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What are the most efficient date systems with temporal symmetry?
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Most date or time systems have an Epoch, or privileged "starting" point. For example, the Epoch of the Gregorian calendar is 1 A.D. The Epoch of Unix time January 1, 1970.
These Epochs introduce a temporal asymmetry of sorts. What would be nice if we had date systems with temporal symmetry. We define temporal symmetry for a date system as follows:
First we will define a date system. For simplicity, we will only worry about years. A date system is a function $d$ that assigns each year (which, for simplicity, is defined as a time period from the midnight of a December 31st exclusive to the midnight of the next December 31st inclusive) to a string in a formal language $L(d)$. For example, the Gregorian calendar would assign the year this post was posted the string "2019 A.D.".
A date system $d$ is said to have temporal symmetry if there exists a permutation $sigma$ on $L(d)$'s alphabet such that
$$forall y. d(y + text{1 year}) = sigma_*(d(y))$$
Where $sigma_*(s)$ applies $sigma$ to each letter of $s$. Note that this implies that $|d(y)|$ is equal to some constant $V(d)$, which we will call the verbosity of $d$. Unfortunately, this also implies that if $V(d)$ and $L(d)$ are finite, then $d$ is periodic. If $d$ has a large enough of period though (such as, say, $10,000$ years) this hopefully won't be a problem. We define $P(d)$ to be the fundamental period of $d$.
For any alphabet $Sigma$, we can define a $d$ with temporal symmetry such that $Sigma$ is the alphabet of $L(d)$. Moreover, $P(d) = g(|Sigma|)$ (where $g$ is Landau's Function). Let $sigma$ be some maximal order permutation of $Sigma$. Say it decomposes into disjoint cycles ${c_1, c_2, dots, c_k}$. Let $s$ be a string such that $|s| = k$ and $s_n in c_n$ for all $n in [1 .. k]$. Also let $y$ be an arbitrary year.
We define $$d(y + text{n years}) := sigma_*^n(s)$$ for all integers $n$. $sigma$ itself witnesses that $d$ has temporal symmetry. The $P(d) = g(|Sigma|)$ since that is defined as the order of $sigma$, which is also the period of $d$. Additionally, $V(d) = k$.
Now, for my question. For a given (finite) alphabet $Sigma$, what are the most efficient temporally symmetric date systems such that $L(d) = Sigma$? We say that a temporally symmetric date system $d$ is most efficient if for all other temporally symmetric date systems $d'$ such that $L(d') = L(d)$, one of the following is true:
- $P(d) gt P(d')$
- $V(d) lt V(d')$
- $P(d) = P(d') land V(d) = V(d')$
Clearly the only important thing about $Sigma$ for the purposes of this question is its size. Something to figure out first is if the date system I gave above is efficient.
group-theory optimization permutations symmetric-groups formal-languages
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
$endgroup$
add a comment |
$begingroup$
Most date or time systems have an Epoch, or privileged "starting" point. For example, the Epoch of the Gregorian calendar is 1 A.D. The Epoch of Unix time January 1, 1970.
These Epochs introduce a temporal asymmetry of sorts. What would be nice if we had date systems with temporal symmetry. We define temporal symmetry for a date system as follows:
First we will define a date system. For simplicity, we will only worry about years. A date system is a function $d$ that assigns each year (which, for simplicity, is defined as a time period from the midnight of a December 31st exclusive to the midnight of the next December 31st inclusive) to a string in a formal language $L(d)$. For example, the Gregorian calendar would assign the year this post was posted the string "2019 A.D.".
A date system $d$ is said to have temporal symmetry if there exists a permutation $sigma$ on $L(d)$'s alphabet such that
$$forall y. d(y + text{1 year}) = sigma_*(d(y))$$
Where $sigma_*(s)$ applies $sigma$ to each letter of $s$. Note that this implies that $|d(y)|$ is equal to some constant $V(d)$, which we will call the verbosity of $d$. Unfortunately, this also implies that if $V(d)$ and $L(d)$ are finite, then $d$ is periodic. If $d$ has a large enough of period though (such as, say, $10,000$ years) this hopefully won't be a problem. We define $P(d)$ to be the fundamental period of $d$.
For any alphabet $Sigma$, we can define a $d$ with temporal symmetry such that $Sigma$ is the alphabet of $L(d)$. Moreover, $P(d) = g(|Sigma|)$ (where $g$ is Landau's Function). Let $sigma$ be some maximal order permutation of $Sigma$. Say it decomposes into disjoint cycles ${c_1, c_2, dots, c_k}$. Let $s$ be a string such that $|s| = k$ and $s_n in c_n$ for all $n in [1 .. k]$. Also let $y$ be an arbitrary year.
We define $$d(y + text{n years}) := sigma_*^n(s)$$ for all integers $n$. $sigma$ itself witnesses that $d$ has temporal symmetry. The $P(d) = g(|Sigma|)$ since that is defined as the order of $sigma$, which is also the period of $d$. Additionally, $V(d) = k$.
Now, for my question. For a given (finite) alphabet $Sigma$, what are the most efficient temporally symmetric date systems such that $L(d) = Sigma$? We say that a temporally symmetric date system $d$ is most efficient if for all other temporally symmetric date systems $d'$ such that $L(d') = L(d)$, one of the following is true:
- $P(d) gt P(d')$
- $V(d) lt V(d')$
- $P(d) = P(d') land V(d) = V(d')$
Clearly the only important thing about $Sigma$ for the purposes of this question is its size. Something to figure out first is if the date system I gave above is efficient.
group-theory optimization permutations symmetric-groups formal-languages
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
$endgroup$
add a comment |
$begingroup$
Most date or time systems have an Epoch, or privileged "starting" point. For example, the Epoch of the Gregorian calendar is 1 A.D. The Epoch of Unix time January 1, 1970.
These Epochs introduce a temporal asymmetry of sorts. What would be nice if we had date systems with temporal symmetry. We define temporal symmetry for a date system as follows:
First we will define a date system. For simplicity, we will only worry about years. A date system is a function $d$ that assigns each year (which, for simplicity, is defined as a time period from the midnight of a December 31st exclusive to the midnight of the next December 31st inclusive) to a string in a formal language $L(d)$. For example, the Gregorian calendar would assign the year this post was posted the string "2019 A.D.".
A date system $d$ is said to have temporal symmetry if there exists a permutation $sigma$ on $L(d)$'s alphabet such that
$$forall y. d(y + text{1 year}) = sigma_*(d(y))$$
Where $sigma_*(s)$ applies $sigma$ to each letter of $s$. Note that this implies that $|d(y)|$ is equal to some constant $V(d)$, which we will call the verbosity of $d$. Unfortunately, this also implies that if $V(d)$ and $L(d)$ are finite, then $d$ is periodic. If $d$ has a large enough of period though (such as, say, $10,000$ years) this hopefully won't be a problem. We define $P(d)$ to be the fundamental period of $d$.
For any alphabet $Sigma$, we can define a $d$ with temporal symmetry such that $Sigma$ is the alphabet of $L(d)$. Moreover, $P(d) = g(|Sigma|)$ (where $g$ is Landau's Function). Let $sigma$ be some maximal order permutation of $Sigma$. Say it decomposes into disjoint cycles ${c_1, c_2, dots, c_k}$. Let $s$ be a string such that $|s| = k$ and $s_n in c_n$ for all $n in [1 .. k]$. Also let $y$ be an arbitrary year.
We define $$d(y + text{n years}) := sigma_*^n(s)$$ for all integers $n$. $sigma$ itself witnesses that $d$ has temporal symmetry. The $P(d) = g(|Sigma|)$ since that is defined as the order of $sigma$, which is also the period of $d$. Additionally, $V(d) = k$.
Now, for my question. For a given (finite) alphabet $Sigma$, what are the most efficient temporally symmetric date systems such that $L(d) = Sigma$? We say that a temporally symmetric date system $d$ is most efficient if for all other temporally symmetric date systems $d'$ such that $L(d') = L(d)$, one of the following is true:
- $P(d) gt P(d')$
- $V(d) lt V(d')$
- $P(d) = P(d') land V(d) = V(d')$
Clearly the only important thing about $Sigma$ for the purposes of this question is its size. Something to figure out first is if the date system I gave above is efficient.
group-theory optimization permutations symmetric-groups formal-languages
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
$endgroup$
Most date or time systems have an Epoch, or privileged "starting" point. For example, the Epoch of the Gregorian calendar is 1 A.D. The Epoch of Unix time January 1, 1970.
These Epochs introduce a temporal asymmetry of sorts. What would be nice if we had date systems with temporal symmetry. We define temporal symmetry for a date system as follows:
First we will define a date system. For simplicity, we will only worry about years. A date system is a function $d$ that assigns each year (which, for simplicity, is defined as a time period from the midnight of a December 31st exclusive to the midnight of the next December 31st inclusive) to a string in a formal language $L(d)$. For example, the Gregorian calendar would assign the year this post was posted the string "2019 A.D.".
A date system $d$ is said to have temporal symmetry if there exists a permutation $sigma$ on $L(d)$'s alphabet such that
$$forall y. d(y + text{1 year}) = sigma_*(d(y))$$
Where $sigma_*(s)$ applies $sigma$ to each letter of $s$. Note that this implies that $|d(y)|$ is equal to some constant $V(d)$, which we will call the verbosity of $d$. Unfortunately, this also implies that if $V(d)$ and $L(d)$ are finite, then $d$ is periodic. If $d$ has a large enough of period though (such as, say, $10,000$ years) this hopefully won't be a problem. We define $P(d)$ to be the fundamental period of $d$.
For any alphabet $Sigma$, we can define a $d$ with temporal symmetry such that $Sigma$ is the alphabet of $L(d)$. Moreover, $P(d) = g(|Sigma|)$ (where $g$ is Landau's Function). Let $sigma$ be some maximal order permutation of $Sigma$. Say it decomposes into disjoint cycles ${c_1, c_2, dots, c_k}$. Let $s$ be a string such that $|s| = k$ and $s_n in c_n$ for all $n in [1 .. k]$. Also let $y$ be an arbitrary year.
We define $$d(y + text{n years}) := sigma_*^n(s)$$ for all integers $n$. $sigma$ itself witnesses that $d$ has temporal symmetry. The $P(d) = g(|Sigma|)$ since that is defined as the order of $sigma$, which is also the period of $d$. Additionally, $V(d) = k$.
Now, for my question. For a given (finite) alphabet $Sigma$, what are the most efficient temporally symmetric date systems such that $L(d) = Sigma$? We say that a temporally symmetric date system $d$ is most efficient if for all other temporally symmetric date systems $d'$ such that $L(d') = L(d)$, one of the following is true:
- $P(d) gt P(d')$
- $V(d) lt V(d')$
- $P(d) = P(d') land V(d) = V(d')$
Clearly the only important thing about $Sigma$ for the purposes of this question is its size. Something to figure out first is if the date system I gave above is efficient.
group-theory optimization permutations symmetric-groups formal-languages
group-theory optimization permutations symmetric-groups formal-languages
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
edited Jan 2 at 0:49
PyRulez
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
asked Jan 2 at 0:40
PyRulezPyRulez
4,63622369
4,63622369
add a comment |
add a comment |
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