Undecidability of Interpolant Existence for Two-Variable First-Order Logic with Two Equivalence Relations

Core Concepts

The interpolant existence problem is undecidable for the two-variable fragment of first-order logic with two equivalence relations, as well as for the two-variable guarded fragment with individual constants and two equivalence relations.

Abstract

The paper investigates the interpolant existence problem (IEP) for fragments of first-order logic (FO) without the Craig interpolation property. The IEP is the problem of deciding, given FO-formulas φ and ψ, whether there exists a formula ι built from the shared symbols of φ and ψ such that φ entails ι and ι entails ψ.
The authors show that the IEP is undecidable for the following fragments:

FO22E: The two-variable fragment of FO with two equivalence relations.
GF22Ec: The two-variable guarded fragment of FO with individual constants and two equivalence relations.

The undecidability is shown by a reduction from the undecidable infinite Post Correspondence Problem (ωPCP). The key idea is to construct FO22E-formulas φ and ¬ψ such that the ωPCP instance has a solution if and only if φ and ¬ψ are satisfied in FO22E(ρ)-bisimilar pointed structures, where ρ is the shared signature of φ and ψ.
The authors also show that the explicit definition existence problem (EDEP) is undecidable for these logics, as the IEP and EDEP are polynomially reducible to each other.

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Key Insights Distilled From

Interpolant Existence is Undecidable for Two-Variable First-Order Logic with Two Equivalence Relations

by Frank Wolter... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02683.pdf

Deeper Inquiries

What is the status of the IEP for the one-variable fragment of first-order logic with one equivalence relation

The status of the Interpolant Existence Problem (IEP) for the one-variable fragment of first-order logic with one equivalence relation is currently undetermined based on the context provided. The undecidability result has been established for the two-variable fragment with two equivalence relations, but specific information regarding the one-variable fragment with one equivalence relation is not provided in the text. Further research or analysis would be needed to determine the status of the IEP for this particular fragment.

Can the undecidability result be extended to other decidable fragments of first-order logic, such as the guarded fragment or the two-variable fragment without equivalence relations

The undecidability result for the IEP in the context provided can potentially be extended to other decidable fragments of first-order logic, such as the guarded fragment or the two-variable fragment without equivalence relations. The key lies in the complexity of the logic and the presence of specific properties like the Craig interpolation property. If these fragments exhibit similar characteristics to the two-variable fragment with two equivalence relations, it is likely that the undecidability result can be extended to them as well.

Are there any decidable fragments of first-order logic for which the IEP is decidable, even though the logic does not enjoy the Craig interpolation property

While the undecidability of the IEP has been demonstrated for certain fragments of first-order logic, there may exist decidable fragments where the IEP is also decidable, even if the logic does not have the Craig interpolation property. The decidable fragments would need to have specific structural properties or restrictions that enable the effective resolution of the interpolant existence problem. Further research into different fragments and their properties would be necessary to identify such decidable cases.