Polynomial Ring Calculus for Modal Logics: A New Semantics and Proof Method for Modalities

Abstract

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.

Formal polynomials, heuristics and proofs in logic

“…The preceding theorem can be strengthened to non-deterministic finite-valued functions as well (and this makes it possible to use polynomial functions with extra-variables to treat non-truth functional logics such as paraconsistent logics and modal logics, (cf. [8] and [1]). Moreover, for fixed p n , there exists a polynomial-time transformation Π that outputs the corresponding polynomial of GF p n [X] for each truth-function, as it can be computed by elementary linear algebra (systems of linear equations) over finite fields.…”

“…Polynomial ring calculus are particularly appropriate for automatic proof systems not only for finitely many-valued logics, but also for non-truth-functional logics, including modal logics (cf. [1]): even logics that have no finite-valued characteristic semantics, as the paraconsistent logics, can be given a two-valued dyadic semantics expressed by multivariable polynomials over the ring…”

“…In [1] a polynomial ring calculus (PRC) for the familiar modal logic S5 was designed, which permits to perform modal deductions through polynomial handling. The paper also investigated the relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics and the Dijkstra-Scholten equationalproof style.…”

Differential and integral calculus for logical operations. A matrix-vector approach

“…Using this kind of polynomial rings has been possible to develop a novel semantic procedure to investigate logical modalities [2].…”

The method of polynomial ring calculus and its potentialities

“…• In certain cases, some constraints on translations will have to be added, as in the cases where modal logic are expressed in polynomial format (see [2]). …”