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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 6, Pages 173–216 (Mi izv8440)  

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

F. N. Pakhomov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The polymodal provability logic $\mathrm{GLP}$ 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 $\mathrm{GLP}$-algebra generated by the constants $\mathbf{0}$, $\mathbf{1}$ is decidable [1]. For every positive integer $n$ we solve the corresponding question for the logics $\mathrm{GLP}_n$ that are the fragments of $\mathrm{GLP}$ with $n$ modalities. We prove that the elementary theory of the free $\mathrm{GLP}_n$-algebra generated by the constants $\mathbf{0}$, $\mathbf{1}$ is decidable for all $n$. We introduce the notion of a linear $\mathrm{GLP}_n$-algebra and prove that all free $\mathrm{GLP}_n$-algebras generated by the constants $\mathbf{0}$, $\mathbf{1}$ are linear. We also consider the more general case of the logics $\mathrm{GLP}_\alpha$ whose modalities are indexed by the elements of a linearly ordered set $\alpha$: we define the notion of a linear algebra and prove the latter result in this case.

Keywords: provability logics, modal algebras, free algebras, elementary theories, Japaridze logic.

Funding Agency Grant Number
Russian Science Foundation 16-11-10252
This work is supported by the Russian Science Foundation under grant 16-11-10252.


DOI: https://doi.org/10.4213/im8440

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English version:
Izvestiya: Mathematics, 2016, 80:6, 1159–1199

Bibliographic databases:

UDC: 512.572
MSC: 03F45, 03B25
Received: 20.05.2016

Citation: F. N. Pakhomov, “Linear $\mathrm{GLP}$-algebras and their elementary theories”, Izv. RAN. Ser. Mat., 80:6 (2016), 173–216; Izv. Math., 80:6 (2016), 1159–1199

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