Dedicated to the Minnowbrook Analytic Reasoning Seminar with special thanks to Kris Micinski and Michael Ballantyne
Suppose you want to write a database. You'd probably start by
implementing relational algebra operators — projection, filter, join,
etc. The easy way is to implement them as functions that take in tables
and return tables, and assemble them into a larger expression. That was
how Prela worked in its first
incarnation. The code was clean, but it was hella slow! Which was not
surprising, because every operator materialized every intermediate
result. The standard solution to this is the iterator
model, where each operator implements an Iterator interface
that streams intermediate tables row by row instead of materializing
them. But implementing the iterator model naively still incurs overhead:
every call to Iterator.next()
triggers a dynamic dispatch,
which costs vtable lookups and destroys cache locality. There are two
standard remedies: vectorization
and compilation. A vectorized database amortizes the overhead by
implementing Iterator.next_batch()
which returns a whole batch of data that can be processed together; a compiled database, well, compiles the incoming query directly to fast machine code that runs without any dynamic dispatch. Either approach takes a lot of very smart people spending their entire working life to build, and it's why systems like DuckDB and Umbra exist. I'm moderately smart but don't have a lot of time, so I was looking for a shortcut. The shortcut I stumbled upon was so beautiful that I literally cried 1 when I finally understood it, and I hope my explanation below will make you cry too :' )
To keep things simple, let's suppose we're just dealing with lists of
numbers, and we want to do two very simple things to them:
inc
adds 1 to every number, and dbl
doubles them. That's pretty easy to write:2
inc(xs) = [x + 1 for x in xs]
dbl(xs) = [2 * x for x in xs]
Now, we can chain them together with dbl(inc(xs))
which
will do two steps in sequence. Problem is, because each function takes
in a list and returns a list, our program produces an
intermediate, namely inc(xs)
. This allocates a new
list only to be thrown away by the call to dbl
. Things only
gets worse when we chain together multiple calls to inc
and
dbl
. A more efficient implementation would fuse together the operations:
inc_n_dbl(xs) = [2 * (x + 1) for x in xs]
Of course, we can't write down every possible combination of operators like this. Is there a way to define each operator modularly, yet still have them compose into tightly fused operations automatically? Yes, if we use a bit of magic from functional compilers — continuation-passing style (CPS).
The key idea of CPS is to define operators that do things
instead of making things. inc
and dbl
as defined above each takes in a list and makes a list.
Instead, the CPS version of each operator takes in a list and an
additional input k
: this k
is a function that
the caller passes in, specifying what it wants to do with each element
after the operator's work is done. k
is called the continuation. Let's look at some code:
function inc(xs, k)
for x in xs
k(x + 1)
end
end
Now suppose k
is the print
function, then
inc
as defined above will add 1 to each number, then print
the result. Note that nothing is returned, and inc
only
does its job (adding 1) then performs what it's told to (apply
k
). As an exercise, you can try and write down
dbl
in CPS style.
But currently each of inc
and dbl
still
takes in a list, and there's no obvious way to compose multiple
operators. To do that, we replace xs
with a "child"
operator op
:
inc(op, k) = op(x -> k(x + 1))
dbl(op, k) = op(x -> k(x * 2))
function scan(xs, k)
for x in xs
k(x)
end
end
Intuitively, inc
now trusts its child op
to
do its job, namely, that op
will apply the continuation it
receives to each item. So instead of iterating over xs
,
inc
simply tags the + 1
step onto the
continuation and passes it to op
. I've also defined a
"source" operator scan
that connects the input list to the operators. Let's see the code in action.
inc(scan(xs), print)
.inc
, this will call
scan(xs, x -> print(x + 1))
scan
, this gets us
for x in xs; print(x + 1); end
So chaining together inc
and scan
indeed
does what we want! Now let's try a longer chain
dbl(inc(scan(xs)), print)
:
dbl
gets us
inc(scan(xs), x -> print(x * 2))
inc
gets us
scan(xs, x -> print((x + 1) * 2))
scan
gets us
for x in xs; print((x + 1) * 2); end
Notice how I used the word expand
— if we annotate every
operator definition with @inline
, the compiler will
actually unfold the code as we did above, and an operator chain gets
compiled down to a fused loop in the end! You can try expanding longer
chains like dbl(inc(dbl(inc(scan(xs)))), print)
to get some
practice thinking about CPS. Julia also has handy tools like
@code_typed
that lets you inspect the compiled code, or the aptly named Cthulhu.jl that does that interactively. In summary, the example shows that if we define operators modularly with CPS, inlining the definitions will automatically produce tightly fused compiled code.
None of these is really new, and have been known to functional programmers for decades by the name of deforestation. But when implemented in Prela, something incredible happens: a clean CPS-style interpreter for Prela automagically recovers fast columnar execution when compiled!
The central design principle behind Prela is "everything is a
binary relation". This means Prela maps cleanly to both a
logical Entity/Relationship data model, as well as to a columnar
physical storage. I won't go into details here, but encourage you to
play with the language to get a feel for that. Practically, this means
we fully normalize every wide table with m attributes to m binary
relations. For example, a table movie
with columns
ID, year, title
becomes:
movie
) over
ID
(you can think of this as a unary table over
ID
)ID
to year
ID
to title
But now the issue is, even for a simple
SELECT * FROM movie
we need to join together 3 different tables! Whereas a column store would simply run:
for i in 0:n
print(id_col[i], year_col[i], title_col[i])
end
In other words, the column store co-iterates the columns in one pass to compute the query.
Let's first look at how Prela used to run this query. The most important operator in Prela is the relation composition \rightarrow, which generalizes function composition the same way relations generalize functions. In standard relational algebra: R \rightarrow S = \pi_{x, z}(R \Join_{R.y = S.y} S) where R's schema is over x and y, and S's schema is over y and z. The second most important Prela operator is the product \times which takes two binary relations and joins them: R \times S = R \Join_{R.x = S.x} S where R's schema is over x, y, and S's schema is over x and z.
So SELECT * FROM movie
is spelled \text{movie} \rightarrow \text{year} \times
\text{title} in Prela. Now, this requires first joining
year
with title
, whose result is joined with
movie
. We can make this a bit cheaper if we can assume the
primary key ID
s are dense and continuous, in which case we
can just store year
as an array of integers and
title
as an array of strings; and for ID
s, we
only need to store one single number n
which says how many IDs there are. But even with this, we still need to do the work to join the tables.
Instead, let's define compose and product in CPS:
compose(lhs, rhs, k) = lhs((x, y) -> rhs(y, (z -> k(x, z))))
product(lhs, rhs, x, k) = lhs(x, (y -> rhs(x, (z -> k((y, z))))))
scan_id(n, k) = for i in 0:n; k(i, i); end
probe(col, i, k) = k(col[i])
Let's go over each line carefully. Semantically compose
is supposed to return a (binary) relation by composing the
lhs
and rhs
relations. In CPS, its job is to
apply k
to every pair in this composition. Its first
argument, lhs
, represents a binary relation and applies the
given continuation to every pair. The rhs
is a little
different: it represents a relation that supports lookup, i.e.,
rhs(key, k)
will look up the values associated with
key
, then apply k
to each such value. Now
going back to compose
— we're saying that, for each
(x, y)
tuple in the LHS, we will look up y
from the RHS, then for each matching z
, we apply
k(x, z)
. In loops this will be:
for (x, y) in lhs
for z in rhs[y]
k(x, z)
end
end
Which is exactly a hash join and a projection that throws away
y
.
Next, product
itself is a relation supporting lookup,
and so are its arguments. To lookup x
in a product, we
first look it up from the lhs
which gets us a bunch of
y
s. Then for each y
, we now look up
x
from the rhs
, getting a bunch of
z
s. Finally, for each (y, z)
pair, we apply
the continuation k((y, z))
. In loops:
for y in lhs[x]
for z in rhs[x]
k((y, z))
end
end
This is what will happen if you look up x
in R \Join_{R.x = S.x} S.
Finally, we have the "source" operators scan_id
and
probe
. scan_id
is scan
but
specialized for a dense ID relation where we only store n
:
it simply increments i
from 0 to n
and applies
k
to (i, i)
. probe
represents an
input relation that supports looking up a primary key i
and
which is backed by a dense vector, so looking up an ID i
simply indexes col[i]
and applies k
to the value.
We're now ready to put everything together and pull the trigger: the
Prela query \text{movie} \rightarrow
\text{year} \times \text{title} desugars to
compose(scan_id(n), product(probe(year), probe(title)))
where n
is the number of movies. Here are the definitions again for reference:
compose(lhs, rhs, k) = lhs((x, y) -> rhs(y, (z -> k(x, z))))
product(lhs, rhs, x, k) = lhs(x, (y -> rhs(x, (z -> k((y, z))))))
scan_id(n, k) = for i in 0:n; k(i, i); end
probe(col, i, k) = k(col[i])
Expand compose
:
scan_id(n, (x, y) -> product(probe(year), probe(title), y, (z -> k(x, z))))
Expand scan_id
:
for i in 0:n
product(probe(year), probe(title), i, (z -> k(i, z)))
end
Expand product
:
for i in 0:n
probe(year, i, (y -> probe(title, i, (z -> k(i, (y, z))))))
end
Expand probe
:
for i in 0:n
k(i, (year[i], title[i]))
end
And taking k = print
, we finally have:
for i in 0:n
print(i, (year[i], title[i]))
end
Spectacular!!
To keep the examples small, I've made several simplifications. The
actual Prela implementation defines two methods drive
and
probe
for each operator which fire depending on how the operator is accessed — scanned or probed. But that's pretty much it, and the complete source is around 1000 lines of Julia code in a single file, supporting select, project, join, groupby, aggregation, CTEs, UDFs, and with performance matching DuckDB on TPCH and Join Order Benchmark. I should note that the performance numbers are riding on lots of assumptions though, the strongest ones being:
Nevertheless, the CPS approach cleanly separates the responsibility of the data engine — which is to map queries to efficient code in some target language — and the responsibility of the compiler which is to produce fast code quickly, and we see there's a lot of room for the compiler to improve. The assumption that PKs are dense can also be relaxed with the help of B-trees, bitmap filters, and other data structures.
But perhaps the biggest strength of the CPS style is that it makes Prela extensible, as users can write their own operator in a natural way with a few lines of code, with the assurance that it will be compiled and fused with the rest of the query for efficient execution.
Long before AI psychosis, there was FP psychosis, clinically defined as the intense psychological response to understanding functional programming concepts like recursion, higher order functions, monads, or in this case, continuation passing style.↩︎
All code in this post is in Julia.↩︎
scan(xs)
stands for
k -> scan(xs, k)
, i.e., it is the curried application of
scan
to xs
. Similarly for
inc(scan(xs))
below which curries inc
with
scan(xs)
.↩︎