Dawar, Anuj and Kopczynski, Eryk and Holm, Bjarki and Grädel, Erich and Pakusa, Wied - Definability of linear equation systems over groups and rings

lmcs:725 - Logical Methods in Computer Science, November 14, 2013, Volume 9, Issue 4
Definability of linear equation systems over groups and rings

Authors: Dawar, Anuj and Kopczynski, Eryk and Holm, Bjarki and Grädel, Erich and Pakusa, Wied

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.


Source : oai:arXiv.org:1204.3022
DOI : 10.2168/LMCS-9(4:12)2013
Volume: Volume 9, Issue 4
Published on: November 14, 2013
Submitted on: January 29, 2013
Keywords: Computer Science - Logic in Computer Science,Computer Science - Computational Complexity


Share

Consultation statistics

This page has been seen 117 times.
This article's PDF has been downloaded 39 times.