# [LLVMdev] New automated decision procedure for path-sensitive analysis

Domagoj Babic babic.domagoj at gmail.com
Sun Apr 8 13:25:16 PDT 2007

```Dear LLVMers,

This email is intended for those interested in path-sensitive analysis,
integer overflow analysis, static analysis, and (perhaps) loop invariant
computation.

Traditionally, such analyses have been considered too expensive to be
practical, and were mostly an academic curiosity. The core of the
problem is the lack of adequate automated decision procedures which
could quickly determine whether a set of constraints is satisfiable or
not, and if it is satisfiable, find a solution.

Recently, I've released Spear -- automated modular arithmetic theorem
prover, which has proven to be very scalable in my setting.

A nice feature of Spear is that it supports all LLVM integral
instructions, including SDIV/UDIV/MUL/..., which makes it really easy to
use in combination with LLVM. However, Spear itself is not LLVM-based
because many people that are interested in such theorem provers do not
use LLVM.

Here I provide two simple examples to give you a flavour of Spear:

------------ Example 1 ---------------

Assume that you want to generate an instance that corresponds to the
following C-like sequential code:

int f(int a, int b) {
int a1;
if (a%2) { a1 = a + 1; }
else { a1 = a; }
// a1 is even
int c = a1 * a1;
// Square of an even number is divisible by 4
assert(c % 4 == 0);
}

You could check the validity of the assertion by checking the
unsatisfiability of the negated formula (corresponds to checking that
the assertion can never be FALSE):

# Checking a simple assertion
v 1.0
d a:i32 a1:i32 c:i32 inca:i32 tmp1:i32 tmp2:i1 tmp3:i32 assert:i1
c inca + a 1:i32
c tmp1 %s a 2:i32
c tmp2 trun tmp1
c a1 ite tmp2 inca a
c c * a1 a1
c tmp3 %s c 4:i32
c assert = tmp3 0:i32
p = assert 0:i1

Spear proves this query to be unsatisfiable in 0.02 sec on AMD 64 X2
4600+ for 32-bit integers, and in 0.08 sec for 64-bit integers with the
default heuristics.

------------ Example 2 ---------------

The last Fermat's theorem says that for integer n>2 the equation
a^n+b^n=c^n has no solutions for non-zero integers a, b, and c.

However, if a,b,c are bounded integers, the equation can have
non-trivial (non-zero) solutions. Let's try to find one such example:

# Finding a solution of a^4 + b^4 = c^4
v 1.0
d a:i64 b:i64 c:i64 a2:i64 b2:i64 c2:i64 a4:i64 b4:i64 c4:i64 sum:i64
c a2 * a a
c b2 * b b
c c2 * c c
c a4 * a2 a2
c b4 * b2 b2
c c4 * c2 c2
c sum + a4 b4
p = sum c4

Spear finds a non-trivial solution of this query in 24.08 sec on AMD
64 X2 4600+ with the default heuristics and in 3.18 sec with
the fh_1_1 heuristic.

--------------------------------------

The prover and the input format specification are available on:
http://www.cs.ubc.ca/~babic/index_spear.htm

Regards,
Domagoj Babic

P.S.
I hesitated to send the email to the list, but several people in the channel
showed interest and encouraged me to send a release notification.

```