Stability | experimental |
---|---|

Maintainer | erkokl@gmail.com |

Safe Haskell | Safe-Infered |

Computing GCD symbolically, and generating C code for it. This example illustrates symbolic termination related issues when programming with SBV, when the termination of a recursive algorithm crucially depends on the value of a symbolic variable. The technique we use is to statically enforce termination by using a recursion depth counter.

# Computing GCD

sgcd :: SWord8 -> SWord8 -> SWord8Source

The symbolic GCD algorithm, over two 8-bit numbers. We define `sgcd a 0`

to
be `a`

for all `a`

, which implies `sgcd 0 0 = 0`

. Note that this is essentially
Euclid's algorithm, except with a recursion depth counter. We need the depth
counter since the algorithm is not *symbolically terminating*, as we don't have
a means of determining that the second argument (`b`

) will eventually reach 0 in a symbolic
context. Hence we stop after 12 iterations. Why 12? We've empirically determined that this
algorithm will recurse at most 12 times for arbitrary 8-bit numbers. Of course, this is
a claim that we shall prove below.

# Verification

We prove that `sgcd`

does indeed compute the common divisor of the given numbers.
Our predicate takes `x`

, `y`

, and `k`

. We show that what `sgcd`

returns is indeed a common divisor,
and it is at least as large as any given `k`

, provided `k`

is a common divisor as well.

# Code generation

Now that we have proof our `sgcd`

implementation is correct, we can go ahead
and generate C code for it.

This call will generate the required C files. The following is the function
body generated for `sgcd`

. (We are not showing the generated header, `Makefile`

,
and the driver programs for brevity.) Note that the generated function is
a constant time algorithm for GCD. It is not necessarily fastest, but it will take
precisely the same amount of time for all values of `x`

and `y`

.

#include <inttypes.h> #include <stdint.h> #include <stdbool.h> #include "sgcd.h" SWord8 sgcd(const SWord8 x, const SWord8 y) { const SWord8 s0 = x; const SWord8 s1 = y; const SBool s3 = s1 == 0; const SWord8 s4 = (s1 == 0) ? s0 : (s0 % s1); const SWord8 s5 = s3 ? s0 : s4; const SBool s6 = 0 == s5; const SWord8 s7 = (s5 == 0) ? s1 : (s1 % s5); const SWord8 s8 = s6 ? s1 : s7; const SBool s9 = 0 == s8; const SWord8 s10 = (s8 == 0) ? s5 : (s5 % s8); const SWord8 s11 = s9 ? s5 : s10; const SBool s12 = 0 == s11; const SWord8 s13 = (s11 == 0) ? s8 : (s8 % s11); const SWord8 s14 = s12 ? s8 : s13; const SBool s15 = 0 == s14; const SWord8 s16 = (s14 == 0) ? s11 : (s11 % s14); const SWord8 s17 = s15 ? s11 : s16; const SBool s18 = 0 == s17; const SWord8 s19 = (s17 == 0) ? s14 : (s14 % s17); const SWord8 s20 = s18 ? s14 : s19; const SBool s21 = 0 == s20; const SWord8 s22 = (s20 == 0) ? s17 : (s17 % s20); const SWord8 s23 = s21 ? s17 : s22; const SBool s24 = 0 == s23; const SWord8 s25 = (s23 == 0) ? s20 : (s20 % s23); const SWord8 s26 = s24 ? s20 : s25; const SBool s27 = 0 == s26; const SWord8 s28 = (s26 == 0) ? s23 : (s23 % s26); const SWord8 s29 = s27 ? s23 : s28; const SBool s30 = 0 == s29; const SWord8 s31 = (s29 == 0) ? s26 : (s26 % s29); const SWord8 s32 = s30 ? s26 : s31; const SBool s33 = 0 == s32; const SWord8 s34 = (s32 == 0) ? s29 : (s29 % s32); const SWord8 s35 = s33 ? s29 : s34; const SBool s36 = 0 == s35; const SWord8 s37 = s36 ? s32 : s35; const SWord8 s38 = s33 ? s29 : s37; const SWord8 s39 = s30 ? s26 : s38; const SWord8 s40 = s27 ? s23 : s39; const SWord8 s41 = s24 ? s20 : s40; const SWord8 s42 = s21 ? s17 : s41; const SWord8 s43 = s18 ? s14 : s42; const SWord8 s44 = s15 ? s11 : s43; const SWord8 s45 = s12 ? s8 : s44; const SWord8 s46 = s9 ? s5 : s45; const SWord8 s47 = s6 ? s1 : s46; const SWord8 s48 = s3 ? s0 : s47; return s48; }