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Deciding some Maltsev conditions in finite idempotent algebras [article]

Alexandr Kazda, Matt Valeriote
2019 arXiv   pre-print
In this paper we investigate the computational complexity of deciding if a given finite algebraic structure satisfies a fixed (strong) Maltsev condition Σ.  ...  Our goal in this paper is to show that Σ-testing can be accomplished in polynomial time when the algebras tested are idempotent and the Maltsev condition Σ can be described using paths.  ...  Path Maltsev conditions In this section, we will show how to express several classical Maltsev conditions using paths and how to efficiently decide them in finite idempotent algebras by checking that they  ... 
arXiv:1704.05928v2 fatcat:7i3kouxnqbbo5hks6z3hzloery

Deciding the existence of minority terms [article]

Alexandr Kazda, Jakub Opršal, Matt Valeriote, Dmitriy Zhuk
2019 arXiv   pre-print
This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation m that satisfies the minority equations m(y,x,x) ≈ m(x,y,x) ≈ m(x,x,y)  ...  We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.  ...  Local Maltsev terms In [5, 7, 8, 13] polynomial-time algorithms are presented for deciding if certain Maltsev conditions hold in the variety generated by a given finite idempotent algebra.  ... 
arXiv:1901.00316v2 fatcat:b5hybyqzujfcpcdnpq7p57mzbu

Algebras and Algorithms

Matt Valeriote
2015 2015 IEEE International Symposium on Multiple-Valued Logic  
When considering computational questions for finite algebras we further assume that they have a finite number of basic operations.  ...  Two algebras are similar if their basic operations are indexed by the same set and the arities of similarly indexed operations are the same. A is a finite algebra if its universe is a finite set.  ...  Some types of idempotent Maltsev conditions are easier to test for over idempotent algebras.  ... 
doi:10.1109/ismvl.2015.45 dblp:conf/ismvl/Valeriote15 fatcat:3eja6v2wvvamre4t6st32us7fm

Deciding the existence of quasi weak near unanimity terms in finite algebras [article]

Alexandr Kazda
2021 arXiv   pre-print
We also observe that the problem of deciding if a given finite algebra has a quasi Taylor operation is solvable in polynomial time by looking, essentially, for local quasi Siggers operations.  ...  We show that for a fixed positive integer k one can efficiently decide if a finite algebra A admits a k-ary weak near unanimity operation by looking at the local behavior of the terms of A.  ...  This is in contrast to the numerous examples of idempotent Maltsev conditions that are EXPTIME-complete to decide in general finite algebras [7, 9] .  ... 
arXiv:2002.06083v3 fatcat:ntok5sh2sjgi3i4nh42j36dmlq

The local-global property for G-invariant terms [article]

Alexandr Kazda, Michael Kompatscher
2021 arXiv   pre-print
For some Maltsev conditions Σ it is enough to check if a finite algebra 𝐀 satisfies Σ locally on subsets of bounded size, in order to decide, whether 𝐀 satisfies Σ (globally).  ...  This local-global property is the main known source of tractability results for deciding Maltsev conditions.  ...  Acknowledgements We thank Matt Valeriote for suggesting the question of studying the conditions Σ .  ... 
arXiv:2109.02065v2 fatcat:sgqbk5xrw5ekldtopmf3wtgl34

Equations implying congruence n-permutability and semidistributivity

Ralph Freese
2013 Algebra Universalis  
In this paper we investigate other types of derivatives that give similar results for congruence n-permutable for some n, and for congruence semidistributivity.  ...  In [3] T. Dent, K. Kearnes andÁ. Szendrei define the derivative, Σ , of a set of equations Σ and show, for idempotent Σ, that Σ implies congruence modularity if Σ is inconsistent (Σ |= x ≈ y).  ...  (The Day terms work.) (3) The converse of the first statement is not true in general but it is true if Σ is linear. (4) For a finite linear, idempotent Σ one can effectively decide if Σ implies CM.  ... 
doi:10.1007/s00012-013-0256-x fatcat:lcd5uw7hnjgxrajgekmvvmqb64

A new algorithm for constraint satisfaction problems with Maltsev templates [article]

Dejan Delic, Aklilu Habte
2017 arXiv   pre-print
In this article, we provide a new algorithm for solving constraint satisfaction problems with Maltsev constraints, based on the new notion of Maltsev consistency.  ...  Simple Idempotent Algebras in Maltsev Varieties. Let A be an algebra.  ...  [15] ) and is closely related to modules over rings of matrices whose entries come from some fixed finite field: A finite idempotent Abelian algebra A is strictly simple if and only if there exist a  ... 
arXiv:1709.08311v2 fatcat:o36fwrdoprfczpqxkh7nlbvaxi

Asking the metaquestions in constraint tractability [article]

Hubie Chen, Benoit Larose
2017 arXiv   pre-print
Among other results, we prove the NP-completeness of deciding a condition conjectured to characterize the tractable problems CSP(H), as well as the NP-completeness of deciding if CSP(H) has bounded width  ...  The CSP is in general NP-hard; a common way to restrict this problem is to fix the second structure H, so that each structure H gives rise to a problem CSP(H).  ...  A strong linear Maltsev condition is a finite set of linear identities {E 1 , . . . , E r }.  ... 
arXiv:1604.00932v2 fatcat:lxq7lwrs2jb3hiajsjrss6klny

Simpler Maltsev conditions for (weak) difference terms in locally finite varieties

Keith A. Kearnes, Ágnes Szendrei, Ross Willard
2017 Algebra Universalis  
This paper is motivated by a practical question: given a finite algebra A in a finite language, how can we best program a computer to decide whether the variety generated by A has a di↵erence term, and  ...  To help address this question we produce a simple Maltsev condition which characterizes di↵erence terms in the class of locally finite varieties. We do the same for weak di↵erence terms.  ...  Simple Maltsev conditions Following [5, Def. 5.2] , given congruences ↵, , of an algebra, define ⌧ (↵, , ) to be the transitive closure of [ (↵ \ ( (↵ \ ) )).  ... 
doi:10.1007/s00012-017-0475-7 fatcat:efrbbwwwyfabnlud3xziovxs7y

SOLVABILITY OF SYSTEMS OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

LÁSZLÓ ZÁDORI
2007 International journal of algebra and computation  
By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition.  ...  We prove that the problem has a dichotomy in the class of finite groupoids with an identity element.  ...  Admitting a non-trivial idempotent Maltsev condition is a decidable property of finite algebras of finite signature, see [6] , and is even in P in the case of idempotent algebras, as shown in [1] .  ... 
doi:10.1142/s0218196707003809 fatcat:4clxdhuqlzdrzabgqhkygwda2m

ON THE COMPLEXITY OF SOME MALTSEV CONDITIONS

RALPH FREESE, MATTHEW A. VALERIOTE
2009 International journal of algebra and computation  
This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity.  ...  For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete.  ...  If A is a finite idempotent algebra and i ∈ typ(V(A)) then there is a finite strictly simple algebra S of type j for some j ≤ i in H S(A).  ... 
doi:10.1142/s0218196709004956 fatcat:tktlwduwabg2diycwowd3cbwzm

Polynomial-Time Tests for Difference Terms in Idempotent Varieties

William Demeo, Ralph Freese, Matthew Valeriote
2019 International journal of algebra and computation  
We consider the following practical question: given a finite algebra A in a finite language, can we efficiently decide whether the variety generated by A has a difference term?  ...  We answer this question (positively) in the idempotent case and then describe algorithms for constructing difference term operations.  ...  If A is a finite idempotent algebra and i ∈ typ(V(A)) then there is a finite strictly simple algebra S of type j for some j i in HS(A).  ... 
doi:10.1142/s021819671950036x fatcat:p5wehl2zj5b27ckcc3qgfw5nui

Random Models of Idempotent Linear Maltsev Conditions. I. Idemprimality [article]

Clifford Bergman, Agnes Szendrei
2019 arXiv   pre-print
We extend a well-known theorem of Murskiǐ to the probability space of finite models of a system M of identities of a strong idempotent linear Maltsev condition.  ...  This implies that even if such an M is distinguishable from another idempotent linear Maltsev condition by a finite model A of M, a random search for a finite model A of M with this property will almost  ...  Idemprimality for random finite models of some familiar strong idempotent linear Maltsev conditionsin the last column of the table.  ... 
arXiv:1901.06316v1 fatcat:674bmhcapzccbapaniehujbrci

Decidability of absorption in relational structures of bounded width

Jakub Bulín
2014 Algebra Universalis  
Absorption theory of Barto and Kozik has proven to be a very useful tool in the algebraic approach to the Constraint Satisfaction Problem and structure of finite algebras in general.  ...  We address the following problem: Given a finite relational structure A and a subset B ⊆ A, is it decidable whether B is an absorbing subuniverse?  ...  Acknowledgements The author would like to thank Libor Barto and Marcin Kozik for their thoughtful comments and helpful discussions and the anonymous reviewer for pointing out an error in the manuscript  ... 
doi:10.1007/s00012-014-0283-2 fatcat:7vxge3redfgi7b5xdgz5ddwlyq

Testing for a Semilattice Term

Ralph Freese, J. B. Nation, Matt Valeriote
2018 Order  
Introduction Many of the poly-time algorithms for testing for the satisfaction of a fixed special Maltsev condition in a given finite idempotent algebra involve checking if there are enough "local versions  ...  The problem of deciding if, given a finite idempotent algebra A and an element 0 ∈ A, that the condition from the previous Lemma holds for A and 0 is in P . Proof.  ... 
doi:10.1007/s11083-018-9455-6 fatcat:spz7ifi7brdpjjwevurfi6akby
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