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Complexity issues of checking identities in finite monoids

Ondřej Klíma
2009 Semigroup Forum  
We study the computational complexity of checking identities in a fixed finite monoid.  ...  We find the smallest monoid for which this problem is coNPcomplete and describe a significant class of finite monoids for which the problem is tractable.  ...  my language errors and suggestions how to improve the presentation of the paper.  ... 
doi:10.1007/s00233-009-9180-y fatcat:psx6schakndj3e72yhhnht4kka

Complexity of the identity checking problem for finite semigroups

J. Almeida, M. V. Volkov, S. V. Goldberg
2009 Journal of Mathematical Sciences  
We prove that the identity checking problem in a finite semigroup S is co-NP-complete whenever S has a nonsolvable subgroup or S is the semigroup of all transformations on a 3-element set.  ...  However an investigation of the computational complexity of this problem has started only recently and has brought rather unexpected results.  ...  However, combining Corollary 1 with some known results, one can completely classify some important series of semigroups with respect to the complexity of identity checking.  ... 
doi:10.1007/s10958-009-9397-z fatcat:qn6axkizirhbxdockdfebpaqhi

A Polynomial Time Algorithm for Left [Right] Local Testability [chapter]

A.N. Trahtman
2003 Lecture Notes in Computer Science  
and for checking the transition graph of the automaton with locally idempotent semigroup.  ...  A semigroup is called semigroup of left [right] zeroes if satises the identity xy = x [xy = y].  ...  (y(n 2 ) time complexity). The whole time and space complexity of the algorithm is y(n 2 ).  ... 
doi:10.1007/3-540-44977-9_19 fatcat:5etk7zvkevht5n7v2cj2tpc7am

Polynomial time algorithm for left [right] local testability [article]

A.N. Trahtman
2020 arXiv   pre-print
the words of length k coincide, (2) the set of segments of length k of the words as well as 3) the order of the first appearance of these segments in prefixes [suffixes] coincide.  ...  Polynomial time algorithm verifies transition graph of automaton with locally idempotent transition semi group.  ...  The whole time and space complexity of the algorithm is O(n 3 ). Γ 2 . (O(n 2 ) time complexity). Let us check the local idempotency (O(n 3 ) time complexity).  ... 
arXiv:2011.04236v1 fatcat:cqqsr4mglfb4zldtlljbkgbx4u

A minimal nonfinitely based semigroup whose variety is polynomially recognizable

M. V. Volkov, S. V. Goldberg, S. I. Kublanovsky
2011 Journal of Mathematical Sciences  
We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.  ...  The first and the second authors acknowledge support from the Federal Education Agency of Russia, project 2.1.1/3537, and from the Russian Foundation for Basic Research, grants 09-01-12142 and 10-01-00524  ...  Under the premise of the lemma, in order to check whether or not a given finite semigroup A belongs to the variety var B, it suffices to check whether or not A satisfies all identities in Σ.  ... 
doi:10.1007/s10958-011-0512-6 fatcat:enlmyvm4krhvtjcircdu7fvxra

The Quantum Query Complexity of Algebraic Properties [article]

Sebastian Doern, Thomas Thierauf
2007 arXiv   pre-print
For a set S and a binary operation on S, we consider the decision problem whether S is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group.  ...  We present quantum query complexity bounds for testing algebraic properties.  ...  To check whether a pair (A, B) is marked, we search for a pair (b, c) Corollary 3. 2 2 The quantum query complexity of the semigroup problem is O(n 5 4 ), if M has constant size.Note that the time complexity  ... 
arXiv:0705.1446v1 fatcat:lrs3hgjszzborl2azodm3wfipy

Identities of the Kauffman Monoid K_4 and of the Jones monoid J_4 [article]

N. V. Kitov, M. V. Volkov
2019 arXiv   pre-print
This leads to a polynomial time algorithm to check whether a given identity holds in K_4. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid J_4.  ...  ., are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids K_3 and K_4 satisfy exactly the same identities.  ...  Thus, it remains to compute the numbers of circles in ξC ·ηC and in (ξη)C and to verify that these numbers are equal, that is, ξC · ηC = (ξη)C .  ... 
arXiv:1910.09190v1 fatcat:7mvdt7zvjngljflhw76hgp7y6a

Semigroups embeddable in hyperplane face monoids

Stuart Margolis, Franco Saliola, Benjamin Steinberg
2013 Semigroup Forum  
A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.  ...  A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup.  ...  Acknowledgements The authors would like to thank Igor Dolinka for helpful correspondence on questions related to this paper and Mark Sapir for explaining in detail the results of his thesis [32] that  ... 
doi:10.1007/s00233-013-9542-3 fatcat:5dzo25vulvbynhqlwdd6l7jdcm

Semigroups embeddable in hyperplane face monoids [article]

Stuart Margolis, Franco Saliola, Benjamin Steinberg
2012 arXiv   pre-print
A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.  ...  A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup.  ...  Acknowledgements The authors would like to thank Igor Dolinka for helpful correspondence on questions related to this paper and Mark Sapir for explaining in detail the results of his thesis [32] that  ... 
arXiv:1212.6683v1 fatcat:u3kbqt5qxzgmnpvxoql2ztlhdi

On the complexity of inverse semigroup conjugacy [article]

Trevor Jack
2021 arXiv   pre-print
We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups.  ...  We describe a connection between checking ~i conjugacy and checking membership in inverse semigroups.  ...  Both of the following problems are in AC 0 : (a) checking if ∼ o is the identity relation on S and (b) checking if ∼ p * is the identity relation on S.Proof.  ... 
arXiv:2111.07551v1 fatcat:4ichaz3qbzfw7ilpyb2guj2ic4

Verification Tools for Checking some kinds of Testability [article]

A.N. Trahtman
2021 arXiv   pre-print
The bounds on order of local testability of transition graph and order of local testability of transition semigroup are also found. For given k, the k-testability of transition graph is verified.  ...  A set of procedures for deciding whether or not a language given by its minimal automaton or by its syntactic semigroup is locally testable, right or left locally testable, threshold locally testable,  ...  So in semigroup case, we have O(22126 2 ) time complexity for both checking the local testability and finding the order.  ... 
arXiv:2106.02312v1 fatcat:offfcgrr7rbjjp7y6fhhkhyeou

THE COMPLEXITY OF CHECKING IDENTITIES OVER FINITE GROUPS

GÁBOR HORVÁTH, CSABA SZABÓ
2006 International journal of algebra and computation  
We analyze the computational complexity of solving a single equation and checking identities over finite meta-abelian groups.  ...  Among others we answer a question of Goldmann and Russel from '98: We prove that it is decidable in polynomial time whether or not an equation over the six element group S 3 has a solution.  ...  Szabó, The computational complexity of checking identites in simple semigroups and matrix semigroups over finite fields, Semigroup Forum (2002) [11] P. Tesson, D.  ... 
doi:10.1142/s0218196706003256 fatcat:eo5ahy36ajaynioq7gbobb4mui

A Package TESTAS for Checking Some Kinds of Testability [chapter]

A. N. Trahtman
2003 Lecture Notes in Computer Science  
The bounds on order of local testability of transition graph and order of local testability of transition semigroup are also found. For given k, the k-testability of transition graph is veried.  ...  We implement a set of procedures for deciding whether or not a language given by its minimal automaton or by its syntactic semigroup is locally testable, right or left locally testable, threshold locally  ...  The graphs of automata with locally idempotent transition semigroup are checked too (y(n 3 ) time complexity).  ... 
doi:10.1007/3-540-44977-9_22 fatcat:ipajbwiuqrciha6xmnnhqcl6nq

On the Complexity of Properties of Transformation Semigroups [article]

Lukas Fleischer, Trevor Jack
2019 arXiv   pre-print
Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is PSPACE-complete.  ...  We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators.  ...  See [1] for background and complexity results on identity checking; in particular, examples of semigroups for which it is coNP-complete.  ... 
arXiv:1811.00060v2 fatcat:l4hopup5ubfodbteg4gqwdwx54

A Note on Complex Algebras of Semigroups [chapter]

Peter Jipsen
2004 Lecture Notes in Computer Science  
The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based.  ...  It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups.  ...  . , A 11 can be embedded in complex algebras of finite semigroups.  ... 
doi:10.1007/978-3-540-24771-5_15 fatcat:j67dj6ishvhuvkudkcgnrfulpa
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