The AI community is increasingly putting its attention towards combining symbolic and neural approaches, as it is often argued that the strengths and weaknesses of these approaches are complementary. One recent trend in the literature are weakly supervised learning techniques that employ operators from fuzzy logics. In particular, these use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks, this background knowledge can be added to regular loss functions, hence making reasoning a part of learning. We study, both formally and empirically, how a large collection of logical operators from the fuzzy logic literature behave in a differentiable learning setting. We find that many of these operators, including some of the most well-known, are highly unsuitable in this setting. A further finding concerns the treatment of implication in these fuzzy logics, and shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and compare how different operators behave in practice. We find that, to achieve the largest performance improvement over a supervised baseline, we have to resort to non-standard combinations of logical operators which perform well in learning, but no longer satisfy the usual logical laws.
The paper aims to develop a framework for coalgebraic fuzzy geometric logic by adding modalities to the language of fuzzy geometric logic. Using the methods of coalgebra, the modal operators are introduced in the language of fuzzy geometric logic. To define the modal operators, we introduce a notion of fuzzy-open predicate lifting. Based on coalgebras for an endofunctor $T$ on the category $\textbf{Fuzzy-Top}$ of fuzzy topological spaces and fuzzy continuous maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations for the defined models are discussed in this work.
Combining symbolic and neural approaches has gained considerable attention in the AI community, as it is often argued that the strengths and weaknesses of these approaches are complementary. One such trend in the literature are weakly supervised learning techniques that employ operators from fuzzy logics. In particular, they use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks (or grounding them), this background knowledge can be added to regular loss functions, henc…
We present a fuzzy version of description logics with concrete domains. Main features are: (i) concept constructors are based on t-norm, t-conorm, negation and implication; (ii) concrete domains are fuzzy sets; (iii) fuzzy modifiers are allowed; and (iv) the reasoning algorithm is based on a mixture of completion rules and bounded mixed integer programming.
Recently there have been some unexpected results concerning Fuzzy Description Logics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike the classical case, the DL ALC with GCIs does not have the finite model property under Lukasiewicz Logic or Product Logic and, specifically, knowledge base satisfiability is an undecidable problem for Product Logic. We complete here the analysis by showing that knowledge base satisfiability is also an undecidable problem for Lukasiewicz Logic.
Spanish Council for Scientific Research (CSIC)（西班牙国家研究委员会）
Within the possibilistic approach to uncertainty modeling, the paper presents a modal logical system to reason about qualitative (comparative) statements of the possibility (and necessity) of fuzzy propositions. We relate this qualitative modal logic to the many--valued analogues MVS5 and MVKD45 of the well known modal logics of knowledge and belief S5 and KD45 respectively. Completeness results are obtained for such logics and therefore, they extend previous existing results for qualitative possibilistic logics in the classical non-fuzzy setting.
Kim Il Sung University（金日成综合大学）；Kim Chaek University of Technology（金策工业综合大学）
In this article, we describe the fuzzy logic, fuzzy language and algorithms as the basis of fuzzy reasoning, one of the intelligent information processing method, and then describe the general fuzzy reasoning method.
In fuzzy propositional logic, to a proposition a partial truth in [0,1] is assigned. It is well known that under certain circumstances, fuzzy logic collapses to classical logic. In this paper, we will show that under dual conditions, fuzzy logic collapses to four-valued (relevance) logic, where propositions have truth-value true, false, unknown, or contradiction. As a consequence, fuzzy entailment may be considered as ``in between'' four-valued (relevance) entailment and classical entailment.
We present a logical framework to represent and reason about fuzzy optimization problems based on fuzzy answer set optimization programming. This is accomplished by allowing fuzzy optimization aggregates, e.g., minimum and maximum in the language of fuzzy answer set optimization programming to allow minimization or maximization of some desired criteria under fuzzy environments. We show the application of the proposed logical fuzzy optimization framework under the fuzzy answer set optimization programming to the fuzzy water allocation optimization problem.
Classical logic has a serious limitation in that it cannot cope with the issues of vagueness and uncertainty into which fall most modes of human reasoning. In order to provide a foundation for human knowledge representation and reasoning in the presence of vagueness, imprecision, and uncertainty, fuzzy logic should have the ability to deal with linguistic hedges, which play a very important role in the modification of fuzzy predicates. In this paper, we extend fuzzy logic in narrow sense with graded syntax, introduced by Novak et al., with many hedge connectives. In one case, each hedge does n…
A fuzzy inference system was developed for predicting the heat index from temperature and relative humidity data. The effectiveness of fuzzy logic in using imprecise mapping of input to output to encode interconnectedness of system variables was exploited to uncover a linguistic model of how the temperature and humidity conditions impact the heat index in a growth room. The developed model achieved an R2 of 0.974 and a RMSE of 0.084 when evaluated on a test set, and the results were statistically significant (F1,5915 = 222900.858, p < 0.001). By providing the advantage of linguistic summarizat…
Artificial Intelligence Research Institute (IIIA), CSIC（西班牙科学研究理事会人工智能研究所）
This paper proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this paper we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.
We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic operators using the smarandache codification of the Venn diagram and a defined vector neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.
We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermo…
We present a unified logical framework for representing and reasoning about both quantitative and qualitative preferences in fuzzy answer set programming, called fuzzy answer set optimization programs. The proposed framework is vital to allow defining quantitative preferences over the possible outcomes of qualitative preferences. We show the application of fuzzy answer set optimization programs to the course scheduling with fuzzy preferences problem. To the best of our knowledge, this development is the first to consider a logical framework for reasoning about quantitative preferences, in gene…
This paper mainly focuses on (1) a generalized treatment of fuzzy sets of type $n$, where $n$ is an integer larger than or equal to $1$, with an example, mathematical discussions, and real-life interpretation of the given mathematical concepts; (2) the potentials and links between fuzzy logic and probability logic that have not been discussed in one document in literature; (3) representation of real-life random and fuzzy uncertainties and ambiguities that arise in data-driven real-life problems, due to uncertain mathematical and vague verbal terms in datasets.
College of Letters and Sciences, National University（美国国立大学文理学院）；Graduate Technological Educational Institute（希腊帕特拉斯技术教育学院）
The article discusses some applications of fuzzy logic ideas to formalizing of the Case-Based Reasoning (CBR) process and to measuring the effectiveness of CBR systems
Fuzzy Description Logics (DLs) are a family of logics which allow the representation of (and the reasoning with) structured knowledge affected by vagueness. Although most of the not very expressive crisp DLs, such as ALC, enjoy the Finite Model Property (FMP), this is not the case once we move into the fuzzy case. In this paper we show that if we allow arbitrary knowledge bases, then the fuzzy DLs ALC under Lukasiewicz and Product fuzzy logics do not verify the FMP even if we restrict to witnessed models; in other words, finite satisfiability and witnessed satisfiability are different for arbi…
Spanish Council for Scientific Research (CSIC)（西班牙科学研究委员会）
In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical two-valued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a {em fuzzy} logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truth-values of some suitable fuzzy propositions associated to the crisp ones.…
Technical Engineering College / Northern Technical University（北方技术大学技术工程学院）
This paper describes the design and development of a prototype technique for artificial intelligence based on the fusion of genetic algorithm, neural network and fuzzy logic. It starts by establishing a relationship between the neural network and fuzzy logic. Then, it combines the genetic algorithm with them. Information fusions are at the confidence level, where matching scores can be reported and discussed. The technique is called the Genetic Neuro-Fuzzy (GNF). It can be used for high accuracy real-time environments.
Institute for Research in Fundamental Sciences (IPM)（基础科学研究所（IPM））；The University of Quebec at Trois-Rivieres（魁北克大学三河分校）
In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy attribute. To do so, we associate some specific crisp random variables to the random experiment. Then, several formulas are presented, which make it easier to compute different conditional probabilities and expected values of these random variables. Also, we provide measure theoretical basis for our probabilistic fuzzy logic framework. Note that in our theory,…
Rift Valley Technical Training Institute（裂谷技术培训学院）
This paper discusses obstacle avoidance using fuzzy logic and shortest path algorithm. This paper also introduces the sliding blades problem and illustrates how a drone can navigate itself through the swinging blade obstacles while tracing a semi-optimal path and also maintaining constant velocity
In recent years, adaptive learning systems rely increasingly on learning hierarchy to customize the educational logic developed in their courses. Most approaches do not consider that the relationships of prerequisites between the skills are fuzzy relationships. In this article, we describe a new approach of a practical application of fuzzy logic techniques to the construction of learning hierarchies. For this, we use a learning hierarchy predefined by one or more experts of a specific field. However, the relationships of prerequisites between the skills in the learning hierarchy are not defini…
This paper discusses the merits and demerits of crisp logic and fuzzy logic with respect to their applicability in intelligent response generation by a human being and by a robot. Intelligent systems must have the capability of taking decisions that are wise and handle situations intelligently. A direct relationship exists between the level of perfection in handling a situation and the level of completeness of the available knowledge or information or data required to handle the situation. The paper concludes that the use of crisp logic with complete knowledge leads to perfection in handling s…
Beginning with a simple semantics for propositions, based on counting observations, it is shown that probabilistic and fuzzy logic correspond to two different heuristic assumptions regarding the combination of propositions whose evidence bases are not currently available. These two different heuristic assumptions lead to two different sets of formulas for propagating quantitative truth values through lattice operations. It is shown that these two sets of formulas provide a natural grounding for the multiplicative and additive operator-sets in linear logic. The standard rules of linear logic th…
This paper discusses a target tracking problem in which no dynamic mathematical model is explicitly assumed. A nonlinear filter based on the fuzzy If-then rules is developed. A comparison with a Kalman filter is made, and empirical results show that the performance of the fuzzy filter is better. Intensive simulations suggest that theoretical justification of the empirical results is possible.
CentraleSupélec（中央理工-高等电力学院）；Laboratoire des Signaux et Systèmes（信号与系统实验室）；Université Paris-Saclay（巴黎-萨克雷大学）；AFSCET（法国系统科学协会）；Conservatoire National des Arts et Métiers（法国国立工艺学院）
This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multiv…
Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e., set of individuals with common properties. The experience in using DLs in applications has shown that in many cases we would like to extend their capabilities. In particular, their use in the context of Multimedia Information Retrieval (MIR) leads to the convincement that such DLs should allow the treatment of the inherent imprecision in multimedia object content representation and retrieval. In this paper we will present a fuzzy e…
National and Technical University of Athens（雅典国立技术大学）；The University of Aberdeen（阿伯丁大学）；The University of Manchester（曼彻斯特大学）
It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). …
Linear representations for a subclass of boolean symmetric functions selected by a parity condition are shown to constitute a generalization of the linear constraints on probabilities introduced by Boole. These linear constraints are necessary to compute probabilities of events with relations between the. arbitrarily specified with propositional calculus boolean formulas.
Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Faculty of Information Technology, Nguyen Tat Thanh University（阮必成大学信息技术学院）
The problem of minimizing finite fuzzy interpretations in fuzzy description logics (FDLs) is worth studying. For example, the structure of a fuzzy/weighted social network can be treated as a fuzzy interpretation in FDLs, where actors are individuals and actions are roles. Minimizing the structure of a fuzzy/weighted social network makes it more compact, thus making network analysis tasks more efficient. In this work, we study the problem of minimizing a finite fuzzy interpretation in a FDL by using the largest crisp auto-bisimulation. The considered FDLs use the Baaz projection operator and th…
A major aspect of human reasoning involves the use of approximations. Particularly in situations where the decision-making process is under stringent time constraints, decisions are based largely on approximate, qualitative assessments of the situations. Our work is concerned with the application of approximate reasoning to real-time control. Because of the stringent processing speed requirements in such applications, hardware implementations of fuzzy logic inferencing are being pursued. We describe a programming environment for translating fuzzy control rules into hardware realizations. Two m…
In this book we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), plithogenic logic (as generalization of classical, fuzzy, intuitionistic fuzzy, and neutrosophic logics), plithogenic probability (as generalization of classical, imprecise, and neutrosophic probabilities), and plithogenic statistics (as generalization of classical, and neutrosophic statistics). Plithogenic Set is a set whose elements are characterized by one or more attributes, and each attribute may have many values. An attribute value v has a corresponding (fuzzy, …
Ithaca is a Fuzzy Logic (FL) plugin for developing artificial intelligence systems within the Unity game engine. Its goal is to provide an intuitive and natural way to build advanced artificial intelligence systems, making the implementation of such a system faster and more affordable. The software is made up by a C\# framework and an Application Programming Interface (API) for writing inference systems, as well as a set of tools for graphic development and debugging. Additionally, a Fuzzy Control Language (FCL) parser is provided in order to import systems previously defined using this standa…
St. Peter’s University（圣彼得大学）；Sahyadri College of Engineering and Management（萨亚德里工程与管理学院）；PA College of Engineering（PA工程学院）
This paper presents a novel fuzzy logic based Adaptive Super-twisting Sliding Mode Controller for the control of dynamic uncertain systems. The proposed controller combines the advantages of Second order Sliding Mode Control, Fuzzy Logic Control and Adaptive Control. The reaching conditions, stability and robustness of the system with the proposed controller are guaranteed. In addition, the proposed controller is well suited for simple design and implementation. The effectiveness of the proposed controller over the first order Sliding Mode Fuzzy Logic controller is illustrated by Matlab based …
National Key Laboratory for Novel Software Technology, Nanjing University（计算机软件新技术全国重点实验室，南京大学）
Neural-symbolic computing aims at integrating robust neural learning and sound symbolic reasoning into a single framework, so as to leverage the complementary strengths of both of these, seemingly unrelated (maybe even contradictory) AI paradigms. The central challenge in neural-symbolic computing is to unify the formulation of neural learning and symbolic reasoning into a single framework with common semantics, that is, to seek a joint representation between a neural model and a logical theory that can support the basic grounding learned by the neural model and also stick to the semantics of …
Ai movement, Center of Artificial Intelligence, Mohammed VI Polytechnic University（穆罕默德六世理工大学人工智能中心 Ai movement）；Lip6, Sorbonne University（索邦大学 Lip6）
Fuzzy logic has been proposed in previous studies for machine diagnosis, to overcome different drawbacks of the traditional diagnostic approaches used. Among these approaches Failure Mode and Effect Critical Analysis method(FMECA) attempts to identify potential modes and treat failures before they occur based on subjective expert judgments. Although several versions of fuzzy logic are used to improve FMECA or to replace it, since it is an extremely cost-intensive approach in terms of failure modes because it evaluates each one of them separately, these propositions have not explicitly focused …
Risk specialists are trying to understand risk better and use complex models for risk assessment, while many risks are not yet well understood. The lack of empirical data and complex causal and outcome relationships make it difficult to estimate the degree to which certain risk types are exposed. Traditional risk models are based on classical set theory. In comparison, fuzzy logic models are built on fuzzy set theory and are useful for analyzing risks with insufficient knowledge or inaccurate data. Fuzzy logic systems help to make large-scale risk management frameworks more simple. For risks t…
Universitat de Lleida (UdL)（莱里达大学）；CSIC（西班牙国家研究委员会）
In this paper we present a propositional logic programming language for reasoning under possibilistic uncertainty and representing vague knowledge. Formulas are represented by pairs (A, c), where A is a many-valued proposition and c is value in the unit interval [0,1] which denotes a lower bound on the belief on A in terms of necessity measures. Belief states are modeled by possibility distributions on the set of all many-valued interpretations. In this framework, (i) we define a syntax and a semantics of the general underlying uncertainty logic; (ii) we provide a modus ponens-style calculus f…
Over past several years, deep learning has achieved huge successes in various applications. However, such a data-driven approach is often criticized for lack of interpretability. Recently, we proposed artificial quadratic neural networks consisting of second-order neurons in potentially many layers. In each second-order neuron, a quadratic function is used in the place of the inner product in a traditional neuron, and then undergoes a nonlinear activation. With a single second-order neuron, any fuzzy logic operation, such as XOR, can be implemented. In this sense, any deep network constructed …
Çankırı Karatekin University（昌克勒卡拉泰金大学）；Gazi University（加齐大学）
The Direct Torque Control (DTC) is well known as an effective control technique for high performance drives in a wide variety of industrial applications and conventional DTC technique uses two constant reference value: torque and stator flux. In this paper, fuzzy logic based stator flux optimization technique for DTC drives that has been proposed. The proposed fuzzy logic based stator flux optimizer self-regulates the stator flux reference using induction motor load situation without need of any motor parameters. Simulation studies have been carried out with Matlab/Simulink to compare the prop…
Mutah University（穆塔大学）；Eotvos Lorand University（厄特沃什·罗兰大学）；University of California Riverside（加州大学河滨分校）；King Abdulaziz City for Science and Technology（阿卜杜勒阿齐兹国王科技城）
The classic win-win has a key flaw in that it cannot offer the parties the right amounts of winning because each party believes they are winners. In reality, one party may win more than the other. This strategy is not limited to a single product or negotiation; it may be applied to a variety of situations in life. We present a novel way to measure the win-win situation in this paper. The proposed method employs Fuzzy logic to create a mathematical model that aids negotiators in quantifying their winning percentages. The model is put to the test on real-life negotiations scenarios such as the I…
The treatment of complex systems often requires the manipulation of vague, imprecise and uncertain information. Indeed, the human being is competent in handling of such systems in a natural way. Instead of thinking in mathematical terms, humans describes the behavior of the system by language proposals. In order to represent this type of information, Zadeh proposed to model the mechanism of human thought by approximate reasoning based on linguistic variables. He introduced the theory of fuzzy sets in 1965, which provides an interface between language and digital worlds. In this paper, we propo…
Indian Institute of Technology Kharagpur（印度理工学院克勒格布尔分校）
A college student's life can be primarily categorized into domains such as education, health, social and other activities which may include daily chores and travelling time. Time management is crucial for every student. A self realisation of one's daily time expenditure in various domains is therefore essential to maximize one's effective output. This paper presents how a mobile application using Fuzzy Logic and Global Positioning System (GPS) analyzes a student's lifestyle and provides recommendations and suggestions based on the results.
Mazandaran Regional Electric Company（马赞达兰地区电力公司）；State University of New York at Binghamton（纽约州立大学宾汉姆顿分校）；University of Hawaii at Manoa（夏威夷大学马诺阿分校）；University of Nebraska（内布拉斯加大学）；Mapna Turbine Engineering and Manufacturing Company（Mapna涡轮工程与制造公司）
Reliability assessment of distribution system, based on historical data and probabilistic methods, leads to an unreliable estimation of reliability indices since the data for the distribution components are usually inaccurate or unavailable. Fuzzy logic is an efficient method to deal with the uncertainty in reliability inputs. In this paper, the ENS index along with other commonly used indices in reliability assessment are evaluated for the distribution system using fuzzy logic. Accordingly, the influential variables on the failure rate and outage duration time of the distribution components, …
The quality of the ranking function is an important factor that determines the quality of the Information Retrieval system. Each document is assigned a score by the ranking function; the score indicates the likelihood of relevance of the document given a query. In the vector space model, the ranking function is defined by a mathematic expression. We propose a fuzzy logic (FL) approach to defining the ranking function. FL provides a convenient way of converting knowledge expressed in a natural language into fuzzy logic rules. The resulting ranking function could be easily viewed, extended, and …
Statistical Hypothesis Testing (SHT) is a class of inference methods whereby one makes use of empirical data to test a hypothesis and often emit a judgment about whether to reject it or not. In this paper we focus on the logical aspect of this strategy, which is largely independent of the adopted school of thought, at least within the various frequentist approaches. We identify SHT as taking the form of an unsound argument from Modus Tollens in classical logic, and, in order to rescue SHT from this difficulty, we propose that it can instead be grounded in t-norm based fuzzy logics. We reformul…
University of Alberta（阿尔伯塔大学）；Carnegie Mellon University（卡内基梅隆大学）
Natural language inference (NLI) aims to determine the logical relationship between two sentences, such as Entailment, Contradiction, and Neutral. In recent years, deep learning models have become a prevailing approach to NLI, but they lack interpretability and explainability. In this work, we address the explainability of NLI by weakly supervised logical reasoning, and propose an Explainable Phrasal Reasoning (EPR) approach. Our model first detects phrases as the semantic unit and aligns corresponding phrases in the two sentences. Then, the model predicts the NLI label for the aligned phrases…
SAPIENZA University of Rome（罗马大学）；Ryerson University（瑞尔森大学）
Bio-inspired algorithms like Genetic Algorithms and Fuzzy Inference Systems (FIS) are nowadays widely adopted as hybrid techniques in commercial and industrial environment. In this paper we present an interesting application of the fuzzy-GA paradigm to Smart Grids. The main aim consists in performing decision making for power flow management tasks in the proposed microgrid model equipped by renewable sources and an energy storage system, taking into account the economical profit in energy trading with the main-grid. In particular, this study focuses on the application of a Hierarchical Genetic…
