A Research On The Simplifications Of Boolean Expression With Integral Relations

The simplification of Boolean expression is an NP problem, and the Systemic algorithm of getting its simplest equivalent formula is still a problem. Simplifications of Boolean expressions can be applied to the optimization of logic circuits, switching algebra and the code of design.In order to reduce the number of input variables and shorten the time delay during hardware implementations, the existing methods in general only consider the simplification of Boolean expression, and there is not any research discovered so far dealing with the simplification of Boolean expression with integer relation expressions.This paper starts with the description and searching for the simplification of Boolean expression with integeral relational expressions, and aiming to find the groups can be simplified and to describe them, then to illustrate the feasibility of the method provided by using examples.The main results of this paper are as follows:Firstly, Boolean expression with addition and subtration can be classified by the relational operators such as "=,≠,≥,>,≤,<". Thus the simplification of the following expression can be realized:f=(R11∧R12∧…∧R1n)∨…∨(Rm1∧Rm2∧…∧Rmn) among which Rij is the intergral relational expression, it represents the formula Aij=Bij, Aij>Bij,Aij≥Bij or Aij≠Bij, as well as Aij and Bij are integral expressions.Secondly, this paper provide certain classes simplification of Boolean expression with multiplicatin, such as,CiXi=±Xj±Cj and Xi*Xj>Xi-Xj+1.Finally, Boolean expression with division such as Ci/Cj=Ck/Cl, Xi/Xj>Xk/Xl and Ci/Xi≥Xi±1 also can be simplified in this paper.Although the results in this paper are only primary and they are further to be studied, we believe that it can be used as an optional method for simplification of the Boolean expression and that it is of positivity to the optimization of Logical Circuit.