Paper Info
Title
Definability of linear equation systems over groups and rings
Abstract
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 a l l 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 from [12, 20, 8] on the logical definability of linear-algebraic problems over finite fields.
Year
DOI
Venue
2013
10.2168/LMCS-9(4:12)2013
LOGICAL METHODS IN COMPUTER SCIENCE
Keywords
DocType
Volume
finite model theory,logics with algebraic operators
Journal
9
Issue
ISSN
Citations 
4
1860-5974
0
PageRank 
References 
Authors
0.34
0
5
Name
Order
Citations
PageRank
anuj dawar188377.18
Erich Grädel21211114.35
Bjarki Holm3262.61
Eryk Kopczynski4649.68
Wied Pakusa5114.18