41 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
Most simple extensions of are undecidable
All known structural extensions of the substructural logic ,
Full Lambek calculus with exchange/commutativity, (corresponding to
subvarieties of commutative residuated lattices axiomatized by -equations) have decidable theoremhood; in particular all the ones defined
by knotted axioms enjoy strong decidability properties (such as the finite
embeddability property). We provide infinitely many such extensions that have
undecidable theoremhood, by encoding machines with undecidable halting problem.
An even bigger class of extensions is shown to have undecidable deducibility
problem (the corresponding varieties of residuated lattices have undecidable
word problem); actually with very few exceptions, such as the knotted axioms
and the other prespinal axioms, we prove that undecidability is ubiquitous.
Known undecidability results for non-commutative extensions use an encoding
that fails in the presence of commutativity, so and-branching counter machines
are employed. Even these machines provide encodings that fail to capture proper
extensions of commutativity, therefore we introduce a new variant that works on
an exponential scale. The correctness of the encoding is established by
employing the theory of residuated frames.Comment: 45 page
Some Undecidable Problems on Representability as Binary Relations
We establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for J onsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions
Defining Recursive Predicates in Graph Orders
We study the first order theory of structures over graphs i.e. structures of
the form () where is the set of all
(isomorphism types of) finite undirected graphs and some vocabulary. We
define the notion of a recursive predicate over graphs using Turing Machine
recognizable string encodings of graphs. We also define the notion of an
arithmetical relation over graphs using a total order on the set
such that () is isomorphic to
().
We introduce the notion of a \textit{capable} structure over graphs, which is
one satisfying the conditions : (1) definability of arithmetic, (2)
definability of cardinality of a graph, and (3) definability of two particular
graph predicates related to vertex labellings of graphs. We then show any
capable structure can define every arithmetical predicate over graphs. As a
corollary, any capable structure also defines every recursive graph relation.
We identify capable structures which are expansions of graph orders, which are
structures of the form () where is a partial order. We
show that the subgraph order i.e. (), induced subgraph
order with one constant i.e. () and an expansion
of the minor order for counting edges i.e. ()
are capable structures. In the course of the proof, we show the definability of
several natural graph theoretic predicates in the subgraph order which may be
of independent interest. We discuss the implications of our results and
connections to Descriptive Complexity