## The Simplest Queue?

My student Jianjun is proving things about ARM executables that handle interrupts. It’s very difficult work, so when I asked him to write up a “simple” case study where an interrupt and the main context communicate through a ring buffer, I thought it would be helpful if I handed him the simplest possible queue that is at all realistic. Since queues are already pretty simple, there seemed to be only two tricks worth playing: force queue size to be a power of two, and waste one slot to simplify full-queue detection. Here’s my code:

```#include "q.h"

#if ((QSIZE)<2)||(!((((QSIZE)|((QSIZE)-1))+1)/2==(QSIZE)))
#error QSIZE must be >1 and also a power of 2
#endif

static int q[QSIZE];

static int
inc (int x)
{
return (x + 1) & MASK;
}

int
full (void)
{
}

int
mt (void)
{
}

int
enq (int item)
{
if (full ())
return 0;
return 1;
}

int
deq (int *loc)
{
if (mt ())
return 0;
*loc = q[tail];
tail = inc (tail);
return 1;
}

int
qents (void)
{
int s = head - tail;
if (s < 0)
s += (QSIZE);
return s;
}
```

The header file simply defines QSIZE and gives prototypes for the functions. Did I miss any good tricks? The code coming out of compilers I have sitting around isn’t quite as clean as I’d have hoped. There’s some redundancy in the code (calling inc() twice in the enqueue and dequeue functions) but eliminating it should be a simple matter for the compiler.

## Negative Correlation Achieved

Recently I reviewed 19 papers that were submitted to CAV 2011. This is the first time I’ve been involved with a pure verification conference, and consequently I greatly enjoyed reading the papers because almost every one contained something new. Each time I submitted a review I looked at the ones that were already submitted for that paper, and kept being surprised at how often I disagreed with the other reviewers. Finally I just computed the correlation between my score and the average of the other reviewers’ scores, and the result was an astonishingly low -0.07.

In contrast, when I review papers at a “systems” venue (doesn’t matter if it’s operating systems, embedded systems, or something else) it’s not at all uncommon for me to give exactly the same score as all, or almost all, of the other reviewers. My guess is that at the last five systems conferences I was involved with, the correlation between my score and the average of the other reviewers’ scores was higher than 0.8.

I’m not sure there’s a take-away message here other than “communities have very different evaluation standards for papers.” However, this does shed a bit of light on why it can be quite difficult to switch areas. Closely related: Doug Comer’s excellent piece on how to insult a computer scientist.

## The Little C Function From Hell

The other day a student and I were trying to understand a subtle part of the C standard. Often, the easiest way to clarify this kind of issue is to recognize that compiler writers have already grappled with it — so just write some code and see what various compilers do with it. I wrote this function:

```int foo (char x) {
char y = x;
return ++x > y;
}
```

Since the expression ++x evaluates to the incremented value of x, it is clear that this function should return “1” for most values of x. The question is: What does it compute for CHAR_MAX?  One possibility is that the function reliably returns “0” for that input, the other possibility is that ++x is undefined on platforms where char is a signed type. For completeness here’s the test harness that prints foo()’s output for all possible inputs:

```int main (void) {
int i;
for (i=CHAR_MIN; i<=CHAR_MAX; i++) {
printf ("%d ", foo(i));
if ((i&31)==31) printf ("\n");
}
return 0;
}
```

This code only works if char is narrower than int, but I’ve never seen a platform where that is not the case.

The first sign that something odd was going on appeared when I compiled the code using Clang:

regehr@home:~\$ clang -O foo.c -o foo
regehr@home:~\$ ./foo
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Not cool — the function is supposed to mostly return true. But then I realized that my default Clang was out of date (2.7) so I tried a very recent Clang snapshot (rev 126534):

regehr@home:~\$ clang -O0 overflow.c -o overflow
regehr@home:~\$ ./overflow
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
regehr@home:~\$ clang -O1 overflow.c -o overflow
regehr@home:~\$ ./overflow
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The result (look at the last character) changed when we changed the optimization level — this is OK if incrementing CHAR_MAX is undefined, and is a compiler bug otherwise.

The Intel C compiler (12.0.2 for x86-64) has similar behavior:

[regehr@bethe ~]\$ icc -O0 foo.c -o foo
[regehr@bethe ~]\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
[regehr@bethe ~]\$ icc -O foo.c -o foo
[regehr@bethe ~]\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A very recent GCC (rev 170512 for x86) gives a consistent output:

regehr@home:~\$ current-gcc -O0 foo.c -o foo
regehr@home:~\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
regehr@home:~\$ current-gcc -O2 foo.c -o foo
regehr@home:~\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

However, the story changes if we ask it not to perform function inlining:

regehr@home:~\$ current-gcc -O2 -fno-inline foo.c -o foo
regehr@home:~\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

So far, Clang 2.7 appears to be wrong, but all other observed behavior is consistent with ++x being undefined when x is CHAR_MAX. Then I tried CompCert and things took a turn for the worse:

regehr@home:~\$ ccomp foo.c -o foo
regehr@home:~\$ ./foo
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

This is very odd because CompCert contains a verified version of C’s tricky implicit casts — the exact thing that foo() was designed to test in the first place.

To make a long story short: when the “char” type is narrower than the “int” type and when x has type “signed char” and value CHAR_MAX, ++x is well-defined by both ANSI C and C99. We know that it is well-defined because:

1. The standard says: “The expression ++E is equivalent to (E+=1).”
2. The standard says: “A compound assignment of the form E1 op= E2 differs from the simple assignment
expression E1 = E1 op (E2) only in that the lvalue E1 is evaluated only once.”
3. The RHS of the simple assignment expression “E1 op E2” is subject to C’s “usual arithmetic conversions.”
4. The usual arithmetic conversions ensure that two operands to a “+” operator of type signed char are both promoted to signed int before the addition is performed.
5. When int is wider than char, there is no possibility of the resulting addition overflowing. Thus, the behavior is well-defined and every correct compiler must emit this output:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

In other words, to evaluate ++x > y when both variables have type signed char and value CHAR_MAX:

1. Signed char 127 is promoted to signed int 127.
2. Signed int 127 is incremented to signed int 128.
3. Signed int 128 is cast to signed char -128, which is the new value for x and also the value of the subexpression ++x. (Update: As Mans points out in comment #5, the result of this type cast is implementation defined. All common C implementations define the behavior to be truncation of a 2’s complement integer.)
4. Signed char -128 is promoted to signed int -128.
5. Signed int -128 > signed int 127 is evaluated to 0.

But what about the fact that none of the four compilers I commonly use reliably returns the correct result?

• In GCC the bug is known and has existed for some time.
• In LLVM/Clang the bug was not known but was fixed in less than 24 hours.
• The Intel compiler is wrong too.
• In CompCert there is no bug. However, there is an unfortunate interaction between its definition of “char” to be unsigned and the signed values for CHAR_MIN and CHAR_MAX found in my Linux machine’s limits.h file. Verifying a compiler is an extremely difficult problem and verifying its entire environment (header files, libraries, assemblers, I/O routines, etc.) is pretty much an open problem. This is why we test.

That’s a lot of trouble being caused by a two-line function. C may be a small language, but it’s not a simple one.

[Thanks to Chucky Ellison from UIUC and to Xavier Leroy for helping me puzzle though all this.]

## Better Testing With Undefined Behavior Coverage

[The bit puzzle results are based on data from Chad Brubaker and the saturating operation results are based on data from Peng Li. They are respectively an undergrad and a grad student in Utah’s CS program.]

Klee is a tool that attempts to generate a collection of test cases inducing path coverage on a system under test. Path coverage means that all feasible control flow paths are executed. It is a strong kind of coverage, but still misses bugs. One way to improve Klee would be to add support for different kinds of coverage metrics: weaker ones like statement coverage would scale to larger programs, and stronger ones such as boundary-value coverage would find more bugs in small codes.

A different way to improve Klee is to continue to target path coverage, but alter the definition of “path.” For example:

• When testing an x86-64 binary containing a cmov instruction, we could make sure to execute both its condition-true path and condition-false path.
• When testing the C expression foo(bar(),baz()), we could make sure to test evaluating foo() and bar() in both orders, instead of just letting the compiler pick one.

This piece proposes undefined behavior coverage, which simply means that for any operation that has conditionally-defined behavior, the well-defined and the undefined behaviors are considered to be separate paths. For example, the C expression 3/y has two paths: one where y is zero and the other where y is non-zero.

Obviously, undefined behavior coverage only makes sense for languages such as C and C++ that admit operations with undefined behavior. An undefined behavior, as defined by the C and C++ standards, is one where the language implementation can do anything it likes. The point is to make the compiler developers’ job easier — they may simply assume that undefined behavior never happens. The tradeoff is that the burden of verification is shifted onto language users.

Undefined behavior coverage makes sense for what I call type 2 functions: those whose behavior is conditionally well-defined.

# An Example

Here’s a simple C function:

```int add_and_shift (int x, int y, int z) {
return (x+y)<<z;
}
```

Due to C’s undefined behaviors, this function has a non-trivial precondition:

0 ≤ z < sizeof(int)*CHAR_BIT
INT_MIN ≤ x+y ≤ INT_MAX

(This is for ANSI C; in C99 the precondition is stronger and quite a bit more complicated, but we won’t worry about that.) If the precondition is not satisfied, the function’s return value is unpredictable. In fact, it’s a bit worse than that: as soon as the program executes an undefined behavior the C implementation is permitted to send email to the developer’s mother (though this hardly ever happens).

The point is that although shift_and_add() seems to admit a single path, it really has a number of additional paths corresponding to failed preconditions for its math operators. If we fail to test these paths, we can miss bugs. Since the precondition checks for math operators in C/C++ are pretty simple, we can just add them in an early phase of the compiler, and that’s exactly what Peng’s hacked version of Clang does.

Without undefined behavior checks, LLVM code for add_and_shift() looks like this:

```define i32 @add_and_shift(i32 %x, i32 %y, i32 %z) nounwind readnone {
entry:
%shl = shl i32 %add, %z
ret i32 %shl
}```

Obviously there’s just one path, and the test case that Klee picks to exercise this path is:

• x = 0, y = 0, z = 0

Next, we compile the same function with undefined behavior checks and run Klee again. This time we get four test cases:

• x = 0, y = 0, z = 0
• x = 0, y = 0, z = 64
• x = -2, y = INT_MIN, z = 0
• x = 2, y = 1, z = 0

The first three tests are exactly the kind of inputs we’d hope to see after looking at the precondition. The 4th input appears to follow the same path as the first. I don’t know what’s going on — perhaps it emerges from some idiosyncrasy of the checked code or maybe Klee simply throws in an extra test case for its own reasons.

Combining Klee with an undefined behavior checker causes Klee to generate additional test cases that — by invoking operations with undefined behavior — should shine some light into dark corners of the system under test. A potential drawback is that all the extra paths are going to cause the path explosion problem to happen sooner than it otherwise would have. However, this should not be serious since we can just run Klee on both versions of the code.

But this is all just talk. The real question is: does this method find more bugs?

# Bit Puzzle Results

The first collection of code is several years’ worth of solutions to an early assignment in Utah’s CS 4400. I already discussed these, so I won’t repeat myself. For each bit puzzle, students receive a reference implementation (which they cannot simply copy since it doesn’t follow the rules for student solutions) and a simple test harness that runs their code against the reference implementation on some inputs, compares the results, and complains about any differences. For each of 10 bit puzzles we have 105 solutions written by students. The automated test suite determines that 84 of these 1050 solutions are faulty. In other words, they return incorrect output for at least one input. Differential testing with Klee finds seven additional buggy functions, for a total of 91.

When the students’ codes are augmented with checks for integer undefined behaviors, Klee finds more paths to explore. The test cases that it generates find the 91 incorrect functions that are already known plus 11 more, for a total of 102 buggy functions. Just to be perfectly clear: a buggy function is one that (after being compiled by GCC) returns the wrong output for an input in a test suite. We are not counting instances of undefined behavior as bugs, we are simply using Klee and the undefined behavior checker to generate a better test suite.

We were able to exhaustively test some of the bit puzzles. In these cases, exhaustive testing failed to find any bugs not found by differential Klee with undefined behavior coverage.

# Saturating Operation Results

The second collection of code is 336 saturating math operations. In this case, the additional tests generated by Klee to satisfy undefined behavior coverage found no additional buggy functions beyond those found using differential whitebox testing. My hypothesis is that:

1. The shift-related undefined behaviors in these functions always involved constant arguments, since shifts were used only to compute values like the maximum and minimum representable integer of a certain width. Since the arguments were constant, Klee had no opportunity to generate additional test cases.
2. The addition and subtraction overflow undefined behaviors were compiled by GCC into modular arithmetic, despite the fact that this behavior is not guaranteed by the standard. This is a natural consequence of generating code using the x86 add and sub instructions. Modular arithmetic was the behavior that people (including me, as described in the previous post) wanted and expected. Therefore, undefined behavior coverage exposed no bugs. Modern C compilers sometimes compile math overflows in a non-modular way (for example, evaluating (x+1)>x to 1), but the saturating arithmetic functions — by chance — do not use code like that.

We were able to exhaustively test saturating operations that take chars (for 16 total bits of input) and short ints (for 32 total bits of input). In these cases, exhaustive testing failed to find any bugs not already found by differential Klee.

# Conclusion

Undefined behavior coverage is a special case of a more interesting code coverage metric that I’ll describe in a subsequent post. We need to try Klee + undefined behavior coverage on some real applications and see what happens; I’m cautiously optimistic that it will be useful.

## Undefined Integer Behaviors in Student Code, Part 2

[This post is based on data gathered by my student Peng Li. He also wrote the undefined behavior checker.]

The other day I posted about undefined integer behaviors in code written by students in a class I used to teach. This post is more of the same, this time from CS 5785, my advanced embedded systems course. Early in the course I often ask the students to implement saturating versions of signed and unsigned addition and subtraction. Their solutions are required to work regardless of whether an integer datatype is defined to be 8, 16, 32, or 64 bits long. Students are not given any kind of test suite, but I repeatedly emphasize that they should write lots of test cases and inspect the output. The assignment is graded by a tester I wrote that uses a combination of a few hundred values that lie near interesting boundaries, and a few thousand random values. The main reason I give this assignment is to get students warmed up a bit and to motivate some material later on in the course. The assignment is here (see part 2).

# Unsigned Saturating Operations

These functions are easy to get right, and also it’s not hard to avoid undefined behavior since the math is unsigned.

 ADD correct wrong no undefined 66 14 undefined 1 3
 SUBTRACT correct wrong no undefined 74 10 undefined 0 0

# Signed Saturating Operations

Although the amount of code required to implement these is small (my solutions are 5 lines each), they’re not completely straightforward.

 ADD correct wrong no undefined 15 10 undefined 28 31
 SUBTRACT correct wrong no undefined 11 4 undefined 20 49

The undefined behaviors were a mix of shift errors and overflowing addition and subtraction. Many students used some sort of shift as part of computing the maximum / minimum representable integer.

# My Code Was Wrong Too

It turns out that my reference solutions (written years ago, before I understood the horror that is undefined behavior in C) contained two signed overflows. Ouch.

# Can We Avoid Undefined Behavior?

90% of the signed operations written by students prior to 2010 executed an undefined integer behavior. In Fall 2010 I included some lecture material on undefined behavior and I also gave students access to Peng’s undefined behavior checker and told them to use it. The result? 46% of the 52 signed functions (written by the 26 students) executed an undefined integer behavior. This is a significant improvement, but still not very good. The real problem — judging from the large number of incorrect solutions — is that many students didn’t test their code very well. Next time I’ll encourage students to graph the functions implemented by their code; this makes some kinds of bugs obvious.