Universal truth of operator statements via ideal membership

TL;DR

This paper presents a novel algebraic framework for verifying statements about linear operators using ideal membership in noncommutative polynomial algebras, enabling automated proofs of operator identities.

Contribution

It introduces a semi-decision procedure that reduces first-order operator statements to ideal membership problems, integrating logical elimination techniques with algebraic computations.

Findings

Framework successfully proves operator identities using ideal membership.

Automated procedure incorporates linearity and benefits from efficient algebraic algorithms.

Illustrative examples demonstrate practical applicability with computer algebra software.

Abstract

We introduce a framework for proving statements about linear operators by verification of ideal membership in a free algebra. More specifically, arbitrary first-order statements about identities of morphisms in preadditive semicategories can be treated. We present a semi-decision procedure for validity of such formulas based on computations with noncommutative polynomials. These algebraic computations automatically incorporate linearity and benefit from efficient ideal membership procedures. In the framework, domains and codomains of operators are modelled using many-sorted first-order logic. To eliminate quantifiers and function symbols from logical formulas, we apply Herbrand's theorem and Ackermann's reduction. The validity of the resulting formulas is shown to be equivalent to finitely many ideal memberships of noncommutative polynomials. We explain all relevant concepts and discuss computational aspects. Furthermore, we illustrate our framework by proving concrete operator statements assisted by our computer algebra software.