RSAR-SAT in Visual Studio .NET

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DPLL(F) Input: F, the formula containing the current set of clauses Output: minimal feature subset (1) if F contains an empty clause (2) return unsatisfiable (3) if F is empty (4) output current assignment (5) return satisfiable (6) if F contains a unit clause {l} (7) F unitPropagate(F) (8) return DPLL(F ) (9) x selectLiteral(F) (10) if DPLL(F {x}) is satisfiable (11) return satisfiable (12) else (13) return DPLL(F { x})
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De nition of the DPLL algorithm
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Branching occurs at lines (9) to (12) via the function selectLiteral(F ). Here the next literal is chosen heuristically from the current formula, assigned the value 1, and the search continues. If this branch eventually results in unsatis ability, the procedure will assign the value 0 to this literal instead and continue the search. The importance of choosing good branching literals is well known different branching heuristics produce drastically different sized search trees for the same basic algorithm, thus signi cantly affecting the ef ciency of the solver. The heuristic currently used within RSAR-SAT is to select the variable that appears in the most clauses in the current set of clauses. Many other heuristics exist for this purpose [414] but are not considered here. A degree of pruning can take place in the search by remembering the size of the currently considered subset and the smallest optimal subset encountered so far. If the number of variables currently assigned 1 equals the number of those in the presently optimal subset, and the satis ability of F is still not known, then any further search down this branch will not result in a smaller optimal subset. Although stochastic methods have been applied to SAT problems [311,137], these are not applicable here as they provide no guarantee of solution minimality. The DPLL-based algorithm will always nd the minimal optimal subset. However, this will come at the expense of time taken to nd it.
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15.1.2 Preprocessing Clauses
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The discernibility function can be simpli ed by replacing those variables that are simultaneously either present or absent in all clauses by single representative variables. For instance, in the formula below, variables a and f can be replaced
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SUPPLEMENTARY DEVELOPMENTS AND INVESTIGATIONS
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by a single variable: {a b c f } {b d} {a d e f } {d c} The rst and third clauses may be considered to be {{a f } b c} and {{a f } d e}, respectively. Replacing {a f } with g results in {g b c} {b d} {g d e} {d c} If a reduct resulting from this discernibility function contains the new variable g, then this variable may be replaced by either a or f . Here {g, d} is a reduct, and so {a, d} and {f, d} are reducts of the original set of clauses. Hence fewer attributes are considered in the reduct-determining process with no loss of information [350]. The complexity of this (optional) preprocessing step is O(a c + a 2 ), where a is the number of attributes and c is the number of clauses. From the generation of the discernibility matrix, the core attributes are immediately determined. These may then be removed from the discernibility function as they will appear in every rough set reduct. Hence, if the union of the core attributes for a dataset results in a reduct, no search is required as this will be the minimal subset.
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15.1.3 Evaluation
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Initial experimentation has been carried out using the algorithm outlined previously [166]. The datasets have been obtained from [38]. Table 15.1 shows the average time taken for the preprocessing of each dataset. For RSAR, this involves constructing partitions for each attribute. For RSAR-SAT, the discernibility matrix is calculated and simpli ed. It can be seen from the table that
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TABLE 15.1 Runtimes for RSAR and RSAR-SAT Dataset Number Number RSAR SAT RSAR of of Setup Setup (s) Clauses Features (s) (s) M-of-N 6 13 0.164 2.333 0.171* 6 13 0.146 2.196 0.304* Exactly Exactly2 10 13 0.136 1.898 0.823* Heart 12 13 0.085 0.380 0.207* 12 16 0.076 0.333 0.170* Vote Credit 200 20 0.148 3.873 1.988* LED 167 24 0.125 68.20 0.097* 57 25 0.019 0.074 0.067* Letters Derm 1126 34 0.187 11.31 0.758* Derm2 1184 34 0.133 6.796 0.897* 6534 38 0.168 87.85 9.590* WQ Lung 171 56 0.032 0.125 0.059 3861 58 0.139 30.40 1.644* DNA
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SAT: Minimal (s) 0.001 0.001 0.001 0.002 0.004 0.077 0.041 0.024 0.094 0.104 0.205 0.023 0.227
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SAT: Full (s) 0.007 0.008 0.008 0.009 0.009 0.094 0.051 0.116 0.456 0.878 116.1 0.786 53.81
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