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Update "search as polynomial" article to new math delimiters

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-05-13 19:06:33 -07:00
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@ -58,7 +58,7 @@ two binomials! Here's the corresponding multiplication:
It's not just binomials that correspond to our combining paths between cities.
We can represent any combination of trips of various lengths as a polynomial.
Each term \\(ax^n\\) represents \\(a\\) trips of length \\(n\\). As we just
Each term \(ax^n\) represents \(a\) trips of length \(n\). As we just
saw, multiplying two polynomials corresponds to "sequencing" the trips they
represent -- matching each trip in one with each of the trips in the other,
and totaling them up.
@ -66,8 +66,8 @@ and totaling them up.
What about adding polynomials, what does that correspond to? The answer there
is actually quite simple: if two polynomials both represent (distinct) lists of
trips from A to B, then adding them just combines the list. If I know one trip
that takes two hours (\\(x^2\\)) and someone else knows a shortcut (\\(x\\\)),
then we can combine that knowledge (\\(x^2+x\\)).
that takes two hours (\(x^2\)) and someone else knows a shortcut (\(x\)),
then we can combine that knowledge (\(x^2+x\)).
{{< dialog >}}
{{< message "question" "reader" >}}
@ -119,16 +119,16 @@ example should do.
The first thing we do is _distribute_ the multiplication over the addition, on
the left. We then do that again, on the right this time. After this, we finally
get some terms, but they aren't properly grouped together; an \\(x\\) is at the
front, and a \\(-x\\) is at the very back. We use the fact that addition is
_commutative_ (\\(a+b=b+a\\)) and _associative_ (\\(a+(b+c)=(a+b)+c\\)) to
rearrange the equation, grouping the \\(x\\) and its negation together. This
gives us \\((1-1)x=0x=0\\). That last step is important: we've used the fact
get some terms, but they aren't properly grouped together; an \(x\) is at the
front, and a \(-x\) is at the very back. We use the fact that addition is
_commutative_ (\(a+b=b+a\)) and _associative_ (\(a+(b+c)=(a+b)+c\)) to
rearrange the equation, grouping the \(x\) and its negation together. This
gives us \((1-1)x=0x=0\). That last step is important: we've used the fact
that multiplication by zero gives zero. Another important property (though
we didn't use it here) is that multiplication has to be associative, too.
So, what if we didn't use numbers, but rather any _thing_ with two
operations, one kind of like \\((\\times)\\) and one kind of like \\((+)\\)?
operations, one kind of like \((\times)\) and one kind of like \((+)\)?
{{< dialog >}}
{{< message "question" "reader" >}}
@ -167,8 +167,8 @@ that "things with addition and multiplication that work in the way we
described" have an established name in math - they're called semirings.
A __semiring__ is a set equipped with two operations, one called
"multiplicative" (and thus carrying the symbol \\(\\times)\\) and one
called "additive" (and thus written as \\(+\\)). Both of these operations
"multiplicative" (and thus carrying the symbol \(\times)\) and one
called "additive" (and thus written as \(+\)). Both of these operations
need to have an "identity element". The identity element for multiplication
is usually
{{< sidenote "right" "written-as-note" "written as \(1\)," >}}
@ -180,10 +180,10 @@ other more "esoteric" things, using numbers to stand for special elements
seems to help use intuition.
{{< /sidenote >}}
and the identity element for addition is written
as \\(0\\). Furthermore, a few equations hold. I'll present them in groups.
First, multiplication is associative and multiplying by \\(1\\) does nothing;
as \(0\). Furthermore, a few equations hold. I'll present them in groups.
First, multiplication is associative and multiplying by \(1\) does nothing;
in mathematical terms, the set forms a [monoid](https://mathworld.wolfram.com/Monoid.html)
with multiplication and \\(1\\).
with multiplication and \(1\).
{{< latex >}}
\begin{array}{cl}
(a\times b)\times c = a\times(b\times c) & \text{(multiplication associative)}\\
@ -191,9 +191,9 @@ with multiplication and \\(1\\).
\end{array}
{{< /latex >}}
Similarly, addition is associative and adding \\(0\\) does nothing.
Similarly, addition is associative and adding \(0\) does nothing.
Addition must also be commutative; in other words, the set forms a
commutative monoid with addition and \\(0\\).
commutative monoid with addition and \(0\).
{{< latex >}}
\begin{array}{cl}
(a+b)+c = a+(b+c) & \text{(addition associative)}\\
@ -219,9 +219,9 @@ another in a particular number of hours. There are, however, other semirings
we can use that yield interesting results, even though we continue to add
and multiply polynomials.
One last thing before we look at other semirings: given a semiring \\(R\\),
the polynomials using that \\(R\\), and written in terms of the variable
\\(x\\), are denoted as \\(R[x]\\).
One last thing before we look at other semirings: given a semiring \(R\),
the polynomials using that \(R\), and written in terms of the variable
\(x\), are denoted as \(R[x]\).
#### The Semiring of Booleans, \\(\\mathbb{B}\\)
@ -239,7 +239,7 @@ is true, and false otherwise.
\end{array}
{{< /latex >}}
For addition, the identity element -- our \\(0\\) -- is \\(\\text{false}\\).
For addition, the identity element -- our \(0\) -- is \(\text{false}\).
Correspondingly, multiplication is the "and" operation (aka `&&`), in which the
result is false if either operand is false, and true otherwise.
@ -252,23 +252,23 @@ result is false if either operand is false, and true otherwise.
\end{array}
{{< /latex >}}
For multiplication, the identity element -- the \\(1\\) -- is \\(\\text{true}\\).
For multiplication, the identity element -- the \(1\) -- is \(\text{true}\).
It's not hard to see that _both_ operations are commutative - the first and
second equations for addition, for instance, can be combined to get
\\(\\text{true}+b=b+\\text{true}\\), and the third equation clearly shows
\(\text{true}+b=b+\text{true}\), and the third equation clearly shows
commutativity when both operands are false. The other properties are
easy enough to verify by simple case analysis (there are 8 cases to consider).
The set of booleans is usually denoted as \\(\\mathbb{B}\\), which means
polynomials using booleans are denoted by \\(\\mathbb{B}[x]\\).
The set of booleans is usually denoted as \(\mathbb{B}\), which means
polynomials using booleans are denoted by \(\mathbb{B}[x]\).
Let's try some examples. We can't count how many ways there are to get from
A to B in a certain number of hours anymore: booleans aren't numbers!
Instead, what we _can_ do is track _whether or not_ there is a way to get
from A to B in a certain number of hours (call it \\(n\\)). If we can,
we write that as \\(\text{true}\ x^n = 1x^n = x^n\\). If we can't, we write
that as \\(\\text{false}\ x^n = 0x^n = 0\\). The polynomials corresponding
to our introductory problem are \\(x^2+x^1\\) and \\(x^3+x^2\\). Multiplying
from A to B in a certain number of hours (call it \(n\)). If we can,
we write that as \(\text{true}\ x^n = 1x^n = x^n\). If we can't, we write
that as \(\text{false}\ x^n = 0x^n = 0\). The polynomials corresponding
to our introductory problem are \(x^2+x^1\) and \(x^3+x^2\). Multiplying
them out gives:
{{< latex >}}
@ -306,7 +306,7 @@ that we lost information, rather than gained it, but switching to
boolean polynomials: we can always recover a boolean polynomial from the
natural number one, but not the other way around.
{{< /sidenote >}}
(which were \\(\\mathbb{N}[x]\\), polynomials over natural numbers \\(\\mathbb{N} = \\{ 0, 1, 2, ... \\}\\)), so it's unclear why we'd prefer them. However,
(which were \(\mathbb{N}[x]\), polynomials over natural numbers \(\mathbb{N} = \{ 0, 1, 2, ... \}\)), so it's unclear why we'd prefer them. However,
we're just warming up - there are more interesting semirings for us to
consider!
@ -316,12 +316,12 @@ equivalent". If we're giving directions, though, we might benefit
from knowing not just that there _is_ a way, but what roads that
way is made up of!
To this end, we define the set of paths, \\(\\Pi\\). This set will consist
of the empty path (which we will denote \\(\\circ\\), why not?), street
names (e.g. \\(\\text{Mullholland Dr.}\\) or \\(\\text{Sunset Blvd.}\\)), and
concatenations of paths, written using \\(\\rightarrow\\). For instance,
a path that first takes us on \\(\\text{Highway}\\) and then on
\\(\\text{Exit 4b}\\) will be written as:
To this end, we define the set of paths, \(\Pi\). This set will consist
of the empty path (which we will denote \(\circ\), why not?), street
names (e.g. \(\text{Mullholland Dr.}\) or \(\text{Sunset Blvd.}\)), and
concatenations of paths, written using \(\rightarrow\). For instance,
a path that first takes us on \(\text{Highway}\) and then on
\(\text{Exit 4b}\) will be written as:
{{< latex >}}
\text{Highway}\rightarrow\text{Exit 4b}
@ -329,7 +329,7 @@ a path that first takes us on \\(\\text{Highway}\\) and then on
Furthermore, it's not too much of a stretch to say that adding an empty path
to the front or the back of another path doesn't change it. If we use
the letter \\(\\pi\\) to denote a path, this means the following equation:
the letter \(\pi\) to denote a path, this means the following equation:
{{< latex >}}
\circ \rightarrow \pi = \pi = \pi \rightarrow \circ
@ -347,8 +347,8 @@ different ways to get from one place to another. This is an excellent
use case for sets!
Our next semiring will be that of _sets of paths_. Some example elements
of this semiring are \\(\\varnothing\\), also known as the empty set,
\\(\\{\\circ\\}\\), the set containing only the empty path, and the set
of this semiring are \(\varnothing\), also known as the empty set,
\(\{\circ\}\), the set containing only the empty path, and the set
containing a path via the highway, and another path via the suburbs:
{{< latex >}}
@ -364,16 +364,16 @@ A + B \triangleq A \cup B
{{< /latex >}}
It's well known (and not hard to verify) that set union is commutative
and associative. The additive identity \\(0\\) is simply the empty set
\\(\\varnothing\\). Intuitively, adding "no paths" to another set of
and associative. The additive identity \(0\) is simply the empty set
\(\varnothing\). Intuitively, adding "no paths" to another set of
paths doesn't add anything, and thus leaves that other set unchanged.
Multiplication is a little bit more interesting, and uses the path
concatenation operation we defined earlier. We will use this
operation to describe path sequencing; given two sets of paths,
\\(A\\) and \\(B\\), we'll create a new set of paths
consisting of each path from \\(A\\) concatenated with each
path from \\(B\\):
\(A\) and \(B\), we'll create a new set of paths
consisting of each path from \(A\) concatenated with each
path from \(B\):
{{< latex >}}
A \times B \triangleq \{ a \rightarrow b\ |\ a \in A, b \in B \}
@ -394,8 +394,8 @@ A \times (B \times C) & = & \{ a \rightarrow (b \rightarrow c)\ |\ a \in A, b \i
What's the multiplicative identity? Well, since multiplication concatenates
all the combinations of paths from two sets, we could try making a set of
elements that don't do anything when concatenating. Sound familiar? It should,
that's \\(\\circ\\), the empty path element! We thus define our multiplicative
identity as \\(\\{\\circ\\}\\), and verify that it is indeed the identity:
that's \(\circ\), the empty path element! We thus define our multiplicative
identity as \(\{\circ\}\), and verify that it is indeed the identity:
{{< latex >}}
\begin{gathered}
@ -409,11 +409,11 @@ sets of paths, either; I won't do that here, though. Finally, let's take
a look at an example. Like before, we'll try make one that corresponds to
our introductory description of paths from A to B and from B to C. Now we need
to be a little bit creative, and come up with names for all these different
roads between our hypothetical cities. Let's say that \\(\\text{Highway A}\\)
and \\(\\text{Highway B}\\) are the two paths from A to B that take two hours
each, and then \\(\\text{Shortcut}\\) is the path that takes one hour. As for
paths from B to C, let's just call them \\(\\text{Long}\\) for the three-hour
path, and \\(\\text{Short}\\) for the two-hour path. Our two polynomials
roads between our hypothetical cities. Let's say that \(\text{Highway A}\)
and \(\text{Highway B}\) are the two paths from A to B that take two hours
each, and then \(\text{Shortcut}\) is the path that takes one hour. As for
paths from B to C, let's just call them \(\text{Long}\) for the three-hour
path, and \(\text{Short}\) for the two-hour path. Our two polynomials
are then:
{{< latex >}}
@ -440,12 +440,12 @@ I only have one last semiring left to show you. It's a fun semiring though,
as even its name might suggest: we'll take a look at a _tropical semiring_.
In this semiring, we go back to numbers; particularly, real numbers (e.g.,
\\(1.34\\), \\(163\\), \\(e\\), that kind of thing). We even use addition --
\(1.34\), \(163\), \(e\), that kind of thing). We even use addition --
sort of. In the tropical semiring, addition serves as the _multiplicative_
operation! This is even confusing to write, so I'm going to switch up notation;
in the rest of this section, I'll use \\(\\otimes\\) to represent the
multiplicative operation in semirings, and \\(\\oplus\\) to represent the
additive one. The symbols \\(\\times\\) and \\(+\\) will be used to represent
in the rest of this section, I'll use \(\otimes\) to represent the
multiplicative operation in semirings, and \(\oplus\) to represent the
additive one. The symbols \(\times\) and \(+\) will be used to represent
the regular operations on real numbers. With that, the operations on our
tropical semiring over real numbers are defined as follows:
@ -461,17 +461,17 @@ the duration of the trip, you'd like to track the distance you must travel for
each route (shorter routes do sometimes have more traffic!). Let's watch what
happens when we add and multiply polynomials over this semiring.
When we add terms with the same power but different coefficients, like
\\(ax\oplus bx\\), we end up with a term \\(\min(a,b)x\\). In other words,
\(ax\oplus bx\), we end up with a term \(\min(a,b)x\). In other words,
for each trip duration, we pick the shortest length. When we multiply two
polynomials, like \\(ax\otimes bx\\), we get \\((a+b)x\\); in other words,
polynomials, like \(ax\otimes bx\), we get \((a+b)x\); in other words,
when sequencing two trips, we add up the distances to get the combined
distance, just like we'd expect.
We can, of course, come up with a polynomial to match our initial example.
Say that the trips from A to B are represented by \\(2.0x^2\oplus1.5x\\\) (the
shortest two-hour trip is \\(2\\) units of distance long, and the one-hour
trip is \\(1.5\\) units long), and that the trips from B to C are represented
by \\(4.0x^3\oplus1.0x^2\\). Multiplying the two polynomials out gives:
Say that the trips from A to B are represented by \(2.0x^2\oplus1.5x\) (the
shortest two-hour trip is \(2\) units of distance long, and the one-hour
trip is \(1.5\) units long), and that the trips from B to C are represented
by \(4.0x^3\oplus1.0x^2\). Multiplying the two polynomials out gives:
{{< latex >}}
\begin{array}{rcl}
@ -484,7 +484,7 @@ The only time we used the additive operation in this case was to pick between
two trips of equal druation but different length (two-hour trip from A to B
followed by a two-hour trip from B to C, or one-hour trip from A to C followed
by a three-hour trip from B to C). The first trip wins out, since it requires
only \\(3.0\\) units of distance.
only \(3.0\) units of distance.
### Anything but Routes
So far, all I've done can be reduced to variations on a theme: keeping track