116 research outputs found
The equational theory of the natural join and inner union is decidable
The natural join and the inner union operations combine relations of a
database. Tropashko and Spight [24] realized that these two operations are the
meet and join operations in a class of lattices, known by now as the relational
lattices. They proposed then lattice theory as an algebraic approach to the
theory of databases, alternative to the relational algebra. Previous works [17,
22] proved that the quasiequational theory of these lattices-that is, the set
of definite Horn sentences valid in all the relational lattices-is undecidable,
even when the signature is restricted to the pure lattice signature. We prove
here that the equational theory of relational lattices is decidable. That, is
we provide an algorithm to decide if two lattice theoretic terms t, s are made
equal under all intepretations in some relational lattice. We achieve this goal
by showing that if an inclusion t s fails in any of these lattices, then
it fails in a relational lattice whose size is bound by a triple exponential
function of the sizes of t and s.Comment: arXiv admin note: text overlap with arXiv:1607.0298
A Nice Labelling for Tree-Like Event Structures of Degree 3
We address the problem of finding nice labellings for event structures
of degree 3. We develop a minimum theory by which we prove that the labelling
number of an event structure of degree 3 is bounded by a linear function of the
height. The main theorem we present in this paper states that event structures
of degree 3 whose causality order is a tree have a nice labelling with 3
colors. Finally, we exemplify how to use this theorem to construct upper bounds
for the labelling number of other event structures of degree 3
A Nice Labelling for Tree-Like Event Structures of Degree 3 (Extended Version)
We address the problem of finding nice labellings for event structures of
degree 3. We develop a minimum theory by which we prove that the labelling
number of an event structure of degree 3 is bounded by a linear function of the
height. The main theorem we present in this paper states that event structures
of degree 3 whose causality order is a tree have a nice labelling with 3
colors. Finally, we exemplify how to use this theorem to construct upper bounds
for the labelling number of other event structures of degree 3
The extended permutohedron on a transitive binary relation
For a given transitive binary relation e on a set E, the transitive closures
of open (i.e., co-transitive in e) sets, called the regular closed subsets,
form an ortholattice Reg(e), the extended permutohedron on e. This
construction, which contains the poset Clop(e) of all clopen sets, is a common
generalization of known notions such as the generalized permutohedron on a
partially ordered set on the one hand, and the bipartition lattice on a set on
the other hand. We obtain a precise description of the completely
join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and
the arrow relations between them. In particular, we prove that (1) Reg(e) is
the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset
of e is a set-theoretic union of completely join-irreducible clopen subsets of
e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen,
iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is
finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is
a bounded homomorphic image of a free lattice, iff e is a disjoint sum of
antisymmetric transitive relations and two-element full relations. We
illustrate the strength of our results by proving that, for n greater than or
equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of
an n-element set is obtained by adding a new top element to a Boolean lattice
with n2^{n-1} atoms. We also determine the factors of the minimal subdirect
decomposition of Bip(n).Comment: 25 page
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
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