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This time, we are in luck. The partial assignment w = 0, x = 1 violates the clause (w x) and can be terminated, thereby pruning a good chunk of the search space. We backtrack out of this cul-de-sac and continue our explorations at one of the two remaining active nodes. In this manner, backtracking explores the space of assignments, growing the tree only at nodes where there is uncertainty about the outcome, and stopping if at any stage a satisfying assignment is encountered. In the case of Boolean satis ability, each node of the search tree can be described either by a partial assignment or by the clauses that remain when those values are plugged into the original formula. For instance, if w = 0 and x = 0 then any clause with w or x is instantly satis ed and any literal w or x is not satis ed and can be removed. What s left is (y z), (y), (y z). Likewise, w = 0 and x = 1 leaves with the empty clause ( ) ruling out satis ability. Thus the nodes of the search tree, representing partial assignments, are themselves SAT subproblems. This alternative representation is helpful for making the two decisions that repeatedly arise: which subproblem to expand next, and which branching variable to use. Since the bene t of backtracking lies in its ability to eliminate portions of the search space, and since this happens only when an empty clause is encountered, it makes sense to choose the subproblem that contains the smallest clause and to then branch on a variable in that clause. If this clause happens to be a singleton, then at least one of the resulting branches will be terminated. (If there is a tie in choosing subproblems, one reasonable policy is to pick the one lowest in the tree, in the hope that it is close to a satisfying assignment.) See Figure 9.1 for the conclusion of our earlier example. More abstractly, a backtracking algorithm requires a test that looks at a subproblem and quickly declares one of three outcomes: 1. Failure: the subproblem has no solution. 2. Success: a solution to the subproblem is found. 3. Uncertainty. 267 (), (y z),
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Figure 38-13. MRI of an adult with congenital hemiplegia. There is severe encephalomalacia mainly in the territory of the right middle cerebral artery.
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