数学代考| Undefined Values 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

Total functions $f: \mathrm{X} \rightarrow \mathrm{Y}$ are functions that are defined for every element in their domain, and total functions are widely used in mathematics. However, there are functions that are undefined for one or more elements in their domain, and one example is the function $y=1 / x$. This function is undefined at $x=0$.

Partial functions arise naturally in computer science, and such functions may fail to be defined for one or more values in their domain. One approach to deal with partial functions is to employ a precondition, which restricts the application of the function to where it is defined. This makes it possible to define a new set (a proper subset of the domain of the function) for which the function is total over the new set.
$16.5$ Undefined Values Undefined terms often $\operatorname{arise}^{1}$ and need to be dealt with. Consider, the example of the square root function $\sqrt{x}$ taken from [4]. The domain of this function is the positive real numbers, and the following expression is undefined:
$$
((x>0) \wedge(y=\sqrt{x})) \vee((x \leq 0) \wedge(y=\sqrt{-} x)) .
$$
The reason this is undefined is since the usual rules for evaluating such an expression involve evaluating each sub-expression and then performing the Boolean operations. However, when $x<0$, the sub-expression $y=\sqrt{x}$ is undefined, whereas when $x>0$, the sub-expression $y=\sqrt{-x}$ is undefined. Clearly, it is desirable that such expressions be handled, and that for the example above, the expression would evaluate to be true.

Classical two-valued logic does not handle this situation adequately, and there have been several proposals to deal with undefined values. Dijkstra’s approach is to use the cand and cor operators in which the value of the left-hand operand determines whether the right-hand operand expression is evaluated or not. Jone’s logic of partial functions [5] uses a three-valued $\operatorname{logic}^{2}$ and Parnas’s $^{3}$ approach is an extension to the predicate calculus to deal with partial functions that preserve the two-valued logic.
16.5.1 Logic of Partial Functions
Jones [5] has proposed the logic of partial functions (LPFs) as an approach to deal with terms that may be undefined. This is a three-valued logic and a logical term may be true, false or undefined (denoted $\perp$ ). The definition of the truth-functional operators used in classical two-valued logic is extended to three-valued logic. The truth tables for conjunction and disjunction are defined in Fig. 16.1.

The conjunction of $P$ and $Q$ is true when both $P$ and $Q$ are true; false if one of $P$ or $Q$ is false; and undefined otherwise. The operation is commutative. The disjunction of $P$ and $Q(P \vee Q)$ is true if one of $P$ or $Q$ is true; false if both $P$ and $Q$ are false; and undefined otherwise. The implication operation $(P \rightarrow Q)$ is true when $P$ is false or when $Q$ is true; false when $P$ is true and $Q$ is false; and undefined otherwise. The equivalence operation $(\mathrm{P} \leftrightarrow Q)$ is true when both $P$ and $Q$ are true or false; it is false when $P$ is true and $Q$ is false (and vice versa); and it is undefined otherwise (Fig. 16.2).

The not operator $(\neg)$ is a unary operator such $\neg A$ is true when $A$ is false, false when $A$ is true and undefined when $A$ is undefined (Fig. 16.3).

图论代考

总函数 $f: \mathrm{X} \rightarrow \mathrm{Y}$ 是为其域中的每个元素定义的函数，并且总函数在数学中被广泛使用。但是，在其域中存在未定义的一个或多个元素的函数，例如函数 $y=1 / x$。此函数在 $x=0$ 处未定义。

偏函数在计算机科学中自然出现，并且此类函数可能无法为其域中的一个或多个值定义。处理偏函数的一种方法是使用前置条件，它将函数的应用限制在定义它的位置。这使得定义一个新集合（函数域的真子集）成为可能，其中函数在新集合上是总的。
$16.5$ 未定义的值未定义的术语通常是 $\operatorname{arise}^{1}$ 并且需要处理。考虑取自 [4] 的平方根函数 $\sqrt{x}$ 的示例。这个函数的域是正实数，下面的表达式是未定义的：
$$
((x>0) \wedge(y=\sqrt{x})) \vee((x \leq 0) \wedge(y=\sqrt{-} x)) 。
$$
这是未定义的原因是因为评估此类表达式的通常规则涉及评估每个子表达式，然后执行布尔运算。但是，当 $x<0$ 时，子表达式 $y=\sqrt{x}$ 未定义，而当 $x>0$ 时，子表达式 $y=\sqrt{-x}$ 未定义。显然，希望处理这样的表达式，并且对于上面的示例，表达式的计算结果为真。

经典的二值逻辑不能充分处理这种情况，并且已经有几个处理未定义值的建议。 Dijkstra 的方法是使用 cand 和 cor 运算符，其中左侧操作数的值决定是否计算右侧操作数表达式。 Jone 的偏函数逻辑 [5] 使用三值 $\operatorname{logic}^{2}$ 和 Parnas 的 $^{3}$ 方法是谓词演算的扩展，以处理保留二值的偏函数有价值的逻辑。
16.5.1 偏函数的逻辑
Jones [5] 提出了偏函数逻辑 (LPF) 作为一种处理可能未定义的术语的方法。这是一个三值逻辑，逻辑项可以是真、假或未定义（表示为 $\perp$ ）。经典二值逻辑中使用的真值泛函运算符的定义扩展到三值逻辑。合取和析取的真值表在图 16.1 中定义。

当 $P$ 和 $Q$ 都为真时，$P$ 和 $Q$ 的合取为真；如果 $P$ 或 $Q$ 之一为假，则为假；否则未定义。该操作是可交换的。如果 $P$ 或 $Q$ 之一为真，则 $P$ 和 $Q(P \vee Q)$ 的析取为真；如果 $P$ 和 $Q$ 都为假，则为假；否则未定义。当$P$ 为假或$Q$ 为真时，蕴含运算$(P \rightarrow Q)$ 为真；当 $P$ 为真且 $Q$ 为假时为假；否则未定义。当 $P$ 和 $Q$ 都为真或假时，等价运算 $(\mathrm{P} \leftrightarrow Q)$ 为真；当 $P$ 为真且 $Q$ 为假时，它为假（反之亦然）；否则它是未定义的（图 16.2）。

非运算符 $(\neg)$ 是一元运算符，当 $A$ 为假时 $\neg A$ 为真，当 $A$ 为真时为假，当 $A$ 未定义时为未定义（图 16.3）。

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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]