标题： 一种关于替代的模态方法 显示英文标题

标题： A modal approach towards substitutions

摘要： 替代在各种上下文中起着至关重要的作用，从分析社会意见的动力学和进行数学计算到进行博弈论分析。 对于许多情况，考虑单步替代通常就足够了。 然而，对于更复杂的情况，迭代替代变得不可或缺。 本文的主要重点是研究同时建模单步和迭代替代的逻辑框架。 我们探讨了这些逻辑的若干性质，包括它们的表达能力、希尔伯特风格的证明系统以及可满足性问题。 此外，我们将所提出的框架与文献中相关的现有框架建立了联系。 例如，我们精确地描述了单步替代与许多经典逻辑中常见的标准语法替换之间的关系。 此外，特别强调了迭代替代。 在此背景下，我们将所提出的框架与涉及迭代推理的现有框架进行了比较，从而突出了我们建议的优势。 显示英文摘要

摘要： Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step substitutions is often adequate. Yet, for more complex cases, iterative substitutions become indispensable. In this article, our primary focus is to study logical frameworks that model both single-step and iterative substitutions. We explore a number of properties of these logics, including their expressive strength, Hilbert-style proof systems, and satisfiability problems. Additionally, we establish connections between our proposed frameworks and relevant existing ones in the literature. For instance, we precisely delineate the relationship between single-step substitutions and the standard syntactic replacements commonly found in many classical logics. Moreover, special emphasis is placed on iterative substitutions. In this context, we compare our proposed framework with existing ones involving iterative reasoning, thereby highlighting the advantages of our proposal.