This paper deals with the problem of recognizability of functions l: Sigma* --> M that map words to values in the support set M of a monoid (M,.,1). These functions are called M-languages. M-languages are studied from the aspect of their recognition by deterministic finite automata whose components take values on M (M-DFAs). The characterization of an M-language l is based on providing a right congruence on Sigma* that is defined through l and a factorization on the set of all M-languages, L(Sigma*,M) (in sort L). A factorization on L is a pair of functions (g,f) such that, for each l in L, g(l). f(l)= l, where g(l) in M and f(l) in L. In essence, a factorization is a form of common factor extraction. A general Myhill-Nerode theorem, which is valid for any L(Sigma*, M), is provided. Basically, l is recognized by an M-DFA if and only if there exists a factorization on L, (g,f), such that the right congruence on Sigma* induced by the factorization (g,f) and f(l), has finite index. This paper shows that the existence of M-DFAs guarantees the existence of natural non-trivial factorizations on L without taking account any additional property on the monoid.
Nominal set plays a central role in a group-theoretic extension of finite automata to those over an infinite set of data values. Moerman et al. proposed an active learning algorithm for nominal word automata with the equality symmetry. In this paper, we introduce deterministic bottom-up nominal tree automata (DBNTA), which operate on trees whose nodes are labelled with elements of an orbit finite nominal set. We then prove a Myhill-Nerode theorem for the class of languages recognized by DBNTA and propose an active learning algorithm for DBNTA. The algorithm can deal with any data symmetry that…
Lecture Notes on Automata, Languages, and Grammars
这些课堂讲稿旨在作为Moore和Mertens的《计算的本质》的补充，或者作为一个独立的资源，任何想要使用它们的人都可以使用。欢迎评论，如果你在课程中使用这些笔记，请告诉我。有61个练习。我强调，自动机是我们可以探索确定性和非确定性计算问题的基本游乐场。与P vs. NP不同的是，我们可以证明非确定性等价于确定性，或者严格地比确定性更强大，分别在有限状态和下推自动机中。我还纠正了一些历史和美学上的不公正：特别是，Myhill-Nerode定理和从前缀的等价类构建最小DFA的想法被恢复到正则语言的泵引理之上。我还讨论了上下文无关语言的泵引理，并简要讨论计数器自动机，队列自动机，以及明确的上下文无关语言和代数生成函数之间的联系。
