294 research outputs found
First steps in synthetic guarded domain theory: step-indexing in the topos of trees
We present the topos S of trees as a model of guarded recursion. We study the
internal dependently-typed higher-order logic of S and show that S models two
modal operators, on predicates and types, which serve as guards in recursive
definitions of terms, predicates, and types. In particular, we show how to
solve recursive type equations involving dependent types. We propose that the
internal logic of S provides the right setting for the synthetic construction
of abstract versions of step-indexed models of programming languages and
program logics. As an example, we show how to construct a model of a
programming language with higher-order store and recursive types entirely
inside the internal logic of S. Moreover, we give an axiomatic categorical
treatment of models of synthetic guarded domain theory and prove that, for any
complete Heyting algebra A with a well-founded basis, the topos of sheaves over
A forms a model of synthetic guarded domain theory, generalizing the results
for S
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
Ultrametric and Generalized Ultrametric in Computational Logic and in Data Analysis
Following a review of metric, ultrametric and generalized ultrametric, we
review their application in data analysis. We show how they allow us to explore
both geometry and topology of information, starting with measured data. Some
themes are then developed based on the use of metric, ultrametric and
generalized ultrametric in logic. In particular we study approximation chains
in an ultrametric or generalized ultrametric context. Our aim in this work is
to extend the scope of data analysis by facilitating reasoning based on the
data analysis; and to show how quantitative and qualitative data analysis can
be incorporated into logic programming.Comment: 19 pp., 5 figures, 3 table
The Haar Wavelet Transform of a Dendrogram: Additional Notes
We consider the wavelet transform of a finite, rooted, node-ranked, -way
tree, focusing on the case of binary () trees. We study a Haar wavelet
transform on this tree. Wavelet transforms allow for multiresolution analysis
through translation and dilation of a wavelet function. We explore how this
works in our tree context.Comment: 37 pp, 1 fig. Supplementary material to "The Haar Wavelet Transform
of a Dendrogram", http://arxiv.org/abs/cs.IR/060810
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