Questions tagged [monoid]
The monoid tag has no usage guidance.
10
questions
0
votes
0
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46
views
Converting a rational star-connected expression to an NFA
For the free monoid, there is Thompson's construction to convert a rational expression to a 'Non-deterministic' Finite Automaton (NFA). One can regard an NFA as a complex (but powerful) way of ...
6
votes
1
answer
154
views
star height of star-free languages
I'm interested in the (restricted) star-height of star free-languages.
Recalling the definitions:
the star height $h(\mathtt{e})$ of a regular expression $\mathtt{e}$ is
$0$ if $\mathtt{e}= \...
- 193
5
votes
1
answer
206
views
Post correspondence problem for finite monoids
The Post correspondence problem has the following version for finite monoids:
Input: a finite monoid $M$ and a finite list $(m_1,m_1'),\ldots, (m_n,m_n')$ of pairs of elements of $M$
Question: is ...
- 51
3
votes
3
answers
141
views
Example of monoid $M$ such that $\operatorname{RAT}(M) \not\subseteq \operatorname{REC}(M)$
Let $M$ be a monoid, the family of rational sets $\operatorname{RAT}(M)$ is defined as the smallest set containing the finite subsets, and closed under union, concatentaion and the star operation. The ...
- 1,947
9
votes
1
answer
190
views
Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid
Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $...
- 1,947
9
votes
1
answer
347
views
Transition monoid membership for DFAs
Given a complete DFA $A=(Q, \Gamma, \delta, F)$, we can define a collection of functions $f_a$ for each $a\in \Gamma$and with $f_a:Q\rightarrow Q$, $f_a(q)=\delta(q, a)$. We can generalize this notion ...
- 1,345
14
votes
3
answers
2k
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On the realisation of monoids as syntactic monoids of languages
Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as
$$
u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L
$$
and the quotient ...
- 1,947
8
votes
0
answers
123
views
What is the optimal binary encoding of the elements of a monoid?
The Question
Let $M$ be a finite monoid. Let $S$ be a generating set of $M$. Say we have a binary encoding of $S$ represented by $\phi:S \rightarrow A^*$ where $A = \{0, 1\}$. This encoding should ...
- 181
9
votes
2
answers
816
views
Formal representation of an abstraction hierarchy
Introduction
I'm writing my PhD thesis on Abstract Delta Modeling (ADM), an abstract algebraic description of modifications (known as deltas) able to act on products (as in 'software products'). This ...
- 231
6
votes
0
answers
191
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Complexity of Roman numeral evaluation
I came up with a result the other day that arbitrary length Roman numeral evaluation can be modeled as a monoid:
https://gist.github.com/4542999
1) Is this a known result?
2) If not, any ...
- 2,331