11 research outputs found
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
AN EXTENSION OF A THEOREM OF ZERMELO
We show that if (M, is an element of(1), is an element of(2)) satisfies the first-order Zermelo-Fraenkel axioms of set theory when the membership relation is is an element of(1) and also when the membership relation is is an element of(2), and in both cases the formulas are allowed to contain both is an element of(1) and is an element of(2), then (M, is an element of(1)) congruent to (M, is an element of(2)), and the isomorphism is definable in (M, is an element of(1 ),( )is an element of(2)). This extends Zermelo's 1930 theorem in [6].Peer reviewe
Diversity, dependence and independence
We propose a very general, unifying framework for the concepts of dependence and independence. For this purpose, we introduce the notion of diversity rank. By means of this diversity rank we identify total determination with the inability to create more diversity, and independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.Peer reviewe
Lindstrom theorems for fragments of first-order logic
Lindstr\"om theorems characterize logics in terms of model-theoretic
conditions such as Compactness and the L\"owenheim-Skolem property. Most
existing characterizations of this kind concern extensions of first-order
logic. But on the other hand, many logics relevant to computer science are
fragments or extensions of fragments of first-order logic, e.g., k-variable
logics and various modal logics. Finding Lindstr\"om theorems for these
languages can be challenging, as most known techniques rely on coding arguments
that seem to require the full expressive power of first-order logic. In this
paper, we provide Lindstr\"om theorems for several fragments of first-order
logic, including the k-variable fragments for k>2, Tarski's relation algebra,
graded modal logic, and the binary guarded fragment. We use two different proof
techniques. One is a modification of the original Lindstr\"om proof. The other
involves the modal concepts of bisimulation, tree unraveling, and finite depth.
Our results also imply semantic preservation theorems