---
title: "Implementing and Verifying \"Static Program Analysis\" in Agda, Part 6: Control Flow Graphs"
series: "Static Program Analysis in Agda"
description: "In this post, I show an Agda definition of Control Flow Graphs and their construction"
date: 2024-11-27T16:26:42-07:00
tags: ["Agda", "Programming Languages"]
---

In the previous section, I've given a formal definition of the programming
language that I've been trying to analyze. This formal definition serves
as the "ground truth" for how our little imperative programs are executed;
however, program analyses (especially in practice) seldom take the formal
semantics as input. Instead, they focus on more pragmatic program
representations from the world of compilers. One such representation are
_Control Flow Graphs (CFGs)_. That's what I want to discuss in this post.

Let's start by building some informal intuition. CFGs are pretty much what
their name suggests: they are a type of [graph](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics));
their edges show how execution might jump from one piece of code to
another (how control might flow).

For example, take the below program.


```
x = ...;
if x {
  x = 1;
} else {
  x = 0;
}
y = x;
```

The CFG might look like this:

{{< figure src="if-cfg.png" label="CFG for simple `if`-`else` code." class="small" >}}

Here, the initialization of `x` with `...`, as well as the `if` condition (just `x`),
are guaranteed to execute one after another, so they occupy a single node. From there,
depending on the condition, the control flow can jump to one of the
branches of the `if` statement: the "then" branch if the condition is truthy,
and the "else" branch if the condition is falsy. As a result, there are two
arrows coming out of the initial node. Once either branch is executed, control
always jumps to the code right after the `if` statement (the `y = x`). Thus,
both the `x = 1` and `x = 0` nodes have a single arrow to the `y = x` node.

As another example, if you had a loop:

```
x = ...;
while x {
  x = x - 1;
}
y = x;
```

The CFG would look like this:

{{< figure src="while-cfg.png" label="CFG for simple `while` code." class="small" >}}

Here, the condition of the loop (`x`) is not always guaranteed to execute together
with the code that initializes `x`. That's because the condition of the loop
is checked after every iteration, whereas the code before the loop is executed
only once. As a result, `x = ...` and `x` occupy distinct CFG nodes. From there,
the control flow can proceed in two different ways, depending on the value
of `x`. If `x` is truthy, the program will proceed to the loop body (decrementing `x`).
If `x` is falsy, the program will skip the loop body altogether, and go to the
code right after the loop (`y = x`). This is indicated by the two arrows
going out of the `x` node. After executing the body, we return to the condition
of the loop to see if we need to run another iteration. Because of this, the
decrementing node has an arrow back to the loop condition.

Now, let's be a bit more precise. Control Flow Graphs are defined as follows:

* __The nodes__ are [_basic blocks_](https://en.wikipedia.org/wiki/Basic_block).
  Paraphrasing Wikipedia's definition, a basic block is a piece of code that
  has only one entry point and one exit point.

  The one-entry-point rule means that it's not possible to jump into the middle
  of the basic block, executing only half of its instructions. The execution of
  a basic block always begins at the top. Symmetrically, the one-exit-point
  rule means that you can't jump away to other code, skipping some instructions.
  The execution of a basic block always ends at the bottom.

  As a result of these constraints, when running a basic block, you are
  guaranteed to execute every instruction in exactly the order they occur in,
  and execute each instruction exactly once.
* __The edges__ are jumps between basic blocks. We've already seen how
  `if` and `while` statements introduce these jumps.

Basic blocks can only be made of code that doesn't jump (otherwise,
we violate the single-exit-point policy). In the previous post,
we defined exactly this kind of code as [simple statements]({{< relref "05_spa_agda_semantics#introduce-simple-statements" >}}).
So, in our control flow graph, nodes will be sequences of simple statements.
{#list-basic-stmts}

### Control Flow Graphs in Agda

#### Basic Definition
At an abstract level, it's easy to say "it's just a graph where X is Y" about
anything. It's much harder to give a precise definition of such a graph,
particularly if you want to rule out invalid graphs (e.g., ones with edges
pointing nowhere). In Agda, I chose the represent a CFG with two lists: one of nodes,
and one of edges. Each node is simply a list of `BasicStmt`s, as
I described in a preceding paragraph. An edge is simply a pair of numbers,
each number encoding the index of the node connected by the edge.

Here's where it gets a little complicated. I don't want to use plain natural
numbers for indices, because that means you can easily introduce "broken" edge.
For example, what if you have 4 nodes, and you have an edge `(5, 5)`? To avoid
this,  I picked the finite natural numbers represented by
[`Fin`](https://agda.github.io/agda-stdlib/v2.0/Data.Fin.Base.html#1154)
as endpoints for edges.

```Agda
data Fin : ℕ → Set where
  zero : Fin (suc n)
  suc  : (i : Fin n) → Fin (suc n)
```

Specifically, `Fin n` is the type of natural numbers less than `n`. Following
this definition, `Fin 3` represents the numbers `0`, `1` and `2`. These are
represented using the same constructors as `Nat`: `zero` and `suc`. The type
of `zero` is `Fin (suc n)` for any `n`; this makes sense because zero is less
than any number plus one. For `suc`, the bound `n` of the input `i` is incremented
by one, leading to another `suc n` in the final type. This makes sense because if
`i < n`, then `i + 1 < n + 1`. I've previously explained this data type
[in another post on this site]({{< relref "01_aoc_coq#aside-vectors-and-finite-mathbbn" >}}).

Here's my definition of `Graph`s written using `Fin`:

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 24 39 >}}

I explicitly used a `size` field, which determines how many nodes are in the
graph, and serves as the upper bound for the edge indices. From there, an
index `Index` into the node list is
{{< sidenote "right" "size-note" "just a natural number less than `size`," >}}
Ther are <code>size</code> natural numbers less than <code>size</code>:<br>
<code>0, 1, ..., size - 1</code>.
{{< /sidenote >}}
and an edge is just a pair of indices. The graph then contains a vector
(exact-length list) `nodes` of all the basic blocks, and then a list of
edges `edges`.

There are two fields here that I have not yet said anything about: `inputs`
and `outputs`. When we have a complete CFG for our programs, these fields are
totally unnecessary. However, as we are _building_ the CFG, these will come
in handy, by telling us how to stitch together smaller sub-graphs that we've
already built. Let's talk about that next.

#### Combining Graphs
Suppose you're building a CFG for a program in the following form:

```
code1;
code2;
```

Where `code1` and `code2` are arbitrary pieces of code, which could include
statements, loops, and pretty much anything else. Besides the fact that they
occur one after another, these pieces of code are unrelated, and we can
build CFGs for each one them independently. However, the fact that `code1` and
`code2` are in sequence means that the full control flow graph for the above
program should have edges going from the nodes in `code1` to the nodes in `code2`.
Of course, not _every_ node in `code1` should have such edges: that would
mean that after executing any "basic" sequence of instructions, you could suddenly
decide to skip the rest of `code1` and move on to executing `code2`.

Thus, we need to be more precise about what edges we need to insert; we want
to insert edges between the "final" nodes in `code1` (where control ends up
after `code1` is finished executing) and the "initial" nodes in `code2` (where
control would begin once we started executing `code2`). Those are the `outputs`
and `inputs`, respectively. When stitching together sequenced control graphs,
we will connect each of the outputs of one to each of the inputs of the other.

This is defined by the operation `g₁ ↦ g₂`, which sequences two graphs `g₁` and `g₂`:

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 72 83 >}}

The definition starts out pretty innocuous, but gets a bit complicated by the
end. The sum of the numbers of nodes in the two operands becomes the new graph
size, and the nodes from the two graphs are all included in the result. Then,
the definitions start making use of various operators like `↑ˡᵉ` and `↑ʳᵉ`;
these deserve an explanation.

The tricky thing is that when we're concatenating lists of nodes, we are changing
some of the indices of the elements within. For instance, in the lists
`[x]` and `[y]`, the indices of both `x` and `y` are `0`; however, in the
concatenated list `[x, y]`, the index of `x` is still `0`, but the index of `y`
is `1`. More generally, when we concatenate two lists `l1` and `l2`, the indices
into `l1` remain unchanged, whereas the indices `l2` are shifted by `length l1`.
{#fin-reindexing}

Actually, that's not all there is to it. The _values_ of the indices into
the left list don't change, but their types do! They start as `Fin (length l1)`,
but for the whole list, these same indices will have type `Fin (length l1 + length l2))`.

To help deal with this, Agda provides the operators
[`↑ˡ`](https://agda.github.io/agda-stdlib/v2.0/Data.Fin.Base.html#2355)
and [`↑ʳ`](https://agda.github.io/agda-stdlib/v2.0/Data.Fin.Base.html#2522)
that implement this re-indexing and re-typing. The former implements "re-indexing
on the left" -- given an index into the left list `l1`, it changes its type
by adding the other list's length to it, but keeps the index value itself
unchanged. The latter implements "re-indexing on the right" -- given an index
into the right list `l2`, it adds the length of the first list to it (shifting it),
and does the same to its type.

The definition leads to the following equations:

```Agda
l1 : Vec A n
l2 : Vec A m

idx1 : Fin n -- index into l1
idx2 : Fin m -- index into l2

l1 [ idx1 ] ≡ (l1 ++ l2) [ idx1 ↑ˡ m ]
l2 [ idx2 ] ≡ (l1 ++ l2) [ n ↑ʳ idx2 ]
```

The operators used in the definition above are just versions of the same
re-indexing operators. The `↑ˡᵉ` operator applies `↑ˡ` to all the (__e__)dges
in a graph, and the `↑ˡⁱ` applies it to all the (__i__)ndices in a list
(like `inputs` and `outputs`).

Given these definitions, hopefully the intent with the rest of the definition
is not too hard to see. The edges in the new graph come from three places:
the graph `g₁` and `g₂`, and from creating a new edge from each of the outputs
of `g₁` to each of the inputs of  `g₂`. We keep the inputs of `g₁` as the
inputs of the whole graph (since `g₁` comes first), and symmetrically we keep
the outputs of `g₂`. Of course, we do have to re-index them to keep them
pointing at the right nodes.

Another operation we will need is "overlaying" two graphs: this will be like
placing them in parallel, without adding jumps between the two. We use this
operation when combining the sub-CFGs of the "if" and "else" branches of an
`if`/`else`, which both follow the condition, and both proceed to the code after
the conditional.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 59 70 >}}

Everything here is just concatenation; we pool together the nodes, edges,
inputs, and outputs, and the main source of complexity is the re-indexing.

The one last operation, which we will use for `while` loops, is looping. This
operation simply connects the outputs of a graph back to its inputs (allowing
looping), and also allows the body to be skipped. This is slightly different
from the graph for `while` loops I showed above; the reason for that is that
I currently don't include the conditional expressions in my CFG. This is a
limitation that I will address in future work.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 85 95 >}}

Given these thee operations, I construct Control Flow Graphs as follows, where
`singleton` creates a new CFG node with the given list of simple statements:

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 122 126 >}}

Throughout this, I've been liberal to include empty CFG nodes as was convenient.
This is a departure from the formal definition I gave above, but it makes
things much simpler.

### Additional Functions

To integrate Control Flow Graphs into our lattice-based program analyses, we'll
need to do a couple of things. First, upon reading the
[reference _Static Program Analysis_ text](https://cs.au.dk/~amoeller/spa/),
one sees a lot of quantification over the predecessors or successors of a
given CFG node. For example, the following equation is from Chapter 5:

{{< latex >}}
\textit{JOIN}(v) = \bigsqcup_{w \in \textit{pred}(v)} \llbracket w \rrbracket
{{< /latex >}}

To compute the \(\textit{JOIN}\) function (which we have not covered yet) for
a given CFG node, we need to iterate over all of its predecessors, and
combine their static information using \(\sqcup\), which I first
[explained several posts ago]({{< relref "01_spa_agda_lattices#least-upper-bound" >}}).
To be able to iterate over them, we need to be able to retrieve the predecessors
of a node from a graph!

Our encoding does not make computing the predecessors particularly easy; to
check if two nodes are connected, we need to check if an `Index`-`Index` pair
corresponding to the nodes is present in the `edges` list. To this end, we need
to be able to compare edges for equality. Fortunately, it's relatively
straightforward to show that our edges can be compared in such a way;
after all, they are just pairs of `Fin`s, and `Fin`s and products support
these comparisons.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 149 152 >}}

Next, if we can compare edges for equality, we can check if an edge is in
a list. Agda provides a built-in function for this:

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 154 154 >}}

To find the predecessors of a particular node, we go through all other nodes
in the graph and see if there's an edge there between those nodes and the
current one. This is preferable to simply iterating over the edges because
we may have duplicates in that list (why not?).

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 165 166 >}}

Above, `indices` is a list of all the node identifiers in the graph. Since the
graph has `size` nodes, the indices of all these nodes are simply the values
`0`, `1`, ..., `size - 1`. I defined a special function `finValues` to compute
this list, together with a proof that this list is unique.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 127 143 >}}

Another important property of `finValues` is that each node identifier is
present in the list, so that our computation written by traversing the node
list do not "miss" nodes.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 145 147 >}}

We can specialize these definitions for a particular graph `g`:

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 156 163 >}}

To recap, we now have:
* A way to build control flow graphs from programs
* A list (unique'd and complete) of all nodes in the control flow graph so that
  we can iterate over them when the algorithm demands.
* A 'predecessors' function, which will be used by our static program analyses,
  implemented as an iteration over the list of nodes.

All that's left is to connect our `predecessors` function to edges in the graph.
The following definitions say that when an edge is in the graph, the starting
node is listed as a predecessor of the ending node, and vise versa.

{{< codelines "Agda" "agda-spa/Language/Graphs.agda" 168 177 >}}

### Connecting Two Distinct Representations

I've described Control Flow Graphs as a compiler-centric representation of the
program. Unlike the formal semantics from the previous post, CFGs do not reason
about the dynamic behavior of the code. Instead, they capture the possible
paths that execution can take through the instructions. In that
sense, they are more of an approximation of what the program will do. This is
good: because of [Rice's theorem](https://en.wikipedia.org/wiki/Rice%27s_theorem),
we can't do anything other than approximating without running the program.

However, an incorrect approximation is of no use at all. Since the CFGs we build
will be the core data type used by our program analyses, it's important that they
are an accurate, if incomplete, representation. Specifically, because most
of our analyses reason about possible outcomes --- we report what sign each
variable __could__ have, for instance --- it's important that we don't accidentally
omit cases that can happen in practice from our CFGs. Formally, this means
that for each possible execution of a program according to its semantics,
{{< sidenote "right" "converse-note" "there exists a corresponding path through the graph." >}}
The converse is desirable too: that the graph has only paths that correspond
to possible executions of the program. One graph that violates this property is
the strongly-connected graph of all basic blocks in a program. Analyzing
such a graph would give us an overly-conservative estimation; since anything
can happen, most of our answers will likely be too general to be of any use. If,
on the other hand, only the necessary graph connections exist, we can be more
precise.<br>
<br>
However, proving this converse property (or even stating it precisely) is much
harder, because our graphs are somewhat conservative already. There exist
programs in which the condition of an <code>if</code>-statement is always
evaluated to <code>false</code>, but our graphs always have edges for both
the "then" and "else" cases. Determining whether a condition is always false
(e.g.) is undecidable thanks to Rice's theorem (again), so we can't rule it out.
Instead, we could broaden "all possible executions"
to "all possible executions where branching conditions can produce arbitrary
results", but this is something else entirely.<br>
<br>
For the time being, I will leave this converse property aside. As a result,
our approximations might be "too careful". However, they will at the very least
be sound.
{{< /sidenote >}}

In the next post, I will prove that this property holds for the graphs shown
here and the formal semantics I defined earlier. I hope to see you there!
