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Logic
Published in Jay Liebowitz, The Handbook of Applied Expert Systems, 2019
The semantics of classical logic assigns meaning to the formulae by defining a mapping between the symbols of the formula and the objects and relationships of the world. This mapping is called an interpretation. An interpretation consists of a domain and a mapping from elements, functions, and relations of the domain to constant, variables, function, and predicate symbols in the formula. Constants, variables, and terms denote elements of the domain. A formula can take the values true or false. A model M is an interpretation that makes a formula α true and is denoted: M = α, which is read “M satisfies α.” A formula α is valid if it is true for every interpretation, and denoted by =α. A formula is satisfiable if it has a model, and is a contradiction if it is false for every interpretation. A set of formulae Γ is satisfiable if there is an interpretation that satisfies every formula in Γ. The fact that every interpretation that satisfies a set of formulae Γ also satisfies a formula a, is denoted by Γ=α and read “Γ logically implies α.” A proof procedure (or calculus) consists of a set of axioms, and a set of rules of inference for deductive reasoning. A formula α which is derived from the axioms by a sequence of inference rules applications is called a theorem and is denoted by: 〈α. If α is derived from a set of formulae Γ, we write Γ 〈 a, which is read α is deduced from Γ. Soundness and completeness are two attributes of a proof procedure with respect to semantics. Soundness means that any theorem deduced by the proof procedure is valid. Completeness means that any valid formula can be proved by the proof procedure. The completeness theorem establishes the relationship between deduction and logical implication:
Adaptive active fault-tolerant controller design for high-speed trains subject to unknown actuator faults
Published in Vehicle System Dynamics, 2018
Xue Lin, Hairong Dong, Xiuming Yao, Baigen Cai
Define . By means of the proof procedure of Theorem 3.1, substituting Equations (8), (22) and (23) into Equation (16) yields Hence, with the application of Lemma 2.2, we can have where According to Equation (25), we obtain . Thus, pre- and post-multiplying both sides of Equation (24) by yields Through applying the Schur complement, Equation (28) can be rewritten as Let . Obviously, on the basis of Lemma 2.1 and , we can have , which implies Equation (27) can be rearranged as where . Based on the bounded signals and , it is obvious that is bounded. The solution of Equation (30) is as follows: which implies that all the signals of the closed-loop system (8) are uniformly ultimately bounded. Furthermore, when , the error between and is So far the proof of Theorem 3.2 has been completed.
An application of adaptive synchronization of uncertain chaotic system in secure communication systems
Published in International Journal of Modelling and Simulation, 2022
Milad Mohadeszadeh, Naser Pariz
where . The proof procedure is the same as the proof of theorem 1; and we omitted here. Finally, one can conclude that the FPS of coupled FO complex chaotic systems is satisfied. Remark 2. The proposed method in this work is robust to the variation of the adaption rate . First, we employ the cost function as , where the time evolution of , the synchronization errors and , the control inputs and and the variation of errors and are replaced by the function . The effectiveness of our scheme can be verified numerically by the variation of the parameter , as shown in Table 1,Remark 3. In practical applications like power systems, the information signal may has low amplitude and high frequency. So, another information signal is chosen to demonstrate the feasibility of this work. The time evolution of and the recovered signal are shown in Figure 9.
New Method of Fuzzy Conditional Inference and Reasoning
Published in Fuzzy Information and Engineering, 2019
We introduced fuzzy natural deduction as an easily proof procedure. The fuzzy intuitions are studied for fuzzy conditional inference with our method for the proposition containing ‘if ··· then ··· ’, ‘if ··· and/or ··· and/or then ··· ’, and ‘if ··· then ··· else ··· ’. All the Fuzzy intuitions are satisfied with our method.