Giorgio Audrito

Enumeration of the adjunctive hierarchy of hereditarily finite sets

Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. This raises the need for efficient representation of such sets, for example by numbers. In 2008, Kirby proposed an adjunctive hierarchy of hereditarily finite sets, based on the fact that they can also be seen as built up from the empty set by repeated adjunction, that is, by the addition of a new single element drawn from the already existing sets to an already existing set. Determining the cardinality

Engineering Resilient Collective Adaptive Systems by Self-Stabilisation

a_n

Optimally-Self-Healing Distributed Gradient Structures Through Bounded Information Speed

of each level of this hierarchy, problem crucial in establishing whether the natural adjunctive hierarchy leads to an efficient encoding by numbers, was left open. In this paper we solve this problem. Our results can be generalized to hereditarily finite sets with atoms, or can be further refined by imposing restrictions on rank, on cardinality, or on the maximum level from where the new adjoined element can be drawn. We also show that

Compositional Blocks for Optimal Self-Healing Gradients

a_n

From Field-Based Coordination to Aggregate Computing.

satisfies the asymptotic formula

GENERIC LARGE CARDINALS AND SYSTEMS OF FILTERS

a_n = C^{2^n} + O(C^{2^{n-1}})

Space-Time Universality of Field Calculus

, for a constant

Optimal single-path information propagation in gradient-based algorithms

C \approx 1.3399

Optimal Skewed Allocation on Multiple Channels for Broadcast in Smart Cities

, which is a too fast asymptotic growth for practical purposes. We thus propose a very natural variant of the adjunctive hierarchy, whose asymptotic behavior we prove to be

Absoluteness via resurrection

\Theta(2^n)