1,317 research outputs found
The decision problem for a three-sorted fragment of set theory with restricted quantification and finite enumerations
We solve the satisfiability problem for a three-sorted fragment of set theory
(denoted ), which admits a restricted form of quantification over
individual and set variables and the finite enumeration operator over individual variables, by showing that it
enjoys a small model property, i.e., any satisfiable formula of
has a finite model whose size depends solely on the length of
itself. Several set-theoretic constructs are expressible by
-formulae, such as some variants of the power set operator and the
unordered Cartesian product. In particular, concerning the unordered Cartesian
product, we show that when finite enumerations are used to represent the
construct, the resulting formula is exponentially shorter than the one that can
be constructed without resorting to such terms
Injecting Abstract Interpretations into Linear Cost Models
We present a semantics based framework for analysing the quantitative
behaviour of programs with regard to resource usage. We start from an
operational semantics equipped with costs. The dioid structure of the set of
costs allows for defining the quantitative semantics as a linear operator. We
then present an abstraction technique inspired from abstract interpretation in
order to effectively compute global cost information from the program.
Abstraction has to take two distinct notions of order into account: the order
on costs and the order on states. We show that our abstraction technique
provides a correct approximation of the concrete cost computations
Modular Construction of Complete Coalgebraic Logics
We present a modular approach to defining logics for a wide variety of state-based systems. The systems are modelled by coalgebras, and we use modal logics to specify their observable properties. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular fashion. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems, for which no complete axiomatisation has been obtained so far
On Tarski's fixed point theorem
A concept of abstract inductive definition on a complete lattice is
formulated and studied. As an application, a constructive and predicative
version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea
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
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