Translation from First-Order Logic to Calculus of Relations

Core Concepts

Linear-size translation from FO= formulas into CoR equations preserving validity and finite validity.

Abstract

This content discusses a translation technique from first-order logic (FO=) into the calculus of relations (CoR). It presents a linear-size translation that preserves both validity and finite validity, providing a conservative reduction. The article covers basic definitions of FO= and CoR, recursive translations, k-tuple structures, and preliminary concepts. It also explores reductions to a more restricted syntax of CoR and Tseitin translations for CoR equations. Additionally, it delves into the Godel class and provides examples of translated equations.

Stats

Our translation gives a linear-size conservative reduction from FO= formulas into formulas of the three-variable fragment of first-order logic. There is a linear-size translation from CoR terms into FO3= formulas with two free variables preserving semantic equivalence. Equations in CoR can be translated into equations without converse or identity preserving validity and finite validity.

Quotes

"There is a recursive translation from FO= formulas into CoR equations preserving validity." "Our translation presents a conservative reduction from FO= formulas to FO3= formulas."

Key Insights Distilled From

Note on a Translation from First-Order Logic into the Calculus of Relations Preserving Validity and Finite Validity

by Yoshiki Naka... at arxiv.org 03-25-2024

https://arxiv.org/pdf/2310.02845.pdf

Deeper Inquiries

How does the elimination of converse and identity impact the expressiveness of translated equations?

The elimination of converse and identity in CoR terms simplifies the structure of the equations, making them more manageable and reducing complexity. By removing these elements, we restrict the operations that can be performed within the equations. This restriction can limit the range of relationships that can be expressed in the translated equations. Converse allows for reversing relations, while identity ensures reflexivity in relations. Without these components, certain types of relationships or properties may not be representable in the translated equations.

What are the implications of reducing alternations of operations in CoR terms?

Reducing alternations of operations in CoR terms has significant implications for both theoretical understanding and practical applications. By limiting how many times different operations like dot-dagger or intersection can alternate within a term, we simplify the structure and potentially improve computational efficiency. This reduction can lead to easier analysis and manipulation of CoR expressions. In practical applications, fewer alternations mean less complex computations, which could result in faster processing times and reduced resource requirements. It also enhances readability and interpretability by streamlining expressions without sacrificing expressive power significantly.

How might the undecidability result for certain classes affect practical applications?

The undecidability result for certain classes implies that there are limitations to what can be algorithmically determined or computed within those classes. In practical applications where decision-making processes rely on automated reasoning or verification using these classes (such as relation algebras), this limitation poses challenges. For instance, tasks like automated theorem proving or model checking may encounter instances where solutions cannot be guaranteed due to undecidability issues. This could lead to incomplete results, requiring manual intervention or alternative approaches to address uncertainties arising from undecidable problems. Understanding these limitations is crucial for designing robust systems that account for potential incompleteness or uncertainty when dealing with problems falling within undecidable classes like relation algebras.