Linear $ \mathrm{GLP}$-algebras and their elementary theories

Abstract

The polymodal provability logic GLP was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free GLP-algebra generated by the constants 0, 1 is decidable [1]. For every positive integer n we solve the corresponding question for the logics GLPn that are the fragments of GLP with n modalities. We prove that the elementary theory of the free GLPn-algebra generated by the constants 0, 1 is decidable for all n. We introduce the notion of a linear GLPn-algebra and prove that all free GLPn-algebras generated by the constants 0, 1 are linear. We also consider the more general case of the logics GLPα whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.

Certain Bounds of Formulas in Free Temporal Algebras

“…We believe that we have introduced a tool that could be useful as part of a demonstration technique. In the future, we will investigate the role of this construction in the field of polymodal algebras (see [11,14,17,19]). However, in the short term, contributing to the study of order and atomicity in temporal varieties along the lines of [7] will be a high priority for us.…”

“…Those with prospective content are divided into two groups: those that are like [1][2][3]13] have been useful for us to be able to capture the fine nuances of astute thinkers and try to imprint them on our algebraic approaches, but their influence on the work lies in the background of our research training. The second group of prospective papers are those that open up fields of technological applications beyond the scope of the present work; they are of the [4,5,14] type. They indicate that the theoretical mathematics we elaborate could underlie certain applications.…”

“…In Section 8, we detail how [8] suggests a field of possible applications of our work. Finally, [11,12,14,[17][18][19] have served us only to intuit a second field of applications or evolution of the techniques developed in this work. Finally, [9,20] are instrumental works where instruments are developed that we now use here, as indicated in the appropriate places; that was their purpose and that is what they were written for.…”