By William McDuff Spears
Paradigms for utilizing neural networks (NNs) and genetic algorithms (GAs) to
heuristically remedy boolean satisfiability (SAT) difficulties are offered. Results
are offered for two-peak and false-peak SAT difficulties. seeing that SAT is NP-Complete,
any different NP-Complete challenge might be reworked into an equivalent
SAT challenge in polynomial time, and solved through both paradigm. This technique
is illustrated for Hamiltonian circuit (HC) difficulties.
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If a j = −1 and all siblings are true (ak wk j = 1), then node i should have an activation ai = a j wi j = −wi j . As before, the truth of children of the parent is a product of the child’s activation and the weight. Similar constraints occur when the parent is an OR. It is important to note a difference between the downstream constraint and upstream constraint definitions. If D (i) is true or false, a real logical constraint is implied. This occurs due to the direction of flow of information. Regardless of the states of the children, the state of the node can be determined unambiguously.
_______________ † Since the solution does in fact start with a 1, the GA is concentrating on the correct subspace. 53 Figure 10: Graph of HC7 Payoff Function for the GA The following table indicates the number of evaluations needed for the GA (using AVE p , where p = 1). Both the mean number of evaluations and the standard deviation are reported. The number of edges in the graph (bits) is denoted by n. ______________________________________________________________ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ n 10 15 21 28 36 45 55 ______________________________________________________________ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ mean ✡ 848 ✡ ✡ 1022 ✡ ✡ ✡ ✡ 5028 ✡ ✡ 21894 ✡ ✡ 70577 ✡ ✡ ✡ 259876 ✡ ✡ 838522 ✡ ✡ ✡ ✡ ✡ sd 2041 1083 2016 36391 49946 227254 123350 ______________________________________________________________ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ Table 9: Performance of GAs on HC Problems Figure 11 presents a graph of the results.
Upstream constraints represent constraints from nodes that are closer to the root, whereas downstream constraints represent constraints from nodes further from the root. Downstream constraints flow from the children of a node. Suppose that some node is a NOT node. Then its activation should be opposite that of its child. An AND node should be true if and only if all of its children are true. An OR node should be true if and only if any of its children is true. Upstream constraints flow from the parent of a node.