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# [Frama-c-discuss] recursive programming vs. declarative programming

• Subject: [Frama-c-discuss] recursive programming vs. declarative programming
• From: Christoph.Weber at first.fraunhofer.de (Christoph Weber)
• Date: Mon Nov 10 09:31:28 2008

```Salut,

today I wonder about the principles of recursive/ axiomatic programming and the things that I could do.

My goal is to define a function called "next_permutation(int* src, int* dest)". In essence, it returns the next greater permutation. Multiple appearances of the same element are allowed.
To prove this, I thought about several things, that must be defined.

1. dest must be  a permutation of src, therefore my custom-made predicate:

/*@ axiomatic IS_permutation {
logic integer is_permutation{L1,L2}(int* t1, int* t2, integer n);
axiom is_permutation_occ{L1,L2}:
\forall int *t1, *t2, n, integer i; i < n ==>
nb_occ{L1}(t1,0,n-1,\at(t1[i],L1)) == nb_occ{L2}(t2,0,n-1,\at(t2[i],L2));
}
*/
(Problems are known)

2. dest must be the next greater permutation src,  therefore, it must be proven, that no permutation exists, which is > src and < dest.
/*@
next_permutation (int* a, int* b)=
is_permutation(a,b)==>a>b &&
\forall x all_permutations(a,x)>a ||
\forall x all_permutations(a,x)<b;
*/

all_permutations has to create permutations of a in dependency of x.

My idea was to implement a recursive function in C, which can do this and derive a possible implementation in ACSL.

C-code:
void all_permutation(int* a, int length, int n, int* number_of_perm)
{
int temp;
if (n == 0)
{
*number_of_perm -= 1;
}
if(*number_of_perm == 0){
return;
}
else
{
for (int k=0; k <= n && *number_of_perm > 0 ; k++)
{
// Swap a[k] und a[n]
temp=a[n];
a[n]=a[k];
a[k]=temp;
all_permutation(a, length, n-1, number_of_perm);
// Swap-back a[k] and a[n]
temp=a[n];
a[n]=a[k];
a[k]=temp;
}
}
}

ACSL:

/*@ axiomatic ALL_permutation {
@
@ logic ??? all_permutation{L}(int* a, integer length, integer n, int* number_of_perm);

@ axiom nth_permutation{L}:
\forall int *a, *number_of_perm, integer length, n; *number_of_perm == 0 ==> ???return??? ;
@ axiom next_permutation{L}:
\forall int *a, *number_of_perm, integer length, n; n == 0 ==> *number_of_perm -= 1;
@ axiom recursive_call{L}:
\forall int *a, *number_of_perm, integer length, n, k, j; 0 <= k < n && n != 0 && *number_of_perm != 0 ==>
swap(a+n, a+k) ==>
all_permutation(a, length, n-1, number_of_perm) ==>
swap(a+n, a+k);
@ }
@*/

You see the problem, this mix of functional and declarative programming is difficult.

I would like a suggestion how this could be solved or if there will be a solution in the near future.

Cheers

Christoph
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