrated approach of Fuzzy Multi-Attribute Decision Making (F-MADM) and Multi-choice Goal Programming (MCGP) with Heterogeneous

Supplier selection problem has gained extensive attention in the prior studies. However, research based on Fuzzy Multi-Attribute Decision Making (F-MADM) approach in ranking resilient suppliers in logistic 4.0 is still in its infancy. Traditional MADM approach fails to address the resilient supplier selection problem in logistic 4.0 primarily because of the large amount of data concerning some attributes that are quantitative, yet difficult to process while making decisions. Besides, some qualitative attributes prevalent in logistic 4.0 entail imprecise perceptual or judgmental decision relevant information, and are substantially different than those considered in traditional suppler selection problems. This study, for the first time, develops a Decision Support System (DSS) that will help the decision maker to incorporate and process such imprecise heterogeneous data in a unified framework to rank a set of resilient suppliers in the logistic 4.0 environment. The proposed framework induces a triangular fuzzy number from large-scale temporal data using probability-possibility consistency principle. Large number of non-temporal data presented graphically are computed by extracting granular information that are imprecise in nature. Fuzzy linguistic variables are used to map the qualitative attributes. Finally, fuzzy based TOPSIS method is adopted to generate the ranking score of alternative suppliers. These ranking scores are used as input in a Multi-Choice Goal Programming (MCGP) model to determine optimal order allocation for respective suppliers. Finally, a sensitivity analysis assesses how the Cost versus Resilience Index (SCRI) changes when differential priorities are set for respective cost and resilience attributes.