Rewrite Systems | 重写系统
Definition | 定义
Rewrite Systems (also known as Rewriting Systems) are formal systems used to express computation or transformation processes by systematically replacing subterms of a formula with other terms. This concept is a generalization of string rewriting, term rewriting, and graph rewriting. A rewrite system consists of a set of rules for transforming expressions into other expressions.
重写系统（也称为重写系统）是用于通过系统地将公式的子项替换为其他项来表达计算或转换过程的形式系统。这一概念是字符串重写、项重写和图重写的推广。重写系统由一组将表达式转换为其他表达式的规则组成。
Components | 组成部分
1. Rules | 规则
A rule in a rewrite system is typically written in the form:
where and are terms, and is called the left-hand side (LHS) and is the right-hand side (RHS). The rule means that any instance of can be replaced by in an expression.
重写系统中的规则通常写成如下形式：
其中， 和 是项， 称为左侧（LHS）， 称为右侧（RHS）。该规则意味着表达式中的任何 实例都可以替换为 。
2. Term | 项
A term is a basic object in a rewrite system that can be replaced according to the rules. Terms can represent variables, constants, operators, or more complex structures like trees.
项是重写系统中的基本对象，可以根据规则进行替换。项可以表示变量、常量、操作符或更复杂的结构（如树）。
3. Application of Rules | 规则的应用
The application of a rule to a term involves finding a subterm that matches the left-hand side of a rule and replacing it with the right-hand side. The process of applying rules repeatedly is known as rewriting.
对项 应用规则涉及找到与规则左侧匹配的子项，并将其替换为右侧。反复应用规则的过程称为重写。
Properties of Rewrite Systems | 重写系统的性质
1. Confluence | 合流性
A rewrite system is confluent if, for any term that can be rewritten in different ways, all sequences of rewriting lead to the same final result, regardless of the order in which the rules are applied.
如果对于可以通过不同方式重写的任何项，所有重写序列都导致相同的最终结果，则该重写系统是合流的，无论规则应用的顺序如何。
2. Termination | 终止性
A rewrite system is terminating if there are no infinite sequences of rewrites. In other words, starting from any term, the rewriting process will eventually reach a term where no rules can be applied.
如果不存在无限的重写序列，则该重写系统是终止的。换句话说，从任何项开始，重写过程最终将达到一个不能应用任何规则的项。
Common Types of Rewrite Systems | 常见的重写系统类型
1. String Rewriting Systems | 字符串重写系统
In string rewriting systems, the terms are strings, and the rules define how substrings can be replaced with other substrings. An example of a string rewriting system is the Thue system.
在字符串重写系统中，项是字符串，规则定义了如何将子字符串替换为其他子字符串。字符串重写系统的一个例子是Thue 系统。
2. Term Rewriting Systems (TRS) | 项重写系统
In term rewriting systems, the terms are structured objects like trees, and the rules describe how subtrees can be replaced. TRS is commonly used in functional programming and symbolic computation.
在项重写系统中，项是结构化的对象（如树），规则描述了如何替换子树。项重写系统常用于函数式编程和符号计算。
3. Graph Rewriting Systems | 图重写系统
In graph rewriting systems, the terms are graphs, and the rules specify how subgraphs can be transformed. This type of system is useful in areas like modeling concurrent computations.
在图重写系统中，项是图，规则指定了如何转换子图。这种系统在建模并发计算等领域中很有用。
Applications | 应用
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Automated theorem proving: Rewrite systems can be used to automatically transform logical expressions into simpler or normal forms, facilitating the proof process.
自动定理证明：重写系统可以自动将逻辑表达式转换为更简单或标准的形式，从而促进证明过程。
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Program transformation and optimization: Rewrite rules can optimize code by systematically replacing inefficient patterns with more efficient ones.
程序转换与优化：通过系统地将低效模式替换为更高效的模式，重写规则可以优化代码。
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Modeling biological and chemical processes: Rewrite systems are used to model the transformations in biological or chemical networks where elements are systematically replaced or transformed.
建模生物和化学过程：重写系统用于建模生物或化学网络中的转换，其中元素被系统地替换或转化。
Conclusion | 总结
Rewrite Systems provide a powerful framework for representing and automating transformations across various domains. The key aspects of rewrite systems include the definition of rules, the application of these rules, and the properties of the system such as confluence and termination. Understanding these systems is essential for applications ranging from theoretical computer science to practical programming optimizations.
重写系统为表示和自动化各个领域的转换提供了强大的框架。重写系统的关键方面包括规则的定义、这些规则的应用以及系统的性质，如合流性和终止性。理解这些系统对于从理论计算机科学到实际编程优化的应用都至关重要。