A generic projection operator for partial cylindrical algebraic decomposition

Abstract

This paper provides a starting point for generic quantifier elimination by partial cylindrical algebraic decomposition pcad. On input of a first-order formula over the reals generic pcad outputs a theory and a quantifier-free formula. The theory is a set of negated equations in the free variables of the input formula. The quantifier-free formula is equivalent to the input for all parameter values satisfying the theory. For obtaining this generic elimination procedure, we derive a generic projection operator from the standard Collins--Hong projection operator. Our operator particularly addresses formulas with many parameters thus filling a gap in the applicability of pcad. It restricts decomposition to a reasonable subset of the entire space. The above-mentioned theory describes this subset. The approach is compatible with other improvements in the framework of pcad. It turns out that the theory contains assumptions that are easily interpretable and that are most often non-degeneracy conditions. The applicability of our generic elimination procedure significantly extends that of the corresponding regular procedure. Our procedure is implemented in the computer logic system redlog.