blog-static/content/blog/search_polynomials.md

title	date	draft	tags
Search as a Polynomial	2022-10-22T14:51:15-07:00	true	Mathematics

Suppose that you're trying to get from city A to city B, and then from city B to city C. Also suppose that your trips are measured in one-hour intervals, and that trips of equal duration are considered equivalent. Given possible routes from A to B, and then given more routes from B to C, what are the possible routes from A to C you can build up?

We can try with an example. Maybe there are two routes from A to B that take two hours each, and one "quick" trip that takes only an hour. On top of this, there's one three-hour trip from B to C, and one two-hour trip. Given these building blocks, the list of possible trips from A to C is as follows.

{{< latex >}} \begin{aligned} \text{two}\ 2h\ \text{trips} \rightarrow \text{one}\ 3h\ \text{trip} = \text{two}\ 5h\ \text{trips}\ \text{two}\ 2h\ \text{trips} \rightarrow \text{one}\ 2h\ \text{trip} = \text{two}\ 4h\ \text{trips}\ \text{one}\ 1h\ \text{trip} \rightarrow \text{one}\ 3h\ \text{trip} = \text{one}\ 4h\ \text{trips}\ \text{one}\ 1h\ \text{trip} \rightarrow \text{one}\ 2h\ \text{trip} = \text{two}\ 3h\ \text{trips}\ \textbf{total:}\ \text{two}\ 5h\ \text{trips}, \text{three}\ 4h\ \text{trips}, \text{one}\ 3h\ \text{trip} \end{aligned} {{< /latex >}}

Does this look a little bit familiar? We're combining every length of trips of A to B with every length of trips from B to C, and then totaling them up. In other words, we're multiplying two binomials!

{{< latex >}} \left(2x^2 + x\right)\left(x^3+x^2\right) = 2x^5 + 2x^4 + x^4 + x^3 = \underline{2x^5+3x^4+x^3} {{< /latex >}}

In fact, they don't have to be binomials. 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 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.

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 (\(x^2+x\)).

Well, that's a neat little thing, and pretty quick to demonstrate, too. But we can push this observation a bit further. To generalize what we've already seen, however, we'll need to figure out "the bare minimum" of what we need to make polynomial multiplication work as we'd expect.

Polynomials over Semirings

Let's watch what happens when we multiply two binomials, paying really close attention to the operations we're performing. The following (concrete) example should do.

{{< latex >}} \begin{aligned} & (x+1)(1-x)\ =\ & (x+1)1+(x+1)(-x)\ =\ & x+1-x^2-x \ =\ & x-x+1-x^2 \ =\ & 1-x^2 \end{aligned} {{< /latex >}}

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 that multiplication by zero gives zero. We didn't use it in this example, but another important property we want is for multiplication to be associative, too.

So, what if we didn't use numbers, but rather anything thing with two operations, one kind of like \((\times)\) and one kind of like \((+)\)? As long as these operations satisfy the properties we have used so far, we should be able to create polynomials using them, and do this same sort of "combining paths" we did earlier. Before we get to that, let me just say 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 need to have an "identity element". The identity element for multiplication is usually {{< sidenote "right" "written-as-note" "written as 1," >}} And I do mean "written as": a semiring need not be over numbers. We could define one over graphs, sets, and many other things! Nevertheless, because most of us learn the properties of addition and multiplication much earlier than we learn about 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; in mathematical terms, the set forms a monoid with multiplication and \(1\). {{< latex >}} \begin{array}{cl} (a\times b)\times c = a\times(b\times c) & \text{(multiplication associative)}\ 1\times a = a = a \times 1 & \text{(1 is multiplicative identity)}\ \end{array} {{< /latex >}}

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 \(0\). {{< latex >}} \begin{array}{cl} (a+b)+c = a+(b+c) & \text{(addition associative)}\ 0+a = a = a+0 & \text{(0 is additive identity)}\ a+b = b+a & \text{(addition is commutative)}\ \end{array} {{< /latex >}}

Finally, a few equations determine how addition and multiplication interact. {{< latex >}} \begin{array}{cl} 0\times a = 0 = a \times 0 & \text{(annihilation)}\ a\times(b+c) = a\times b + a\times c & \text{(left distribution)}\ (a+b)\times c = a\times c + b\times c & \text{(right distribution)}\ \end{array} {{< /latex >}}

That's it, we've defined a semiring. First, notice that numbers do indeed form a semiring; all the equations above should be quite familiar from algebra class. When using polynomials with numbers to do our city path finding, we end up tracking how many different ways there are to get from one place to 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 \(R[x]\).

The Semiring of Booleans, \(\mathbb{B}\)

Alright, it's time for our first non-number example. It will be a simple one, though - booleans (that's right, true and false from your favorite programming language!) form a semiring. In this case, addition is the "or" operation (aka ||), in which the result is true if either operand is true, and false otherwise.

{{< latex >}} \begin{array}{c} \text{true} + b = \text{true}\ b + \text{true} = \text{true}\ \text{false} + \text{false} = \text{false} \end{array} {{< /latex >}}

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.

{{< latex >}} \begin{array}{c} \text{false} \times b = \text{false}\ b \times \text{false} = \text{false}\ \text{true} \times \text{true} = \text{true} \end{array} {{< /latex >}}

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 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 \(\mathbb{B}[x]\).

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 them out gives:

{{< latex >}} (x^2+x^1)(x^3+x^2) = x^5 + x^4 + x^4 + x^3 = x^5 + x^4 + x^2 {{< /latex >}}

And that's right; if it's possible to get from A to B in either two hours or one hour, and then from B to C in either three hours or two hours, then it's possible to get from A to C in either five, four, or three hours. In a way, polynomials like this give us less information than our original ones (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!

Polynomials over Sets of Paths, \(\mathcal{P}(\Pi)\)

Until now, we explicitly said that "all paths of the same length are 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:

{{< latex >}} \text{Highway}\rightarrow\text{Exit 4b} {{< /latex >}}

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:

{{< latex >}} \circ \rightarrow \pi = \pi = \pi \rightarrow \circ {{< /latex >}}

{{< sidenote "right" "paths-monoid-note" "So those are paths." >}} In fact, if you clicked through the monoid link earlier, you might be interested to know that paths as defined here form a monoid with concatenation \rightarrow and the empty path \circ as a unit. {{< /sidenote >}} Paths alone, though, aren't enough for our polynomials; we're tracking 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 elements 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 >}} {\text{Highway}\rightarrow\text{Exit 4b}, \text{Suburb Rd.}} {{< /latex >}}

So what are the addition and multiplication on sets of paths? Addition is the easier one: it's just the union of sets:

{{< latex >}} 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 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 \(B\):

{{< latex >}} A \times B = { a \rightarrow b\ |\ a \in A, b \in B } {{< /latex >}}

The fact that this definition of multiplication on sets is associative relies on the associativity of path concatenation; if path concatenation weren't associative, the second equality below would not hold.

{{< latex >}} \begin{array}{rcl} A \times (B \times C) & = & { a \rightarrow (b \rightarrow c)\ |\ a \in A, b \in B, c \in C } \ & \stackrel{?}{=} & { (a \rightarrow b) \rightarrow c \ |\ a \in A, b \in B, c \in C } \ & = & (A \times B) \times C \end{array} {{< /latex >}}

What's the multiplicative identity? Well, since multiplication concatenates all the combination 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:

{{< latex >}} \begin{gathered} {\circ} \times A = { \circ \rightarrow a\ |\ a \rightarrow A } = { a \ |\ a \in A } = A \ A \times {\circ}= { a\rightarrow \circ \ |\ a \rightarrow A } = { a \ |\ a \in A } = A \end{gathered} {{< /latex >}}

It's not too difficult to verify the annihilation and distribution laws for 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 are then:

{{< latex >}} \begin{array}{rcl} P_1 & = & {\text{Highway A}, \text{Highway B}}x^2 + {\text{Shortcut}}x \ P_2 & = & {\text{Long}}x^3 + {\text{Short}}x^2 \end{array} {{< /latex >}}

Multiplying them gives: {{< latex >}} \begin{array}{rl} & {\text{Highway A} \rightarrow \text{Long}, \text{Highway B} \rightarrow \text{Long}}x^5\ + & {\text{Highway A} \rightarrow \text{Short}, \text{Highway B} \rightarrow \text{Short}, \text{Shortcut} \rightarrow \text{Long}}x^4\ + & {\text{Shortcut} \rightarrow \text{Short}}x^3 \end{array} {{< /latex >}}

This resulting polynomial gives us all the paths from city A to city C, grouped by their length!