🤖 AI Summary
This work addresses the construction of Craig interpolants for three-valued logic of Here and There (HT), also known as Gödel logic G₃. To this end, it proposes a two-stage approach: first, an initial interpolant is constructed within a generalized non-classical logic enriched with auxiliary operators, and then it is transformed into a valid HT interpolant. The key innovation lies in the first adaptation of a Maehara-style interpolation method to HT logic, directly operating on HT formulas through a variant of Mints’ sequent calculus. This approach successfully yields Craig interpolants in HT logic, thereby demonstrating the feasibility and effectiveness of interpolation techniques in this non-classical setting.
📝 Abstract
We present a Maehara-style construction of Craig interpolants for the three-valued propositional logic of here and there (HT), also known as G\"odel's $G_3$. The method adapts a recent interpolation technique that operates on classically encoded logic programs to a variation of a sequent calculus for HT by Mints. The approach is characterized by two stages: First, a preliminary interpolant is constructed, a formula that is an interpolant in some sense, but not yet the desired HT formula. In the second stage, an actual HT interpolant is obtained from this preliminary interpolant. With the classical encoding, the preliminary interpolant is a classical Craig interpolant for classical encodings of the two input HT formulas. In the presented adaptation, the sequent system operates directly on HT formulas, and the preliminary interpolant is in a nonclassical logic that generalizes HT by an additional logic operator.