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Commit ffbc54f

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Merge pull request #47 from justinmeiners/10-clarifications
10 clarifications
2 parents bbe934c + b9ae73d commit ffbc54f

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‎05_swap.html

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@@ -210,7 +210,7 @@ <h3>Relative and absolute efficiency</h3>
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I still program in C++ because as far as I could ascertain it&rsquo;s the only language which allows me
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generality and absolute efficiency.
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I can program as general as I like.
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I can talk about things like <a href="https://en.wikipedia.org/wiki/Monoid">monoids</a> and <a href="https://en.wikipedia.org/wiki/Semigroup">semi-groups</a>.
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I can talk about things like <a href="https://en.wikipedia.org/wiki/Monoid">monoids</a> and <a href="https://en.wikipedia.org/wiki/Semigroup">semi-groups</a><supid="fnref:1"><ahref="#fn:1" rel="footnote">1</a></sup>.
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When it compiles I could look at assembly code and see it is good.
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It is absolutely efficient.</p>
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sequence, you swap.
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So it is very important practically.
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But it also happens to be very important
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theoretically, because a long time ago when people were starting group theory<supid="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup>
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theoretically, because a long time ago when people were starting <a href="https://en.wikipedia.org/wiki/Group_theory">group theory</a>
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they discovered that any permutation of a sequence could be generated out of swap<sup id="fnref:2"><a href="#fn:2" rel="footnote">2</a></sup>.
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Swap is the most primitive operation.
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The reason is sequence. And any other
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<hr/>
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<ol>
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<li id="fn:1">
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<p><a href="https://en.wikipedia.org/wiki/Group_theory">Group theory</a>
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is one of the main subjects of abstract algebra.
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The key idea is that many mathematical structures behave similarly.
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You can add and subtract numbers, you can add and subtract vectors and matrices.
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Can we study all structures which can add and subtract all together?
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It turns out you can, and one such structure is a group.</p>
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<p>Alex&rsquo;s ideas about generic programming are inspired
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by abstract algebra.</p><a href="#fnref:1" rev="footnote">&#8617;</a></li>
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<p>Groups, monoids, and rings are a few of the subjects of abstract algebra,
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a field which studies the fundamental properties of mathematical structures.
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The key idea is that many different mathematical objects appear to function similarly.
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Vectors and matrices can be &ldquo;added&rdquo; and &ldquo;subtracted&rdquo; just like integers.
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In what ways are they fundamentally the same?
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One explanation is that all of them form a group.
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Below is a formal definition:</p>
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<p>A <strong>group</strong> is a set <code>G</code> with a binary operation <code>* : G x G -&gt; G</code> such that:</p>
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<ol>
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<li><code>G</code> contains an identity element <code>e</code> in <code>G</code> such that <code>e * x = x * e = x</code> for all <code>x</code> in <code>G</code>.</li>
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<li>The operation <code>*</code> is associative. So <code>((x * y) * z) = (x * (y * z))</code> for all <code>x, y, z</code> in <code>G</code>.</li>
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<li>Every element <code>x</code> in <code>G</code> has an inverse element <code>y</code> such that x * y = y * x = e.</li>
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</ol>
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<p>For example integers are a group with the operation of addition and the identity element 0.</p>
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<ol>
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<li><code>0 + x = x + 0 = x</code></li>
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<li><code>((x + y) + z) = (x + (y + z))</code>.</li>
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<li><code>x + (-x) = (-x) + x = 0</code>.</li>
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</ol>
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<p>The process of discovering and applying generic concepts is very similar.
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Alex introduces the basics of abstract algebra, from a programmers perspective,
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in his book &ldquo;From Mathematics to Generic Programming&rdquo;.</p><a href="#fnref:1" rev="footnote">&#8617;</a></li>
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<li id="fn:2">
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<p>A <a href="https://en.wikipedia.org/wiki/Permutation_group">permutation</a>
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is a bijection (1-1, onto) map from a set to itself.
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<li id="fn:4">
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Alex himself uses Tropical semi-rings to describe
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several algorithms in his book &ldquo;From Mathematics to Generic Programming&rdquo; (See chapter 8.6).
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So his issue here is not algebraic abstractions, but pursuing abstraction
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with enormous cost.<a href="#fnref:4" rev="footnote">&#8617;</a></li>
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So his issue here is not abstraction itself, rather that it can become too costly.<a href="#fnref:4" rev="footnote">&#8617;</a></li>
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<li id="fn:5">
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<p>The <code>^</code> symbol is bitwise <a href="https://en.wikipedia.org/wiki/Exclusive_or">exclusive or</a>.
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The expression <code>a ^ b</code> means <code>a</code> is true, or <code>b</code> is true, but not both.

‎05_swap.md

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@@ -80,7 +80,7 @@ After all, I didn't start with C++.
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I still program in C++ because as far as I could ascertain it's the only language which allows me
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generality and absolute efficiency.
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I can program as general as I like.
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I can talk about things like [monoids][monoid] and [semi-groups][semi-group].
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I can talk about things like [monoids][monoid] and [semi-groups][semi-group][^about-group-theory].
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When it compiles I could look at assembly code and see it is good.
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It is absolutely efficient.
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sequence, you swap.
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So it is very important practically.
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But it also happens to be very important
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theoretically, because a long time ago when people were starting group theory[^group-theory]
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theoretically, because a long time ago when people were starting [group theory][group-theory]
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they discovered that any permutation of a sequence could be generated out of swap[^permutation].
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Swap is the most primitive operation.
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The reason is sequence. And any other
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what do you do?
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You deny the utility of swap.
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[^group-theory]: [Group theory](https://en.wikipedia.org/wiki/Group_theory)
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is one of the main subjects of abstract algebra.
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The key idea is that many mathematical structures behave similarly.
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You can add and subtract numbers, you can add and subtract vectors and matrices.
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Can we study all structures which can add and subtract all together?
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It turns out you can, and one such structure is a group.
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[group-theory]: https://en.wikipedia.org/wiki/Group_theory
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Alex's ideas about generic programming are inspired
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by abstract algebra.
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[^about-group-theory]: Groups, monoids, and rings are a few of the subjects of abstract algebra,
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a field which studies the fundamental properties of mathematical structures.
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The key idea is that many different mathematical objects appear to function similarly.
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Vectors and matrices can be "added" and "subtracted" just like integers.
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In what ways are they fundamentally the same?
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One explanation is that all of them form a group.
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Below is a formal definition:
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A **group** is a set `G` with a binary operation `* : G x G -> G` such that:
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1. `G` contains an identity element `e` in `G` such that `e * x = x * e = x` for all `x` in `G`.
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2. The operation `*` is associative. So `((x * y) * z) = (x * (y * z))` for all `x, y, z` in `G`.
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3. Every element `x` in `G` has an inverse element `y` such that x * y = y * x = e.
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For example integers are a group with the operation of addition and the identity element 0.
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1. `0 + x = x + 0 = x`
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2. `((x + y) + z) = (x + (y + z))`.
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3. `x + (-x) = (-x) + x = 0`.
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The process of discovering and applying generic concepts is very similar.
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Alex introduces the basics of abstract algebra, from a programmers perspective,
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in his book "From Mathematics to Generic Programming".
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[^permutation]: A [permutation](https://en.wikipedia.org/wiki/Permutation_group)
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is a bijection (1-1, onto) map from a set to itself.
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[^tropical]: Alex himself uses Tropical semi-rings to describe
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several algorithms in his book "From Mathematics to Generic Programming" (See chapter 8.6).
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So his issue here is not algebraic abstractions, but pursuing abstraction
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with enormous cost.
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So his issue here is not abstraction itself, rather that it can become too costly.
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[^move]:
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Since C++11 this issue has been addressed by [move semantics](https://en.cppreference.com/w/cpp/language/move_constructor)

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