+ // (e|a)a*a = a+
+ // aa*(e|a) = a+
+ // a(e|a)*(e|a) = a+
+ // (e|a)a*a = a+
+ else
+ {
+ eps_check =
+ (has_epsilon (R_last_ik) + has_epsilon (R_last_kk) +
+ has_epsilon (R_last_kj));
+
+ if (eps_check == 1)
+ {
+ if (needs_parentheses (R_temp_kk))
+ GNUNET_asprintf (&R_cur_r, "(%s)+", R_temp_kk);
+ else
+ GNUNET_asprintf (&R_cur_r, "%s+", R_temp_kk);
+ }
+ }
+ }
+ // aa*b = a+b
+ // (e|a)(e|a)*b = a*b
+ else if (0 == strcmp (R_temp_ik, R_temp_kk))
+ {
+ if (has_epsilon (R_last_ik))
+ {
+ if (needs_parentheses (R_temp_kk))
+ GNUNET_asprintf (&R_cur_r, "(%s)*%s", R_temp_kk, R_last_kj);
+ else
+ GNUNET_asprintf (&R_cur_r, "%s*%s", R_temp_kk, R_last_kj);
+ }
+ else
+ {
+ if (needs_parentheses (R_temp_kk))
+ GNUNET_asprintf (&R_cur_r, "(%s)+%s", R_temp_kk, R_last_kj);
+ else
+ GNUNET_asprintf (&R_cur_r, "%s+%s", R_temp_kk, R_last_kj);
+ }
+ }
+ // ba*a = ba+
+ // b(e|a)*(e|a) = ba*
+ else if (0 == strcmp (R_temp_kk, R_temp_kj))
+ {
+ if (has_epsilon (R_last_kj))
+ {
+ if (needs_parentheses (R_temp_kk))
+ GNUNET_asprintf (&R_cur_r, "%s(%s)*", R_last_ik, R_temp_kk);
+ else
+ GNUNET_asprintf (&R_cur_r, "%s%s*", R_last_ik, R_temp_kk);
+ }
+ else
+ {
+ if (needs_parentheses (R_temp_kk))
+ GNUNET_asprintf (&R_cur_r, "(%s)+%s", R_last_ik, R_temp_kk);
+ else
+ GNUNET_asprintf (&R_cur_r, "%s+%s", R_last_ik, R_temp_kk);
+ }
+ }
+ else
+ {
+ if (strlen (R_temp_kk) > 0)
+ {
+ if (needs_parentheses (R_temp_kk))
+ {
+ GNUNET_asprintf (&R_cur_r, "%s(%s)*%s", R_last_ik, R_temp_kk,
+ R_last_kj);
+ }
+ else
+ {
+ GNUNET_asprintf (&R_cur_r, "%s%s*%s", R_last_ik, R_temp_kk,
+ R_last_kj);
+ }
+ }
+ else
+ {
+ GNUNET_asprintf (&R_cur_r, "%s%s", R_last_ik, R_last_kj);
+ }
+ }
+ }
+
+ GNUNET_free_non_null (R_temp_ik);
+ GNUNET_free_non_null (R_temp_kk);
+ GNUNET_free_non_null (R_temp_kj);
+
+ if (NULL == R_cur_l && NULL == R_cur_r)
+ {
+ *R_cur_ij = NULL;
+ return;
+ }
+
+ if (NULL != R_cur_l && NULL == R_cur_r)
+ {
+ *R_cur_ij = R_cur_l;
+ return;
+ }
+
+ if (NULL == R_cur_l && NULL != R_cur_r)
+ {
+ *R_cur_ij = R_cur_r;
+ return;
+ }
+
+ if (0 == nullstrcmp (R_cur_l, R_cur_r))
+ {
+ *R_cur_ij = R_cur_l;
+ GNUNET_free (R_cur_r);
+ return;
+ }
+
+ GNUNET_asprintf (R_cur_ij, "(%s|%s)", R_cur_l, R_cur_r);
+
+ GNUNET_free (R_cur_l);
+ GNUNET_free (R_cur_r);
+}
+
+
+/**
+ * create proofs for all states in the given automaton. Implementation of the
+ * algorithm descriped in chapter 3.2.1 of "Automata Theory, Languages, and
+ * Computation 3rd Edition" by Hopcroft, Motwani and Ullman.
+ *
+ * @param a automaton.
+ */
+static void
+automaton_create_proofs (struct GNUNET_REGEX_Automaton *a)
+{
+ unsigned int n = a->state_count;
+ struct GNUNET_REGEX_State *states[n];
+ char *R_last[n][n];
+ char *R_cur[n][n];
+ char *temp;
+ struct GNUNET_REGEX_Transition *t;
+ char *complete_regex;
+ unsigned int i;
+ unsigned int j;
+ unsigned int k;
+
+
+ /* create depth-first numbering of the states, initializes 'state' */
+ GNUNET_REGEX_automaton_traverse (a, &number_states, states);
+
+ /* Compute regular expressions of length "1" between each pair of states */
+ for (i = 0; i < n; i++)
+ {
+ for (j = 0; j < n; j++)
+ {
+ R_cur[i][j] = NULL;
+ R_last[i][j] = NULL;
+ }
+ for (t = states[i]->transitions_head; NULL != t; t = t->next)
+ {
+ j = t->to_state->proof_id;
+ if (NULL == R_last[i][j])
+ GNUNET_asprintf (&R_last[i][j], "%c", t->label);
+ else
+ {
+ temp = R_last[i][j];
+ GNUNET_asprintf (&R_last[i][j], "%s|%c", R_last[i][j], t->label);
+ GNUNET_free (temp);
+ }
+ }
+ if (NULL == R_last[i][i])
+ GNUNET_asprintf (&R_last[i][i], "");
+ else
+ {
+ temp = R_last[i][i];
+ GNUNET_asprintf (&R_last[i][i], "(|%s)", R_last[i][i]);
+ GNUNET_free (temp);
+ }
+ }
+ for (i = 0; i < n; i++)
+ for (j = 0; j < n; j++)
+ if (needs_parentheses (R_last[i][j]))
+ {
+ temp = R_last[i][j];
+ GNUNET_asprintf (&R_last[i][j], "(%s)", R_last[i][j]);
+ GNUNET_free (temp);
+ }
+
+ /* Compute regular expressions of length "k" between each pair of states per induction */
+ for (k = 0; k < n; k++)
+ {
+ for (i = 0; i < n; i++)
+ {
+ for (j = 0; j < n; j++)
+ {
+ // Basis for the recursion:
+ // $R^{(k)}_{ij} = R^{(k-1)}_{ij} | R^{(k-1)}_{ik} ( R^{(k-1)}_{kk} )^* R^{(k-1)}_{kj}
+ // R_last == R^{(k-1)}, R_cur == R^{(k)}
+
+ // Create R_cur[i][j] and simplify the expression
+ automaton_create_proofs_simplify (R_last[i][j], R_last[i][k],
+ R_last[k][k], R_last[k][j],
+ &R_cur[i][j]);
+ }
+ }
+
+ // set R_last = R_cur
+ for (i = 0; i < n; i++)
+ {
+ for (j = 0; j < n; j++)