Rough Set & Riemannian Covariance Matrix Theory for Mining The

Rough Set & Riemannian Covariance Matrix Theory for Mining the Multidimensionality of Artificial Consciousness

Rory Lewis Department of Computer Science University of Colorado Colorado Springs

ABSTRACT criteria were typically associated with a specific scenario. One pub- This paper presents a means to analyze the multidimensionality licly available scenario, being researched by artificial consciousness of human consciousness as it interacts with the brain by utilizing military laboratories in the US, Russia, and China is as follows: Rough Set Theory and Riemannian Covariance Matrices. We math- A mother is standing in front of her family’s house that has ematically define the infantile state of a robot’s operating system been burning for some time. Her brain has calculated that running artificial consciousness, which operates mutually exclu- the burning roof will collapse at any second. She hears her sively to the operating system for its AI and locomotor functions. two infant children screaming out for her from within the house. She yearns to rescue them, but her brain is telling CCS CONCEPTS her she will die—the roof is about to collapse. The mother, • Theory of computation → Semantics and reasoning; • Math- of course, overrides the obvious mandate delivered by her ematics of computing → Information theory. brain, runs into the house, grabs both children, and then runs out. Put another way, the mother’s consciousness not KEYWORDS only overrode her brain, it ordered the brain to move all necessary muscles in her body to immediately rescue her Rough Sets, Riemannian Theory, Artificial Consciousness. children. ACM Reference Format: Similarly, a military humanoid may risk its $20M body to rescue Rory Lewis. 2020. Rough Set & Riemannian Covariance Matrix Theory for a wounded human. To do this, we code the operating system (OS) Mining the Multidimensionality of Artificial Consciousness. In The 10th running its artificial consciousness to “roughly” assess when to International Conference on Web Intelligence, Mining and Semantics (WIMS override the OS running its artificial intelligence (AI). As seen in 2020), June 30-July 3, 2020, Biarritz, France. ACM, Biarritz, France, 4 pages. race between the unclassified Russian FEDOR Skybot F-850 and the https://doi.org/10.1145/3405962.3405974 US’ Atlas humanoids, has captivated the public and culminated in 2019 with FEDOR working on the International Space Station [5]. 1 INTRODUCTION It is easy to realize how, at the top-secret-levels of the Chinese, The 17th Century French philosopher René Descartes postulated Russian and US’ militaries, there exist incredibly sophisticated, that I think, therefore I am, inferring that the mere act of lethal, humanoids that can drive cars, pick locks, shoot weapons, thinking about one’s existence proves there is someone there doing parachute from Low Earth Orbit and recharge themselves. But, even the thinking [4]. Nowadays, there are many established theories of if Russia were able to avoid US Space Intelligence and drop 5,000 hypothetical self-consciousness phenomenology in the domains of philosophy, military humanoids, into a US city, there is a fatal psychopathology and neuroscience [9]. Even though it is generally problem. No matter how many uniforms and weapons of enemy vs. accepted that human consciousness is separate to human intellect, foe are in its database, there will always be unknowns, eg., when when pressed, humans disagree as to where, when, or if the brain it sees humans in clothing not in its database, it would not know and consciousness part ways. However, most advanced societies whether to shoot or greet them, pick a lock and hide or drive a I know I mandate that no matter how brilliant or intellectually challenged a car and runaway. This is because i) it cannot realize don’t know person is, each soul is afforded an equality of being. Recently, the , and ii) it cannot invoke human sentient feelings from author spent a sabbatical working on classified artificial conscious- the way these people act as to who they probably are. In fact, there human ness at the United States Pentagon & Department of Defense (DoD). are millions of sentient, common sense abilities they cannot During this period, the research team studied a plurality of criteria perform. Hence, there will be a Russian human, remotely connected inherent in humans’ brains versus humans’ consciousness. These to each humanoid to make human decisions. By disabling these communications, all these Russian humanoids would be disabled. Consider that, at the top-secret levels of the Chinese, Russian, and Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed US militaries, there exist incredibly sophisticated, lethal, humanoids for profit or commercial advantage and that copies bear this notice and the full citation that can drive cars, pick locks, shoot weapons, parachute from Low on the first page. Copyrights for components of this work owned by others than ACM Earth Orbit, and recharge themselves. But, even if Russia were able must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a to avoid US Space Intelligence and drop dozens of “hypothetical” fee. Request permissions from [email protected]. military humanoids into a US city, there is a fatal problem. No matter WIMS 2020, June 30-July 3, 2020, Biarritz, France how many uniforms and weapons of the enemy (i.e., US citizens, © 2020 Association for Computing Machinery. ACM ISBN 978-1-4503-7542-9/20/06...$15.00 soldiers, and AI military humanoid) are in their database, there will https://doi.org/10.1145/3405962.3405974 always be unknowns with which to contend. For example, when a

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Figure 2: Covariance Matrix. x1Murder :y − axis) is the point they (a) (b) would murder, and x2Murder _S : x − axis is the environment.

newborns to prevent detection of the birth [14]. Beginning in 1943, Figure 1: Psychoanalytical Covariance: (a) Linear and (b) non linear some babies were given to Nazi couples as “Aryan” babies under relationships. Nazi Germany’s Lebensborn Program, Again, the mothers would kill their babies rather than hand them over to the Nazis [1]. Russian humanoid sees humans in clothing not in its database, it The covariant, linear, relationship of this tragic, phenomena is il- would not know whether to shoot or greet them, pick a lock and lustrated in Fig. 1 (a) by computing the covariance and homogeneity hide, or drive a car and run away. This is because A) it cannot realize of the psychoanalysis of consciousness. A non linearly covariance “I know I don’t know” and B) it cannot interpret human “sentient is illustrated in Fig. 1 (b), whereas the variables of financial morality feelings and actions” in order to discern who they probably are. In and financial state have no covariance. There is no pattern; some fact, there are millions of sentient, common sense “human” abilities indigents are saints, while others use their financial plight to justify they can neither perform nor duplicate. Therefore, there must be a stealing. Some billionaires are philanthropists, while others use Russian human controller, remotely connected to each humanoid in their fortunes to sow corruption and deceit. order to make human decisions. By disabling their means of remote communication, the ‘hypothetical’ Russian humanoids would be Í Í(x − x¯)(y − y¯) (xi − µx )(yi − µy ) rendered useless. S = i i σ = (1) xy n − 1 xy N 2 THE IMMEDIATE NEED FOR ARTIFICIAL Eq. 1, Sxy is used to calculate the sample covariance of the cohort answering how they would consciously act in a given situation CONSCIOUSNESS while σxy calculates the population covariance. For the sample co- US military generals know that the billions of dollars invested in variance we take each data point for x and subtract the mean x¯, humanoids is wasted unless some headway into artificial conscious- multiply them by (yi −y¯), add them and divide by n − 1, the amount ness and sentient common sense is being made. These commanders of samples minus 1. For the population covariance µ we use the have mandated that the operating system of any installed artificial entire population and divide by N . consciousness must operate mutually exclusively to its artificial To illustrate how Knowledge Discovery, over a plurality of dimen- intelligence and locomotor functions. One of the Pentagon officials sions, is a non-trivial undertaking, we observe data from a previous put it this way: It’s really simple Dr. Lewis, our humanoids set of psychoanalytical experiments. A cohort of N = 53 subjects un- will be destroyed by the humanoids of the first country dertook three sets of conscious morality test. The first test queried to create artificial consciousness. when they would morally justify murder as shown in the Covari- ance Matrix Table 1, where x1Murder :y − axis) is the point they 2.1 Mathematically Defining Consciousness would murder, and x2Murder_S : x − axis is the environment. Fig. Single Dimension. Defining consciousness often starts with the 3 a shows this covariance was positively linear where the MeanX primary dimension of morality. It is generally accepted that people was 58.925, MeanY was 64.170, the PopulationsCovariance(X, Y) have a bivariate perspective: when life is stable and good, the odds yielded 450.89961 and the SampleCovariance(X, Y) yielded 459.57075. of committing an immoral act are very low. Conversely, when life The cohort was asked similar questions in regards to coveting and is perilous and unpredictable, the odds are higher. For example, a stealing. A partial covariance matrix is seen as a scatter plot illus- healthy and nurturing mother will not consider killing her new- trating the covariance between the linearity of the murder question born baby. However, in the Nazi concentration camps of WWII, and the non linearity of the coveting questions. The entire covari- thousands of Jewish women were conditioned to go against this ance matrix, as a scatter plot, is seen in Fig. 3 c. fundamental commandment. Many became pregnant from the SS The Issue is the complexity of consciousness is complex. Even guards 1) raping them, or 2) allowing and watching the male pris- the aforementioned 3-D experiments show the inability to mine oners rape the women [12] or 3) by being forced to work in the three dimensions adequately. The Challenge is to data mine and concentration camp brothels [2]. extract classifiers across hundreds of dimensions so as to recreate Stanislawa Leszczyńsk, the Midwife of Auschwitz delivered over consciousness; artificial consciousness. The Hypothesis Riemann- 3,000 babies that, despite her efforts, were immediately drowned ian Covariance Matrices are specifically made to handle an infinite in front of the mothers [1]. Prior to 1943, pregnant prisoners were amount of dimensions. We propose Rough Sets to emulate how immediately killed unless they secretly aborted or killed their own human conscious decision making always has some variance, gray

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(a) (b) (c)

Figure 5: Tangent Space. T × M for manifold M with tangent vector Covariance Matrixes]: (a) Positively linear covariance Figure 3: (Wikipedia). [6] for conscious Murder analysis. (b) Non-linear covariance matrix of conscious Covetousness versus Murder . (c) Non-linear covariance the comma , between these two surfaces indicates the intersection Covetousness Stealinд Murder matrix of conscious & versus of these two surfaces,that creates a curve in 3-D. Hence, a parame- terized curve in a multidimensional space Rn is a map γ that takes a real parameter t and maps it to an ordered n tuple of real numbers as defined in Eq. 4:

γ : t → Rn, t ∈ R α < t < β (4) Where both α and β are also real numbers. To express a param- eterized mapping as a vector γ (t) where all of its elements representing a separate coordinate, which is critical as we move towards Figure 4: A Series of Rough Set Dimensions of Consciousness in a many dimensions later in this paper, where γ (t) is represented as: Euclidean Domain. (a) The first two Rough Set Covariance Matrices of a finite set of dimensions. (b) Additional dimensions.(c) Last three γ (t) = [γ (t),γ (t),γ (t),...] (5) dimensions of the finite set.(d & e) Two, of many, vectors traversing 1 2 3 all the dimensions. Where we now take γ (t) in Eq. 5 as a function of time t and then as time increases we trace the parameter γ (t) = [x(t),y(t), z)t)]. area and subjectivity. Additionally, it has a manageable means for Now to smooth the function, we first look at a single function. To machine learning because of the graphing of the upper bounds and make a single function infinitely differentiable, we smooth the func- ( , ) n lower bounds in an x y clustered setting. ( ) ( ) d f tion f t on the open interval α, β so long as its derivative dt n exists for all n ≥ 1 ≥ and t ∈ (α, β). Now looking at smoothing out 3 RIEMANNIAN COVARIANCE MATRICES & the curve, we take Eq. 5 γ (t) = [γ1(t),γ2(t),γ3(t),...] and differen- DIFFERENTIAL GEOMETRY tiate it by differentiating all its individual components. So, ifwe The goal here is that, when given a large set of Rough Set sam- have, for example 200 dimensions of rough set representations of ples comprised of x and y coordinates each representing a singu- consciousness, the γ = 200, as set forth in Eq. 6: lar dimension (d) of one’s consciousness, as illustrated in Fig. 4 , , ..., − , − , ) n n n n 1d 2d N 2d N 1d N , we want to trace how dependent d γ d γ1 d γ2 d γ3 ∈ ∈ γ (t) = [γ1(t),γ2(t),γ3(t),...] −→ = , , ,... variables, y Y relate to independent variable, x X as depicted dtn dtn dtn dtn by the two vectors traversing all the dimensions in Fig. 6 (d) and (6) (e). Instinctively we, like the author initially did, will fall back on Transferring from curves to manifolds, a manifold is a topologi- ( )N ⊂ × training the system using xi,yxi i=1 X Y and seek to find the cal space that locally resembles Euclidean space near each point. classifiers8 [ ]. To which one may well say is regression, and this is More precisely, each point of an n-dimensional manifold has a correct, if the data is comprised of real vectors. The issue is that we neighborhood that is homeomorphic to the Euclidean space of di- want to do this in a Euclidean setting where we can analyze any mension n. In this more precise terminology, a manifold is referred finite number of Rough Set dimensions of one’s consciousness. to as an n-manifold. To calculate the distance on a manifold, we To connect a finite plurality of Rough Set dimensions to the use tangent space TxM of a manifold M at a point x. As one curves on a 3-D Euclidean space we first consider the 2-D curve hypothetically travels on the surface of the manifold’s curvature y − y(x) ⇔ f (x,y) = K [7] as shown in Eq. 2: we pass through the point x with vectors of magnitude and direction, C = {(x,y) ∈ R2 | f (x,y) = K} (2) and these vectors are tangent to the manifold [6] as illustrated in Fig. 5. At this point the author is using the simplest of manifolds which Where the 2-D curve C is defined by a set of ordered pair of real is called a topological manifold, and even though by definition, all R2 numbers squared . For a 3-D curve we use an ordered triple as manifolds are topological manifolds, it usually is interpreted as set forth in Eq. 3: being a manifold that lacks additional structure. We now smooth the parametric curve γ (t → M that maps t ∈ [α, β] to M as it C = {(x,y, z) ∈ R3 | f (x,y, z) = K , f (x,y, z) = K } (3) 1 1 2 2 traverses through p and this sets up a smooth surface area curving Where z in the f1 equation represents a first surface z = д1(x,y), over the manifold. As we continue along the manifold: φ◦γ : t → R. and the z in the f2 equation represents a second surface z = д2(x,y). Labeling the local coordinates as x = φ ◦γ (t) to represent the value

- 250 - WIMS 2020, June 30-July 3, 2020, Biarritz, France Rory Lewis of a single parameter, as the position, on the manifold, as vector as a function of time t hence making t = t0 when at p.

3.1 Riemannian Kernels In 2014, Chen et al., [11] compared the viability Riemannian classification using three types of classifiers, kernel SVM, logistic regression, and partial least squares. The motivation was, as sought in this paper’s investigation, to exploit learning methods for multidimensional vector space on Riemannian manifold. The justification herein is that Riemannian kernels enable their classifiers to operate in an extrinsic feature space without having to compute the coordinates in original space. In our case, we have already calculated all necessary locations of the upper and lower bounds using Rough Sets before we pass each singular dimension into the Riemannian system, thus making our SVM classifier in the kernel space is

−→∗ ∗ f (x) = w T ϕ(x) + b , (7) −→ Where w ∗ is the weight vector, b∗, is the bias, and ϕ(x) is the Figure 6: A Series of Rough Set Dimensions of Consciousness in a mapping ϕProj & ϕIP and generates the kernel function k(·, ·) by Euclidean Domain. [15] T Ki,j = κ(xi,yj ) = ϕ(xi)) ϕ(xij ). (8) unique vectors and tangents, that the Riemannian metric tensors Where it should be noted that for the aforementioned work, are the first candidates we will test. Chen et al., [11] employed Chang et al’s., LibSV M library for support vector machines [3] implementation on the pre-calculated γ (t) = {[γ1(t),γ2(t),дp ] : Tp M × Tp M −→ R,p ∈ M} (11) Riemannian kernel matrices for classification. Wherep −→ дp (X(p), Y(p)) for any two tangent vectors X(p), Y(p) is a smooth function of p. Rough Set Riemannian Kernels. Considering that the author has modeled each rough set dimension as a collection of linear REFERENCES subspaces [11] as illustrated in Fig. 4, ()1d , 2d , ..., N −2d , N −1d , N ), [1] J. Beal. Stanislawa leszczynska: the midwife of auschwitz. Midwifery today with which correspond to points lying on the Riemannian manifold M international midwife, (102):30–30, 2012. N [2] J. Caplan. Gender and the concentration camps. In Concentration Camps in Nazi as illustrated in Fig. 5, denoted by P = {Pi }i=1, where N is the Germany, pages 94–119. Routledge, 2009. number of Rough Set dimensions of consciousness. The distance [3] C.-C. Chang and C.-J. Lin. Libsvm: A library for support vector machines. ACM U U transactions on intelligent systems and technology (TIST), 2(3):1–27, 2011. between two Rough Set upper bounds Pi and Pj and the same [4] R. Descartes. René Descartes: Meditations on first philosophy: With selections from two Rough Sets lower bounds PL and PL is calculated by mapping the objections and replies. Cambridge University Press, 2013. i j [5] E. Garcia. The militarization of artificial intelligence: a wake-up call for the global the Riemannian manifold to Euclidean space using Chen et al’s south. Available at SSRN 3452323, 2019. Mercer kernels and Lovell et al’s Projection kernels [10] yields the [6] B. Keng. Manifolds: A gentle introduction. https://is.gd/IdirQm, 2018-04-17 06:24 2018. (Accessed on 01/18/2019). covariance between between the two subspaces: [7] O. Khan. Introduction to differential geometry: Curves. https://t.ly/7JOxE, 2018- 06-10 2018. (Accessed on 01/12/2019). Proj .−Poly. ( · ∥ T ∥2 )α [8] H. J. Kim, N. Adluru, M. D. Collins, M. K. Chung, B. B. Bendlin, S. C. Johnson, R. J. Ki,j = γ Pi Pj F , (9) Davidson, and V. Singh. Multivariate general linear models (mglm) on riemannian Proj .−Poly. manifolds with applications to statistical analysis of diffusion weighted images. Where Ki,j is an element in the kernel matrix. The cor- In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, T pages 2705–2712, 2014. responding mapping is ϕProj . = Pi Pi . Now implementing Chel- [9] T. Kircher and A. S. David. Self-consciousness: an integrative approach from lappa et al’s kernel methodology [13] we yield a radial basis function philosophy, psychopathology and the neurosciences. The self in neuroscience and (RBF) kernel [10] using ϕ by: psychiatry, 1, 2003. Proj . [10] M. Liu, R. Wang, Z. Huang, S. Shan, and X. Chen. Partial least squares regression on grassmannian manifold for emotion recognition. In Proceedings of the 15th Proj .−RBF 2 ACM on International conference on multimodal interaction, pages 525–530, 2013. K = exp(γ · ∥ϕProj .Pi − ϕProj .Pj ∥ ) (10) i,j F [11] M. Liu, R. Wang, S. Li, S. Shan, Z. Huang, and X. Chen. Combining multiple kernel methods on riemannian manifold for emotion recognition in the wild. In 3.2 Riemannian Manifolds Proceedings of the 16th International Conference on multimodal interaction, pages 494–501, 2014. Drawing your attention to compare the manifold of Fig. 5 and the [12] J. Ringelheim. Women and the holocaust: A reconsideration of research. Signs, rough sets manifold in Fig. 6 we see that the plain is now comprised 10(4):741–761, 1985. [13] R. Vemulapalli, J. K. Pillai, and R. Chellappa. Kernel learning for extrinsic classi- of a rough set diagram. In reality this is a little misleading as it fication of manifold features. In Proceedings of the IEEE Conference on Computer does not visually look like this What is happening, and what we Vision and Pattern Recognition, pages 1782–1789, 2013. are trying to illustrate is that each vector at each p is being set out, [14] G. M. Weisz and K. Kwiet. Managing pregnancy in nazi concentration camps: The role of two jewish doctors. Rambam Maimonides medical journal, 9(3), 2018. when all taken together, make up the rough set chart in Fig. 6. It is [15] F. Yger and M. Sugiyama. Supervised logeuclidean metric learning for symmetric here that the author has determined that on order to create these positive definite matrices. arXiv preprint arXiv:1502.03505, 2015.

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