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Figure 11. (a) Elimination of substitution operator vertex. (b) Elimination of existential quantification operator vertex.

Figure 11 (a) Elimination of substitution operator vertex. (b) Elimination of existential quantification operator vertex.

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The BED for this expression is shown in figure 1. The low-edges are drawn using dashed lines and all edges are implicitly directed downwards. Notice that vertices representing the same Boolean sub-function are shared. A key operation on BEDs is the up operation which moves a variable vertex up above an operator vertex. Let op be an arbitrary binary Boolean operator, let x be a Boolean variable, and let f; and fj (i = 0,1) be arbitrary Boolean expressions. It is simple to verify that
The BED for this expression is shown in figure 1. The low-edges are drawn using dashed lines and all edges are implicitly directed downwards. Notice that vertices representing the same Boolean sub-function are shared. A key operation on BEDs is the up operation which moves a variable vertex up above an operator vertex. Let op be an arbitrary binary Boolean operator, let x be a Boolean variable, and let f; and fj (i = 0,1) be arbitrary Boolean expressions. It is simple to verify that
below the operator vertex before performing the up- step. In fact, this is the only way the size of the BED can increase. Figure 3. Proving the distributive law. (a) 21 is moved above the three conjunctions using three up- steps. (b) Conjunctions with children that are con- stant vertices are removed. (c) 2; is moved above the disjunction to the right. (d) The disjunction with both children equal to 0 is removed. (e-f) Identifying equivalent vertices. At this point the two children of the biimplication operator are identical and the BED is reduced to 1, proving the tautology.
below the operator vertex before performing the up- step. In fact, this is the only way the size of the BED can increase. Figure 3. Proving the distributive law. (a) 21 is moved above the three conjunctions using three up- steps. (b) Conjunctions with children that are con- stant vertices are removed. (c) 2; is moved above the disjunction to the right. (d) The disjunction with both children equal to 0 is removed. (e-f) Identifying equivalent vertices. At this point the two children of the biimplication operator are identical and the BED is reduced to 1, proving the tautology.
Figure 2. The left and right BED are equivalent.
Figure 2. The left and right BED are equivalent.
Figure 4. Performing an up-step on the vertex v. As described in the introduction, the key operation on BEDs, called up, exchanges variables with other variables or operators, see figure 4. This transforma- tion pulls a variable up one level. The vertices in the
Figure 4. Performing an up-step on the vertex v. As described in the introduction, the key operation on BEDs, called up, exchanges variables with other variables or operators, see figure 4. This transforma- tion pulls a variable up one level. The vertices in the
Figure 6. The data structures used to represent a BED. to v and thus implements an inverse of G used by mk. Finally, the mark field is used to mark visited vertices when traversing the graph. This is used in the up_one and up-_all algorithms presented in the next section and can also be used in mark-scan garbage collection algo- rithms. The total memory requirements are six words per vertex when using linking and five words when us- ing indirection. Using these data structures, it is not difficult to implement mk as a constant time operation.
Figure 6. The data structures used to represent a BED. to v and thus implements an inverse of G used by mk. Finally, the mark field is used to mark visited vertices when traversing the graph. This is used in the up_one and up-_all algorithms presented in the next section and can also be used in mark-scan garbage collection algo- rithms. The total memory requirements are six words per vertex when using linking and five words when us- ing indirection. Using these data structures, it is not difficult to implement mk as a constant time operation.
_ Up-one and up_all are significantly different strate- gies for building an ROBDD. Figure 10 illustrates how the operator vertices are converted into variable ver- tices by the two algorithms. Figure 10. Converting a BED to an ROBDD. To the left, using up_one repeatedly, to the right, during an up-all call. Grey areas represent variables and white areas represent operators.
_ Up-one and up_all are significantly different strate- gies for building an ROBDD. Figure 10 illustrates how the operator vertices are converted into variable ver- tices by the two algorithms. Figure 10. Converting a BED to an ROBDD. To the left, using up_one repeatedly, to the right, during an up-all call. Grey areas represent variables and white areas represent operators.
‘Garbage collection is used in this example, making this number the maximum number rather than the total number of BED vertices. Table 1. Experimental results for the ISCAS 85 benchmark. Nota is the total number of BED nodes for the verification. The CPU times are in seconds on a Sun Ultra-SPARC 1.
‘Garbage collection is used in this example, making this number the maximum number rather than the total number of BED vertices. Table 1. Experimental results for the ISCAS 85 benchmark. Nota is the total number of BED nodes for the verification. The CPU times are in seconds on a Sun Ultra-SPARC 1.
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