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Therefore, taking an action in a state results in two (possibly absorbing) new states. This
one-to-two state transition violates the standard MDP definition, but it can be thought of as
cloning the MDP and creating one copy for each transition. The two copies continue
independently thereafter. The immediate cost paid for choosing a branching rule is computed in
two different ways. The total number of search nodes that results from unit propagations,
between the original and the two new states, is the node cost paid for the decision at the original
node. That is the immediate cost for choosing the branching rules MAXO, MOMS, MAMS, and
JW. The other three branching rules perform trial partial searches to determine the branching
variable. The total number of nodes created during these trial searches is added to the above
node cost for branching rules UP, GUP, and SUP. The definition of the cost function implies that
the total accumulated cost after a complete run of the DPLL procedure will be the size of the
entire search tree (in terms of nodes) plus the total count of nodes created during the trial
searches. Minimizing this total cost is a way to minimize the total running time, which is
proportional to the total number of nodes explored.
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