数学代考| Temporal Logic 离散数学代写代写 - 代考代写：100%准时可靠您的作业代写专家

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.

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Temporal logic is concerned with the expression of properties that have time dependencies, and the various temporal logics can express facts about the past, present and future. Temporal logic has been applied to specify temporal properties of natural language, artificial intelligence as well as the specification and verification of program and system behaviour. It provides a language to encode temporal knowledge in artificial intelligence applications, and it plays a useful role in the knowledge in artificial intelligence applications, and it plays a useful role in the formal specification and verification of temporal properties (e.g. liveness and fairness) in safety- critical systems.

The statements made in temporal logic can have a truth-value that varies over time. In other words, sometimes the statement is true and sometimes it is false, but it is never true or false at the same time. The two main types of temporal logics are linear time logics (reasoning about a single timeline) and branching time logics (reasoning about multiple timelines).
The roots of temporal logic lie in work done by Aristotle in the fourth century B. C. when he considered whether a truth-value should be given to a statement about a future event that may or may not occur. For example, what truth value (if any) should be given to the statement that “There will be a sea battle tomorrow”. Aristotle argued against assigning a truth-value to such statements in the present time.
Newtonian mechanics assumes an absolute concept of time independent of space, and this viewpoint remained dominant until the development of the theory of relativity in the early twentieth century (when space-time became the dominant paradigm). Arthur Prior began analyzing and formalizing the truth-values of statements concerning future events in the $1950 \mathrm{~s}$, and he introduced Tense Logic (a temporal logic) in the parly 1960s. Tense logic contains four modal operators (strong and weak) that express events in the future or in the past:
$-P$ (It has at some time been the case that),

$F$ (It will be at some time be the case that),
$H$ (It has always been the case that),
$G$ (It will always be the case that).
272
16 Advanced Topics in Logic
The $P$ and $F$ operators are known as weak tense operators, while the $H$ and $G$ operators are known as strong tense operators. The two pairs of operators are interdefinable via the equivalences:
$$
\begin{aligned}
H \phi, & \cong \neg P \neg \phi \
F \phi & \cong \neg G \neg \phi \
G \phi, & \cong \neg F \neg \phi
\end{aligned}
$$
$$
P \phi \cong \neg H \neg \phi
$$
The set of formulae in Prior’s temporal logic may be defined recursively, and they include the connectives used in classical logic (e.g. $\neg, \wedge, \vee, \rightarrow, \leftrightarrow) .$ We can express a property $\phi$ that is always true as $A \phi \cong H \phi \wedge \phi \wedge G \phi$, and a property that is sometimes true as $E \phi \cong P \phi \vee \phi \vee F \phi$. Various extensions of Prior’s tense logic have been proposed to enhance its expressiveness, including the binary since temporal operator ‘ $S$ ‘, and the binary until temporal operator ‘ $U$ ‘. For example, the meaning of $\phi S \psi$ is that $\phi$ has been true since a time when $\psi$ was true.

Temporal logics are applicable in the specification of computer systems, and a For example, a faimess property may state that it will always be the case that a certain property will hold sometime in the future. The specification of temporal properties often involves the use of special temporal operators.

We discuss common temporal operators that are used, including an operator to express properties that will always be true, properties that will eventually be true and a property that will be true in the next time instance. For example
$\square P \quad P$ is always true.
$P P$ will be true sometime in the future.
$O P \quad P$ is true in the next time instant (discrete time).
Linear Temporal Logic (LTL) was introduced by Pnueli in the late- $1970 \mathrm{~s}$. This linear time logic is useful in expressing safety and liveness properties. Branching time logics assume a non-deterministic branching future for time (with a deterministic, linear past). Computation Tree Logic (CTL and CTL*) were introduced in the early $1980 \mathrm{~s}$ by Emerson and others.

It is also possible to express temporal operations directly in classical mathematics, and the well-known computer scientist, Parnas, prefers this approach. He is sical mathematics already possesses the ability to do this. For example, the value of

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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]