For completeness, before getting into any of the crazy (and crazy cool!) modern technologies, I want to give a crash-course overview of the basics of computer architecture. How do computers work at the logical level?
Most posts in this series are going to be deep dives into a particular technology, how it works, and how it might be implemented. However, I don’t feel justified in getting into that without having an accompanying crash course on the pre-requisites.
This post is going to zoom through what should be a one-semester undergraduate university course. It won’t be comprehensive, but should get the ideas across.
It’s easy to see computers as magical black boxes. One common joke is that “Computers are just rocks we tricked into thinking.” And that’s actually sort of true. But computers don’t think the way we do. Really, computers are nothing more than glorified calculators. The images you see on your screen are the results of (sometimes complex) calculations to produce location and color data for each pixel. Everything comes down to moving around data and performing arithmetic on that data.
A typical definition of a computer is “a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically.” We might ask what can be computed in this way; namely: if I give you a function definition, and some inputs to that function, what types of computers are capable of evaluating the function on those inputs? The study of different types of computers is called models of computation. The highest class of model of computation contains “computers that can compute anything which can be computed.”^{1} Those are what we’re concerned with here.
A computer has to be programmable by the definition above. Computers will get their programs as a sequence of steps to perform. Each step tells the computer to use a single one of its capabilities. Computers have some short-term storage called registers. They can operate directly only on values that are in registers.
The most basic capabilities of a modern computer all fall into one of the following categories:
After performing one step, we move on to the next step in the sequence.
Any particular computer has a particular set of instructions that it can understand, called an “instruction set,” or ISA^{2}. Of course, it doesn’t understand them written in English. They have to be encoded in binary in a consistent way which depends on the particular computer. We’re not going to concern ourselves with that encoding here. We’ll assume the existence of a circuit called a “decoder,” which translates the binary-encoded instruction into a bunch of “control signals” which control what the rest of the computer does.
We’re also not going to worry about modelling a computer that can compute quickly. Computing at all will do for now, and we’ll worry about being fast later.
Common instructions available in most instruction sets are:
So what do we need to implement all of this?
Here’s a very basic way we could combine the above pieces into something resembling a computer.
The decoder controls what everything else does. It tells memory to load or store (or do nothing), it tells the register file which registers to read and write, the ALU what operation to perform, the test unit what logical test to use, and the program counter whether or not it should overwrite and with what value (possibly depending on the result of the test, which comes from the test unit).
This is actually a perfectly good basic model of a computer. Modern computers don’t look like this, but they have components that are individually recognizable as components of this model.
A computer based on this model is going to be bad. Why?
Each “cycle” of the computer is controlled by a clock. When the clock ticks, we move on to the next step according to the program counter. We need to give each step enough time for all of the control and data signals to propogate through the whole system. For some of these signals, that might be slow^{5}. Say the slowest signal chain starts from the program counter, goes through the program, decoder, register file, ALU, and back to the register file (perhaps for an operation like division, which is fairly slow to perform). This slowest chain is called the critical path. If it takes half a second for a signal to get all the way through the critical path, then we cannot run our clock any faster than 0.5s/cycle (alternatively, 2 cycles per second or 2 Hz).
Even though some, or even most, instructions won’t use the critical path, any instruction might be the instruction that does. So the critical path is the limiting factor to our clock speed. The clock speed is a significant factor when considering the speed of a computer. A faster clock will almost always mean a faster computer^{6}.
So the first way we can think of to improve the clock speed is to make the critical path shorter.
The major improvement to the basic design, used in all modern systems, is called pipelining. We don’t want to limit our system to only working on a single instruction at a time, and being stuck until that instruction is done. We also want to make the critical path shorter. We can hit both birds with one stone: split up the execution of an instruction into several “stages,” where each stage takes one clock cycle.
Since an instruction can only occupy one stage at a time, if we have 3 stages then we can also be “executing” 3 instructions at once. We also run the clock faster^{7}, but this is negated by the fact that it also takes more cycles for an individual instruction to finish.
These are the relevant numbers concerning a pipeline:
For our basic model here, the throughput should stay at 1, even though the latency has increased. Since we’re running the clock faster, that means we’re finishing more instructions per second, so our computer is going faster. In an ideal world, the latency triples, the clock speed gets cut into 1/3, and the throughput remains 1. That works out to our computer running about three times faster, without any changes to the program!
To implement pipelining, we use “pipeline registers,” also called “fences,” to separate each stage. We can modify the basic design to something like this, for a 3-stage example.
This gives us a mostly clean division between each stage of the pipeline - fetch, execute, and writeback. But hang on - stage 3 doesn’t appear to exist at all!
Stage 3 is the “writeback” stage, where the results are written to where they have to go, which is either the register file or program counter (memory writes are handled in the second stage, which is called “execute”). What logic there is to handle in this stage would be built into the register file and program counter, so there’s no new units here. But that does mean there isn’t a clear separation between stages. As you might expect, that causes major problems.
Consider the instruction sequence that adds register A to register B, storing to register C, and then adds register C to register D, storing back to register A. We can visualize what’s going on with a timing table as follows:
\[\begin{array}{|c|c|c|c|} \hline \text{stage} & \text{cycle 1} & \text{cycle 2} & \text{cycle 3} & \text{cycle 4} \\ \hline \text{Fetch} & \mathtt{ADD\ A\ B\ C} & \tt{ADD\ C\ D\ A} & - & - \\ \hline \text{Execute} & - & \tt{ADD\ A\ B\ C} & \tt{ADD\ C\ D\ A} & - \\ \hline \text{Write} & - & - & \tt{ADD\ A\ B\ C} & \tt{ADD\ C\ D\ A} \\ \hline \end{array}\]This table shows us the instruction in each stage at any given cycle. We could also write these as instructions vs time, instead of stage vs time, but I find this way a bit easier.
There’s a big problem with this program. The result of the first instruction won’t be available in the register file until after the cycle that the instruction writes back. But looking at cycle 3 in the table, the second instruction needs to read register C as input on the same cycle!
So what gives, and how can we fix it?
The above problem is known as a data hazard, specifically of the read-after-write kind. This is often abbreviated RAW. Hazards describe types of data dependencies, where the order of instructions in the program implicitly connect instructions which write and read the same data from a register. A RAW hazard means that we must not read the register until after the write has completed.
There are other types of hazards as well. If we have a write-after-read hazard, then we must ensure that the instruction which reads is able to read the data before it is overwritten by the write. There are also write-after-write hazards, which require us to enforce that results are written in the correct order so that the correct data is in registers after the instruction sequence.
Since our model executes instructions in order^{8}, WAR and WAW hazards simply cannot happen.
There are two ways we could try to resolve the RAW hazard in the above program.
The first, and most obvious solution, is to force ADD C D A
to wait until ADD A B C
completes
writeback before allowing it to execute. This is tricky to implement, however, because up to
this point we have always assumed that instructions move through the pipeline at exactly one
stage per cycle - no more, no less. However it’s certainly possible and this approach is known as
a “pipeline stall.”
The better solution is to provide a shortcut for ADD C D A
’s writeback so that it is accessible
to an executing instruction in the same cycle. This technique is called data forwarding.
Either way, we have to detect that two instructions have a hazard; this is generally easy as we can just remember which register(s) are used by every instruction and compare the registers being read in the execute stage to the registers being written in the write-back stage.
Using the first approach, we need to modify our first pipeline fence so that it gets an extra
control signal. This signal describes if it should output the stored instruction at all. If it
shouldn’t, it should instead output a NOP
instruction, which is short for “no operation.”
NOP
s which enter the pipeline due to stalls are commonly called pipeline bubbles, since they
behave identically to air bubbles in a water pipe.
Additionally, if we stall, we need to prevent the program counter from advancing, or else we will lose the stalled instruction.
Adding a hazard control unit to our model, we can come up with a computer design like the following.
With this approach, the same program experiences the following timing table:
\[\begin{array}{|c|c|c|c|} \hline \text{stage} & \text{cycle 1} & \text{cycle 2} & \text{cycle 3} & \text{cycle 4} & \text{cycle 5}\\ \hline \text{Fetch} & \mathtt{ADD\ A\ B\ C} & \tt{ADD\ C\ D\ A} & \tt{ADD\ C\ D\ A} & - & - \\ \hline \text{Execute} & - & \tt{ADD\ A\ B\ C} & \tt{NOP} & \tt{ADD\ C\ D\ A} & - \\ \hline \text{Write} & - & - & \tt{ADD\ A\ B\ C} & \tt{NOP} & \tt{ADD\ C\ D\ A} \\ \hline \end{array}\]I think once these tables get complicated, they are easier to read if we put the pipeline on the horizontal axis, to match the pipeline diagrams. Let’s do that from now on; here’s the same table transposed.
\[\begin{array}{|c|c|c|c|} \hline \text{cycle} & \text{Fetch} & \text{Execute} & \text{Write} \\ \hline 1 & \tt{ADD\ A\ B\ C} & - & - \\ \hline 2 & \tt{ADD\ C\ D\ A} & \tt{ADD\ A\ B\ C} & - \\ \hline 3 & \tt{ADD\ C\ D\ A} & \tt{NOP} & \tt{ADD\ A\ B\ C} \\ \hline 4 & - & \tt{ADD\ C\ D\ A} & \tt{NOP} \\ \hline 5 & - & - & \tt{ADD\ C\ D\ A} \\ \hline \end{array}\]We can see that the NOP
bubble causes the whole program to take an extra cycle, as expected.
Instead, let’s use a forwarding unit behind the register file to detect these hazards, and forward data from the writeback stage. We get a design that looks as follows.
Now we get a timing table which is the same as the original one, except this time it works!
There is a potential complication with this design though. Often, the critical path of the system with this type of design is already in the execute stage (although this is by no means a guarantee). The Forwarding Unit, while cheap, does introduce some extra signal delay and can make the critical path longer, slowing down the clock.
Additionally, this plan only works if the writeback stage is immediately after the stage performing the read. If retrieving instructions from the program is fast, we may opt to place the fetch-execute fence so that the register file read happens in the fetch stage instead. This would mean there is a two cycle gap between register reads and register writes, so two adjacent instructions with a RAW dependency cannot use this forwarding scheme directly. Multi-stage separation like this is pretty much always the case in real designs. Ideally, we combine both approaches. We stall exactly long enough for the data to be available for forwarding.
That simple example program didn’t make any use of the control flow capabilities of the computer - (conditionally) jumping. Let’s look at what happens if we execute a simple program like this.
ADD A, B, C
JMP 2 # jump forward two instructions
ADD B, C, D # this should be skipped
SUB C, B, A
Remember that the control signals for jump instructions go through the Test unit, which is in the execute stage. Perhaps you can already see where this is going! Here’s the timing table.
\[\begin{array}{|c|c|c|c|} \hline \text{cycle} & \text{Fetch} & \text{Execute} & \text{Write} \\ \hline 1 & \tt{ADD\ A,\ B,\ C} & - & - \\ \hline 2 & \tt{JMP\ 2} & \tt{ADD\ A,\ B,\ C} & - \\ \hline 3 & \tt{ADD\ B,\ C,\ D} & \tt{JMP\ 2} & \tt{ADD\ A,\ B,\ C} \\ \hline 4 & \tt{SUB\ C,\ B,\ A} & \tt{ADD\ B,\ C,\ D} & \tt{JMP\ 2} \\ \hline 5 & \tt{SUB\ C,\ B,\ A} & \tt{SUB\ C,\ B,\ A} & \tt{ADD\ B,\ C,\ D} \\ \hline 6 & - & \tt{SUB\ C,\ B,\ A} & \tt{SUB\ C,\ B,\ A} \\ \hline 7 & - & - & \tt{SUB\ C,\ B,\ A} \\ \hline \end{array}\]Oh no! The ADD B, C, D
instruction and even an extra copy of the SUB
instruction,
snuck into the pipeline before we realized we were supposed to skip forward!
What gives, and how do we fix it?
The fact that there is time between fetching a jump (or branch) instruction, and actually redirecting the program counter, means that there will always be a chance for instructions that should have been skipped to sneak into the pipeline. This is called a control hazard.
Since we’re not omniscient, and neither is our computer, there’s not really anything we can do to make the program go faster in every case, unlike with the RAW hazards. In future posts, we’ll see some methods that can work in most cases.
The simplest approach is after decoding a jump (or branch) instruction, we simply stall until it completes execution and then continue after being redirected. This, obviously, introduces large pipeline bubbles and is generally not ideal.
An easy potential improvement to that is to have separate data paths (signal paths through the circuit) for jump and branch instructions, since jumps can be detected and executed much more easily. In our simple computer architecture, we can most likely detect jump instructions directly in the decoder and execute them in the Fetch stage without causing critical path issues.
A harder improvement is to consider that, sometimes, the instruction that would sneak into the
pipeline actually should be the next instruction. We don’t know it yet, but we could hope to
get lucky. When the branch actually executes, if the branch is in fact taken, we have to track
down those “hopeful” instructions and remove them from the pipeline. For our simple pipelines,
this is easy - it’s every instruction in the pipeline in an earlier stage than the branch.
We remove those instructions by replacing them with NOP
s, which is called squashing the pipeline.
We can implement that in our pipeline registers by adding some control ability. When we need to
stall, the fence currently (1) does not read new input from the previous stage, and (2) does
not send its stored instruction to the next stage (sending a NOP
instead). In order to squash,
the fence should replace the stored instruction with a NOP
instead of reading new input from
the previous stage. Next cycle, after the program counter redirect, the Fetch stage will contain
the correct next instruction, prepared to send it into the first fence. The squashed instructions
have all become NOP
s.
An architecture block diagram and timing table for the above program with this scheme could look like this.
\[\begin{array}{|c|c|c|c|} \hline \text{cycle} & \text{Fetch} & \text{Execute} & \text{Write} \\ \hline 1 & \tt{ADD\ A,\ B,\ C} & - & - \\ \hline 2 & \tt{JMP\ 2} & \tt{ADD\ A,\ B,\ C} & - \\ \hline 3 & \tt{SUB\ C,\ B,\ A} & \tt{NOP} & \tt{ADD\ A,\ B,\ C} \\ \hline 4 & - & \tt{SUB\ C,\ B,\ A} & \tt{NOP} \\ \hline 5 & - & - & \tt{SUB\ C,\ B,\ A} \\ \hline \end{array}\]We executed the unconditional jump immediately in the fetch stage (and have the decoder
pass on a NOP
instead of the now-useless JMP
). As a result, the erroneous instruction
never enters the pipeline at all, and we even finish two cycles faster. Nice!
What if that unconditional jump was a conditional branch? Let’s not worry about types of
conditional branches here; let’s just assume that the same JMP 2
instruction now needs
to use the Test unit. We get a timing table like this one.
As before, we still let the erroneous ADD
and SUB
instructions into the pipeline,
but now we squash them when the JMP
instruction goes to write back. Since they never
reach the writeback stage themselves, the values they computed are never stored in the
register file. It’s like the instructions never existed at all.
A more advanced technique called branch prediction attempts to guess whether or not a branch will be taken as soon as it is decoded, and gets the next instruction from the predicted location in the program. What we’re doing is the same as predicting that all branches are not taken.
It turns out that accurate branch prediction is extremely important for performance in even a moderately deep pipeline, because mispredicting a branch introduces a bubble in the pipeline of length equal to the number of stages between the program counter and whichever stage is able to detect the misprediction (in our case, this is writeback, but with some effort, complex designs can do it in the execute stage). A modern x86 processor has a pipeline with around 12 stages in the relevant portion of the pipeline, so the misprediction penalty is huge.
Branch prediction is a difficult problem, with an incredibly rich field of results and techniques. Modern branch predictors are incredibly accurate, achieving prediction accuracies in the neighborhood of 99%. Methods of branch prediction will be the topic of several future posts!
There’s a common joke around the internet about how Internet Explorer takes 10x as long as every other browser to serve a webpage. Memory is like the Internet Explorer of execution units. It takes much longer to read or write to RAM than to execute any other type of instruction.^{9} Given that we need to access RAM whenever we finish each small unit of computation, it’s completely unacceptable to spend over 99% of program execution time sitting around waiting for RAM.
The idea of having registers as short-term, fast storage is so good that we can abuse it to solve this problem too.
Instead of talking directly to RAM whenever we need to access it, we use a middle-man memory unit called a cache. Just like caching in your web browser, the cache in a CPU remembers data that it recently had to get from memory. Since the cache is smaller and closer to the CPU than RAM, it is much faster to access. If the memory we’re looking for is already in the cache, we can retrieve it in usually just a single cycle. This is called a “cache hit.” In the event of a cache miss, we only have to pay the huge penalty for accessing RAM one time, and then future references to the same data will go through the cache.
When a program access some memory, it will almost always access the same memory or nearby memory soon after. Programs accessing nearby memory is called “spatial locality,” while the fact that those accesses are typically soon after each other is called “temporal locality.” Programs written to maximize spatial and temporal locality of memory accesses are likely to perform better if the computer has a cache.
Cache design is a whole can of worms; figuring out the interaction size between cache and RAM (how much data to retrieve surrounding the requested data, assuming it will be needed soon), as well as the size of the cache itself, and when to evict cache entries to make room for new ones, are all important considerations. Additionally, most designs will use multiple layers of cache, keeping a smaller, extremely fast cache directly next to the execution circuitry, and a larger but somewhat slower cache near the edge of the CPU core. Multi-core systems often have a third layer, with a significantly larger and slower cache shared by all or several cores.
We’re not going to get into it here, but it is important to be aware that caches exist and that they are not a one-size-fits-all solution to slow memory problems. On real hardware, writing programs in a “cache-aware” fashion, but without changing the underlying algorithm, can create performance improvements large enough to be noticeable by a human.
From the outside, a computer is simply a black box which takes in an algorithm, executes it step-by-step, and spits out a result. In actuality, there are a huge host of techniques we can use to maintain this outside appearance, but achieve the same goals much, much faster.
Pipelining lets us re-use existing hardware on several instructions at the same time, slowing down individual instructions but speeding up the clock and therefore also how many instructions complete per second. It introduces challenges in maintaining the data-flow and control-flow of the original program, but these challenges can be overcome without too much difficulty.
We’ve also seen how we can use even basic branch prediction techniques to eliminate branch stalls if we’re able to “get lucky,” and hinted at the possibility that if we try very, very hard, we can “get lucky” almost every time.
Finally, we’ve briefly discussed how slow memory is, and how we can use caching techniques to eliminate the large stalls associated with waiting for memory accesses.
This crash-course overview covers more or less the same topics as a one-semester undergraduate computer architecture course. To avoid getting too mathematical, I covered caches in significantly less detail than such a course would.
In future posts, we will take deep dives into particular advanced computer architecture techniques. We’ll look at other pipeline designs, particularly ones that can execute instructions out of order. We’ll also see various techniques for branch prediction, and discuss the design and construction of caches in significantly more detail.
See you next time!
It is surprisingly easy to prove that not everything can be computed, by constructing an explicit example of a function which cannot be computed by any algorithm. The YouTube channel “udiprod” has a nice approachable video on this which I can highly recommend. ↩
The A stands for Architecture, because an ISA is generally considered half the design of a computer. The other half is the hardware that interprets the instructions and does what they say. ↩
Depending on the ISA, the register may be specified by the ISA itself (for example, defining LDI to always place the value into register 0) or it may be specified as part of the particular instruction. The same goes for all references to “specified register” here. ↩
Where to jump to can be specified as an “absolute,” for example “go to step 4,” or as a relative, for example “go back 3 steps.” Which one is used depends on the particular ISA, and many ISAs use both for different instructions. ↩
In particular, accessing memory is very slow, but we’re not going to worry about that yet. ↩
Up to a point. At some point, the fact that memory is so slow becomes the limiting factor to how fast we can operate on data. The computer would be able to work on any data as it comes from memory and be done before the next data is ready from memory. Empirically, this happens in the neighborhood of 4 billion cycles per second, or 4 GHz. Modern computers have clocks running at about that speed. ↩
Up to 3 times faster, though in practice the critical paths of the individual stages will usually not each be exactly one third of the critical path of the non-pipelined design. ↩
The implication being that there are models which execute instructions out of order, and that is precisely why I’m writing this series. ↩
In the neighborhood of 200 clock cycles. ↩
When we looked at groupoids, we saw how the combination of restrictions can lead to generalized results about all groupoids, like the inverse of the inverse theorem, and the inverse of product theorem.
In this post, we’re going to start looking at monoids, which have a set of properties satisfied by, well, nearly everything. In particular, monoids appear everywhere in programs.
Categories didn’t have a requirement that multiplication be total. For many pairs of arrows, it doesn’t make sense to chain them together. It only works if they have the same intermediate node in the graph.
Let’s require the multiplication to be total.
For that to work, every pair of arrows is going to have to go to and from the same graph node. As a result, there is only one graph node (and there are multiple arrows from that node to itself).
This gives us a view of monoids as categories. We have identites (every arrow is an identity, in this view).
However this view isn’t super useful, because every arrow is the same. This model kind of sucks. We could modify how we look at the model to make it more interesting. Or… we could let the same thing happen by starting somewhere else.
Semigroups have associative, total multiplication, but they don’t have to have identities. Our example of a finite semigroup was to start with some arbitrary set \(A\). Then we considered a set of functions, \(F\), where each function \(f \in F\) is from \(A\) to \(A\). Function composition in any set is total and associative.
But the only function which is an identity under composition is the function that maps any \(a : A\) to itself, which we write as either \(f(a) = a\), or \(a \mapsto a\). We didn’t require that this function be in \(F\) when we considered semigroups.
For monoids, we’re going to require it.
The identity function is a function that “does nothing.” This is spectacularly useful. Not only do we know that there is an identity that works for every function in \(F\), we know what it is. And if we need to conjure up an element of our monoid to use somewhere, we can always safely use this identity because multiplication by it will never “mess up” another value.
Starting from an operation which has an identity and which is total, we can get a monoid by also requiring it to be associative. I unfortunately don’t have a good way to visualize this change. Please let me know if you do!
Correspondingly, a structure which is both a unital magma and a semigroup will always be a monoid.
Let’s consider some examples. Let’s start with the set \(\mathbb B = \{ true, false \}\). Let’s pick the common operation “or”:
\[\begin{aligned} false &\| false &= false \\ false &\| true &= true \\ true &\| false &= true \\ true &\| true &= true \\ \end{aligned}\]As we can see, \(false\) certainly acts as the identity on either side. Operating false on false gives false, false on true gives true. We can check that it’s associative (exercise). And, the above table shows that it’s total.
This gives us the monoid \((\mathbb B, false, \|)\), or “booleans with or.”
Sometimes, there’s more than one way to imbue a set with a monoid structure. What if we pick a different operation?
\[\begin{aligned} false\ & \&\&\ false &= false \\ false\ & \&\&\ true &= false \\ true\ & \&\&\ false &= false \\ true\ & \&\&\ true &= true \\ \end{aligned}\]Now \(true\) is an identity^{1}. This gives us a different monoid over the same domain, \((\mathbb B, true, \&\&)\), or “booleans with and.”
Notice that we can recover a semigroup from any monoid by simply “forgetting” the identity element. If we neglect to point out that \(true\) is the identity of \(\&\&\), we get the semigroup \((\mathbb B, \&\&)\).
There’s a way that we can describe functions between structures - the function from monoids to semigroups, defined by \(F(D, e, *) = (D, *)\), is known as a forgetful functor. I’ll probably have some future posts exploring this idea further, as it can be combined with category theory to do some pretty general things.
The nicest thing it can do is be reversed. If we have some semigroup, we can attach a new element \(e\) to the domain of the semigroup and define \(e\) to be the identity. This new \(e\) is called a “point,”^{2} and it might not be distinguishable from every element already in the domain. If the semigroup already had an identity which was forgotten, then we can prove that \(e\) and the identity are one and the same.
Regardless, sometimes we have a semigroup without an identity, and this construction allows us to summon one out of thin air.
Let’s consider a little bit of C code for a moment.
struct SomeSemigroup; // opaque
SomeSemigroup *times(SomeSemigroup x, SomeSemigroup y);
By assumption, times
here is an associative operation, which may or may not have
an identity element. We don’t know. But we can extend the operation with a new
identity as follows;
SomeSemigroup *id = NULL;
SomeSemigroup *times_extension(SomeSemigroup *x, SomeSemigroup *y) {
if (x == id) {
return y;
}
else if (y == id) {
return x;
}
else {
return times(*x, *y);
}
}
This new semigroup is now a monoid. The identity is NULL
. It may be that there was
already some other identity in semigroup; in this case, NULL
is functionally
indistinguishable from that other identity within the semigroup structure.
A common example of a semigroup in programming languages is non-empty array(list)s. We can append any two arrays to get a third, and this append operation is associative. We have an additional (useful) structure that we can lookup any element in an array and get something meaningful back, since they are never empty.
But sometimes, when initializing an array for an algorithm, for example, it’s truly useful to be able to say that it is initially empty. We can recover a structure where that is allowed by appending a new point to our array semigroup, the null pointer, and extending the concatenation operation as above.
Now even if the language allowed empty arrays (as most do), the empty array and the null pointer are functionally indistinguishable. You can’t lookup values from either of them, and appending either one to another array will result in that other array.
We say that the null pointer and the empty array are isomorphic. They carry the same
information. In this case, that is no information at all! We can easily write a pair
of functions to witness the isomorphism^{3}. One of them sends the empty array to NULL
,
the other sends NULL
to the empty array. While we generally have a structural
definition of equality in programs, what we often truly want is informational equality,
where two objects are equal if they are isomorphic.
The exact meaning of isomorphic can differ with respect to context. Sometimes the structure is information that we care about (for example, trees). Thinking about what exactly equality means for a particular datatype and implementing it correctly can avoid major headache- inducing bugs. The concept of monoids (indeed, unital magma) tells us that if we have two different objects which are both the identities of a type’s core operation, then we should probably be writing a normalizing function, which replaces one with the other, because they must be functionally indistinguishable.
What if we have some unknown type \(T\), and we want to imbue it with a monoid structure? Perhaps we want to be able to put off deciding what binary operation to use until later, if there are several choices. Or perhaps there’s no reasonable choice at all, but we still want to be able to put things together.
One common example of this is validation, or parsing. We may encounter an error while validating a piece of data, but we don’t want our program to just fail immediately.
Instead, we want to combine all the errors that we discover during the entire process of validation (or at least, as far as we can go), so that the user can get as much relevant information as possible back from our tool. It’s not really meaningful to “combine” errors. We could append the callstacks and append the messages, but that doesn’t tell anyone anything. If anything, it’s actively confusing. Instead, we just want some monoid structure that leaves individual errors alone.
We can always construct such a structure. Instead of considering elements of \(T\) itself,
we consider elements of a new type, \(L(T)\). \(L(T)\) is the type of lists containing
elements of type \(T\). Each element of \(T\) corresponds exactly to a singleton list
containing it. We call the function from \(T \to L(T)\) an injection, because it “injects”
elements of \(T\) into this monoid structure. For example, the definition might be as
simple as inject(t) = List.singleton(t)
, in a language where this syntax makes sense.
Now the monoid multiplication is list concatenation, which as above is associative. If we need to conjure up an element from nowhere, we have to be able to do that without relying on the existence of things in \(T\). Perhaps we are initializing our parser and we need some initial value for the set of errors. It’s perfectly safe to choose the empty list, because it is the identity of this monoid and the concatenation with non-empty lists later won’t cause weird things to happen to our structure.
This is a very real example of why being able to conjure up a “nothing” element of a type is extremely useful!
If you’re a programmer, you’ve probably written loops like this before:
bool flag = false;
while (someCondition) {
do_some_work();
flag = flag || checkFlag();
}
if (flag) {
clean_up();
}
flag
here is acting in the \((\mathbb B, false, \|)\) monoid. If the body of the loop
is also implementing some type of monoidal behavior^{4}, we could combine these two
monoids and possibly clean up our code.
I want to leave off this overview of monoids with a quick proof that every program can be described by the structure of some monoid.
Firstly, it is well known that any program can be represented by a Turing Machine. It’s generally fairly easy to find such a machine, because the semantics of the programming language are usually defined in terms of an abstract machine, which is then implemented in terms of some assembly language, which is itself (usually) quite close to a Turing Machine on its own.
A Turing Machine is its own kind of structure, generally written \((Q, \Gamma, b, \Sigma, \delta, q_0, F)\). In order, we have a set of states, a set of symbols that we can write to some kind of memory, a designated “blank symbol,” which appears in any memory cell that has not yet been written to, a subset of \(\Gamma\) representing the symbols which can be in memory before we start (the “program input”), a set of transitions, an initial state, and a set of final states.
The Turing Machine starts in the “initial configuration,” where the memory has some input symbols in it, the machine is pointed at a particular memory cell, and is in the initial state. It executes by using the transitions, which describe for each state, how to use the value of memory currently being looked at to decide how to modify the current memory cell and switch to a different one, as well as possibly also switching states.
In any configuration of the turing machine, we can apply some sequence of transitions to get to a new configuration. A record of configurations that we move through between the initial state and a final state is called an accepting history.
A Turing Machine accepts an input (computes a result for that input) if and only if an accepting history exists. An accepting history is a chain of states, all of which are valid. So given any two states in an accepting history, we can find a (composition of) transition(s) which moves one to the other.
This structure is monoidal. We have the set \(C\) of configurations for our machine \(M\). We can define, for any input \(w \in L(\Sigma)\), the function \(T_w : C \to C\). \(T_w(c)\) tells us which state \(c'\) we will end up in if the “input” in memory begins with \(w\); if it does not begin with \(w\), then the result is whichever state \(M\) uses to reject an input.
The set \(T = \{ T_w \ | \ w \in L(\Sigma) \}\) is the domain of our monoid. The multiplication is the composition of these transition operators. The identity transition is \(T_\emptyset\), because every input can be seen as beginning with the empty string. Any two of these can be composed, and composition is associative, hence we have a monoid.
This is called the transition monoid of a machine. It formalizes the notion that any computation can be broken down into steps which are independent of each other, and the results of those steps can be combined according to some monoidal structure to produce the result of the program.
Why is that useful? After all, the construction is, well, rather obtuse. It does tell us exactly how to find such a monoid, but that monoid doesn’t exactly lend itself to pretty code. An example of the constructed monoid might look like this, in a dialect of C extended with nested functions that are safe to return pointers to^{5}.
typedef void*(*step)(void*) ConstructedMonoid;
void *id_step(void *state) {
return state;
}
step *times_step(step *step1, step *step2) {
step *new_step(void *state) {
return step2(step1(state))
}
return new_step;
}
step *id = id_step;
step *(*times)(step*,step*) = times_step;
Here the monoid is the set of all functions which take a pointer and return a pointer. The pointer is presumed to point to any relevant information that the program needs to continue. The identity is the function which returns the state unchanged, and the multiplication composes two steps.
Naively, this is stupid. We don’t gain anything by writing our program this way, and really it just makes the program more opaque. Chances are we were already breaking the problem down into smaller steps in a more readable fashion.
This idea becomes useful when the monoid is isomorphic to a simpler monoid. In that case, we can disregard the elements-are-functions notion, and replace the functions with the elements of the simpler monoid. Each step of our program becomes as simple as generating the next monoid element, with regards to the globally-available input but without caring about the results or side effects of other steps. Then we multiply all the resulting monoid elements together and the result falls out.
We used this pattern when writing an autograder for a course taught in Haskell, which requires some functions to be a written in a certain recursive style. The grader is implemented as a a function converts a stack of nested function definitions into a flattened list of bindings along with all the other names in scope. This transformation is implemented by (recusively) translating each individual definition into an element of a monoid and then multiplying them together as we recurse back out. Each element of the flattened list of bindings is then checked individually for violations of the desired property and the results are monoidally combined into the final result.
Any monoid corresponding to a turing machine in this way ends up being isomorphic to another monoid which is an instance of a special kind of monoid called a monad. Monads are studied in category theory, which we’re not going to get into here, but some programming languages make them explicitly representable. This makes it possible to express programs by describing how to translate inputs into elements of the monoid and combining them together.
A monoid in this way represents the structure of the programming environment. What that means is that it represents what side effects can be performed by the functions which return elements of the monoid - throwing errors, manipulating a shared state, things like that.
By generalizing such a program from a specific monoid to “any monoid which contains that monoid as a substructure,” we can generalize the environments in which a function will work. Rather than having a function which works “with my database,” we can have a function which works “with some database.” We can then easily test our programs with a mock database, while using a real database for our actual business environment. Since it’s the same code running in both cases, we can be confident that the tests passing means our logic is sound, even though the underlying database is different.
This is related to other programming patterns like dependency injection. There’s a very rich space of programming constructs and patterns that can be explored here to find ways to write cleaner programs. Haskell has a collection of libraries known as algebraic effect libraries which provide implementations of this idea.
We’ll explore this concept further in future posts, I hope, but now back to the math.
A group is a monoid with inverses. Alternatively, it is a total groupoid, or an associative loop.
A monoid \((D, e, *)\) is a group if for any \(x \in D\), there is an element \(x^{-1} \in D\) such that \(x * x^{-1} = e\). We can easily prove that inverses are unique, which is a good exercise.
Many, many things that we say in day-to-day mathematics form groups. For example, the monoid \((\mathbb Z, 0, +)\) is also a group, because for any \(x \in \mathbb Z\), we have \(-x \in \mathbb Z\), and \(x + (-x) = 0\).
However, the monoid \((\mathbb N, 0, +)\) is not a group, because negative numbers aren’t in \(\mathbb N\). The monoid \((\mathbb Z, 1, *)\) is also not a group, because for example, there’s no inverse of 2. There’s a mechanism by which we could extend this monoid into a group, and the result would be \(\mathbb Q\). Perhaps in the future I’ll explore this construction in another series of posts, which would probably build up to how we can define \(\mathbb R\) algebraicly.^{6}
As discussed in the last post, Rubik’s Cubes are associated with a group where every element of the group is a sequence of moves. We can get this group by starting with all of the 90 degree clockwise rotations of a single face (there are 6 such moves) and then “closing” the set under concatenation: add to the set every move or sequence of moves which can be obtained by concatenating two (sequences of) moves in the set. Repeat until nothing new gets added.
We can check that inverses exist in this group. Call the clockwise rotation of the front face \(F\). We can check that \(F^{-1} = F * F * F\), which we can also write \(F^3\). This means that \(F\) can be undone by performing \(F\) three more times; \(F^{-1} * F = F^4 = e\). The groupoid properties then tell us that (since products are always defined in a group), if we have the sequence of moves \(A * B\), then \((A * B)^{-1}\) is nothing other than \(B^{-1} * A^{-1}\). This corresponds to undoing sequence \(AB\) by first undoing \(B\), and then undoing \(A\). That’s exactly what we expect!
From here, we could develop a rich theory of groups, called group theory, and apply it to the study of a huge variety of real groups that appear in mathematics. Eventually, I’d like to develop some of that theory in a sequence of posts, and build up to an understandable proof of the fact that quintic polynomials are not solvable in general in terms of addition, multiplication, and \(n\)th roots.
One other property that we frequently ask of binary operations is that they commute. That is, they satisfy the restriction that \(x * y = y * x\).
If we have a monoid with this property, we call it a commutative monoid. If we have a group with this property, we call it an abelian group. Most of the examples we’ve discussed so far are commutative, including the boolean monoids, and the additive group. The list monoid is not commutative, because order matters in lists. In contrast, the set monoid is commutative, because sets are unordered. The transition monoids of state machines are not commutative.
If you have a Rubik’s Cube handy, check that the Rubik’s cube group is not abelian. We could loosen our restriction a little bit, and ask the following. Given some group \((G, e, *)\), what is the largest subset \(C \subseteq G\) such that for any \(z \in C, g \in G\), we have that \(z * g = g * z\)? We say that every element of \(C\) commutes with every element of \(G\).
We can prove that this set \(C\), called the center of \(G\), is itself a group:
so \(y * z\) is also in \(C\).
Since \(C\) is a subset of \(G\), and \((C, e, *)\) is still a group, we call \((C, e, *)\) a subgroup of \((G, e, *)\) and we write \(C < G\).
The center of a group is nice because it is the largest subgroup which is abelian. In some groups, the center is trivial, meaning it contains only the identity.
I’ll end this post with something that I consider to be a good thought exercise: is the center of the Rubik’s Cube group trivial? If not, how big is it?
By developing some group theory, we could fairly easily prove an answer to this question.
In this series of posts, we introduced the general concepts of each of the major single-sorted, binary-operator algebraic structures. We saw how they arise from adding increasingly stringent constraints to the binary operator, and attached some examples to each of the weird names.
I hope that abstract algebra feels approachable from here. We can use it to achieve general results which we can then apply to specific scenarios to get results for free. A further treatment of abstract algebra would explore some more important structures, notably rings, fields, and vector fields. These structures are nothing more than adding some new requirements to our set.
We can improve programs, by implementing the results as very general functions and then applying the general functions to specific scenarios, achieving great degrees of code re-use and understandability. Simply by seeing a reference to a common general function, we can immediately understand the broad strokes of what the function is doing, even if we don’t yet understand all the specifics of the structure we’re applying it on.
Most likely, my next posts will be about computer hardware, specifically superscalar out-of-order processing and how we can use the visual digital logic simulator Turing Complete to explore the different components of such systems and how they work together.
Further down the line, future posts will explore some more group theory, and some more direct application of abstract algebra to programming in Haskell. I want to explore so-called free structures, and how we can use them to describe computations that build other computations, a technique called higher-order programming. We’ve seen some free structures in these posts already, though I didn’t explicitly call them out^{7}. I also alluded to higher-order programming via effects earlier in this post, but there are other neat things we can do with the technique that don’t seem to get much exposure.
Remember from the last post, we proved that identites are unique in unital magma. Since monoids are unital magma, identities are unique in monoids too. So without checking this fact, we immediately know that \(true\) is the only identity. For the remainder of the post, I’ll say “the identity” of a monoid, instead of “an identity.” ↩
In the original post in this overview series, I mentioned that distinguished elements of structures are always called points, and structures that have any are called “pointed.” ↩
As discussed in Setting the Stage ↩
We’ll see in a moment that it always is. ↩
C++ has such a construct, but I very strongly dislike C++, so this is what we get. In a later post, we will review these concepts through the lens of Haskell, where this structure will turn out to be… quite familiar. ↩
That is, in terms of the properties that the set should have. What might those properties even be, and how can we construct a set that has those properties if it is necessarily uncoutably infinite? ↩
Specifically, our example of magma was the free magma on binary tree nodes, linked lists (the semigroups that we got by making the magma associative) are the free semigroups on linked list nodes, and general lists containing things of type \(T\) are the free monoids on \(T\).
In a deep sense, the composition of functions that we saw when expressing programs as monoids described how to take a list of steps to perform and flatten it into a single long step that composes all the elements of the list. By generalizing over more interesting composition operators (which tend to arise from the “more interesting” monoids that we may be lucky enough to be isomorphic to), we can again arrive at a free monoid, this time over programs themselves. These free monoids are called free monads and form the basis for programming via effects. ↩
In this post we’ll wrap up our exploration of the bottom half with unital magma and quasigroups. Then we’ll talk about loops and groupoids. Groupoids are where we really start getting enough interesting structure to make interesting statements about all groupoids at once.
The possible properties we’ve put on the operator so far include totality, association, and identities^{1}. We got semigroups by starting from magma and requiring the operation to be associative. What if instead, we require the operation to have identities?
Specifically, the property we’re going to add is slightly different than what we had before. Before we assumed that there was an identity for each arrow in a category, and that the left and right identities could be different. However, this was only really necessary because categories did not require the operation to be total.
The reason this is undesirable is because that if I give you an element of the set \(S\) and ask you for the (left or right) identity element, you can’t easily give it to me. You’d have to find the correct identity for that element, and this could be quite difficult. So we’re going to add a stronger requirement instead, which makes it easy to find the correct identity.
\[\text{The Identity Axiom} \\ \text{There is an } e \text{ such that for all } x, e \cdot x = x \cdot e = x\]Now it’s easy to find the correct identity, because it’s always the same.
We do have to be careful though. The axiom doesn’t say that there is only one \(e \in S\) with this property, just that there is at least one. We cannot (naively) assume that the identity is unique - we’ll come back to this in a moment.
Recall that we defined \(\cdot(A, B)\) for our model of magma to be the operation that makes a new tree node whose children are \(A\) and \(B\). The carrier set for this model is the set of (rooted) binary trees. But I implicitly disregarded the existence of empty trees - we assumed that every tree in the set has at least one node. This was effectively required because every node needs to have either zero or two children; \(\emptyset \cdot A\) would only have one child which is not allowed.
But we can still give a definition for \(\emptyset \cdot A\) - it can just be \(A\) again! By definition, \(\emptyset\) (the empty tree) would be our left identity.
We can similarly define \(A \cdot \emptyset = A\), and now it’s also the right identity^{2}.
This lets \(\emptyset\) satisfy our definition of the identity element.
Now recall that we got here by adding \(\emptyset\) to our carrier set. It wasn’t there before, but we’ve added it in and defined \(\cdot\) for it. What’s stopping us from also adding a new element, 🙂, and defining it to also be an identity?
Seemingly, the answer is nothing. And this takes us to our first highly-general application of the ideas of abstract algebra.
Let’s suppose that we have a unital magma with two distinct elements \(e_1, e_2\) which both satisfy the identity law. What happens if we try and compute \(e_1 \cdot e_2\)?
\[\begin{aligned} e_1 &= e_1 \cdot e_2 \\ &= e_2 \end{aligned}\]We can prove that \(e_1 = e_2\)! This happens because both identities are both left and right identities.
The amazing thing is that this proof only uses the identity law. Whenever we have a structure with this identity law, we get for free that the identities are unique!
This applies to anything that is a unital magma - unital magmas, but also monoids, loops, groups, and more.
Therefore, from now on, we will say the identity instead of an identity.
Latin Squares are a generalization of Sudoku puzzles. We have an \(n \times n\) grid, and \(n\) distinct symbols. We place each element in the grid such that each one appears exactly once in every row an column. Here’s an example one with 4 elements - we say it has order 4.
\[\begin{array}{|c|c|c|c|} \hline b & d & c & a \\ \hline a & c & d & b \\ \hline c & b & a & d \\ \hline d & a & b & c \\ \hline \end{array}\]It’s not very hard to check that this is indeed a latin square. Take a few moments and check!
What’s interesting, though, is that this looks suspiciously like a multiplication table. All we have to do is label the rows and columns with the operands and then declare that \(x \cdot y\) is the value in the row labeled \(x\) and column labeled \(y\).
\[\begin{array}{c||c|c|c|c|} & a & b & c & d \\ \hline\hline a & b & d & c & a \\ \hline b & a & c & d & b \\ \hline c & c & b & a & d \\ \hline d & d & a & b & c \\ \hline \end{array}\]We could describe any magma in this form, but our running example so far has used the domain of binary trees, which is an infinite domain. Putting that in table form would be pretty tricky! However the magma described by this table is finite. The carrier set is just \(D = \{1,2,3,4\}\).
\(D\), along with the multiplication table to define \(\cdot(-,-)\)^{3}, form a magma. Since the operation is defined by a multiplication table, we could instead call the operation multiplication. Let’s do that. While we’re at it, we might as well use the multiplication symbol \(*\).
For example, we have \(a * b = d\). This pair of \((D, *)\) gives us a “finite magma.”
Take a moment and check: is this magma unital? Answer^{4}
We should also check if this magma is associative, which would make it a semigroup. Randomly picking elements \(a, b, d\), we can check:
\[\begin{aligned} (a * b) * d &= d * d \\ &= c \\ \\ a * (b * d) &= a * b \\ &= d \\ \end{aligned}\]It’s not associative!
However, the fact that the multiplication table is a latin square gives us an interesting property. The multiplication is invertible. That is, we have the following property.
\[\text{The Divisibility Axiom} \\ \text{For any } a,b \in D, \text{ there are unique } x,y \in D \text{ such that} \\ a * x = b \\ y * a = b\]We write that \(x = a \backslash b\) and \(y = b / a\). I read these as “x is a under b” and “y is b over a.” Respectively, we call these operations “left division” and “right division.”
If a magma has the divisibility property, we could call the magma divisible. But more commonly, we call these structures quasigroups.
We can check, from the definitions, that all of the following properties hold:
\[\begin{aligned} y &= x * (x \backslash y) \\ y &= x \backslash (x * y) \\ y &= (y / x) * x \\ y &= (y * x) / x \\ \end{aligned}\]These identities say that multiplication and division on the same side, by the same element, in either order, have no effect. We’d expect that of operations called “multiplication” and “division,” and we get these identities for free from the definitions even though we didn’t require them explicitly!
These properties will hold in any quasigroup. Checking the diagram, groups are (unsurprisingly) quasigroups, and groups are very prevalent. These properties will also hold in groups.
The most common quasigroups are numbers with subtraction, for example \((\mathbb Z, -)\) or \((\mathbb R, -)\). Subtraction is total, and invertible on either side. We don’t typically think of subtraction as “multiplication,” but it fits the definition. And indeed, the above properties hold in these quasigroups.
If we add the identity axiom to our requirements for a quasigroup, we get a structure called a loop. One example of a loop (which is not also associative) is described by the following table.
\[\begin{array}{c|c|c|c|c} 1 & 2 & 3 & 4 & 5 \\ \hline 2 & 4 & 1 & 5 & 3 \\ \hline 3 & 5 & 4 & 2 & 1 \\ \hline 4 & 1 & 5 & 3 & 2 \\ \hline 5 & 3 & 2 & 1 & 4 \\ \end{array}\]There are some more subclassifications of loops, but I’m not going to get into them here. I just wanted to mention that this structure has a name, and it’s going to be one of the possible stepping stones to groups.
Continuing our exploration of inverses, what happens if we add inverses to a category? That is, for every arrow \(A \to B\) in a category, we’ll add the arrow \(B \to A\) if it doesn’t already exist.
The exact properties we’ll add (which agree with our notion of division from quasigroups) are
The third property is a version of the identity property that works despite the fact that groupoids don’t actually have to have identities. It says that whatever the result of \(a * a^{-1}\) is, it is an identity for any values with which multiplication is defined. However, not every element necessarily has multiplication defined with an identity!
In every groupoid, we can prove the following useful theorems:
The proofs require some strong symbolic manipulation. This is pretty common in abstract algebra (hence the “abstract”) but the exchange is that the results are very general and powerful. Let’s get in the practice.
The right multiplication is justified because associativity and axiom 2 tell us that the relevant product is defined.
This proof looks very similar to the last one. We start with the only relevant equality that we know, and then we rearrange things in the only two ways possible and the equality we want just falls out.
Given that \(a * b\) exists, we know that \((a * b)^{-1}\) exists. Let’s call it \(e\). Similarly to the last proof, we can justify that \(a * b * b^{-1} * a^{-1}\) exists.
\[\begin{aligned} e &= (a * b)^{-1} & \text{Definition of } e \\ e * (a * b) * (b^{-1} * a^{-1}) &= (a * b)^{-1} * (a * b) * (b^{-1} * a^{-1}) & \\ e * (a * b) * (b^{-1} * a^{-1}) &= (a * b)^{-1} * (a * (b * b^{-1})) * a^{-1} & \text{Associativity} \\ e * (a * b) * (b^{-1} * a^{-1}) &= (a * b)^{-1} * a * a^{-1} & \text{Axiom 3} \\ e * (a * b) * (b^{-1} * a^{-1}) &= (a * b)^{-1} & \text{Axiom 3} \\ (a * b)^{-1} * (a * b) * (b^{-1} * a^{-1}) &= (a * b)^{-1} & \text{Definition of } e \\ (b^{-1} * a^{-1}) &= (a * b)^{-1} & \text{Axiom 3} \\ \end{aligned}\]Both of these properties are easier to prove on proper groups, but it’s interesting that we don’t actually need the existence of an identity to prove them.
Returning to our category model, a groupoid is a category with inverses. Using our function example, each node of the graph is a set. Each arrow \(A \to B\) is a function from \(A\) to \(B\). For simplicity, we’re only going to pick one function for each arrow.
If we have an arrow \(f : A \to B\), we also have an arrow \(g : B \to A\). For each pair, we know that both products \(f * g\) and \(g * f\) are defined. This is our Axiom 2 of inverses above, and we can easily check that the model meets it: \((A \to B) * (B \to A) = (A \to A)\) and vice versa.
We can also check the properties that we proved in the last couple sections, but I’ll leave that as an exercise.
You may have heard people say that group theory can be applied to study Rubik’s Cubes. This is true, and part of the reason is that any two moves on a Rubik’s Cube can be composed. Each move is an element of the “Rubik’s Cube Group,” and products in a group always exist. Of course, this group is also a groupoid.
But we could find a different puzzle where we can’t always compose any moves, for example, fifteen puzzles. Fifteen puzzles are commonly studied as groups, but they can be more naturally represented by the groupoid of sequences of moves. The product of two sequences means doing one and then the other (and since this is only possible if the hole is in the correct place in the intermediate configuration, the product does not always exist).
We’ll still have all of the properties that we’ve shown hold for general groupoids. We have partial identities (do nothing, with the hole in a particular place), inverses, and composition associates. We could then apply groupoid theory to the fifteen puzzle to discover various properties of the “15 puzzle groupoid” and learn things about the nature of the puzzle.
One common fact about 15 puzzles is that exactly half of the possible configurations are solvable. Any solvable configuration can be transformed to the solved configuration by applying some element of the 15-puzzle groupoid^{5} (a sequence of moves). It’s easy to prove, using the groupoid axioms, that any two solvable configurations can be transformed into each other (exercise). If we went and developed some groupoid theory, we could show that these facts imply that any two unsolvable configurations can also be transformed into each other, which is much less obvious.^{6}
The development of such a theory may be the topic of future posts, but this hopefully teases some of the power of abstract algebra. We get general results which we can then apply to learn specific things about specific models.
An extremely common structure in the real world is a torsor. Torsors are sets, equiped with an invertible binary operation (often written \(+\)), but without a notion of “zero.” This corresponds to the notion of a groupoid, although if we make it more precise we’ll see that I’m waving my hands fairly vigorously at the moment.^{7}
But these posts are about conceptually understanding structures, so, I’ll keep waving my hands for the moment.
Consider musical notes. We have notes like A, A#, D, Fb, etc. There’s a notion
of a “next note,” and of a “previous note,” next(A) = A#
, prev(A) = Ab
. These
operations are inverses. If we see the notes as the nodes of a graph, then the
next
and prev
functions are arrows between the nodes, and we have a groupoid.
Using these functions, we can measure distances between notes. It takes 4 steps
of the next
function to get from A to C#. We can say that the distance between
these notes is 4. By measuring distances, we can recover something that looks
a lot like addition. However, we don’t have values that we can assign to notes themselves,
other than arbitrary names, and we don’t have a notion of a “zero” note. So we
can add a note to a distance, for example B + (C# - A) = D#
(why?), but we can’t
add notes to other notes.
Torsors are pretty common in the real world. In physics, energy is a torsor. There’s no notion of “zero” energy. If we examine the same scenario in two different reference frames, we will almost certainly measure different amounts of energy for every object involved. Which measurement of “zero” is the right one? Neither! They are both equally meaningless. Yet both analyses will come to exactly the same conclusions, because they will measure the same differences in energy, and this is what matters.
Sometimes we think something is a torsor and later find out that there is a true zero. Temperatures are a good example. Temperatures were thought to be a torsor until absolute zero was discovered. Absolute zero is the true zero of temperatures. No matter what scale we use to measure temperatures (analogous to a reference frame), we will always agree on the meaning of absolute zero.
This concludes the introduction to the basic algebraic structures, and motivating some of the things that they gives us the language to talk about. In the next post, we will talk about monoids, which are by far the most prevalent mathematical structure in programs that you’ve never heard of^{8}. We’ll also talk about groups, which are a prevalent mathematical structure.
Recall from the previous post’s section on categories, we say “the identity law” to mean the existence of both left and right identities. ↩
The identity laws for a small category require that an identity exists for each element of the carrier set. They do not require that the identities be unique or the same for every element. ↩
This notation means “the operation \(\cdot\) which takes two unspecified arguments.” I noticed that plain \(\cdot\) is a bit awkward to read on the rendered page, so I’ll use this from now on when referring to an operation that we don’t have a better name for. ↩
No, it is not. There’s no left or right identity, let alone something which is both. ↩
I didn’t define what it means to apply a sequence of moves to the puzzle, but intuitively we know what it means. However it’s worth noticing that this is itself a binary operation. Rather than being from \(D \times D \to D\), though, it is from \(F \times D \to F\), where \(F\) is the set of 15-puzzle configurations. Such operations are called groupoid actions. These are closely related to group actions, and many of the same results apply to both. ↩
Group theorists will recognize this as the statement that the groupoid action which applies a sequence of moves to a configuration has two orbits, with equal cardinality. ↩
Torsors are usually analyzed as a group acting on a different set \(X\). Using the group action and the group multiplication (which does have an identity), we can obtain a subtraction operation on \(X\) which measures distances as elements of the group. The 0 distance corresponds to the identity of the group. Yet the set \(X\) does not have a designated zero. If we pick an arbitrary element of \(X\) and declare it to be zero, we recover the group itself, and this is called “trivializing” the torsor. But seeing things as a torsor is useful when we want to work directly and only on the distances between things, and I find this conceptually close to groupoids. Your mileage may vary with this one. ↩
Unless you enjoy programming in Haskell, in which case you probably recognize that just about everything in programs is a monoid 🙂 ↩
As a reminder, we’ll be starting to explore the various single-domain, single binary operator algebraic structures in the following hierarchy.
We’ll start at the bottom and work our way up, talking about each arrow individually. Let’s build some intuition!
Experienced mathematicians should feel comfortable skipping this section. I will introduce some basic concepts and notation, then move on.
As mentioned in the last post, before we can have structure, we need to have things to put structure on. This is the domain of the structure, and it will almost always be a set of some kind.
Sets aren’t themselves very interesting. They are collections of things. They can be any things. Even other sets! We write sets as a list of things inside curly braces, for example \(\{1, 2, 3\}\). Even though we write them as a list, keep in mind that sets have no structure at all. \(\{3, 2, 1\}\) is the same set, because the order doesn’t matter (order would be structure). All that matters is that the elements are the same.
However within a set, the elements do not have to look like each other. For example, we can easily write \(\{1, cat, \text{:)}\}\). I’m not sure how this set is useful, but it is a set.
There are several well-known sets with special names. For example, the natural numbers are written \(\mathbb N\). Here’s a quick list of some common sets of numbers.
If we want to say that some thing \(x\) belongs to a set \(S\), we write \(x \in S\). When we have a set containing things that are alike, we can call the set a “type” and say that its elements “are that type.” For example, there is a type of natural numbers, and 0 is a natural number. When viewing a set this way, we would instead write \(x : S\).
There’s also a set usually written 1, which is the set containing a single element. It doesn’t really matter what that element is, because you can easily view any set with only one element through the lens of any other set with only one element.
Wait. What does that mean?
In the last post we discussed how we can view natural numbers and stacks of pennies as being “the same thing.” We did that by describing how we could view both of them as modeling the same set of properties. Here, what we’re trying to say is that if you have a set with one element, say \(\{1\}\), then anything you say about that set will immediately apply to another set, say \(\{\text{dog}\}\), which also only has one element. We make it apply by replacing any reference to \(1\) with a reference to “dog.”
What that describes is a way to translate from \(\{1\}\) to \(\{\text{dog}\}\). This is a function! If we call the function \(\tt{translate}\), then we can define it as \(\tt{translate}(1) = \text{dog}\). We would say that the type of this function is from \(\{1\}\) to \(\{\text{dog}\}\). The notation for this is simply \(\tt{translate} : \{1\} \to \{\text{dog}\}\).
We can also go the other way, using \(\tt{translate'} : \{\text{dog}\} \to \{1\}\), in the obvious way. Since we can go in both directions without losing or gaining any information, then conceptually, both sets must contain exactly the same information to begin with!
This concept of “containing the same information” is at the core of what we can do with abstract algebra. We can prove properties about some model by showing that it contains the same information as some other model where those properties are easier to prove - and proving the properties on that model instead.^{1}
That’s a bit of an abstract concept, so let’s work an example.
Let’s suppose we have a computer program which reads a simple expression, like “1 + 1,” as a string, and evaluates it. We’ll let the expressions contain a pair of natural numbers and either “+” or “*”.
It’s pretty hard to do any meaningful work on a string. Turning “1 + 1” into “2” is not so simple!
However most programming languages provide functions like parseInt : String -> int
and
toString : int -> String
. If we restrict ourselves to strings that actually contain integers,
then this pair of functions “witnesses” that those strings and the set of integers contain the
same information, just like translate
and translate'
witnessed that \(\{1\}\) and
\(\{\text{dog}\}\) do.
So lets use these functions to translate our strings into a domain we can work on more easily:
class Op(Enum):
Add = 1
Times = 2
def parseExp(input: str) -> Tuple[int, Op, int]:
left, opstr, right = input.split()
op = Add if opstr == "+" else Times # for simplicity
return int(left), op, int(right)
def toString(result: Tuple[int, Op, int]) -> str:
return [ str(result[0])
, "+" if result[1] == Add else "*"
, str(result[2])
].join(' ')
These two functions witness that (well-formed!) expressions contain the same information whether we represent them as strings or as our little custom datatype.
These types of bidirectional transformations are extremely common in programming. Hopefully that motivates the power of abstract algebra to help us think about and improve our programs.
So let’s finish this out. Now to show our property (that we can evaluate expressions-as-strings), we have to show that our new custom datatype has the same property (that we can evaluate them). Now it’s easy;
def eval(exp: Tuple[int, Op, int]) -> int:
match exp[1]:
case Add:
return exp[0] + exp[2]
case Times:
return exp[0] * exp[2]
And of course we can recover the result as a string with str
, if we want to.
This kind of bidirectional transformation is called a bijection. In order to know that we have the same information both ways, we need to know that the transformations in each direction preserve the structure we are working with. Bijections don’t have to preserve structure, but sets don’t have any structure to preserve anyway. A bijection that does preserve structure is called an isomorphism, from the Greek iso-, meaning “same” and morphous meaning “form.”
For sets, all bijections are isomorphisms. In abstract algebra, we care much more about isomorphisms than general bijections.
The section title is mostly a joke about the more common usage of the term “magma,” don’t try and read too much into it.
There’s only one arrow in the diagram to talk about here - we get magmas from sets.
In order to get a magma from a set, we need to add an operation that operates on two elements of the set, and produces a third. This operation is typically written \(\cdot\).
If our carrier set is \(S\), we can write the type of \(\cdot\) as \(\cdot : S \times S \to S\). For those unfamiliar, \(S \times S\) means the type of two things in \(S\), or a pair of things in \(S\). For example, \((1, 2) \in \mathbb N \times \mathbb N\). If \(x,y \in S\), we write \(\cdot(x, y)\) or equivalently (and preferably) \(x \cdot y\).
The only property that a magma imposes on the operation is its type. It must work for any two elements of \(S\), and the result must also be an element of \(S\). The operation does not have to be associative, commutative, or anything else.
Most of the structures we will look at later will subsume magmas, meaning that anything which models them is also a model of a magma. For the sake of example, though, let’s consider something which is a magma and only a magma: binary trees.
A binary tree is a data structure where at any point, we either refer to two smaller binary trees, or to nothing. The operation \(\cdot\) combines its arguments as subtrees under a new node. Some examples of this magma:
Get out a piece of paper and consider what \(A \cdot B\) looks like. What about \(B \cdot A\)?
They aren’t the same! That means that \(\cdot\) does not commute.
What about \((A \cdot B) \cdot C\)? Compare that to \(A \cdot (B \cdot C)\). They also aren’t the same! This means that \(\cdot\) does not associate.
The only property that we are guaranteed by a magma is closure: if \(a\) and \(b\) are in the magma, then \(a \cdot b\) exists and is also in the magma.
Despite the section header, these are worth talking about if only as a stepping stone.
Once again, we’re going to imbue our carrier set \(S\) with a binary operation \(\cdot : S \times S \to S\). However, this time, we’re going to put a different requirement on it. Instead of requiring it to be total, we require it to associate. What that means is that there might be some elements of \(S\) for which \(\cdot\) is undefined.
However, if \(a \cdot b\) exists, and \((a \cdot b) \cdot c\) exists, then both \(b \cdot c\) and \(a \cdot (b \cdot c)\) must also exist. Furthermore, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
We can see a semigroupoid as a directed graph by looking at the graph in a different way than we did for magma.
This time, consider the nodes of the graph. There are arrows between them; let the set of all those arrows form our carrier set. Remember, sets can be of strange things! An arrow from node \(A\) to node \(B\) gives us a “path” to get from \(A\) to \(B\). There are a lot of notations used for this by different authors, and honestly I find most of them obscure^{2}. So let’s use a simple one: The arrow from \(A\) to \(B\) will literally be written \(A \to B\) :).
Now we can define \(x \cdot y\) for two arrows \(x\) and \(y\). First of all, remember that \(\cdot\) does not have to be defined for every pair of arrows. So since arrows describe paths between nodes, let’s let \(\cdot\) be the operator which combines paths. That seems pretty natural, right?
We define \((A \to B) \cdot (B \to C) = A \to C\). In order for a particular graph to to model a semigroupoid in this way, the arrow \(A \to C\) must exist. That’s not a property of semigroupoids though. It’s just a property of the way we have defined this model. Here’s a pair of examples of such semigroupoids.
A very important facet of abstract algebra is apparent here: there are many things that fit the mold of “semigroupoids,” and two of them are shown here, but these two semigroupoids are not the same. Proving something about the first would not necessarily prove the same thing about the second, because they don’t contain the same information! In math-speak, these semigroupoids are not isomorphic.
We could easily take some arbitrary set \(\{\) 🙂, 🙁, 😃 \(\}\) and define, again arbitrarily, that 😃 \(\cdot\) 🙁 = 🙂. The result is certainly a semigroupoid (there aren’t enough equations to even trigger the associativity requirement). But this semigroupoid is isomorphic to the first one above (why?). Since they carry the same information, we only have to talk about one of them. Graphs are easier to visualize, so we’re going to use them!
In the second semigroupoid above, notice that \(2 \to 2\) and \(4 \to 4\) exist, because of the pair of arrows \(2 \to 4\) and \(4 \to 2\). We can take \((2 \to 4) \cdot (4 \to 2)\) to “round-trip” back to \(2\), and according to our definition, that means \(2 \to 2\) has to exist.
We can also use the second semigroupoid to check that the way we defined the operation is actually associative. You can check that
\[\begin{aligned} ((1 \to 2) \cdot (2 \to 4)) \cdot (4 \to 3) &= (1 \to 4) \cdot (4 \to 3)\\ &= 1 \to 3\\ \\ (1 \to 2) \cdot ((2 \to 4) \cdot (4 \to 3)) &= (1 \to 2) \cdot (2 \to 3)\\ &= 1 \to 3\\ \end{aligned}\]Indeed, those are the same.
However, it’s also quite apparent that \(\cdot\) does not commute. In fact, it fails to commute in the most spectacular fashion imaginable: not only does \((B \to C) \cdot (A \to B)\) fail to equal \((A \to B) \cdot (B \to C)\), it doesn’t even exist! We didn’t define the operation for arrows that don’t share an intermediate node, and the semigroupoid laws don’t say we have to.
Since arrows describe paths between nodes, it’s pretty common to only draw the minimal number of arrows and leave the combined paths as implied. In the complex graph, we could omit arrows \(2 \to 2, 4 \to 4, 1 \to 3, 2 \to 3,\) and \(1 \to 4\).
Small categories are where things really start getting interesting. For simplicity, I’m just going to call these categories. This is conceptually fine, but you can see the footnote to see what “small” means here if you want.^{3}
With semigroupoids, we saw that it was ocassionally necessary to have an arrow \(a \to a\), in order to make sure that the product of two arrows exists. When we have these arrows, we get the nice pair of equations
\[\begin{aligned} (a \to a) \cdot (a \to b) &= a \to b \\ (a \to b) \cdot (b \to b) &= a \to b \\ \end{aligned}\]These are respectively called the left identity equation and the right identity equation. For the arrow \(a \to b\), the arrow \(a \to a\) is called a left identity and the arrow \(b \to b\) is called a right identity.
A category is what we get if we require that these identity arrows exist. We add a pair of requirements to our operation:
In our graph model^{4}, recall that we only required arrows to exist if they were necessary to “combine paths.” These two new requirements additionally require arrows to exist from every node to itself. Collectively, these requirements are called the “left and right identity laws.” We’re never going to want one without the other, so we’ll shorten that to just “the identity law,” and use that phrase to mean the pair of them together.
Categories have the nice property that for any node in the graph, there is definitely an arrow pointing away from it, and an arrow pointing towards it. That means that when proving things about categories, we can freely say “pick an arrow pointing to \(b\)” or “pick an arrow from \(a\),” without running into the problems that this could cause with a semigroupoid.
Other than our graph model, a pretty typical example of a category is that of functions from some type to itself. If we consider the collection of all functions \(f : S \to S\), we get a category. The operation is function composition. The (singular) identity is simply \(f(x) = x\), which is both a left and right identity to every other function in the category. Such functions are often referred to as endomorphisms, from the Greek root endo- meaning “within.” This structure is incidentally more than just a category.
Every arrow in our original hierarchy^{5} represents adding a single new restriction to the operation \(\cdot\). This time, we’re taking a semigroupoid (which requires that the operation associate) and additionally require that the operation be total.
In our graph view, this means we have to somehow define \((a \to b) \cdot (c \to d)\), and it’s not obvious how to do that. In fact, it seems there isn’t any reasonable way to do it at all!
There’s one exception. If we allow our graph to have multiple arrows between the same nodes, then a graph with only one node and many arrows from that node to itself forms a semigroup. Since every arrow looks like \((a \to a)\), we can always define \(\cdot\). But now it’s not so obvious how this is useful.
We could imagine that the singular node represents some set, and that each arrow represents some arbitrary function from that set to itself. This is extremely general, but it is a semigroup. Really, it’s a whole family of semigroups, because it will form a semigroup no matter which functions we choose from the set to itself. Notably, unlike the similar category example, we don’t have to have the identity function in our collection!
Since semigroups are semigroupoids (and magmas), anything that we can prove about semigroupoids (or magmas) will also apply to semigroups, and therefore applies to this amazingly general structure of “composing functions from a set to itself.”
We can get an alternate view of semigroups by coming from Magmas. Magmas required the operation to be total. Semigroups additionally require the operation to associate.
Looking back at the section on magmas, we called two trees equivalent if they looked the same. This was the reason the operation didn’t associate. But what if we redefine equality, specifically so that \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\)? No one says we can’t do this. What structure do we end up with?
Since we can now shift the parentheses around arbitrarily, we can always move them as far to the left as we want. The result will be binary trees that look like this:
So we now say two trees are equivalent if we can rearrange their nodes (but without swapping the order - semigroups still aren’t commutative!) to move all the “interesting” nodes to the left, and they look the same after we’re done. If we wanted to worry about order, we could label the nodes at the bottom with numbers, but let’s keep it simple here and leave them as dots.
We can see an easier criteria for equality based on that. Two of these semigroups are “the same” if they have the same number of nodes. That’s it!
And since we can always adjust such a tree to whatever (tree) structure we like, the “structure” that the semigroup knows is just the order of the leaves at the bottom. That carries exactly the same information as an ordered list of the leaves!^{6} Notice though, that this model doesn’t support empty lists (yet).
So by coming at semigroups from different angles, we are able to see two completely different views of the object. And yet amazingly, both views describe exactly the same thing. This is the power of abstract algebra at work. We can prove things about semigroups, mentally reasoning about them as nonempty lists, and as long as in practice we only use the semigroup laws, we will get the same results about semigroups as graphs for free! Including the semigroup of function compositions that we saw in the last section, which is seemingly entirely unrelated to the semigroups we are looking at now. In fact, they are one and the same!
This relationship can be daunting. How can we conceptualize that they have the same structure?
I think it is best to simply not try and find deeper meaning here. We’ve seen that what matters is the properties that they have in common. In both cases, we have a set, and a binary operation which is total and associative. This is all the relationship that they need.
Rather than be confused how they can be the same, I think it suffices to see this core property that they share and simply sigh in awe.
I’m going to leave this post off here to avoid it getting too long. In the next post we will take a brief look at unital magma and quasigroups. Then we’ll get to the interesting stuff that comes up in programming and in life all the time - monoids and groups.
It is really important that we show the models contain the same information. We could have two different models of a structure which do not contain the same information, and that difference in information might make the properties true in one model but false in the other! ↩
The worst offender for newcomers is \(Hom(A, B)\) for \(A \to B\). This comes from category theory and I won’t be using it here. But if you do see it in the wild, mentally rewrite it to \(A \to B\). ↩
Essentially, there’s a problem with the way we hand-wavily defined a set as a collection of anything. We could define a set as “the collection of all objects with some property,” for example all numbers that are natural, or all integers divisible by 3. But even worse, we can define sets of sets in some ways that are straight-up impossible. The quintessential example is called Russel’s Paradox. The set of “all sets that do not contain themselves” cannot exist. If it doesn’t contain itself, it should. But if it contains itself, then it shouldn’t! This was a major problem in the formalization of math, and modern set theory systems define sets in a way that prevents this. But category theorsists really want to be able to talk about categories of categories in the most generic way possible. To solve this problem, we distinguish between “small” categories, whose arrows (and objects) form sets in the modern set-theory way, and “large” categories, whose arrows (and objects) are just defined by some property. The latter collections are called “proper classes,” and generally a “class” is a collection of things which are not classes. All this to say, the word “small” here solves a problem that we didn’t even have to think about, so I’m just going to drop it in this post. ↩
The graph model is actually the standard model of categories. Categories are usually defined as having two carrier sets, the graph nodes and the graph arrows, considered separately. But for our purposes it’s really enough to consider just the arrows and assume that they point between distinguishable things. Either way, if you had a different model of a category, it would be possible to transform it to the graph model and only lose information, not gain anything. Category theorists say the graph model is “universal,” but in our language, we’ll say graphs are “the free categories on graph nodes.” It will be a while before we get into free structures, but it is one of my eventual goals in this post category. ↩
Does that sound suspiciously like a category? Because it is! ↩
From a programming perspective, this relationship is what lets us turn programs (strings of words) into richer parse trees. The programming language dictates what additional non-associative tree structure to imbue the text with, but the original program needn’t care. It’s just a list! ↩
Is that because it truly is scary, or simply because they didn’t have access to good resources to learn it? We’re going to find out!
In this mini-series I hope to approach Abstract Algebra from a simple angle, in terms of motivating examples and building up desirable properties. There will be little-to-no advanced math here. Equational reasoning skills will help, as may some previous exposure to proofs. This post is aimed at hobbyist or experienced programmers with a less formal mathematics background, or at interested pre-university mathematicians. Many motivating examples will be in terms of programming.
This long introductory post aims to introduce the general concept of abstract algebra and motivate why we would want to use it.
With all that out of the way, let’s get into the meat of it!
Indeed, why math? Mathematics gives us the tools to talk about various patterns we encounter every day. Algebra (the non-abstract kind) gives us the tools to talk about everyday constraints, like “What time do I have to leave my house to be at work at 9:00?” It can also talk about much more complicated, but still everyday problems, related to money, logistics, cooking, and just about anywhere else you might encounter an equals sign.
Arithmetic gives us the tools to talk about various types of counting. For example, I know I have 12 cookies and 3 friends, and I want to count how many cookies each friend can get.
Calculus gives us the tools to talk about rates and totals. How much does the circumference of a circle increase if you change the radius? What happens if you total up the lengths of the circumferences for circles of various radii?
Geometry lets us talk about shapes without (necessarily) worrying about manipulating numbers.
All of these things let us talk about concrete things, which we can touch and move around, or at least construct physical examples. Abstract algebra is tricky to view this way, because it is abstract by nature. Abstract algebra gives us the tools to talk about structure - what happens when we enforce certain rules on mathematical objects.
But what does that even mean? Consider the natural numbers, \(\mathbb N\). If we don’t give them structure, we have a collection of weird symbols like \(3\) and \(42\) that we can’t manipulate. But we can give them a structure by declaring that they have an order. \(2\) comes after \(1\), \(3\) comes after \(2\), etc. Is there something special about this structure in particular? No, there isn’t. We could easily have put a different number first, for example. Or we could use completely different symbols, like emoji. Or why even stop at symbols - a stack of pennies has the same structure!
So what we really gain from abstract algebra is the language to talk about three things:
The last one is where I find the real beauty in abstract algebra, and it lets us tie all sorts of different types of math together.
Now that we know what we’re talking about at a high level, let’s get a bit more specific. Obviously, as programmers or as mathematicians, we see these types of structures all the time. Programmers will recognize the stack of pennies as a linked list (or perhaps some other kind of list). But we can also talk about other shapes of structures entirely, so let’s get a bit more precise.
Before we can talk about something have structure, we need to have, well, something. In our running example, the “something” is either \(\mathbb N\) or a collection of pennies. In either case, we have a collection of things - a set.
Every “algebraic structure” is going to start with a set. This set is called the domain of the structure, or the carrier set. There are structures that have more than one domain, but we’re not going to talk about those in this post. In this post, we will only discuss single-domain structures.
Then, we put structure on the set by defining (arbitrarily!) a relationship between the objects in the set. For our “natural number structure,” that relationship is the followed-by relation. \(2\) is followed by \(1\), etc. Now we ask ourselves what properties this structure has. We could list a few:
In fact, these 3 properties exactly describe our structure. Here, we’re using our language in way #1 from above - we’re talking about a specific structure.
Since these 3 properties exactly describe our structure, we should be able to state the same 3 properties for stacks of pennies:
These are the same 3 rules, just stated with slightly different wording. And they describe exactly the same structure, merely with a different carrier set.
That means that anything we prove about stacks of pennies immediately applies to natural numbers as well - or to anything else with this structure. Thus what really matters is the properties that define our structure, and not the structure itself. Let’s rephrase our properties in a more general way.
Now we refer to some generic “domain,” but we don’t actually care what it is. We only care that it has the structure dictated by these properties. These properties are commonly known as the “Peano axioms”^{1}.
This structure is generically called the natural numbers, whether we use \(\mathbb N\) as the carrier set or stacks of pennies. I want to emphasize that our conceptual understanding of the structure comes from the properties, and not from the carrier set.
The technical designation for this structure comes from the following facts about it:
So we would call the natural numbers a “pointed unary system.”
We’ve now taken a step away from concrete objects, and described how we talk about two things having the same structure. We define the structure’s properties, and show that both things satisfy those properties. We call these properties the axioms of the structure. When something concrete satisfies the axioms of a structure, we call it a model of that structure. It’s also pretty common to just call it the structure itself, so the set of pennies along with the “add a penny to the stack” operation may just be called the “penny system,” even though it’s the same system as the natural numbers.
Whenever we prove something generic about the axioms, we’ve also proved that thing about every model of the axioms. Let’s see an example.
The Peano Axioms we’ve been looking at describe counting. \(\sigma\), conceptually, is “count to the next number.” It turns out that anything with this structure also has a much more interesting structure. Using counting, we can describe arithmetic.
Let’s propose some additional properties for addition:
Let’s suppose we have any model \((D, 0, \sigma)\) of the Peano axioms. This notation means that the carrier set is \(D\), along with the designated value \(0\) and the operation \(\sigma\) for counting. We can define addition as follows:
\[\begin{align} a + 0 &= a, \\ a + \sigma(b) &= \sigma(a + b). \end{align}\]Now we want to check that it satisfies the properties we asked for. First, let’s suppose \(x,y \in D\). One of the two patterns above will fit \(x + y\), so we definitely have a definition for \(x + y\). Since (by axiom 1) \(\sigma(a + b)\) will always again be a natural number, we have that \(x + y\) does really exist.
What about the second rule, which is usually called commutivity?
First, let’s show that \(a + 0 = 0 + a\). If \(a = 0\), we’re done. So let’s assume that \(a \neq 0\).
\[\begin{aligned} 0 + a &= 0 + \sigma(b) & \text{By axiom 3} \\ &= \sigma(0 + b) & \text{By the second pattern} \\ &= \sigma(b + 0) & \text{Explained below} \\ &= \sigma(b) & \text{By the first pattern} \\ &= a & \text{By definition of } b\\ &= a + 0 & \text{By the first pattern, applied in reverse} \end{aligned}\]What about that step in the middle, where we used the fact that \(\sigma(0 + b) = \sigma(b + 0)\)? Doesn’t that rely on what we’re trying to prove? Well it does - but \(b\) is closer to \(0\) in the \(\sigma\)-chain than \(a\) is. So we could “unroll” the proof by writing out the steps for \(b\), and for \(b-1\), until we eventually terminate at \(0\). We know that we will eventually terminate, because our Axiom 3, or the uniqueness axiom, tells us that eventually, we will have to reach 0, and we know that our claim holds for 0. Formally, this reasoning is called induction.^{2}
To prove the general claim of \(x + y = y + x\), we would use a similar inductive step to show that \(x + \sigma(y) = \sigma(x) + y\), and then we would be able to “move” layers of \(\sigma\) from one side to the other until we get the balance correct. The details are left as an exercise.
I’ll also leave proving the 3rd property of addition, associativity, as an exercise. It’s less annoying than commutativity.
So we’ve shown that any system that supports counting also supports adding. Why is that important?
Addition is a binary operator. It takes two arguments, both natural numbers in this case, and produces a third natural number. Addition puts a much richer structure on the domain. Instead of only being able to say that \(3 \to 4\), \(4 \to 5\), and \(5 \to 6\), we can now make much broader, more comprehensive statements like \((3, 4) \to 7\). And this doesn’t only apply to numbers, but to any set that it makes sense to “count.” That’s a very powerful result!
Some of the richest - and most practical - structures arise from single-domain, binary-operator systems. Mathematicians define a whole hierarchy of such structures, starting with no properties at all (a set with no operation) and gradually increasing the requirements of the operation to become more and more specific. The significant parts of this hierarchy look like this.
The structure we’ve been talking about here is a commutative monoid. As we continue to develop the language of abstract algebra, we will learn conceptually exactly what that means.
Hopefully I’ve piqued your interest! In the next few posts we will describe each type of structure in the diagram and the arrows that it points to. That is, I want to describe each structure in terms of what it adds to something that we are already familiar with. As we get further down the line, we will see how identifying these types of structures can improve programs, clarify ideas, and help us make deep observations about the nature of (mathematical) existence.
For simplicity, I’ve combined the fact that \(0\) exists with its special property. But technically these are two separate properties. If we don’t dictate that \(0\) must exist, then the empty set would satisfy all of the other properties! So the Peano axioms actually have 4 properties, with the first one being “0 is a natural number.” ↩
Typically, induction would be stated as a Peano Axiom. But the typical statements are all very difficult to parse and not conceptually enlightening. However, my handwavy description of “unrolling” the proof can be formalized. It turns out that defining that 0 is the “the smallest natural number,” in the way that we did, is equivalent to giving induction as an axiom. The details of the proof are outside the scope of this post. ↩