7172030
Information-theoretic inequalities for contoured probability distributions
We show that for a special class of probability distributions that we call contoured distributions, information-theoretic invariants and inequalities are equivalent to geometric invariants and inequalities of bodies in Euclidean space associated with the distributions. Using this, we obtain characterizations of contoured distributions with extremal Shannon and Renyi entropy. We also obtain a new reverse information-theoretic inequality for contoured distributions.
8455393
Moment-Entropy Inequalities for a Random Vector
The pth moment matrix is defined for a real random vector, generalizing the classical covariance matrix. Sharp inequalities relating the pth moment and Renyi entropy are established, generalizing the classical inequality relating the second moment and the Shannon entropy. The extremal distributions for these inequalities are completely characterized
16863176
The Largest Cartesian Closed Category of Stable Domains
Berry's category of dI-domains with stable functions is a relatively intricate, yet elegant, framework for semantics of programming languages. Despite over 15 years of work in the area, the exact reasons for distributivity (Axiom d) and finitariness (Axiom I) have not been fully explicated. This paper shows that Axiom d and Axiom I are important when one works within the realm of Scott-domains. In particular, it has been shown that (i) if [D →s D] has a countable basis, then D must be finitary, for any Scott-domain D; (ii) if [D →s D] is bounded complete, then D must be distributive, for any finitary Scott-domain D. Therefore, the category of dI-domains is the largest cartesian closed category within omega-algebraic, bounded complete domains, with the exponential being the stable function space.
15701939
Disjunctive Systems and L-Domains
Disjunctive systems are a representation of L-domains. They use sequents of the form X⊢Y, with X finite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an L-domain. On the other hand, via the notion of stable neighborhoods, every L-domain can be represented as a disjunctive system. More generally, we have a categorical equivalence between the category of disjunctive systems and the category of L-domains. A natural classification of domains is obtained in terms of the style of the entailment: when ¦X¦=2 and ¦Y¦=0 disjunctive systems determine coherent spaces; when ¦Y¦≤1 they represent Scott domains; when either ¦X¦=1 or ¦Y¦=0 the associated cpos are distributive Scott domains; and finally, without any restriction, disjunctive systems give rise to L-domains.
29934128
A Representation of SFP
This chapter introduces a representation of the category of SFP domains. It uses structures similar to but more general than Scott’s information systems. Distilled from Gentzen’s sequent calculi, a basic structure called a sequent structure is produced. Sequent structures determine a major part of the axioms of information systems. A category of special kind of sequent structures called the strongly finite ones is shown to be equivalent to the category of SFP domains. Constructions like the Plotkin power domain and the function space are given, as well as a complete partial order of such structures to give solutions to recursively defined systems.
15029835
Quasi-Prime Algebraic Domains
Abstract The purpose of this paper is to introduce a new class of domains called quasi-prime algebraic domains. We show that this class is a good candidate for the purpose of denotational semantics of programming languages. This is achieved by exhibiting many constructions usually needed for denotational semantics on quasi-prime algebraic domains. In the first part we motivate the notion of quasi-primes and quasi-prime algebraic domains by studying the effect of linear functions on domains that are not prime algebraic. In the second part we study categories of quasi-prime algebraic domains by representing them as irreducible information systems. We introduce a symmetric monoidal closed category of quasi-prime algebraic domains with quasi-linear functions. We show further that with the usual Scott continuous functions as morphisms, quasi-prime algebraic domains are cartesian closed. In the third part we show the existence of a saturated quasi-prime algebraic domain, and introduce a framework for solving domain equations in quasi-prime algebraic domains. It is then possible to give, for example, a model for the un-typed lambda calculus with a reflexive quasi-prime algebraic domain.
11563132
Weakly Distributive Domains
In our previous work [17] we have shown that for any ω-algebraic meet-cpo D, if all higher-order stable function spaces built from D are ω-algebraic, then D is finitary. This accomplishes the first of a possible, two-step process in solving the problem raised in [1,2]: whether the category of stable bifinite domains of Amadio-Droste-Gobel [1,6] is the largest cartesian closed full sub-category within the category of ω-algebraic meet-cpos with stable functions. This paper presents results on the second step, which is to show that for any ω-algebraic meet-cpo D satisfying axioms M and | to be contained in a cartesian closed full sub-category using ω-algebraic meet-cpos with stable functions, it must not violate MI∞. We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category using ω-algebraic meet-cpos, property MI∞ must not be violated. We further demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff's M3 and N5 [5] are weakly distributive (but non-distributive). We introduce also the notion of meet-generators in constructing stable functions and show that if an ω-algebraic meet-cpo D contains an infinite number of meet-generators, then [D → D] fails I. However, the original problem of Amadio and Curien remains open.