Efficient Decision Procedures for RNmatrix Semantics

2026-05-03

AI Summary

This work addresses the longstanding absence of efficient decision procedures based on restricted nondeterministic matrix (RNmatrix) semantics, which has hindered automated theorem proving for paraconsistent, intuitionistic, and modal logics. The authors encode RNmatrix semantics—along with its row-elimination criteria—as SMT problems, leveraging off-the-shelf SMT solvers to decide formula validity and construct countermodels, thereby yielding a general-purpose automated prover. Their approach achieves the first complete implementation of a decision procedure for the entire $C_n$ hierarchy of paraconsistent logics, outperforming the state-of-the-art tools in this domain while attaining competitive performance in intuitionistic and modal logics. This breakthrough effectively overcomes the historical bottleneck of lacking scalable and practical reasoning mechanisms grounded in RNmatrix semantics.

Abstract

Restricted non-deterministic matrices (RNmatrices) impose constraints on the rows of non-deterministic matrices (Nmatrices), filtering out ``unsound" rows and retaining only ``valid" ones. This yields a more expressive framework than standard Nmatrices. Although this approach enables sound and complete semantics for a broad class of logics, \eg, paraconsistent logics, propositional intuitionistic logic, and the fifteen normal modal logics of the modal cube, no {\em efficient} decision procedures based on these semantics have been proposed. In this paper, we implement the RNmatrix framework to develop a new suite of automated theorem provers for these logics. By encoding RNmatrices and their elimination criteria as Satisfiability Modulo Theories (SMT) problems, we leverage SMT solvers to decide formula validity and construct countermodels. We illustrate the method for paraconsistent logics, where our prover outperforms the current state-of-the-art and provides the first implementation for the entire $C_n$ hierarchy, as well as for intuitionistic and modal logics, where our general-purpose prover achieves competitive performance.

Problem

• RNmatrices
• decision procedures
• non-deterministic matrices
• automated theorem proving
• logical semantics

Innovation

• RNmatrices
• SMT encoding
• automated theorem proving
• paraconsistent logics
• decision procedures