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# Proving the correctness of a binary search procedure with SPARK/Ada Riccardo Bernardini
I graduated in 1990 in Electrical Engineering and since then I have been in university, doing research in the field of DSP. To me programming is more a tool than a job.
Updated on ・9 min read

# Introduction

SPARK/Ada is a language derived from Ada that allows for a formal checking (i.e., mathematically prove the correctness) of the software. Several types of checks can be done: from the absence of runtime exceptions to no use of uninitialized variables, up to the formal proof that a given procedure/function fulfills its contract.

I like formal checking because it can actually prove that your program is correct (or that, at least, some kind of errors are absent), something that is usually not possible with testing. Of course, formal checking cannot applied to every program, but when it is possible it is a very powerful tool.

SPARK/Ada is definitively on my list of stuff that I want to learn.

Recently I had to write a procedure for a binary search in an ordered array. I thought that it could be an interesting exercise to write it in SPARK/Ada in order to have it formally verified. This is a brief summary of this experience, written with a tutorial spirit.

# The requirements

The problem I want to solve is the following

• INPUTS
• `Table`: an array of `Float` sorted in strictly increasing order (that is, `Table(n+1) > Table(n)` for every `n`).
• `What` : a `Float` such that
• it is included between the array extremes, that is, `Table(Table'First) ≤ What ≤ Table(Table'Last)`, but
• it is not necessarily in `Table`, that is it is not required that there is `n` such that `Table(n) = What`
• OUTPUTS
• The unique index `n` such that
``````Table(n) ≤ What < Table(n+1)
``````

# The solution

## The spec file

### First basic draft of specs

Let's first flesh out the spec file (roughly [very roughly] similar to a `*.h` file for you C people).

``````pragma SPARK_Mode (On);

package Searching is
subtype Element_Type is Float;
subtype Index_Type is Natural;

type Array_Type is
array (Index_Type range <>) of Element_Type;

function Find (What  : Element_Type;
Table : Array_Type)
return Index_Type;
end Searching;
``````

### Entering SPARK Mode

The first line we encounter is

``````pragma SPARK_Mode (On);
``````

This claims that this package will be compatible with SPARK restrictions and conventions. The lines

``````   subtype Element_Type is Float;
subtype Index_Type is Natural;
``````

define `Element_Type` and `Index_Type` as synominous of `Float` and `Index_Type`, respectively. This is not strictly necessary, but it can make it easier to change the types in a future

Function declaration

`````` function Find (What  : Element_Type;
Table : Array_Type)
return Index_Type;
``````

should be clear enough.

### Introducing contracts

In order for SPARK to be able to proof that our implementation of `Find` is correct, we need to describe to SPARK what we expect `Find` to do. This is done by means of a contract. Like a normal contract between people, a function contract usually has two parts: preconditions and postconditions.

The idea is that if you (the caller) do your part (i.e. you meet the preconditions when you call me), then I, the function, promise to do my part, that is, that the result will satisfy the post-conditions.

If a contract is given, I can ask SPARK to prove that the post-conditions follow from the pre-conditions. If SPARK succeeds, I know that the code I wrote is correct (in the sense that it respects the contract) without doing any test: I have a mathematical proof for it.

BTW, contracts are very useful even if you are not using SPARK. First, they are a wonderful "bug trap." By using the right options, you can ask the compiler to insert code that checks the pre-conditions when the function is called and the post-conditions when the function ends. If pre/post-conditions are not met, an exception is raised.

Moreover, contracts document in a formal and unambiguous way the behavior of the function and, differently from usual comment-based documentation, cannot go out of sync with the code.

OK, so contracts are cool. How do we write the contract for our function? Well, let's start with the precondition and let's check the specs. It is said that (i) `Table` must be sorted and (ii) `What` must be between `Table` extrema; we just translate that in Ada in this way

``````    function Find (What  : Element_Type;
Table : Array_Type)
return Index_Type
with
Pre =>
Is_Sorted (Table)
and Table (Table'Last) >= What
and What >= Table (Table'First);
``````

where `Is_Sorted` is a function that checks if its argument is sorted and defined as follows

``````function Is_Sorted (Table : Array_Type) return Boolean
is (for all L in Table'Range =>
(for all M in Table'Range =>
(if L > M then Table (L) > Table (M))))
with Ghost;
``````

The body of the function (this form is called expression function) is the translation in Ada of the definition of monotonic array.

Note the `with Ghost` in the definition. This says that `Is_Sorted` is a ghost function, that is, a function that can be used only in contracts or assertions; if used in other places ("real code") an error is raised. (I love this kind of protections ♥)

The preconditions look right and they represent the actual specs, but if we try to run SPARK we get an error

`medium: array index check might fail`

at line

``````      and What >= Table (Table'First);
``````

This means that `Table'First`, that is the first index of the array, can be ... outside array bounds? The first index is by definition within the bounds, right?

Well... Actually, SPARK is right. If you ask for a counterexample you get

``````e.g. when Table'First=1 and Table'Last=0
``````

Now, when in the world can be possible that the last index is smaller than the first one?!? Well, if `Table` is empty... Empty arrays in Ada have the last index smaller than the first one. Therefore, trying to access `Table (Table'First)` would cause an error.
Well, SPARK, good catch... Well, I agree (although it could happen because of an error), but SPARK does not know it. To make SPARK happy it suffices to add `Table'Length > 0` to the precondition to get

``````    function Find (What  : Element_Type;
Table : Array_Type)
return Index_Type
with
Pre => Table'Length > 0
and then (Is_Sorted (Table)
and Table (Table'Last) >= What
and What >= Table (Table'First));
``````

Now SPARK is happy and even the casual user sees that you cannot call the function with an empty table.

#### The postcondition

Also for the post-conditions we translate directly the requirement from English to Ada. It is required that `What` is between `Table(n)` and `Table(n+1)` where `n` is the value returned by the function. In Ada we get

``````         (if Find'Result = Table'Last then
Table (Find'Result) = What
else
Table (Find'Result) <= What
and What < Table (Find'Result + 1));
``````

Note that `Find'Result` is the value returned by `Find` and that we consider as special case `Find'Result = Table'Last` since in this case there is no "following entry" in `Table`. Overall, the function declaration with the full contract is

``````   function Find (What  : Element_Type;
Table : Array_Type)
return Index_Type
with
Pre =>
Table'Length > 0
and then (Is_Sorted (Table)
and Table (Table'Last) >= What
and What >= Table (Table'First)),
Post =>
(if Find'Result = Table'Last then
Table (Find'Result) = What
else
Table (Find'Result) <= What
and What < Table (Find'Result + 1));
``````

## The body file (implementation)

### The algorithm

The algorithm for a binary search is well known and exemplified in the following picture Basically, we keep two "cursors" in the table: `Bottom` and `Top` with the condition that it must always be

``````Table(Bottom) ≤ What < Top(top)
``````

We get the `Middle` point between `Top` and `Bottom` and if `Table(Middle)` is too large we move `Top` to `Middle`, otherwise we move `Bottom`. We iterate this until on of the following two conditions holds

• `Table(Bottom) = What`, that is, we actually found `What` in `Table`
• `Top = Bottom+1` there are no intermediate entries between `Top` and `Bottom`

In both cases we return `Bottom`

### The actual code

Let's start with a basic skeleton of the procedure.

``````function Find (What : Element_Type; Table : Array_Type) return Index_Type
is
Bottom : Index_Type;
Top    : Index_Type;
Middle : Index_Type;
begin
if Table (Table'Last) = What then
return Table'Last;
end if;

Bottom := Table'First;
Top    := Table'Last;

pragma Assert (Table (Top) > What and What >= Table(Bottom));

while Table (Bottom) < What and Top - Bottom > 1 loop
Middle := (Bottom + Top)/2;

if Table (Middle) > What then
Top := Middle;
else
Bottom := Middle;
end if;
end loop;

return Bottom;
end Find;

``````

It is just a translation of the algorithm informally described above. Note how the special case `Table (Table'Last) = What` is handled separately; in this way we know that

``````Table (Top) > What and What >= Table(Bottom)
``````

holds (see the `pragma Assert`).

Now, in order to make it easier for SPARK to prove the correctness of the code we insert in the code some `pragma Assert` claiming properties that hold true in different points of the code.

Automatic theorem proving is not easy and some hint to the prover can help. Moreover, I love spreading generously my code with assertion since they help understanding what the code does and which properties I expect to be true. They are also formidable "bug traps".

The function body with all the assertions looks like

``````function Find (What : Element_Type; Table : Array_Type) return Index_Type
is
Bottom : Index_Type;
Top    : Index_Type;
Middle : Index_Type;
begin
if Table (Table'Last) = What then
return Table'Last;
end if;

pragma Assert (Table (Table'Last) > What);

Bottom := Table'First;
Top    := Table'Last;

pragma Assert (Bottom < Top);
pragma Assert (Table (Bottom) <= What and What < Table (Top));

while Table (Bottom) < What and Top - Bottom > 1 loop

pragma Loop_Invariant (Bottom >= Table'First);
pragma Loop_Invariant (Top <= Table'Last);
pragma Loop_Invariant (Top > Bottom);
pragma Loop_Invariant (Table (Bottom) <= What and What < Table (Top));
pragma Loop_Variant (Decreases => Top - Bottom);

Middle := (Bottom + Top)/2;
pragma Assert (Bottom < Middle and Middle < Top);

if Table (Middle) > What then
Top := Middle;
else
Bottom := Middle;
end if;
end loop;

pragma Assert (Table (Bottom) <= What and Table (Bottom + 1) > What);
return Bottom;
end Find;

``````

There are just two points worth describing; the first is this sequence of `pragma`s inside the loop

``````pragma Loop_Invariant (Bottom >= Table'First);
pragma Loop_Invariant (Top <= Table'Last);
pragma Loop_Invariant (Top > Bottom);
pragma Loop_Invariant (Table (Bottom) <= What
and What < Table (Top));
pragma Loop_Variant (Decreases => Top - Bottom);

``````

`pragma`s `Loop_Invariant` and `Loop_Variant` are specific to SPARK. A loop invariant is a condition that is true at every iteration and you can see that this loop invariant is a formalization of what we said before: at every iteration `What` is between `Top` and `Bottom`. Loop invariants are important in the proof of correctness of `while` loops.

A `while` loop can potentially go on forever; a technique that allows us to prove that the loop will terminate is to search for some value that always increases (or decreases) at every iteration; if we know a bound for this value (e.g., it is always positive and it always decreases) we can deduce that the loop will terminate. The `pragma` `Loop_Variant` allows us to declare said value. In this case the distance between `Top` and `Bottom` is halved at every iteration, therefore it is a good variant. Since `Top-Bottom` cannot be smaller than 2 (see condition in the while), we deduce that the loop will terminate.

A second observation is about seemly innocuous line

`````` Middle := (Top + Bottom) / 2;
``````

Here SPARK complains with

``````medium: overflow check might fail
(e.g. when Bottom = Index_Type'Last-4 and Top = Index_Type'Last-2)
``````

Good catch! Here SPARK is observing that although theoretically `Middle` should be in the range of representable integers if `Top` and `Bottom` are, there is a possibility of an overflow while doing the sum. Since on my PC `Index_Type'Last = 2^63-1`, it is unlikely that one would work with tables so big, nevertheless...

We have two solutions: (i) allow a smaller range of integers (that is, up to `Index_Type'Last/2`) or (ii) compute the mean with function

``````function Mean (Lo, Hi : Index_Type) return Index_Type
is (Lo + (Hi - Lo) / 2)
with
Pre  => Lo < Index_Type'Last and then Hi > Lo + 1,
Post => Hi > Mean'Result and Mean'Result > Lo;
``````

that is guaranteed to have no overflow. Note that in the precondition we require that `Hi > Lo + 1`, this is guaranteed by the loop condition and it is necessary in order to guarantee that the post conditions hold which in turn guarantees that the loop variant decreases at every iteration.

# Finally...

OK, now if we run SPARK we get

``````Summary of SPARK analysis
=========================

-------------------------------------------------------------------------------------------------------
SPARK Analysis results        Total      Flow   Interval   CodePeer      Provers   Justified   Unproved
-------------------------------------------------------------------------------------------------------
Data Dependencies                 .         .          .          .            .           .          .
Flow Dependencies                 .         .          .          .            .           .          .
Initialization                    3         3          .          .            .           .          .
Non-Aliasing                      .         .          .          .            .           .          .
Run-time Checks                  29         .          .          .    29 (CVC4)           .          .
Assertions                       14         .          .          .    14 (CVC4)           .          .
Functional Contracts              3         .          .          .     3 (CVC4)           .          .
LSP Verification                  .         .          .          .            .           .          .
-------------------------------------------------------------------------------------------------------
Total                            49    3 (6%)          .          .     46 (94%)           .          .

``````

Do you see on the extreme right the column `Unproved` with no entries? That is what I want to see... SPARK was able to prove everything, so now you can kick back, relax and enjoy your binary search function with confidence: you made terra bruciata (scorched earth) around it and no bug can survive. 