10 research outputs found
The subpower membership problem for semigroups
Fix a finite semigroup and let be tuples in a direct
power . The subpower membership problem (SMP) asks whether can be
generated by . If is a finite group, then there is a
folklore algorithm that decides this problem in time polynomial in . For
semigroups this problem always lies in PSPACE. We show that the SMP for a full
transformation semigroup on 3 letters or more is actually PSPACE-complete,
while on 2 letters it is in P. For commutative semigroups, we provide a
dichotomy result: if a commutative semigroup embeds into a direct product
of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P;
otherwise it is NP-complete
The Subpower Membership Problem for Finite Algebras with Cube Terms
The subalgebra membership problem is the problem of deciding if a given
element belongs to an algebra given by a set of generators. This is one of the
best established computational problems in algebra. We consider a variant of
this problem, which is motivated by recent progress in the Constraint
Satisfaction Problem, and is often referred to as the Subpower Membership
Problem (SMP). In the SMP we are given a set of tuples in a direct product of
algebras from a fixed finite set of finite algebras, and are
asked whether or not a given tuple belongs to the subalgebra of the direct
product generated by a given set.
Our main result is that the subpower membership problem SMP() is
in P if is a finite set of finite algebras with a cube term,
provided is contained in a residually small variety. We also
prove that for any finite set of finite algebras in a variety
with a cube term, each one of the problems SMP(), SMP(), and finding compact representations for subpowers in
, is polynomial time reducible to any of the others, and the first
two lie in NP
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
The subpower membership problem of 2-nilpotent algebras
The subpower membership problem SMP(A) of a finite algebraic structure A asks
whether a given partial function from A^k to A can be interpolated by a term
operation of A, or not. While this problem can be EXPTIME-complete in general,
Willard asked whether it is always solvable in polynomial time if A is a
Mal'tsev algebras. In particular, this includes many important structures
studied in abstract algebra, such as groups, quasigroups, rings, Boolean
algebras. In this paper we give an affirmative answer to Willard's question for
a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that
might be essential in answering the question for general nilpotent Mal'tsev
algebras in the future.Comment: 17 pages (including appendix
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Hardness Results for the Subpower Membership Problem
We first provide an example of a finite algebra with a Taylor term whose subpower membership problem is NP-hard. We then prove that for any consistent strong linear Maltsev condition M which does not imply the existence of a cube term, there exists a finite algebra satisfying M whose subpower membership problem is EXPTIME-complete. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show as a corollary that there are finite algebras which generate congruence distributive and congruence k-permutable (k ≥ 3) varieties whose subpower membership problem is EXPTIME-complete. Finally, we show that the spectrum of complexities of the problems SMP() for finite algebras in varieties which are congruence distributive and congruence k-permutable (k ≥ 3) is fuller than P and EXPTIME-complete by giving examples of finite algebras in such a variety whose subpower membership problems are NP-complete and PSPACE-complete, respectively
Aichinger equation on commutative semigroups
We consider Aichinger's equation
for functions defined on commutative
semigroups which take values on commutative groups. The solutions of this
equation are, under very mild hypotheses, generalized polynomials. We use the
canonical form of generalized polynomials to prove that compositions and
products of generalized polynomials are again generalized polynomials and that
the bounds for the degrees are, in this new context, the natural ones. In some
cases, we also show that a polynomial function defined on a semigroup can
uniquely be extended to a polynomial function defined on a larger group. For
example, if solves Aichinger's equation under the additional restriction
that , then there exists a unique
polynomial function defined on such that
. In particular, if is also bounded on a set
with positive Lebesgue measure then its unique
polynomial extension is an ordinary polynomial of variables with total
degree , and the functions are also restrictions to
of ordinary polynomials of total degree defined on
.Comment: 8 pages; submitted to a journal. This second version eliminates
theorem 10 from the previous one, since it was not righ