MetiTarski: An Automatic Prover for the Elementary Functions

Abstract

Abstract. Many inequalities involving the functions ln, exp, sin, cos, etc., can be proved automatically by MetiTarski: a resolution theorem prover (Metis) modified to call a decision procedure (QEPCAD) for the theory of real closed fields. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts, while deleting as redundant clauses that follow algebraically from other clauses. MetiTarski includes special code to simplify arithmetic expressions.

MetiTarski’s Menagerie of Cooperating Systems

“…Instead of having to write an entire theorem prover, we would merely need to write some interface code and modify certain standard aspects of resolution. Arithmetic simplification obviously had to be introduced (for example, to identify 2x + y with x + y + 0 + x), and the standard mechanisms for selecting the most promising clause and literal were tuned to our application [1,2]. Early versions of MetiTarski performed well despite having only a modest amount of specialist code.…”

MetiTarski: Past and Future

“…Unfortunately, we found [2] that the Cohen-Hörmander procedure couldn't cope with polynomials of degree larger than five, which ruled out the use of accurate bounds. The next version of MetiTarski [3] invoked an external decision procedure, QEPCAD [12], which implements a much more powerful technique: cylindrical algebraic decomposition (CAD). More recently, we have integrated MetiTarski with the computer algebra system Mathematica, which contains a family of highly advanced RCF decision procedures.…”

Applications of MetiTarski in the Verification of Control and Hybrid Systems

“…We have been able to prove safety properties of the systems such as Room Heating and Navigation, which cannot be verified by HSolver. 3 We are planning to extend our case studies to cover nonlinear cases by finding methods of solving systems of polynomial nonlinear ordinary differential equations analytically in terms of elementary and special functions. An example of such method is the Prelle-Singer procedure [12], extensions of which are also implemented in computer algebra systems such as REDUCE (the PSODE package [13]) and Maple (the PSsolver package [8]).…”

“…MetiTarski [1][2][3] is a new automatic theorem prover for special functions over the real numbers. It consists of a resolution theorem prover (Metis) combined with a decision procedure (QEPCAD) for the theory of real closed fields.…”