TACAS'11::Optimal Base Encodings for Pseudo-Boolean Constraints
This paper talk about a new method to translate the PSB problem into SAT.
Of course there are some direct solving methods, such as that of PBS solver proposed by Fadi Aloul .
But it seems that translating into SAT is also very interesting. it refer to some early papers:
1. A Translation of Pseudo-Boolean Constraints to SAT:
This paper just iterate though all coefficient, and case split on they to consider the two cases of them, and substract the correspond value from the right handside of (dis)equation.
In this way, an SAT instance is encoded.
2. Translating Pseudo-Boolean Constraints into SAT
it propose three methods
a. BDD based: similar to above method
b. adder network: it express the coefficient with a set of 2-exp, and add them together with adders.but because there are XOR gates in these adders, it is Not arc-consistent.
c. sorting network : for small coefficients n, it use n length bit vector to express it.Assume again that the sum of all coe is N, then a sorter of size N is instanced.For variables x and its coe n, x is connected to n inputs of the sorter. For right hand side x, the 0-th to x-th output of sorter is asserted to be 1.
To deal with larger coe, a set of base must be founds, such that each sorter can be used to express larger number that 1.
To minimize the size of generated SAT instance, a set of optimal base must be found, this is the motivation of this TACAS11 paper.
This TACAS11 paper propose three cost function to evaluate the total cost that need to be minimized:
a. summing the digits used to express coes: only bases with only prime numbers are need to be considered.
b. also taking into acount of the carry bits between sorters: non-prime based must be consider
c. consider the number of comparators in sorters.
With these cost function, it proposes three searching methods, they approximate the cost function and perform branch and bound searching to find a minimized cost base.