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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -106,4 +106,5 @@ Here are the posts that I’ve written so far for this series:
* {{< draftlink "Our Programming Language" "05_spa_agda_semantics" >}}
* {{< draftlink "Control Flow Graphs" "06_spa_agda_cfg" >}}
* {{< draftlink "Connecting Semantics and Control Flow Graphs" "07_spa_agda_semantics_and_cfg" >}}
* {{< draftlink "A Verified Forward Analysis" "08_spa_forward" >}}
* {{< draftlink "Forward Analysis" "08_spa_forward" >}}
* {{< draftlink "Verifying the Forward Analysis" "09_spa_verified_forward" >}}

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@ -79,6 +79,7 @@ a less specific output! The more you know going in, the more you should know
coming out. Similarly, when given less specific / vaguer information, the
analysis shouldn't produce a more specific answer -- how could it do that?
This leads us to come up with the following rule:
{#define-monotonicity}
{{< latex >}}
\textbf{if}\ \text{input}_1 \le \text{input}_2,

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---
title: "Implementing and Verifying \"Static Program Analysis\" in Agda, Part 8: Forward Analysis"
series: "Static Program Analysis in Agda"
description: "In this post, I use the monotone lattice framework and verified CFGs to define a sign analysis"
date: 2024-12-01T15:09:07-08:00
tags: ["Agda", "Programming Languages"]
draft: true
---
In the previous post, I showed that the Control Flow graphs we built of our
programs match how they are really executed. This means that we can rely
on these graphs to compute program information. In this post, we finally
get to compute that information. Let's jump right into it!
### Choosing a Lattice
A lot of this time, we have been [talking about lattices]({{< relref "01_spa_agda_lattices" >}}),
particularly [lattices of finite height]({{< relref "03_spa_agda_fixed_height" >}}).
These structures represent things we know about the program, and provide operators
like \((\sqcup)\) and \((\sqcap)\) that help us combine such knowledge.
The forward analysis code I present here will work with any finite-height
lattice, with the additional constraint that equivalence of lattices
is decidable, which comes from [the implementation of the fixed-point algorithm]({{< relref "04_spa_agda_fixedpoint" >}}),
in which we routinely check if a function's output is the same as its input.
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 4 8 >}}
The finite-height lattice `L` is intended to describe the state of a single
variable.
One example of a lattice that can be used as `L` is our
sign lattice. We've been using the sign lattice in our examples [from the very beginning]({{< relref "01_spa_agda_lattices#lattices" >}}),
and we will stick with it for the purposes of this explanation. However, this
lattice alone does not describe our program, since it only talks about a single
sign; programs have lots of variables, all of which can have different signs!
So, we might go one step further and define a map lattice from variables to
their signs:
{{< latex >}}
\text{Variable} \to \text{Sign}
{{< /latex >}}
We [have seen]({{< relref "02_spa_agda_combining_lattices#the-map-lattice" >}})
that we can turn any lattice \(L\) into a map lattice \(A \to L\), for any
type of keys \(A\). In this case, we will define \(A \triangleq \text{Variable}\),
and \(L \triangleq \text{Sign}\). The
[sign lattice has a finite height]({{< relref "02_spa_agda_combining_lattices#the-map-lattice" >}}),
and I've proven that, as long as we pick a finite set of keys, [map lattices
\(A \to L\) have a finite height if \(L\) has a finite height]({{< relref "03_spa_agda_fixed_height#fixed-height-of-the-map-lattice" >}}).
Since a program's text is finite, \(\text{Variable}\) is a finite set, and
we have ourselves a finite-height lattice \(\text{Variable} \to \text{Sign}\).
We're on the right track, but even the lattice we have so far is not sufficient.
That's because variables have different signs at different points in the program!
You might initialize a variable with `x = 1`, making it positive, and then
go on to compute some arbitrary function using loops and conditionals. For
each variable, we need to keep track of its sign at various points in the code.
When we [defined Control Flow Graphs]({{< relref "06_spa_agda_cfg" >}}), we
split our programs into sequences of statements that are guaranteed to execute
together --- basic blocks. For our analysis, we'll keep per-variable for
each basic block in the program. Since basic blocks are nodes in the Control Flow
Graph of our program, our whole lattice will be as follows:
{{< latex >}}
\text{Info} \triangleq \text{NodeId} \to (\text{Variable} \to \text{Sign})
{{< /latex >}}
We follow the same logic we just did for the variable-sign lattice; since
\(\text{Variable} \to \text{Sign}\) is a lattice of finite height, and since
\(\text{NodeId}\) is a finite set, the whole \(\text{Info}\) map will be
a lattice with a finite height.
Notice that both the sets of \(\text{Variable}\) and \(\text{NodeId}\) depend
on the program in question. The lattice we use is slightly different for
each input program! We can use Agda's parameterized modules to automaitcally
parameterize all our functions over programs:
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 36 37 >}}
Now, let's make the informal descriptions above into code, by instantiating
our map lattice modules. First, I invoked the code for the smaller variable-sign
lattice. This ended up being quite long, so that I could rename variables I
brought into scope. I will collapse the relevant code block; suffice to say
that I used the suffix `v` (e.g., renaming `_⊔_` to `_⊔ᵛ_`) for properties
and operators to do with variable-sign maps (in Agda: `VariableValuesFiniteMap`).
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 41 82 "" "**(Click here to expand the module uses for variable-sign maps)**" >}}
I then used this lattice as an argument to the map module again, to
construct the top-level \(\text{Info}\) lattice (in Agda: `StateVariablesFiniteMap`).
This also required a fair bit of code, most of it to do with renaming.
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 85 112 "" "**(Click here to expand the module uses for the top-level lattice)**" >}}
### Constructing a Monotone Function
We now have a lattice in hand; the next step is to define a function over
this lattice. For us to be able to use the fixed-point algorithm on this
function, it will need to be [monotonic]({{< relref "01_spa_agda_lattices#define-monotonicity" >}}).
Our goal with static analysis is to compute information about our program; that's
what we want the function to do. When the lattice we're using is the sign lattice,
we're trying to determine the signs of each of the variables in various parts
of the program. How do we go about this?
Each piece of code in the program might change a variable's sign. For instance,
if `x` has sign \(0\), and we run the statement `x = x - 1`, the sign of
`x` will be \(-\). If we have an expression `y + z`, we can use the signs of
`y` and `z` to compute the sign of the whole thing. This is a form
of [abstract interpretation](https://en.wikipedia.org/wiki/Abstract_interpretation),
in which we almost-run the program, but forget some details (e.g., the
exact values of `x`, `y`, and `z`, leaving only their signs). The exact details
of how this partial evaluation is done are analysis-specific; in general, we
simply require an analysis to provide an evaluator. We will define
[an evaluator for the sign lattice below](#instantiating-with-the-sign-lattice).
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 166 167 >}}
From this, we know how each statement and basic block will change variables
in the function. But we have described them process as "if a variable has
sign X, it becomes sign Y" -- how do we know what sign a variable has _before_
the code runs? Fortunately, the Control Flow Graph tells us exactly
what code could be executed before any given basic block. Recall that edges
in the graph describe all possible jumps that could occur; thus, for any
node, the incoming edges describe all possible blocks that can precede it.
This is why we spent all that time [defining the `predecessors` function]({{< relref "06_spa_agda_cfg#additional-functions" >}}).
We proceed as follows: for any given node, find its predecessors. By accessing
our \(\text{Info}\) map for each predecessor, we can determine our current
best guess of variable signs at that point, in the form of a \(\text{Variable} \to \text{Sign}\)
map (more generally, \(\text{Variable} \to L\) map in an arbitrary analysis).
We know that any of these predecessors could've been the previous point of
execution; if a variable `x` has sign \(+\) in one predecessor and \(-\)
in another, it can be either one or the other when we start executing the
current block. Early on, we saw that [the \((\sqcup)\) operator models disjunction
("A or B")]({{< relref "01_spa_agda_lattices#lub-glub-or-and" >}}). So, we apply
\((\sqcup)\) to the variable-sign maps of all predecessors. The
[reference _Static Program Analysis_ text](https://cs.au.dk/~amoeller/spa/)
calls this operation \(\text{JOIN}\):
{{< latex >}}
\textit{JOIN}(v) = \bigsqcup_{w \in \textit{pred}(v)} \llbracket w \rrbracket
{{< /latex >}}
The Agda implementation uses a `foldr`:
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 139 140 >}}
Computing the "combined incoming states" for any node is a monotonic function.
This follows from the monotonicity of \((\sqcup)\) --- in both arguments ---
and the definition of `foldr`.
{{< codelines "agda" "agda-spa/Lattice.agda" 143 151 "" "**(Click here to expand the general proof)**" >}}
From this, we can formally state that \(\text{JOIN}\) is monotonic. Note that
the input and output lattices are different: the input lattice is the lattice
of variable states at each block, and the output lattice is a single variable-sign
map, representing the combined preceding state at a given node.
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 145 149 >}}
Above, the `m₁≼m₂⇒m₁[ks]≼m₂[ks]` lemma states that for two maps with the same
keys, where one map is less than another, all the values for any subset of keys
`ks` are pairwise less than each other (i.e. `m₁[k]≼m₂[k]`, and `m₁[l]≼m₂[l]`, etc.).
This follows from the definition of "less than" for maps.
{#less-than-lemma}
So those are the two pieces: first, join all the preceding states, then use
the abstract interpretation function. I opted to do both of these in bulk:
1. Take an initial \(\text{Info}\) map, and update every basic block's entry
to be the join of its predecessors.
2. In the new joined map, each key now contains the variable state at
the beginning of the block; so, apply the abstract interpretation function
via `eval` to each key, computing the state at the end of the block.
I chose to do these in bulk because this way, after each application of
the function, we have updated each block with exactly one round of information.
The alternative --- which is specified in the reference text --- is to update
one key at a time. The difference there is that updates to later keys might be
"tainted" by updates to keys that came before them. This is probably fine
(and perhaps more efficient, in that it "moves faster"), but it's harder to
reason about.
#### Generalized Update
To implement bulk assignment, I needed to implement the source text's
Exercise 4.26:
> __Exercise 4.26__: Recall that \(f[a \leftarrow x]\) denotes the function that is identical to
> \(f\) except that it maps \(a\) to \(x\). Assume \(f : L_1 \to (A \to L_2)\)
> and \(g : L_1 \to L_2\) are monotone functions where \(L_1\) and \(L_2\) are
> lattices and \(A\) is a set, and let \(a \in A\). (Note that the codomain of
> \(f\) is a map lattice.)
>
> Show that the function \(h : L_1 \to (A \to L_2)\)
> defined by \(h(x) = f(x)[a \leftarrow g(x)]\) is monotone.
In fact, I generalized this statement to update several keys at once, as follows:
{{< latex >}}
h(x) = f(x)[a_1 \leftarrow g(a_1, x),\ ...,\ a_n \leftarrow g(a_n, x)]
{{< /latex >}}
I called this operation "generalized update".
At first, the exercise may not obviously correspond to the bulk operation
I've described. Particularly confusing is the fact that it has two lattices,
\(L_1\) and \(L_2\). In fact, the exercise results in a very general theorem;
we can exploit a more concrete version of the theorem by setting
\(L_1 \triangleq A \to L_2\), resulting in an overal signature for \(f\) and \(h\):
{{< latex >}}
f : (A \to L_2) \to (A \to L_2)
{{< /latex >}}
In other words, if we give the entire operation in Exercise 4.26 a type,
it would look like this:
{{< latex >}}
\text{ex}_{4.26} : \underbrace{K}_{\text{value of}\ a} \to \underbrace{(\text{Map} \to V)}_{\text{updater}} \to \underbrace{\text{Map} \to \text{Map}}_{f} \to \underbrace{\text{Map} \to \text{Map}}_{h}
{{< /latex >}}
That's still more general than we need it. This here allows us to modify
any map-to-map function by updating a certain key in that function. If we
_just_ want to update keys (as we do for the purposes of static analysis),
we can recover a simpler version by setting \(f \triangleq id\), which
results in an updater \(h(x) = x[a \leftarrow g(x)]\), and a signature for
the exercise:
{{< latex >}}
\text{ex}_{4.26} : \underbrace{K}_{\text{value of}\ a} \to \underbrace{(\text{Map} \to V)}_{\text{updater}\ g} \to \underbrace{\text{Map}}_{\text{old map}} \to \underbrace{\text{Map}}_{\text{updated map}}
{{< /latex >}}
This looks just like Haskell's [`Data.Map.adjust` function](https://hackage.haskell.org/package/containers-0.4.0.0/docs/src/Data-Map.html#adjust), except that it
can take the entire map into consideration when updating a key.
My generalized version takes in a list of keys to update, and makes the updater
accept a key so that its behavior can be specialized for each entry it changes.
The sketch of the implementation is in the `_updating_via_` function from
the `Map` module, and its helper `transform`. Here, I collapse its definition,
since it's not particularly important.
{{< codelines "agda" "agda-spa/Lattice/Map.agda" 926 931 "" "**(Click here to see the definition of `transform`)**" >}}
The proof of monotonicity --- which is the solution to the exercise --- is
actually quite complicated. I will omit its description, and show it here
in another collapsed block.
{{< codelines "agda" "agda-spa/Lattice/Map.agda" 1042 1105 "" "**(Click here to see the proof of monotonicity of \(h\))**" >}}
Given a proof of the exercise, all that's left is to instantiate the theorem
with the argument I described. Specifically:
* \(L_1 \triangleq \text{Info} \triangleq \text{NodeId} \to (\text{Variable} \to \text{Sign})\)
* \(L_2 \triangleq \text{Variable} \to \text{Sign} \)
* \(A \triangleq \text{NodeId}\)
* \(f \triangleq \text{id} \triangleq x \mapsto x\)
* \(g(k, m) = \text{JOIN}(k, m)\)
In the equation for \(g\), I explicitly insert the map \(m\) instead of leaving
it implicit as the textbook does. In Agda, this instantiation for joining
all predecessor looks like this (using `states` as the list of keys to update,
indicating that we should update _every_ key):
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 152 157 >}}
And the one for evaluating all programs looks like this:
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 215 220 >}}
Actually, we haven't yet seen that `updateVariablesFromStmt`. This is
a function that we can define using the user-provided abtract interpretation
`eval`. Specifically, it handles the job of updating the sign of a variable
once it has been assigned to (or doing nothing if the statement is a no-op).
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 191 193 >}}
The `updateVariablesFromExpression` is now new, and it is yet another map update,
which changes the sign of a variable `k` to be the one we get from running
`eval` on it. Map updates are instances of the generalized update; this
time, the updater \(g\) is `eval`. The exercise requires the updater to be
monotonic, which constrains the user-provided evaluation function to be
monotonic too.
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 173 181 >}}
We finally write the `analyze` function as the composition of the two bulk updates:
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 226 232 >}}
### Instantiating with the Sign Lattice
Thus far, I've been talking about the sign lattice throughout, but implementing
the Agda code in terms of a general lattice `L` and evaluation function `eval`.
In order to actually run the Agda code, we do need to provide an `eval` function,
which implements the logic we used above, in which a zero-sign variable \(x\)
minus one was determined to be negative. For binary operators specifically,
I've used the table provided in the textbook; here they are:
{{< figure src="plusminus.png" caption="Cayley tables for abstract interpretation of plus and minus" >}}
These are pretty much common sense:
* A positive plus a positive is still positive, so \(+\ \hat{+}\ + = +\)
* A positive plus any sign could be any sign still, so \(+\ \hat{+}\ \top = \top\)
* Any sign plus "impossible" is impossible, so \(\top\ \hat{+} \bot = \bot\).
* etc.
The Agda encoding for the plus function is as follows, and the one for minus
is similar.
{{< codelines "agda" "agda-spa/Analysis/Sign.agda" 76 94 >}}
As the comment in the block says, it would be incredibly tedious to verify
the monotonicity of these tables, since you would have to consider roughly
125 cases _per argument_: for each (fixed) sign \(s\) and two other signs
\(s_1 \le s_2\), we'd need to show that \(s\ \hat{+}\ s_1 \le s\ \hat{+}\ s_2\).
I therefore commit the _faux pas_ of using `postulate`. Fortunately, the proof
of monotonicity is not used for the execution of the program, so we will
get away with this, barring any meddling kids.
From this, all that's left is to show that for any expression `e`, the
evaluation function:
{{< latex >}}
\text{eval} : \text{Expr} \to (\text{Variable} \to \text{Sign}) \to \text{Sign}
{{< /latex >}}
is monotonic. It's defined straightforwardly and very much like an evaluator /
interpreter, suggesting that "abstract interpretation" is the correct term here.
{{< codelines "agda" "agda-spa/Analysis/Sign.agda" 176 184 >}}
Thought it won't happen, it was easier to just handle the case where there's
an undefined variable; I give it "any sign". Otherwise, the function simply
consults the sign tables for `+` or `-`, as well as the known signs of the
variables. For natural number literals, it assigns `0` the "zero" sign, and
any other natural number the "\(+\)".
To prove monotonicity, we need to consider two variable maps (one less than
the other), and show that the abstract interpretation respects that ordering.
This boils down to the fact that the `plus` and `minus` tables are monotonic
in both arguments (thus, if their sub-expressions are evaluated monotonically
given an environment, then so is the whole addition or subtraction), and
to the fact that for two maps `m₁ ≼ m₂`, the values at corresponding keys
are similarly ordered: `m₁[k] ≼ m₂[k]`. We [saw that above](#less-than-lemma).
{{< codelines "agda" "agda-spa/Analysis/Sign.agda" 186 223 "" "**(Click to expand the proof that the evaluation function for signs is monotonic)**" >}}
That's all we need. With this, I just instantiate the `Forward` module we have
been working with, and make use of the `result`. I also used a `show`
function (which I defined) to stringify that output.
{{< codelines "agda" "agda-spa/Analysis/Sign.agda" 225 229 >}}
But wait, `result`? We haven't seen a result just yet. That's the last piece,
and it involves finally making use of the fixed-point algorithm.
### Invoking the Fixed Point Algorithm
Our \(\text{Info}\) lattice is of finite height, and the function we have defined
is monotonic (by virtue of being constructed only from map updates, which
are monotonic by Exercise 4.26, and from function composition, which preserves
monotonicity). We can therefore apply the fixed-point-algorithm, and compute
the least fixed point:
{{< codelines "agda" "agda-spa/Analysis/Forward.agda" 235 238 >}}
With this, `analyze` is the result of our forward analysis!
In a `Main.agda` file, I invoked this analysis on a sample program:
```Agda
testCode : Stmt
testCode =
⟨ "zero" ← (# 0) ⟩ then
⟨ "pos" ← ((` "zero") Expr.+ (# 1)) ⟩ then
⟨ "neg" ← ((` "zero") Expr.- (# 1)) ⟩ then
⟨ "unknown" ← ((` "pos") Expr.+ (` "neg")) ⟩
testProgram : Program
testProgram = record
{ rootStmt = testCode
}
open WithProg testProgram using (output; analyze-correct)
main = run {0ℓ} (putStrLn output)
```
The result is verbose, since it shows variable signs for each statement
in the program. However, the key is the last basic block, which shows
the variables at the end of the program. It reads:
```
{"neg" ↦ -, "pos" ↦ +, "unknown" ↦ ⊤, "zero" ↦ 0, }
```
### Verifying the Analysis
We now have a general framework for running forward analyses: you provide
an abstract interpretation function for expressions, as well as a proof
that this function is monotonic, and you get an Agda function that takes
a program and tells you the variable states at every point. If your abstract
interpretation function is for determining the signs of expressions, the
final result is an analysis that determines all possible signs for all variables,
anywhere in the code. It's pretty easy to instantiate this framework with
another type of forward analysis --- in fact, by switching the
`plus` function to one that uses `AboveBelow ℤ`, rather than `AboveBelow Sign`:
```Agda
plus : ConstLattice → ConstLattice → ConstLattice
plus ⊥ᶜ _ = ⊥ᶜ
plus _ ⊥ᶜ = ⊥ᶜ
plus ⊤ᶜ _ = ⊤ᶜ
plus _ ⊤ᶜ = ⊤ᶜ
plus [ z₁ ]ᶜ [ z₂ ]ᶜ = [ z₁ Int.+ z₂ ]ᶜ
```
we can defined a constant-propagation analysis.
However, we haven't proved our analysis correct, and we haven't yet made use of
the CFG-semantics equivalence that we
[proved in the previous section]({{< relref "07_spa_agda_semantics_and_cfg" >}}).
I was hoping to get to it in this post, but there was just too much to
cover. So, I will get to that in the next post, where we will make use
of the remaining machinery to demonstrate that the output of our analyzer
matches reality.

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