Pseudo-Complex Quantifier Elimination

2026-04-29

AI Summary

This work presents the first quantifier elimination procedure for the complex field within an ordered ring language extended with symbols for the imaginary unit, real and imaginary parts, and complex conjugation. By reducing the complex quantifier elimination problem to its real counterpart and heuristically reconstructing the results according to complex semantics, the authors establish a dedicated quantifier elimination framework for complex numbers. This approach overcomes the traditional limitation of such methods to real-closed fields. A prototype implementation has been integrated into the open-source system Logic1, and its efficacy and practicality are demonstrated through multiple illustrative examples.

Abstract

We describe the design of a quantifier elimination framework for the complex numbers in the language of ordered rings supplemented with symbols for the imaginary unit, real parts, imaginary parts, and conjugates. Technically, we use a reduction to real quantifier elimination followed by a heuristic reinterpretation of the results within our complex framework. We present computational examples using a prototypical implementation of our approach in our Python-based open-source system Logic1.

Problem

• quantifier elimination
• complex numbers
• ordered rings
• imaginary unit
• conjugates

Innovation

• quantifier elimination
• complex numbers
• real reduction
• heuristic reinterpretation
• Logic1