### Table 1: Minimization of APL for individual BDDs

"... In PAGE 12: ... 6.1 Binary Case Table1 compares the number of nodes and APL of BDDs optimized using four different methods: (a) exact minimization of the number of nodes; (b) exact minimization of the APL; (c) the algorithm in [16]; and (d) the heuristic APL minimization algorithm presented in this paper. In the table, Name lists the names of benchmark functions.... In PAGE 12: ...sing Walsh spectrum described in Section 4.5. The BDDs in this table use complemented edges. Table1 includes the same benchmark functions as the experiment in [16] except for incompletely speci ed functions. We omitted incompletely... In PAGE 15: ... To make our results compatible with the results in [16], we optimized each output of the multiple-output benchmark functions independently, and obtained the sum of the values over all outputs. Thus, the number of nodes and APL in Table1 are different from those of the SBDD. Two rounds of sifting are performed in all experiments.... In PAGE 15: ... Tables 1 and 2 show that the proposed heuristic minimization minimizes the APL in short computation time. For small benchmark functions in Table1 , the heuristic minimization could obtain BDDs with near-minimum APLs. For large bench- mark functions in Table 2, the heuristic algorithm reduces APLs to 84% on the average.... In PAGE 15: ... The SBDDs and SMDDs in this table do not use complemented edges. Note that the values (Nodes, APL, Time) of BDDs in this table are different from the values in Table1... ..."

### Table 2: Exact Vs Heuristic

"... In PAGE 19: ...o the heuristic (Section 4.4). The exact method ran out of memory much faster, and hence we were only able to deal with small examples with the exact method. However, Table2 shows that for examples where the exact method could complete, the heuristic answers were almost always the same. 6 Conclusions and Future Work We have implemented the matching algorithms, and demonstrated that on BDD building the incremental procedures take less time than the non-incremental.... ..."

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### Table 2: Performance of exact and heuristic algorithm

### TABLE 5.2 BENCHMARK FUNCTIONSTHAT ARE ORTHODOX. Circuit No. No. No. No.Non No.

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### Table 2: Clock period minimization with Algorithm RSMINP and an exact MILP-based branch-and-bound algo- rithm.

2001

"... In PAGE 21: ... To evaluate the relative speed and efficiency of our retiming and clock scheduling heuristic, we independently developed a MILP-based branch-and-bound solver. Table2 compares the runtimes and output clock periods of the two programs for a subset of our test suite. In general, the CPU requirements of the MILP-based optimizer grow very fast, due to the high computational complexity of mixed-integer linear programming.... In PAGE 21: ... With a 48-hour timeout, the MILP-solver runs out of time on most circuits, without having discovered a better solution than the heuristic scheme. The last column of Table2 shows the relative clock period improvement achieved by the MILP- based solver over our heuristic. Except for daio, the fastest circuit computed by the heuristic is as good as that of the MILP solver.... ..."

### Table 1. State machine

"... In PAGE 3: ...dentical i.e. there is no state change produced by an input, the state bits in Q are not deactivated. Let us illustrate the proposed approach by designing the state machine ( an MCNC benchmark circuit ) specified in Table1 . We consider only the next states, the inclusion of the output logic is straightforward.... ..."

### Table 6: Results for incomplete specified functions name d vars clauses time res

2007

"... In PAGE 5: ... In contrast the result for graycode6 was computed very fast. The results for incompletely specified functions are shown in Table6 . Several options for embedding these functions Table 6: Results for incomplete specified functions name d vars clauses time res... In PAGE 6: ... For example, for the function decod24 [4] we can chose the values for the con- stant inputs from {00, 01, 10, 11}. The results are shown as decod24-v0 to decod24-v3 in Table6 . It is apparent, that all four functions exhibit very similar properties, but for one the minimal Toffoli realization requires one more gate.... ..."

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### Table 12: Event table for Incomplete state. Incomplete

1999

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### Table 1 - XML Algorithms Purpose Algorithm Specified as

"... In PAGE 10: ... Table1 above provides an overview of the algorithms specified in the XML-Signature and XML-Encryption standards. Required algorithms are the minimal to comply with the standard; while recommended is just that, recommended.... In PAGE 11: ...11 A message digest is an encoded message, a cryptographic checksum of an octet stream [WS-Security, 2002], which is created using an algorithm. The SHA-1 algorithm [NIST, 1993], in Table1 , is required in the XML Signature specification. The only parameter is the element to be signed.... ..."