Deciding polynomial-transcendental problems

This paper presents a decision procedure for a certain class of sentences of first order logic involving integral polynomials and a certain specific analytic transcendental function trans(x) in which the variables range over the real numbers. The list of transcendental functions to which our decision method directly applies includes exp(x)exp(x), the exponential function with respect to base ee, ln(x)ln(x), the natural logarithm of xx, and arctan(x)arctan(x), the inverse tangent function. The inputs to the decision procedure are prenex sentences in which only the outermost quantified variable can occur in the transcendental function. In the case trans(x)=exp(x), the decision procedure has been implemented in the computer logic system REDLOG. It is shown how to transform a sentence involving a transcendental function from a much wider collection of functions (such as hyperbolic and Gaussian functions, and trigonometric functions on a certain bounded interval) into a sentence to which our decision method directly applies. Closely related work is reported by Anai and Weispfenning (2000), Collins (1998), Maignan (1998), Richardson (1991), Strzebonski (in press) and Weispfenning (2000).

► A decision procedure for classes of formulae of real algebra extended by transcendental functions.
► Additional functions include exp,arctanexp,arctan, and loglog, hyperbolic and trigonometric functions on suitable intervals.
► For the exponential function the decision procedure is implemented in the REDLOG-package of REDUCE.
► Proofs involve Lindemann’s theorem, but not Schanuel’s conjecture.