arXiv subjects

Daniel J. Clouse

Publications and source records attributed to Daniel J. Clouse.

6 recordsLinked to original sources

arXiv2022

A Note on Rough Set Algebra and Core Regular Double Stone Algebras

Rough Set Theory (RST), first introduced by Pawlak in 1982, is an approach for dealing with information systems where knowledge is uncertain or incomplete.\cite{Pawlak} It is of fundamental importance in many subfields of artificial intelligence and cognitive science.\cite{RSTppf} Given a universe $U$ with an equivalence relation $θ$, the pair $\langle U,θ\rangle$ is referred to as an information system and we denote its collection of rough sets $R_θ$. In our main Theorem we show $R_θ$ with $|θ_u| > 1\ \forall\ u \in U$ to be isomorphic to core regular double Stone algebras, CRDSA, that are complete and atomic, and that the crisp, or definable, sets form a complete atomistic Boolean algebra. These guarantees of infimum/supremeum for arbitrary subsets and formulations in terms of fundamental elements are likely useful if dealing with equivalence relations with an infinite number of partitions, such as projective Hilbert spaces. We further derive that every CRDSA is isomorphic to a subalgebra of a principal rough set algebra, $R_θ$, for some approximation space $\langle U,θ\rangle$. In our main Corollary we show explicitly how to embed $R_θ$ into the CRDSA and first demonstrate by extending the culminating finite example of \cite{RCRDSA}. As our capstone, we consider the projective Hilbert space of complex numbers, $\mathbb{C}$ and show, among other things, the power set of the set of pure states is a complete, atomistic Boolean algebra. In closing, we suggest other Quantum relevant applications that may be useful, such as Hilbert spaces of operators

math.RA↗

arXiv2020

Advancing the Research and Development of Assured Artificial Intelligence and Machine Learning Capabilities

Artificial intelligence (AI) and machine learning (ML) have become increasingly vital in the development of novel defense and intelligence capabilities across all domains of warfare. An adversarial AI (A2I) and adversarial ML (AML) attack seeks to deceive and manipulate AI/ML models. It is imperative that AI/ML models can defend against these attacks. A2I/AML defenses will help provide the necessary assurance of these advanced capabilities that use AI/ML models. The A2I Working Group (A2IWG) seeks to advance the research and development of assured AI/ML capabilities via new A2I/AML defenses by fostering a collaborative environment across the U.S. Department of Defense and U.S. Intelligence Community. The A2IWG aims to identify specific challenges that it can help solve or address more directly, with initial focus on three topics: AI Trusted Robustness, AI System Security, and AI/ML Architecture Vulnerabilities.

cs.LG↗

arXiv2018

The Nearly Boolean Nature of Core Regular Double Sone Algebras, CRDSA (Ternary Set Partitions, CRDSA, Embeddings and Dual Equivalences)

In "Centre of Core Regular Double Stone Algebra" (CRDSA), many useful results are shown that begin to indicate the nearly Boolean nature of CRDSA which we focus on here. We define the node set lattice through the well known binary operations of ternary set partitions and show the resultant lattice is isomorphic to C3^J where C3 is the 3 element chain CRDSA. We derive that every CRDSA is a subdirect product of C3 similarly as for Boolean algebras and C2. We use these results to show that every Boolean algebra is the center of some CRDSA. Next we show that C3 is primal implying that the variety generated by it is dually equivalent to the category of Boolean algebras. In some sense this is a last step towards our goal of establishing CRDSA as nearly Boolean, but leaves us a bit dissatisfied. Hence we continue by establishing a duality between the category of CRDSA and specifically crafted bi-topological spaces. Towards this end we first establish necessary and sufficient conditions on a pairwise zero-dimensional space such that it will have a CRDSA base B1. We note that these conditions are indicative of how nearly Boolean CRDSA are. For example, if u in B1 is not clopen/complemented then Cl(u) is in B1 and is clopen/complemented. Then we establish necessary and sufficient conditions for a bi-continuous map to have an inverse that is a CRDSA homomorphism again indicating how nearly Boolean CRDSA are, these inverses must respect the appropriate conditions on the boundary of non-clopen elements of B1. In culmination we show the category of core regular double Stone algebras is dually equivalent to the category of what we call core regular double pairwise Stone spaces. We note that the conditions for this duality can easily be relaxed to yield a duality for a less rigid class lattices than CRDSA, bounded distributive pseudo-complemented lattices for example.

math.RA↗

arXiv2018

Exploring Core Regular Double Stone Algebras, CRDSA, II. Moving Towards Duality

This is the second in a series of three notes on an investigation into core regular double Stone algebras, CRDSA, which are meant to be read in order. This note begins our investigation of duality for CRDSA through bi-topological spaces. More succinctly, duality through refinement of a pre- established duality of pairwise Stone spaces and bounded distributive lattices. In this note we establish necessary and sufficient conditions on a pairwise zero-dimensional space such that it will have a core regular double Stone algebra base. We note that these conditions can easily be relaxed to that which is necessary for a pairwise zero-dimensional space to have a base that is not as rigid as a CRDSA such as bounded distributive-pseudo complemented lattices for example. Furthermore, these conditions give a topological representation indicative of just how "nearly Boolean" CRDSA are, the closure of non-clopen/complemented base elements are clopen/complemented base elements. For the purposes of this note only we will call any pairwise zero-dimensional space with a core regular double Stone algebra base a core regular double pairwise zero-dimensional space. From that result we gain confidence to claim that the category of core regular double Stone algebras is dually equivalent to what we refer to in this note as the category of core regular double pairwise Stone spaces. We validate this claim in the next note, Exploring Core Regular Double Stone Algebras, CRDSA, III. Establishing Duality.

math.RA↗

arXiv2018

Exploring Core Regular Double Stone Algebras, CRDSA, III. Establishing Duality

This is the last in a series of three notes on an investigation into core regular double Stone algebras, CRDSA, which are meant to be read in order. This note ends our initial investigation of duality for CRDSA through bi-topological spaces. More succinctly, duality through refinement of a pre-established duality of pairwise Stone spaces and bounded distributive lattices. In this note we show that the pairwise Stone derived from a CRDSA L has a base that is a CRDSA isomorphic to L. For the purposes of this note only we will call any such pairwise zero-dimensional space a core regular double pairwise zero-dimensional space and similarly for the corresponding pairwise Stone spaces. Then we establish necessary and sufficient conditions for a bi-continuous map to have an inverse that is a CRDSA homomorphism. These results are topologically indactive of just how "nearly Boolean" CRDSA are, these inverses must respect the appropriate conditions on the boundary of non-clopen elements of the bases of the bi-topological space. From that result we validate our earlier claim that the category of core regular double Stone algebras is dually equivalent to the category of core regular double pairwise Stone spaces. We note that the conditions for this duality should be able to be easily relaxed to yield a duality for a less rigid subclass of bounded distributive lattices than CRDSA, bounded distributive pseudo-complemented lattices for example.

math.RA↗

arXiv2008

The dual geometry of Boolean semirings

It is well known that the variety of Boolean semirings, which is generated by the three element semiring S, is dual to the category of partially Stone spaces. We place this duality in the context of natural dualities. We begin by introducing a topological structure \uS and obtain an optimal natural duality between the quasi-variety ISP(S) and the category IS_cP+(\uS). Then we construct an optimal and very small structure \uS_os that yields a strong duality. The geometry of some of the partially Stone spaces that take part in these dualities is presented, and we call them "hairy cubes", as they are n-dimensional cubes with unique incomparable covers for each element of the cube. We also obtain a polynomial representation for the elements of the hairy cube.

math.CT↗