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arxiv: 2604.26400 · v1 · submitted 2026-04-29 · 💻 cs.SC · cs.LO

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Pseudo-Complex Quantifier Elimination

Nicolas Faro{\ss}, Thomas Sturm

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Pith reviewed 2026-05-07 12:39 UTC · model grok-4.3

classification 💻 cs.SC cs.LO
keywords quantifier eliminationcomplex numbersordered ringsreal quantifier eliminationheuristic reinterpretationsymbolic computationdecision procedures
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The pith

Quantifier elimination for the complex numbers reduces to real quantifier elimination followed by heuristic reinterpretation in a language with imaginary unit, real parts, imaginary parts, and conjugates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a framework for eliminating quantifiers from formulas over the complex numbers. The language is that of ordered rings extended with symbols for the imaginary unit, real and imaginary parts, and complex conjugates. The method first translates the input to an equivalent real quantifier elimination problem, solves it there, and then applies a heuristic step to rewrite the output back into the original complex language. If the reinterpretation works reliably, existing real quantifier elimination procedures become usable for complex decision problems without building entirely new complex-specific algorithms.

Core claim

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.

What carries the argument

Reduction of pseudo-complex formulas to real quantifier elimination followed by heuristic reinterpretation that restores the imaginary unit, real/imaginary parts, and conjugates.

If this is right

  • Decision procedures for properties of complex polynomials and expressions become available by reusing any real quantifier elimination engine.
  • Formulas involving the imaginary unit i, Re, Im, and conjugate operations can be handled directly in the input language.
  • The approach yields a working prototype that produces concrete elimination results for sample problems.
  • Extension to other algebraic domains becomes possible by repeating the same reduction-and-reinterpretation pattern.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could shorten development time for complex symbolic solvers by leveraging the maturity of real quantifier elimination implementations.
  • If the heuristic covers all common cases that arise in practice, the framework would support automated verification tasks in control theory or signal processing that mix real and imaginary components.
  • Similar reduction strategies might apply to other number systems equipped with an involution, such as quaternions or Clifford algebras.

Load-bearing premise

The heuristic reinterpretation step correctly and completely maps results from real quantifier elimination back into the pseudo-complex language without introducing errors or incompleteness.

What would settle it

A concrete complex formula with quantifiers whose elimination result, when checked by direct substitution or an independent complex solver, differs from the output produced by the reduction-plus-reinterpretation procedure.

Figures

Figures reproduced from arXiv: 2604.26400 by Nicolas Faro{\ss}, Thomas Sturm.

Figure 1
Figure 1. Figure 1: Circuit diagram of a passive RC high-pass filter Example 10 (Gain of passive RC high-pass filter) view at source ↗
Figure 2
Figure 2. Figure 2: Circuit diagram of an active RC filter; see [18, Example 1] normal form used time (s) 3 Cartesian coordinates conjugate 0.04 4 Roots of unity, φ1 conjugate 0.03 4 Roots of unity, φ2 conjugate 1.23 5 Counterexample for geometry provers conjugate 0.07 6 Orthogonality (n = 3) conjugate 0.2 7 Cauchy-Schwarz inequality (n = 4) conjugate 24.57 8 Self-adjoint matrices, φ1 conjugate 1.28 8 Self-adjoint matrices, φ… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper describes 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. It reduces complex QE problems to real quantifier elimination followed by a heuristic reinterpretation of the results back into the pseudo-complex language, and illustrates the approach with computational examples from a prototypical implementation in the Python-based Logic1 system.

Significance. If the heuristic reinterpretation can be formalized and validated, the reduction to established real QE procedures would offer a practical route to QE over complexes in this extended language, with potential applications in symbolic computation. The open-source Logic1 implementation and reported examples provide a concrete basis for reproducibility and empirical testing, which is a positive aspect of the work.

major comments (1)
  1. [Abstract] Abstract: the central claim that the procedure constitutes a quantifier elimination framework rests on the heuristic reinterpretation step correctly and completely mapping real QE outputs back into the target language. No soundness or completeness argument, decision procedure, or exhaustive case analysis for this step is indicated, and the description explicitly labels it heuristic while reporting only prototypical examples; this is load-bearing because a counterexample (e.g., involving conjugates and ordering) would falsify the claim that quantifiers are eliminated correctly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance and reproducibility. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the procedure constitutes a quantifier elimination framework rests on the heuristic reinterpretation step correctly and completely mapping real QE outputs back into the target language. No soundness or completeness argument, decision procedure, or exhaustive case analysis for this step is indicated, and the description explicitly labels it heuristic while reporting only prototypical examples; this is load-bearing because a counterexample (e.g., involving conjugates and ordering) would falsify the claim that quantifiers are eliminated correctly.

    Authors: We agree that the reinterpretation step is heuristic and that the manuscript does not provide a soundness or completeness proof for it. The abstract and body of the paper explicitly describe the method as a reduction from complex QE to real QE followed by heuristic reinterpretation, with the work centered on the design of this reduction and its prototypical implementation in Logic1, illustrated by computational examples. We do not claim that the current procedure is a complete, proven decision procedure for the full language; the framework is presented as a practical reduction technique whose reinterpretation component is heuristic at this stage. A counterexample would indeed show the limits of the heuristic, which is why we report only examples rather than a general guarantee. In the revised manuscript we will (i) rephrase the abstract to emphasize that the framework relies on heuristic reinterpretation and (ii) add a short discussion of its scope and limitations in the introduction and conclusion. We believe these clarifications address the concern while preserving the paper’s focus on the reduction approach and the open-source implementation. revision: partial

Circularity Check

0 steps flagged

No circularity: reduction to real QE plus heuristic reinterpretation is a design proposal, not a self-referential derivation

full rationale

The paper presents a framework that reduces quantifier elimination over pseudo-complex numbers to real quantifier elimination followed by a heuristic reinterpretation step, supported by prototypical examples in Logic1. No step in the described chain reduces by construction to its own inputs, renames a fitted parameter as a prediction, or relies on self-citation for a load-bearing uniqueness claim. The method is offered as an engineering approach with acknowledged heuristic elements rather than a closed first-principles derivation, making it self-contained against external real QE benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard algebraic properties of the complex numbers and the correctness of existing real quantifier elimination algorithms; the heuristic reinterpretation is an unproven domain assumption.

axioms (2)
  • domain assumption Complex numbers can be treated in the language of ordered rings extended by symbols for i, Re, Im, and conjugate, with standard algebraic identities holding.
    Invoked in the definition of the target language for QE.
  • standard math Real quantifier elimination procedures are sound and complete for the real ordered field.
    Used as the base for the reduction step.

pith-pipeline@v0.9.0 · 5346 in / 1216 out tokens · 37922 ms · 2026-05-07T12:39:25.325716+00:00 · methodology

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Reference graph

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