Relations on words

带正则约束的 E-模式语言的等价问题不可判定

The Equivalence Problem of E-Pattern Languages with Regular Constraints is Undecidable

本文研究了带正则约束的擦除与非擦除模式语言的成员资格、包含和等价问题，证明在允许正则约束和变量相等的情况下，擦除等价问题变得不可判定。

Abelian Combinatorics on Words: a Survey

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these res…

Testing k-binomial equivalence

Two words $w_1$ and $w_2$ are said to be $k$-binomial equivalent if every non-empty word $x$ of length at most $k$ over the alphabet of $w_1$ and $w_2$ appears as a scattered factor of $w_1$ exactly as many times as it appears as a scattered factor of $w_2$. We give two different polynomial-time algorithms testing the $k$-binomial equivalence of two words. The first one is deterministic (but the degree of the corresponding polynomial is too high) and the second one is randomised (it is more direct and more efficient). These are the first known algorithms for the problem which run in polynomial…

On the Complexity of Equivalence and Minimisation for Q-weighted Automata

This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomi…

带长度约束的E-模式语言等价问题是不可判定的

The Equivalence Problem of E-Pattern Languages with Length Constraints is Undecidable

本文研究带长度约束的模式语言，证明擦除等价问题不可判定，并推广了无终结符包含问题及正则约束下非擦除等价问题的不可判定性。

有限与双无限词上有限传感器的完全组合语法

Complete Compositional Syntax for Finite Transducers on Finite and Bi-Infinite Words

本文提出以有限状态非确定性传感器统一自动机理论、迁移系统与sofic子移位，并给出带可靠改写规则的组合图解语法，证明其等式理论完全性。

Equivalence Checking and Intersection of Deterministic Timed Finite State Machines

There has been a growing interest in defining models of automata enriched with time, such as finite automata extended with clocks (timed automata). In this paper, we study deterministic timed finite state machines (TFSMs), i.e., finite state machines with a single clock, timed guards and timeouts which transduce timed input words into timed output words. We solve the problem of equivalence checking by defining a bisimulation from timed FSMs to untimed ones and viceversa. Moreover, we apply these bisimulation relations to build the intersection of two timed finite state machines by untiming the…

Maximum Number of Distinct and Nonequivalent Nonstandard Squares in a Word

The combinatorics of squares in a word depends on how the equivalence of halves of the square is defined. We consider Abelian squares, parameterized squares, and order-preserving squares. The word $uv$ is an Abelian (parameterized, order-preserving) square if $u$ and $v$ are equivalent in the Abelian (parameterized, order-preserving) sense. The maximum number of ordinary squares in a word is known to be asymptotically linear, but the exact bound is still investigated. We present several results on the maximum number of distinct squares for nonstandard subword equivalence relations. Let $\mathi…