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Xn ) is defined as follows: For all ω ∈ EN V with ω(xi ) = vi : if I(x = e, ω) = ω , then R(ω (x), v1 , . . , vn ) is true, otherwise false. Lemma 3. Given a program ΠSSA in SSA form and its corresponding CSP representation CSP (ΠSSA ). For all ω ∈ EN V : I(ΠSSA , ω) = ω iff ω ∪ ω is a solution of CSP (ΠSSA ). Using this lemma we can finally conclude that the whole conversion process is correct: Theorem 4. , Π = ΠLF = ΠSSA = CSP (Π). This theorem follows directly from Lemma 1 to 3. Example: From the SSA form which is depicted in Fig.

6 depicts the hypertree evolutions of five different programs. It can be seen that in all of these cases the hypertree width reaches an upper bound, as indicated in Theorem 6. On the Complexity of Program Debugging Using Constraints 31 5 Conclusion Debugging is considered a computationally very hard problem. This is not surprising, given the fact that model-based diagnosis is NP-complete. However, more surprising is that debugging remains a hard problem even when considering single-faults only, at least when using models which utilize the entire semantics of the program in order to obtain precise results.

In order to alleviate as far as possible, these problems, several variations of the classical gradient descent algorithm and new methods have been proposed, such as, adaptive step size, appropriate weights initialization, rescaling of variables, or even second order methods and methods based on linear least-squares. In [2] we ﬁnd a description of the Sensitivity-Based Linear Learning Method (SBLLM), P. Meseguer, L. M. ): CAEPIA 2009, LNAI 5988, pp. 42–50, 2010. c Springer-Verlag Berlin Heidelberg 2010 An Incremental Learning Method for Neural Networks 43 a novel supervised learning method for two-layer feedforward neural networks.