Tuples and Memory Allocation
1 Computing with arbitrary-sized data
2 Adding pairs to our language
2.1 Syntax and examples
2.2 Representing pairs
2.2.1 Introducing pointers
2.2.2 Tagging pair values
2.2.3 Allocating space on the heap
2.3 ANF transformations
2.4 Compiling pairs and pair accesses
3 Generalizing to tuples
3.1 Syntax and examples
3.2 Representing tuples
3.3 Compiling tuples and tuple accesses
4 Memory Allocation

Tuples and Memory Allocation

Based on these notes by Ben Lerner. Small edits, bug fix regarding heap initialization, changes related to concrete syntax, ANF, and simplified to skip static typing.

1 Computing with arbitrary-sized data

So far, our compiler supports a language with arbitrarily complex arithmetic and logical expressions, function definitions, and function applications. However, our only values are two primitives: integers and booleans. To make our language more fully-featured, we need the ability to handle structured, arbitrarily sized data.

At a high level, we know that there are three kinds of composite data:

The first kind of composite data really only manifests itself in the type system, where we need to be able to say, “These values are part of this enumeration, while those values are not.” Since enumerated data is merely a finite set of constants, we can simulate them merely by using some distinct set of integers.

Since union data is essentially an enumeration of structured data, if we had the ability to create structured data types, then we could encode union data types by adding a field to each value identifying which variant of the union this value belongs to.

But structured data does not admit such a simple encoding. It requires the ability to group several values into one, and use that compound thing as a value: it should be created by some expressions, used by others, and passed to and returned from functions. In other words, structured data is a fundamentally new thing in our language.

We can consider various kinds of structured data such as records with named fields (as in C structs or Racket define-structs), or indexed data like arrays. But the simplest form of structured data is the humble pair, and its bigger cousins the tuples.

Standard reminder: Every time we enhance our source language, we need to consider several things:

  1. Its impact on the concrete syntax of the language

  2. Examples using the new enhancements, so we build intuition of them

  3. Its impact on the abstract syntax and semantics of the language

  4. Any new or changed transformations needed to process the new forms

  5. Executable tests to confirm the enhancement works as intended

2 Adding pairs to our language

2.1 Syntax and examples

We’ll write pairs as

‹expr›: ... | ( cons ‹expr› ‹expr› )

For example, (cons 4 5) is a pair consisting of two numbers. The flexibility of our grammar, though, allows us to nest pairs inside one another, as in (cons 3 (cons (or true false) false)) or (cons (cons (cons 1 2) 3) (+ 4 5)), or any other combination we might attempt. However, in order to use our pairs, we need to be able to access their components. We’ll do so with two new primitive operators

‹expr›: ... | ( fst ‹expr› ) | ( snd ‹expr› )

The intended semantics are that these two primitives project out the first or second components of the pairs: (fst (cons 3 4)) == 3 and (snd (cons 4 (cons true 5))) == (cons true 5).

We can add these expression forms to our AST easily enough, modelling the new accessors as unary primitives:

type expr = ...
  | Pair of expr * expr
type prim1 = ... | Fst | Snd

2.2 Representing pairs

Introducing pairs into our language is a much more invasive change than it might appear. For the first time, we have values in our language that are too big to fit into a register—after all, each component of a pair filled a register on its own! So where can we put them? One natural thought might be simply to put the two halves of the pair adjacent to each other in two consecutive stack slots, and perhaps some careful bookkeeping might remember that those two slots “go together”. But this fails for several reasons:

Instead, we’re going to have to look elsewhere to store our pairs: we’re going to use the heap.

2.2.1 Introducing pointers

The heap is another region of memory, at numerically-smaller addresses than our stack, where we can potentially allocate space for whatever values we want. We’ll defer for a moment how exactly to do that, but suppose that we can. Then for each pair we have to evaluate, we can request space for two consecutive words of memory, and store the two halves of the pair in those two words. Crucially for us, the address of (the first of) those two words is unique: no other pair will exist at that exact address. And, even better, one address is small enough to fit into a register. We can use this address as the representation of pairs: it is small enough to pass into and out of functions, and small enough to be placed into the components of another pair. By adding this indirection, we’ve solved all three of the failings above.

2.2.2 Tagging pair values

Of course, now that we have a new kind of value, we need a tag to go with it to distinguish pairs from booleans or integers. If we use three bits for tagging, we have 4 tags available (recall that all even numbers are encoding integers), of which we use one for booleans, so we have three tags remaining for marking pairs. We can arbitrarily choose one of them, say 0x1, to mark our pointers as pairs. This seems risky, though — unlike integers and booleans, whose representations we could choose for our own convenience, memory addresses are interpreted by the computer, so we can’t capriciously change bits in them! However, we only need three bits, which means if we can ensure that our values always end with three known bits, then we can carefully replace our tag bits with those three known bits before asking the computer to reinterpret our value as a pointer. (Note carefully the emphasis in wording here: a given 64-bit quantity is a value in our language while it is tagged, and becomes interpretable as a pointer when we replace the tag bits with whatever those bits should have been.)

The easiest set of known bits to work with is 0b000. Can we ensure that every memory address we allocate always ends in three zero bits? Yes! “Ends with three zero bits” is the same thing as saying “multiple of 8” Conveniently, our pairs are two words wide, which means they’re exactly 16 bytes long. If we can ensure that every allocation we make is aligned to an 8-byte boundary, then all our addresses will end in 0x000, which means we are free to use those three bits for whatever tag we need.

2.2.3 Allocating space on the heap

Where can we actually obtain memory from? One possibility is “just use malloc”, but that just defers the problem. Besides, we have no guarantees that malloc will enforce our 8-byte-alignment constraint. Instead, we’ll use malloc once, to obtain a big buffer of space to use, and we’ll manipulate that buffer directly from our assembly. (See below for how malloc actually gets itself started.)

Here is one possible strategy for handling memory; there are many others. Let’s devote one register, R15, to always store a pointer to the next available heap location. Then “allocating a pair” amounts to storing two values into [R15] and [R15 + 8], and then incrementing R15 by 16. (We’ll ignore out-of-memory errors and garbage collection, for now.) Once again, if R15 started off as a multiple of 8, then after this process it remains a multiple of 8, so our alignment invariant still holds.

All that remains is to initialize R15 appropriately, which requires collaboration between main and our_code_starts_here. We need to allocate a buffer of memory from main, using calloc to allocate-and-clear the memory, and then pass that pointer in to our code. In other words, our_code_starts_here now really is a function that takes in arguments—in this case, the “start of the heap”.

int main() {
  uint64_t* HEAP = calloc(1024, sizeof(uint64_t)); // Allocate 8KB of memory for now
  uint64_t result = our_code_starts_here(HEAP);
  return 0;

Do Now!

Why must the call to free(HEAP) happen after the call to print(result)?

On the callee side, in our_code_starts_here, we need to store this provided address into R15, but we first need to ensure that it is a multiple of 8.1Multiples of 8 are actually guaranteed for us, but it is worthwhile to ensure this for ourselves, especially if for some reason we need a more stringent alignment guarantee later on. In particular, we need to round the address up to the nearest multiple of 8 (because rounding down might give us an unallocated address). The easiest way to achive this is to add 7 to the address, then round down to the nearest multiple of 8:

  ...                           ;; basic function prologue as usual
  mov R15, RDI                  ;; Load R15 with the passed-in pointer
  add R15, 7                    ;; \ add 7 to get above the next multiple of 8
  mov R11, 0xfffffffffffffff8L  ;; | (R11 is just some scratch register)
  and R15, R11                  ;; / then round back down.

2.3 ANF transformations

If you are using ANF, you need to update the transformation to account for pairs.

Given that evaluating a pair actually performs a memory allocation, we cannot treat pairs as immediate values: the value simply isn’t immediately ready. Obviously we do want to be able to bind them to variables, so they must be compound expressions. This leads to a simple design:

type 'a cexpr =
  | CPair of 'a immexpr * 'a immexpr * 'a

We force the two components of the pair to be immediate, so that the only computational step happening here is the memory allocation itself.

2.4 Compiling pairs and pair accesses

We now have all the tools we need to generate code for all our pair-related expressions. To construct a pair,

... assume the two parts of the pair are already evaluated ...
mov [R15],     <first part of pair>
mov [R15 + 8], <second part of pair>
mov RAX, R15   ;; Start creating the pair value itself
add RAX, 0x1   ;; tag the pair
add R15, 16    ;; bump the heap pointer

The order of execution here is important: we must fully evaluate the two parts of the pair prior to creating the pair, or else the evaluation of each component might modify R15, leading to non-consecutive memory addresses for our pair. (ANF conversion ensures this for us; if you don’t use ANF, then make sure to fully evaluate the pair component expresssions first.) Next we must save the current value of R15 as our result RAX, so that we can tag it correctly.

To access the first element of a pair,

mov RAX, <the pair value>
<check that RAX is indeed a pair>
sub RAX, 0x1       ;; untag it
mov RAX, [RAX + 0] ;; treat RAX as a pointer, and get its first word

Accessing the second element uses mov RAX, [RAX + 8] instead.

3 Generalizing to tuples

Tuples are simply longer pairs. We could consider representing tuples as a linked-list of pairs, but that would be quite inefficient. Instead, we’ll generalize everything above that hard-coded “two elements” to become a list of elements. This has some consequences for our representation, but it mostly goes through smoothly.

3.1 Syntax and examples

We’ll write tuples as

‹expr›: ... | ( tup ‹expr› ... ‹expr› )

A tuple can have zero or more fields, as in (tup 3 4), (tup), or (tup true false 5). Accessing elements of a tuple can’t be restricted to two unary primitive operators now, because we don’t know in advance how large our tuples will be. Instead, we’ll add a more general expression

‹expr›: ... | ( get ‹expr› ‹expr› )

The first expression must evaluate to a tuple, and the second to a number that denotes the position we want to access. Of course, in order to ensure safety, we’ll need to check at runtime that the index is a positive number no greater than the size of the tuple.

We’ll represent tuples and tuple accesses as

type expr =
  | Tuple of expr list (* components of the tuple *)
  | Get of expr * expr (* tuple, desired index *)

3.2 Representing tuples

We can’t merely store all the items of the tuple consecutively in memory; we need to know how many items there are. Accordingly, we’ll add a header word at the beginning of our tuple in memory, that will store the size of the tuple:


Note carefully that the first word is an actual integer; it is not an encoded value in our language.

Unlike pairs, when we allocate tuples we might easily violate our alignment invariant. In fact, simply allocating a pair will do so, because we now need three words in total: the header word plus the two component words. We can resolve this in one of two ways: we could either add a padding word at the end of our tuple to bring it back into alignment if necessary, or we could increment R15 once more to bring it back into alignment.

Do Now!

What are the possible tradeoffs of these two approaches? Are there any observable differences between them, given our language so far?

Again, we need to tag our tuple values, just as we did above for pairs. Since tuples generalize pairs, the simplest approach is to eliminate pairs as a special case, and just use tuples everywhere — and then reuse the 0x1 tag to now mean tuples. It’s important to note that we could use both representations in our compiler — but if we use two distinct representations, we need to use two different tag values to distinguish them!

Do Now!

What are the possible tradeoffs of using just one representation for all tuples, vs using two representations for pairs and tuples separately?

3.3 Compiling tuples and tuple accesses

Again we generalize, assuming that all parts of the tuple have been evaluated.

... ;; assume the parts of the tuple are already evaluated
mov [R15 + 0], <size of the tuple>
mov [R15 + 8 * 1], <first part of tuple>
mov [R15 + 8 * 2], <second part of tuple>
mov [R15 + 8 * n], <last part of tuple>
mov RAX, R15           ;; Start creating the tuple value itself
add RAX, 0x1           ;; tag the tuple
add R15, 8 * (n + 1)   ;; bump the heap pointer
add R15, <padding>     ;; realign the heap pointer if needed

To implement a tuple access, again we generalize, being careful to account for the header word:

mov RAX, <the tuple value>
<check that RAX is indeed a tuple>
sub RAX, 0x1                 ;; untag it
cmp n, 0                     ;; \ make sure the index
jl  index_too_low            ;; / is non-negative
cmp n, [RAX]                 ;; \ make sure the index is
jge index_too_high           ;; / within the size of the tuple
mov RAX, [RAX + 8 * (n + 1)] ;; treat RAX as a pointer, and get its nth word

4 Memory Allocation

If we’re using R15 to store the memory address where our available heap space begins, where do we get its initial value from?

1Multiples of 8 are actually guaranteed for us, but it is worthwhile to ensure this for ourselves, especially if for some reason we need a more stringent alignment guarantee later on.