Simple CAD Construction and its Applications

The method of quantifier elimination by cylindrical algebraic decomposition (CAD) takes a formula from the elementary theory of real closed fields as input, and constructs a CAD of the space of unquantified variables. This decomposition is truth invariant with respect to the input formula, meaning that the formula is either identically true or identically false in each cell of the decomposition. The method determines the truth of the input formula for each cell of the CAD, and then uses the CAD to construct a solution formula—a quantifier-free formula that is equivalent to the input formula (see Collins, 1975; Collins and Hong, 1991; Hong, 1992). Simple equivalent formulas can be constructed from a truth-invariant CAD (see Hong, 1992; Brown and Collins, 1996), which motivates the consideration of both quantified and unquantified input formulas. There are other uses for this truth-invariant CAD as well, such as determining the topology of the input formula’s solution space or producing a visualization of the solution space. Often there will exist a “simpler” truth-invariant CAD than the one produced by the method.