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based on degrees of truth, rather than on the standard binary truth-values of ‘true or false’ (1 or 0) of propositional logic. That is, while statements made in propositional logic are either true or false ( 1 or 0$)$, the truth-value of a statement made in fuzzy logic is a value between 0 and 1. Its value expresses the extent to which the statement is true, with a value of 1 expressing absolute truth, and a value of 0 expressing absolute falsity.

Fuzzy logic uses degrees of truth as a mathematical model for vagueness, and this is useful since statements made in natural language are often vague and have a certain (rather than an absolute) degree of truth. It is an extension of classical logic to handle the concept of partial truth, where the truth-value lies between completely true and completely false. Lofti Zadeh developed fuzzy logic at Berkeley in the $1960 \mathrm{~s}$, and it has been successfully applied to Expert Systems and other areas of Artificial Intelligence.

For example, consider the statement “John is tall”. If John were 6 foot, 4 inches, then we would say that this is a true statement (with a truth value of 1) since John is well above average height. However, if John is 5 feet, 9 inches tall (around average height), then this statement has a degree of truth, and this could be indicated by a fuzzy truth valued of $0.6$. Similarly, the statement that today is sunny may be assigned a truth-value of 1 if there are no clouds, $0.8$ if there are a small number of clouds and 0 if it is raining all day.

Propositions in fuzzy logic may be combined together to form compound propositions. Suppose $X$ and $Y$ are propositions in fuzzy logic, then compound propositions may be formed from the conjunction, disjunction and implication operators. The usual definition in fuzzy logic of the truth-values of the compound propositions formed from $X$ and $Y$ is given by
$$
\begin{aligned}
&\text { Truth }(\neg X)=1 \text {-Truth }(X), \
&(X \text { and } Y)=\min (\text { Truth }(X), \operatorname{Truth}(Y)),
\end{aligned}
$$
Truth $(X$ and $Y)=\min ($ Truth $(X)$, Truth $(Y))$, Truth $(X$ or $Y)=\max (\operatorname{Truth}(X)$, Truth $(Y))$,
Truth $(X \rightarrow Y)=\operatorname{Truth}(\neg X$ or $Y)$
There is another way in which the operators may be defined in terms of multiplication:
$$
\text { Truth }(X \text { and } Y)=\operatorname{Truth}(X) * \operatorname{Truth}(Y),
$$
Truth $(X$ or $Y)=1-(1-\operatorname{Truth}(X)) *(1-\operatorname{Truth}(Y))$
Truth $(X \rightarrow Y)=\max {z \mid \operatorname{Truth}(X) * z \leq \operatorname{Truth}(Y)}$ where $0 \leq z \leq 1$
$16.2$ Fuzzy Logic
271
Under these definitions, fuzzy logic is an extension of classical two-valued logic, which preserves the usual meaning of the logical connectives of propositional logic when the fuzzy values are just ${0,1}$.

Fuzzy logic has been useful in the expert system and artificial intelligence applications, and it has also been applied to the aerospace and automotive sectors, and to the medical, robotics and transport sectors. The first fuzzy logic controller was developed in England, in the mid-1970s.

图论代考

基于真实程度，而不是基于命题逻辑的“真或假”（1 或 0）的标准二进制真值。也就是说，虽然命题逻辑中的陈述是真或假（1 或 0$）$，但模糊逻辑中的陈述的真值是介于 0 和 1 之间的值。它的值表示该陈述的程度为真，值为 1 表示绝对真，值为 0 表示绝对假。

模糊逻辑使用真实度作为模糊性的数学模型，这很有用，因为用自然语言做出的陈述通常是模糊的，并且具有一定的（而不是绝对的）真实度。它是处理部分真理概念的经典逻辑的扩展，其中真值介于完全正确和完全错误之间。 Lofti Zadeh 在伯克利以 1960 美元开发了模糊逻辑，并已成功应用于专家系统和人工智能的其他领域。

例如，考虑语句“John is high”。如果约翰身高 6 英尺 4 英寸，那么我们会说这是一个真实的陈述（真值为 1），因为约翰的身高远高于平均身高。然而，如果约翰身高 5 英尺 9 英寸（大约是平均身高），那么这个陈述就有一定程度的真实性，这可以通过价值 0.6 美元的模糊真实性来表示。类似地，如果没有云，则可以为今天是晴天的语句分配真值 1，如果有少量云，则为 $0.8$，如果一整天都在下雨，则可以分配真值 0。

模糊逻辑中的命题可以组合在一起形成复合命题。假设$X$和$Y$是模糊逻辑中的命题，那么复合命题可以由合取、析取和蕴涵算子构成。由 $X$ 和 $Y$ 形成的复合命题的真值的模糊逻辑的通常定义由下式给出
$$
\开始{对齐}
&\text { 真相 }(\neg X)=1 \text {-真相 }(X), \
&(X \text { and } Y)=\min (\text { Truth }(X), \operatorname{Truth}(Y)),
\end{对齐}
$$
真值 $(X$ 和 $Y)=\min ($ 真值 $(X)$, 真值 $(Y))$, 真值 $(X$ 或 $Y)=\max (\operatorname{真值}(X) $, 真相 $(Y))$,
真相 $(X \rightarrow Y)=\operatorname{真相}(\neg X$ or $Y)$
还有另一种方法可以用乘法来定义运算符：
$$
\text { 真相 }(X \text { 和 } Y)=\operatorname{真相}(X) * \operatorname{真相}(Y),
$$
真相 $(X$ 或 $Y)=1-(1-\operatorname{Truth}(X)) *(1-\operatorname{Truth}(Y))$
真相 $(X \rightarrow Y)=\max {z \mid \operatorname{Truth}(X) * z \leq \operatorname{Truth}(Y)}$ 其中 $0 \leq z \leq 1$
$16.2$ 模糊逻辑
271
在这些定义下，模糊逻辑是经典二值逻辑的扩展，当模糊值只是 ${0,1}$ 时，它保留了命题逻辑的逻辑连接词的通常含义。

模糊逻辑在专家系统和人工智能应用中非常有用，它也被应用于航空航天和汽车领域，以及医疗、机器人和运输领域。第一个模糊逻辑控制器是在 1970 年代中期在英国开发的。

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Cryptosystem
A system that describes how to encrypt or decrypt messages
Plaintext
Message in its original form
Ciphertext
Message in its encrypted form
Cryptographer
Invents encryption algorithms
Cryptanalyst
Breaks encryption algorithms or implementations

编码理论代写

编码理论（英语：Coding theory）是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩、加密、纠错，最近也用于网络编码中。不同学科（如信息论、电机工程学、数学、语言学以及计算机科学）都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正（或检测）数据传输中的错误。

编码共分四类：^[1]