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Update SPA article on Lattices with 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 18:22:21 -07:00
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@ -198,7 +198,7 @@ but what we're looking for is something similar. Instead, we simply require a
similar function for our signs. We call this function "[least upper bound](https://en.wikipedia.org/wiki/Least-upper-bound_property)", similar function for our signs. We call this function "[least upper bound](https://en.wikipedia.org/wiki/Least-upper-bound_property)",
since it is the "least (most specific) since it is the "least (most specific)
element that's greater (less specific) than either `s1` or `s2`". Conventionally, element that's greater (less specific) than either `s1` or `s2`". Conventionally,
this function is written as \\(a \\sqcup b\\) (or in our case, \\(s_1 \\sqcup s_2\\)). this function is written as \(a \sqcup b\) (or in our case, \(s_1 \sqcup s_2\)).
We can define it for our signs so far using the following [Cayley table](https://en.wikipedia.org/wiki/Cayley_table). We can define it for our signs so far using the following [Cayley table](https://en.wikipedia.org/wiki/Cayley_table).
{{< latex >}} {{< latex >}}
@ -212,7 +212,7 @@ We can define it for our signs so far using the following [Cayley table](https:/
\end{array} \end{array}
{{< /latex >}} {{< /latex >}}
By using the above table, we can see that \\((+\ \\sqcup\ -)\ =\ ?\\) (aka `unknown`). By using the above table, we can see that \((+\ \sqcup\ -)\ =\ ?\) (aka `unknown`).
This is correct; given the four signs we're working with, that's the most we can say. This is correct; given the four signs we're working with, that's the most we can say.
Let's explore the analogy to the `max` function a little bit more, by observing Let's explore the analogy to the `max` function a little bit more, by observing
that this function has certain properties: that this function has certain properties:
@ -220,18 +220,18 @@ that this function has certain properties:
* `max(a, a) = a`. The maximum of one number is just that number. * `max(a, a) = a`. The maximum of one number is just that number.
Mathematically, this property is called _idempotence_. Note that Mathematically, this property is called _idempotence_. Note that
by inspecting the diagonal of the above table, we can confirm that our by inspecting the diagonal of the above table, we can confirm that our
\\((\\sqcup)\\) function is idempotent. \((\sqcup)\) function is idempotent.
* `max(a, b) = max(b, a)`. If you're taking the maximum of two numbers, * `max(a, b) = max(b, a)`. If you're taking the maximum of two numbers,
it doesn't matter which one you consider first. This property is called it doesn't matter which one you consider first. This property is called
_commutativity_. Note that if you mirror the table along the diagonal, _commutativity_. Note that if you mirror the table along the diagonal,
it doesn't change; this shows that our \\((\\sqcup)\\) function is it doesn't change; this shows that our \((\sqcup)\) function is
commutative. commutative.
* `max(a, max(b, c)) = max(max(a, b), c)`. When you have three numbers, * `max(a, max(b, c)) = max(max(a, b), c)`. When you have three numbers,
and you're determining the maximum value, it doesn't matter which pair of and you're determining the maximum value, it doesn't matter which pair of
numbers you compare first. This property is called _associativity_. You numbers you compare first. This property is called _associativity_. You
can use the table above to verify the \\((\\sqcup)\\) is associative, too. can use the table above to verify the \((\sqcup)\) is associative, too.
A set that has a binary operation (like `max` or \\((\\sqcup)\\)) that A set that has a binary operation (like `max` or \((\sqcup)\)) that
satisfies the above properties is called a [semilattice](https://en.wikipedia.org/wiki/Semilattice). In Agda, we can write this definition roughly as follows: satisfies the above properties is called a [semilattice](https://en.wikipedia.org/wiki/Semilattice). In Agda, we can write this definition roughly as follows:
```Agda ```Agda
@ -260,10 +260,10 @@ and the "least upper bound" function can be constructed from one another.
As it turns out, the `min` function has very similar properties to `max`: As it turns out, the `min` function has very similar properties to `max`:
it's idempotent, commutative, and associative. For a partial order like it's idempotent, commutative, and associative. For a partial order like
ours, the analog to `min` is "greatest lower bound", or "the largest value ours, the analog to `min` is "greatest lower bound", or "the largest value
that's smaller than both inputs". Such a function is denoted as \\(a\\sqcap b\\). that's smaller than both inputs". Such a function is denoted as \(a\sqcap b\).
As for what it means, where \\(s_1 \\sqcup s_2\\) means "combine two signs where As for what it means, where \(s_1 \sqcup s_2\) means "combine two signs where
you don't know which one will be used" (like in an `if`/`else`), you don't know which one will be used" (like in an `if`/`else`),
\\(s_1 \\sqcap s_2\\) means "combine two signs where you know \(s_1 \sqcap s_2\) means "combine two signs where you know
{{< sidenote "right" "or-join-note" "both of them to be true" -7 >}} {{< sidenote "right" "or-join-note" "both of them to be true" -7 >}}
If you're familiar with <a href="https://en.wikipedia.org/wiki/Boolean_algebra"> If you're familiar with <a href="https://en.wikipedia.org/wiki/Boolean_algebra">
Boolean algebra</a>, this might look a little bit familiar to you. In fact, Boolean algebra</a>, this might look a little bit familiar to you. In fact,
@ -277,7 +277,7 @@ In fact, booleans with \((\lor)\) and \((\land)\) satisfy the semilattice
laws we've been discussing, and together form a lattice (to which I'm building laws we've been discussing, and together form a lattice (to which I'm building
to in the main body of the text). The same is true for the set union and to in the main body of the text). The same is true for the set union and
intersection operations, \((\cup)\) and \((\cap)\). intersection operations, \((\cup)\) and \((\cap)\).
{{< /sidenote >}}". For example, \\((+\ \\sqcap\ ?)\ =\ +\\), because a variable {{< /sidenote >}}". For example, \((+\ \sqcap\ ?)\ =\ +\), because a variable
that's both "any sign" and "positive" must be positive. that's both "any sign" and "positive" must be positive.
There's just one hiccup: what's the greatest lower bound of `+` and `-`? There's just one hiccup: what's the greatest lower bound of `+` and `-`?
@ -289,8 +289,8 @@ out that this "impossible" value is the least element of our set (we added
it to be the lower bound of `+` and co., which in turn are less than `unknown`). it to be the lower bound of `+` and co., which in turn are less than `unknown`).
Similarly, `unknown` is the largest element of our set, since it's greater Similarly, `unknown` is the largest element of our set, since it's greater
than `+` and co, and transitively greater than `impossible`. In mathematics, than `+` and co, and transitively greater than `impossible`. In mathematics,
it's not uncommon to define the least element as \\(\\bot\\) (read "bottom"), and the it's not uncommon to define the least element as \(\bot\) (read "bottom"), and the
greatest element as \\(\\top\\) (read "top"). With that in mind, the greatest element as \(\top\) (read "top"). With that in mind, the
following are the updated Cayley tables for our operations. following are the updated Cayley tables for our operations.
{{< latex >}} {{< latex >}}
@ -338,7 +338,7 @@ In Agda, we can therefore write a lattice as follows:
### Concrete Example: Natural Numbers ### Concrete Example: Natural Numbers
Since we've been talking about `min` and `max` as motivators for properties Since we've been talking about `min` and `max` as motivators for properties
of \\((\\sqcap)\\) and \\((\\sqcup)\\), it might not be all that surprising of \((\sqcap)\) and \((\sqcup)\), it might not be all that surprising
that natural numbers form a lattice with `min` and `max` as the two binary that natural numbers form a lattice with `min` and `max` as the two binary
operations. In fact, the Agda standard library writes `min` as `_⊓_` and operations. In fact, the Agda standard library writes `min` as `_⊓_` and
`max` as `_⊔_`! We can make use of the already-proven properties of these `max` as `_⊔_`! We can make use of the already-proven properties of these