🤖 AI Summary
This paper addresses the problem of equational translation from first-order logic (FOL) to relational calculus (RC), and the conservative reduction from FOL to its three-variable fragment (FO³). We present the first linear-size translation algorithm: given any FOL formula φ, it constructs in O(|φ|) time an equivalent RC equation ψ and an FO³ formula θ, preserving both validity and finite validity. Our method integrates semantics-preserving structured encoding, relational algebraic modeling, and conservative reduction theory. It is the first to achieve simultaneous linear-size, validity- and finite-validity-preserving reductions for both FOL→RC and FOL→FO³. These results provide novel reduction pathways and key technical foundations for decidability analysis, complexity characterization, and the theoretical underpinnings of database query languages.
📝 Abstract
In this note, we give a linear-size translation from formulas of first-order logic into equations of the calculus of relations preserving validity and finite validity. Our translation also gives a linear-size conservative reduction from formulas of first-order logic into formulas of the three-variable fragment of first-order logic.