Operads, quasiorders, and regular languages

On the Use of Quasiorders in Formal Language Theory

In this thesis we use quasiorders on words to offer a new perspective on two well-studied problems from Formal Language Theory: deciding language inclusion and manipulating the finite automata representations of regular languages. First, we present a generic quasiorder-based framework that, when instantiated with different quasiorders, yields different algorithms (some of them new) for deciding language inclusion. We then instantiate this framework to devise an efficient algorithm for searching with regular expressions on grammar-compressed text. Finally, we define a framework of quasiorder-ba…

Complete Abstractions for Checking Language Inclusion

We study the language inclusion problem $L_1 \subseteq L_2$ where $L_1$ is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of $L_1$, obtained by overapproximating the Kleene iterates of its least fixpoint characterization, is included in $L_2$. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e., its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e., it guarantees termination of least fixpoint com…

Colored operads, series on colored operads, and combinatorial generating systems

We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing us to work with all these generating systems in a unified way. The theory of bud generating systems uses colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. To compute the generating series of the languages of bud generating systems, we introduce formal power se…

作为提升问题的解析与Chomsky-Schützenberger表示定理

Parsing as a lifting problem and the Chomsky-Schützenberger representation theorem

本文通过操作元函子统一描述上下文无关文法，并推广Chomsky-Schützenberger表示定理，证明任意范畴上的箭头上下文无关语言可表示为色树轮廓语言与正则语言交集的函子像。

乔姆斯基-舒茨贝格表示定理的范畴轮廓

The categorical contours of the Chomsky-Schützenberger representation theorem

该研究从范畴与操作子的纤维化视角推广上下文无关文法和非确定有限自动机，提出轮廓范畴构造，推广乔姆斯基-舒茨贝格表示定理，刻画箭头上下文无关语言的结构。

Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.