An Interior Point Approach to Boolean Vector Function Synthesis

Table 6: 8 input Boolean functions SAT m k A vars clauses itr time We conclude this paper by reporting on preliminary computational testing of the integer programming algorithm described in 10] and 11] on instances of the Boolean Vector Function Synthesis Problem. We use the standard integer programming formulation for the Satissability problem 8]. The runs were carried out on a 33 MHz MIPS3000 Silicon Graphics Indigo workstation. The integer programming code was compiled with Fortran compiler f77 and the C language compiler cc using compiler ags-O2-Olimit 800. CPU times were measured with the system call times(). We rst consider the 2-Bit Adder (y = x + z) with the 4-input, 3-output speciication given in Table 5. The Satissa-bility problem formulation for this adder, using at most k = 12 distinct product terms has 408 variables and 1751 clauses. The interior point code took 66 iterations, producing the 11 product term vector function y1 = x1x2z2 + x2z1z2 + x1z1 y2 = x1x2 z1z2 + x1 x2z1 + x1x2z1z2 + x1 x2 z1 + x1x2z1 z2 + x1x2 z1 z2 y3 = x2 z2 + x2z2 in 34.3 CPU seconds. Now consider 8-input Boolean vector functions. Table 6 summarizes the functions, Satissability problems and the experimental results. For each instance, the table shows the number of output variables (m), the number of product terms (k), the number of i/o-pairs in the truth table (A), the number of variables (vars) and clauses of the Satissability problem and the number of iterations and CPU time for the interior point algorithm to produce the Boolean function. Finally, in Table 7, we summarize results for three larger Boolean functions. In addition to the parameters shown in Table 6, this table gives the number of input variables (n) of the instance. This preliminary computational experiment was limited to a narrow set of test problems. Nevertheless, it indicates that the interior point approach can synthesize complex Boolean vector functions in reasonable time. We believe this approach holds much promise for solving real-world instances. In a forthcoming paper, we extend the computational study to a wider set of problems, including some real-world circuit design problems .