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Self-referential Boltzmann machine
Journal Pre-proof Self-referential Boltzmann machine Yong Tao
PII: DOI: Reference:
S0378-4371(19)32102-8 https://doi.org/10.1016/j.physa.2019.123775 PHYSA 123775
To appear in:
Physica A
Received date : 1 March 2019 Revised date : 31 October 2019 Please cite this article as: Y. Tao, Self-referential Boltzmann machine, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123775. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Self-referential Boltzmann Machine Yong Tao†
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College of Economics and Management, Southwest University, Chongqing, China Department of Management, Technology and Economics, ETH Zurich, Switzerland
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Abstract: We recently reported that the income structure of the low and middle class (about 95% of populations) in a well-functioning free-market country would follow a Boltzmann-like distribution that has a self-referential entropy [Physica A 502, 436-446 (2018)]. The empirical evidences cover 66 free-market countries and the Hong Kong SAR. By contrast, the entropy of a physical system is not self-referential. This finding implies that the self-reference may be a potential difference between biological and lifeless-physical systems. In this paper, we argue that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a self-referential Boltzmann machine (SRBM), where each person plays the role of a neuron. Because of the self-reference of the entropy, we show that the SRBM always has a positive energy even if all neurons are inactive. This implies the presence of a kind of positive zero-point energy. Based on such a positive zero-point energy, we further show that the SRBM may be a self-motivated system with a biological sense. Our finding supports that a human society functions like a kind of complex system (or organism) with potential self-motivations, just as motivated by an “invisible hand” coined by Adam Smith. As a simple application, we apply the self-motive of the SRBM to perform the task of searching images.
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Keywords: Boltzmann machine; Self-reference; Self-motive; Self-awareness; Gibbs term PACS numbers: 87.85.dq; 87.19.lv; 89.75.Fb
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Correspondence to: [email protected] or [email protected] 1
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1. Introduction
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Understanding the origin of self-awareness is one of the most challenging and fundamental questions in 21st century science [1]. Recently, some scholars have observed the potential connections between consciousness, intelligence, and entropy. [2-7]. On the one hand, Wissner-Gross and Freer [2] proposed a deep connection between individual intelligence and entropy maximization. On the other hand, Perez Velazquez et al [3-7] suggested that consciousness can be considered as a consequence of the most probable distribution that maximizes entropy of brain functional networks. In the equilibrium statistical physics, the most probable distribution related to the entropy maximization is the Boltzmann distribution. Interestingly, the Boltzmann distribution has been employed to construct Boltzmann machines, which are stochastic neural networks [8], assuming that the energy distribution of neurons obeys the Boltzmann distribution [9]. For years, because of the endeavor of Hinton [10], Boltzmann machines have contributed to the opening of the field of deep learning architectures [11]. As an application, deep learning architectures were recently used to construct AlphaGo [12], a computer program, which beat the world’s No.1 ranked player of the game Go. A remarkable merit of Boltzmann machines is due to the unsupervised-learning function, which captures the feature of human-like learning. Despite these advances, we still do not know how to construct a neural network that exhibits human-like thinking. The most significant feature of human thinking is the emergence of self-awareness. Motivated by Godel’s Incompleteness Theorem, Neumann [13] and Hofstadter [14] conjectured that the emergence of self-awareness might be due to the “self-reference” of neural networks in the brain. Unfortunately, no one has clarified to date how to construct a neural network that exhibits self-reference. Recently, Tao theoretically showed [15-18] that a well-functioning free-market society evolves according to the rule of entropy maximization and the household income structure will obey a Boltzmann-like distribution. By analyzing datasets of household income from 66 countries and Hong Kong SAR, Tao et al confirmed that [19], for all the countries, the income structure for lower and middle classes (about 95% of populations) exactly follows the Boltzmann-like distribution. More recently, by doing an in-depth investigation for the Boltzmann-like distribution, Tao showed that [20], different from the entropy in a physical system, the entropy in a human society is self-referential. As a human society is obviously a biological system, Tao identified the “self-reference” of entropy as a central characteristic for distinguishing between biological and lifelessphysical systems [20]. The main purpose of this paper is to show that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a kind of Boltzmann machine, which we call the “self-referential Boltzmann machine” (SRBM). Because of the self-reference of the entropy, we will find that the SRBM always has a positive energy, even if all neurons are inactive. This singular feature leads to the possibility that the SRBM may be a self-motivated system with a biological sense. There is a large body of psychological literature arguing that self-awareness consists of 2
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2. Boltzmann machines
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different self-motives [21-24]. Thus, the SRBM provides a potential perspective for exploring the origin of self-awareness, which bridges the literature between neural networks and psychology. In particular, due to the self-motivation feature of the SRBM, we argue that a human society should be a kind of complex system (or organism) [25]. This finding contradicts the long-standing reductionism that is popular in the neoclassical economics. The remainder of this paper is organized as follows. Section 2 briefly introduces Boltzmann machines. In section 3, we show that the income structure of a wellfunctioning free-market society will obey a Boltzmann-like distribution, and that, different from Boltzmann’s physical systems, the entropy in a human society exhibits self-reference. In section 4, we show that if a human society obeys such a Boltzmannlike income distribution, it will spontaneously form a SRBM, where each person plays the role of a neuron. In section 5, we show that the SRBM always has a positive energy even if all neurons are inactive. In section 6, we show that this singular feature leads to the possibility that the SRBM may be a self-motivated system with a biological sense. Furthermore, we design an algorithm for the SRBM. In section 7, we apply the selfmotive of the SRBM to perform the task of searching images. In section 8, our conclusion follows.
𝒗,𝒉
𝑃 𝒗, 𝒉
𝒗,𝒉
,
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∑𝒗,𝒉
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For simplicity, we only introduce the Restricted Boltzmann Machine (RBM), which is a special Boltzmann machine [10, 26]. A RBM consists of 𝑚 visible units 𝒗 𝑣 ⋯ 𝑣 , representing observable data, and 𝑛 hidden units 𝒉 ℎ ⋯ ℎ , to capture the dependencies between observed variables [26]. The random variables and the joint probability distribution 𝒗 𝒉 take values belonging to 0,1 under the model is given by the Boltzmann distribution [26]: (1)
with the energy function: 𝐻 𝒗, 𝒉
∑
∑
𝜔 ℎ𝑣
∑
𝑏𝑣
∑
𝑐ℎ,
(2)
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where 𝑇 denotes the temperature. For all 𝑖 ∈ 1, … , 𝑛 and 𝑗 ∈ 1, … , 𝑚 , 𝜔 is a real valued weight associated with the edge between units 𝑣 and ℎ , and 𝑏 and 𝑐 are real valued bias terms associated with the 𝑗 th visible and the 𝑖 th hidden variable, respectively [26]. The summation ∑𝒗,𝒉 in equation (1) runs over all 2 states, where 𝑚 𝑛 𝑁 . The RBM is shown in Figure 1. The main purpose of the RBM is to maximize the log-likelihood [26]: ∗
, ∗, ∗
: 𝑙𝑛 ∑𝒉 𝑃 𝒗, 𝒉 ,
where the summation ∑𝒉 runs over 2
(3) states. 3
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ℎ ℎ ℎ 𝑐 𝑐 ⋯ 𝑐
𝜔
𝑏 𝑏 ⋯ 𝑏 𝑣 𝑣 𝑣
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Figure 1. The RBM with 𝑛 hidden and 𝑚 visible neurons
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The optimal parameters 𝜔∗ , 𝑏 ∗ , 𝑐 ∗ determine the “learning memory” of neural networks [10, 26]. In general, it is impossible to find the analytical solution to the optimal problem (3). Therefore, the existing literature focused on seeking the numerical approximation of the solution 𝜔∗ , 𝑏∗ , 𝑐 ∗ [10, 26]. Because the optimal problem (3)
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is to find the maximum likelihood parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ , the learning obeys a kind
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of unsupervised process. This feature makes the RBM approach the human-like learning. However, there is no evidence showing that the RBM exhibits human-like thinking. The most significant feature of the human thinking is the emergence of selfawareness. The existing psychological literature [21-24] argued that self-awareness might consist of different self-motives. In this paper, we will show, different from the RBM, that the SRBM may be a self-motivated system with a biological sense. To this end, let us first introduce the Boltzmann-like distribution that has been found in human societies [15-20].
3. Self-referential entropy
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Different from physical systems, Tao showed that [20] the entropy of a human society would exhibit self-reference. To confirm this, let us consider a human society consisting of 𝑁 self-interested agents [15-20], where the production and consumption behaviors among 𝑁 agents obey the Arrow-Debreu general equilibrium model (ADGEM) [18-20], which describes the ideally free markets. If one employs the symbol to denote a potential income distribution in which there are 𝑎 agents, each 𝑎 of which obtains 𝜀 units of income and 𝜀 𝜀 ⋯ 𝜀 , then Tao showed that, due to the constraint of ADGEM, the most probable income distribution 𝑎∗ obeys a Boltzmann-like pattern [18-20]: 𝑎∗
,
(4)
𝑘 1,2, … , 𝑙, where 𝑔 denotes the number of industries, and 𝛼 and 𝛽 are Lagrange multipliers [15-20]. 4
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By using equation (4), the total number of agents, 𝑁, and gross domestic product (GDP), 𝐸, can be written in the form [18, 20]: 𝑁
∑
,
(5)
𝐸
∑
.
(6)
sense in the economics1 [15-20]. Therefore, we employ
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Without loss of generality, we assume that 𝜀 is a function of 𝑋 ; that is, 𝜀 𝜀 𝑋 . In the statistical physics, 𝑋 denotes the volume [27]; however, 𝑋 makes no 0 to describe a human
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0 to describe a physical system [27].
society [15-20] and employ
By using equations (5) and (6), one can obtain: 𝑑𝐸
𝑑𝑁
𝑑 𝑁
𝛼𝑁
∑
𝛽𝐸
𝑑𝑋.
(7)
The derivation for equation (7) can be found in Appendix A. We discuss equation 0 and
0, where 𝑘
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(7) in terms of two cases; that is,
1,2, … , 𝑙.
Let us define the entropy as: 𝑆 𝑁 𝛼𝑁 𝛽𝐸. In section 4, we will verify that 𝑆 indeed stands for the entropy. In this paper, we mainly explore the case
(8)
0, where we will show that the
entropy in a human society is self-referential. This differs from the entropy in a physical system. To verify this, let us introduce the following proposition.
,
𝑁
𝑆
0 for 𝑘
1,2, … , 𝑙 and if
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Proposition 1: If
𝑁
,
𝐸 𝑁, 𝑆
(9)
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is solvable, then 𝑆 is independent of 𝑁.
Before verifying Proposition 1, we prove a lemma: Lemma 1: The partial differential equation (9) has a general solution: Φ
,
𝑙𝑛𝑁
0,
(10)
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where Φ 𝑎, 𝑏 is a smooth function of 𝑎 and 𝑏. Proof. See Appendix A.
We cannot find a variable with an economic meaning in the neoclassical economics that corresponds to 𝑋.
1
5
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𝑬 𝜶
𝒅𝑬
𝜷
𝒅𝑵
𝟏 𝜷
𝒅 𝑵
𝜶𝑵
𝑬 𝑵, 𝑺 𝜷𝑬 .
𝑵
𝜶𝑵
𝜷𝑬 𝑵, 𝑺 .
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𝑺
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Definition of Entropy
Self-reference
Figure 2. The entropy of a human society, 𝑺, is defined by itself.
Proof of Proposition 1: Obviously, if 𝑑𝐸
𝑑𝑁
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Now we start to prove Proposition 1.
0 for 𝑘 𝑑𝑆.
1,2, … , 𝑙, equation (7) yields: (11)
, .
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Thus, 𝑆 is independent of 𝑁 if and only if:
(12) (13)
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Substituting equations (12) and (13) into equation (8) we obtain equation (9). Therefore, 𝑆 is independent of 𝑁 if and only if equation (9) is solvable. By Lemma 1, we complete the proof. □
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By the Proposition 1, it is easy to verify that the entropy 𝑆 is defined by itself. Let us show it. When Proposition 1 holds, 𝑆 is independent of 𝑁; thus, 𝐸 is a function of 𝑆 and 𝑁, i.e., 𝐸 𝐸 𝑁, 𝑆 . Substituting 𝐸 𝐸 𝑁, 𝑆 into equation (8), which is the definition of entropy, we obtain: 𝑆 𝑁 𝛼𝑁 𝛽𝐸 𝑁, 𝑆 , (14) where 𝑆 is defined by 𝑆, indicating a self-reference. We also plot Figure 2 to show such a self-reference. However, the self-reference of the entropy does not occur in the case
0,
which corresponds to Boltzmann’s physical systems [27], where 𝑋 denotes the volume.
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0 in Appendix B, where we explain why the
We carefully discuss the case
entropy in a physical system is not self-referential. 0, which ensures the self-reference of
Henceforth, we only consider the case
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the entropy. In this case, equations (4) and (9) constitute a Boltzmann-like distribution that has a self-referential entropy. In next section, we employ it to construct a SRBM.
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4. Self-referential Boltzmann machine
Substituting equations (12) and (13) into equation (4), we get Tao’s Boltzmann-like distribution [20]: 𝑎∗ ,
𝑁𝐸 𝑆 𝑁 𝐸 𝑘 1,2, … , 𝑙 , 𝐸
where 𝐸
(15)
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𝐸
∑
and 𝐸
𝑎∗ 𝜀 . Here, we have considered equation
(9), which ensures the self-reference of the entropy. Now we show that if a human society obeys the Boltzmann-like income distribution (15), it will spontaneously form a kind of Boltzmann machine, which we call the SRBM. To this end, we assume that the income distribution among 𝑁 agents obeys equation (15); thus, the joint probability distribution among 𝑁 agents, 𝑃 𝒗, 𝒉 , can be written as: 𝒗,𝒉
𝒗, 𝒉
𝒗,𝒉
∑𝒗,𝒉
𝒗,𝒉
∙ 𝒗,𝒉
𝒗,𝒉
,
(16)
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𝑃
∙ 𝒗,𝒉
𝜽
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where each agent plays the role of a neuron. Therefore, we do not distinguish between agents and neurons2 in this paper. Without loss of generality, we use 𝒗 to denote the visible neurons and 𝒉 the hidden neurons. The derivation for equation (16) can be found in Appendix C. Like the RBM (3), the probability that the neural network assigns to the visible vector 𝒗 is given by summing over all possible hidden vectors: ∑𝒉 𝑃 𝒉, 𝒗 ; therefore, the SRBM is determined by the maximum likelihood estimate: : 𝑙𝑛 ∑𝒉 𝑃
𝒗, 𝒉 ,
(17)
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𝑆 𝑁 𝐸 𝒗, 𝒉 , (18) 𝑠. 𝑡. 𝐸 𝒗, 𝒉 𝑁𝐸 𝒗, 𝒉 where 𝜽 denotes the maximum likelihood parameters. Different from the traditional Boltzmann machines, there is a constraint equation (18) in the SRBM. Furthermore, the maximum likelihood formulation of the optimal problem (17) indicates that the Boltzmann-like distribution (15) “spontaneously” induces the human society to form a SRBM. From this meaning, if a human society obeys the Boltzmann-like income 2
Likewise, we do not distinguish between income and energy. 7
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distribution (15), it can be regarded as a SRBM. Now we show that the SRBM (17) can be simplified to a refined form. To this end, let us write down the average values of 𝑆 𝑁, 𝐸 𝒗, 𝒉 ∑𝒗,𝒉 𝑆 𝑁, 𝐸 𝒗, 𝒉
∙𝑃
𝐸 ∑𝒗,𝒉 𝐸 𝒗, 𝒉 ∙ 𝑃 If we order: 𝛺
𝒗,𝒉
∑𝒗,𝒉 𝑒
𝒗, 𝒉 ,
(19)
𝒗, 𝒉 .
∙ 𝒗,𝒉
(20)
of
𝑆̅
𝒗,𝒉
,
𝑙𝑛𝛺,
𝐸
𝑙𝑛𝛺,
where 𝐸 𝒗, 𝒉
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then one has: 𝑁
and 𝐸 𝒗, 𝒉 :
and 𝐸 𝒗, 𝒉
(21) (22) (23)
, see equations (12) and (13).
𝑑𝑁
𝑑𝐸
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Using equations (22) and (23), it is easy to obtain: 𝑑 𝑙𝑛𝛺
𝛼
𝑙𝑛𝛺
𝛽
𝑙𝑛𝛺 ∙ 𝑑𝛼
where we have used d𝑙𝑛𝛺
𝑙𝑛𝛺 ,
(24)
𝑙𝑛𝛺 ∙ 𝑑𝛽.
On the other hand, equation (11) can be written in the form: 𝑑𝑆̅.
𝑑𝑁
𝑑𝐸
(25)
Comparing equations (24) and (25), we have: 𝑙𝑛𝛺
𝛼
𝑙𝑛𝛺
𝛽
𝑙𝑛𝛺.
(26)
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𝑆̅
𝑃
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The consistency between equations (19) and (26) will be justified immediately. Substituting equation (21) into equation (26) one has: ∑𝒗,𝒉 𝑃 𝒗, 𝒉 ∙ 𝑙𝑛𝑃 𝒗, 𝒉 . (27) 𝑆̅ Now, we start to simplify equation (16). Substituting equation (18) into equation (16) one has: ,
𝒗, 𝒉
𝒗,𝒉
,
∑𝒗,𝒉
𝒗,𝒉
.
(28)
Combining equations (19) and (27) we have: ∑𝒗,𝒉 𝑆 𝑁, 𝐸 𝒗, 𝒉
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∙𝑃
𝒗, 𝒉
∑𝒗,𝒉 𝑃
𝒗, 𝒉 ∙ 𝑙𝑛𝑃
𝒗, 𝒉 .
(29)
Substituting equation (28) into equation (29) yields:
∑𝒗,𝒉 𝑒
,
𝒗,𝒉
1,
(30)
which implies that equation (28) can be simplified to: 𝑃
𝒗, 𝒉
𝑒
,
𝒗,𝒉
.
(31) 8
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From equations (27) and (31), we verify that 𝑆 indeed stands for the entropy3. The consistency between equations (19) and (26) is ensured by equation (30). Using equations (30) and (31), the SRBM (17) can be rewritten in the form: ,
: 𝑙𝑛 ∑𝒉 𝑒 ,
𝑠. 𝑡. ∑𝒗,𝒉 𝑒 where 𝑆 𝑁, 𝐸 𝒗, 𝒉
𝒗,𝒉
𝒗,𝒉
1
,
(32)
is determined by equation (18). Equation (32) is the standard
of
𝜽
form of the SRBM.
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5. Central feature of the SRBM
The general solution of equation (18) has been found to be equation (10). Thus, any function 𝑆 𝑁, 𝐸 𝒗, 𝒉
satisfying equation (10) can be used to construct the SRBM. 𝒗,𝒉
𝑆 𝑁, 𝐸 𝒗, 𝒉
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Here, we only consider the solution: 𝑁 ∙ 𝑙𝑛𝑁,
(33)
which satisfies the well-known thermodynamic relation 𝑑𝑆
as 𝑁 is a constant.
Generally, 𝐸 𝒗, 𝒉 can be expanded as the Taylor’s series: 𝐸 𝒗, 𝒉 ∑
∑
𝐸 𝟎, 𝟎 ∑
𝜌 𝑣𝑣
∑ 𝑏𝑣
∑
𝜔 ℎ𝑣
∑
∑
∑
𝜎 ℎℎ
𝑐ℎ,
(34)
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where we only expand the series up to the two-order terms. For the SRBM, we have the following theorem.
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Theorem 1: Assume that equation (9) holds. If 𝑁 ≫ 1 and 𝐸 𝒗, 𝒉 𝐸 𝒗, 𝒉 0. Proof. We assume 𝐸 𝒗, 𝒉 Φ 0,
𝑙𝑛𝑁
0 . Substituting it into equation (10) yields
0, which can be rewritten as
𝑙𝑛𝑁
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. Since 𝑁 ≫ 1,
𝑁𝑙𝑛𝑁 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ∙ 𝑁 0. Therefore, by equation (31) we have 1, contradicting 0 𝑃 𝒗, 𝒉 1. □
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we conclude 𝑆 𝑃 𝒗, 𝒉 𝑒
0, one has
Theorem 1 implies that the SRBM should always have a positive energy. Here, we further explain that 𝐸 𝒗, 𝒉 0 indeed holds for the SRBM. To this end, substituting equations (33) and (34) into equation (32) one has: Strictly speaking, 𝑆 stands for the amount of information, and the average value 𝑆̅ stands for the entropy, i.e., the average amount of information. However, we do not distinguish the amount of information and entropy in this paper. 3
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,
,
, ,
𝐸
∑
𝜌 𝑣𝑣
∑
,
∙
𝑠. 𝑡. ∑𝒗,𝒉 𝑒 where
∙
: 𝑙𝑛 ∑𝒉 𝑒
𝐸 𝟎, 𝟎 ∑
∑
𝐸
and ∑
𝑏𝑣 ∙
By using ∑𝒗,𝒉 𝑒
(35)
1 ∑
∑
𝜔 ℎ𝑣
𝑐ℎ.
∑
𝜎 ℎℎ
of
,
1, it is easy to verify 𝐸 𝒗, 𝒉
𝐸
𝐸
0. In
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particular, for 𝑁 1 , we have 𝐸 𝒗, 𝒉 0 , which implies a positive zero-point energy 𝐸 𝟎, 𝟎 0 . This central feature significantly differs from the traditional Boltzmann machines. For example, by equation (2), it is easy to check that the RBM leads to 𝐻 𝟎, 𝟎 0. Here, we also attempt to explain the possible underlying origin why the SRBM has a positive zero-point energy. To this end, we observe that if 𝑁 ≫ 𝒗,𝒉
1, equation (33) can be rewritten as 𝑆
𝒗,𝒉
𝑁 ∙ 𝑙𝑛𝑁
ln𝑁!, where an
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additional Gibbs term ln𝑁! emerges. Such a Gibbs term is usually related to quantum effects in physics, which requires the presence of a kind of zero-point energy, that is, 𝐸 𝟎, 𝟎 0. Because equation (33) is the solution of equation (9) that guarantees the self-reference of the entropy 𝑆 , we think that the presence of a positive zero-point energy is actually due to self-reference. From the perspective of Neumann [13] and Hofstadter [14], self-reference may be a central difference of distinguishing between a biological organism and a lifeless-physical system. For example, both authors [13-14] conjectured that a self-loop or self-replication of a biological organism makes sense if and only if a kind of self-reference occurs. Now, we show that a kind of self-loop with a biological sense may occur in the SRBM because it has a positive zero-point energy. To verify this, we can simply observe that a SRBM can activate its neurons by consuming its zero-point energy; in other words, a SRBM can activate its neurons by itself. This will lead to a kind of self-loop with a biological sense, which we call the “self-motive”. To assess this, we design the algorithm for the SRBM (35).
6. Algorithm
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0 and 𝜌 0 . This corresponds to the For simplicity, we order 𝑇 1 , 𝜎 case of the RBM. Thus, equation (35) can be simplified to: ,
, ,
𝑠. 𝑡. 𝑙𝑛 ∑𝒗,𝒉 𝑒
where 𝐸
∙
: 𝑙𝑛 ∑𝒉 𝑒
∑
∑
,
∙
0
𝜔 ℎ𝑣
∑
(36) 𝑏𝑣
∑
𝑐ℎ.
For the optimal problem (36), we can construct a Lagrange function: 𝐿 𝜽|𝒗
𝑙𝑛 ∑𝒉 𝑒
∙
𝜆 ∙ 𝑙𝑛 ∑𝒗,𝒉 𝑒 10
∙
,
(37)
Journal Pre-proof
where 𝜆 denotes the Lagrange multiplier and 𝜽
𝜔 ,𝑏 ,𝑐 ,𝐸 .
By applying the same technique developed by [26] into equation (37), it is easy to obtain the optimal conditions as below: 𝜽𝒗 𝑃 ℎ 1|𝝂 ∙ 𝜈 𝜆 ∙ ∑𝒗 𝑃 𝒗 ∙ 𝑃 ℎ 1|𝝂 ∙ 𝜈 , (38)
𝜽𝒗 𝜽𝒗
𝑃
ℎ
𝜆
1,
𝜽𝒗 where
𝒗 ∙𝜈,
1|𝝂
𝜆 ∙ ∑𝒗 𝑃
∙
𝑙𝑛 ∑𝒗,𝒉 𝑒
𝑃
𝜎 𝑡
𝜆 ∙ ∑𝒗 𝑃
𝒗
∑𝒉 𝑃
(40) (41)
and
𝑃
ℎ
ℎ
1|𝝂
𝜎 ∑
𝜔 𝜈
(42) 𝑐
with
Pr e-
.
𝒗, 𝒉
(39)
1|𝝂 ,
𝒗 ∙𝑃
,
of
𝜈
p ro
𝜽𝒗
Equation (42) is due to equation (30), which ensures the self-reference of the entropy, see the derivation for equation (30) in section 4. Computing equation (42) requires examining all potential combinations 𝒗, 𝒉 . This leads to a huge amount of computation, which implies an unavoidable cost of producing self-reference. By using equations (38)-(42), the optimal parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ , 𝐸 ∗ can be found
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al
according to the following iterative algorithm: 𝜽𝒗 𝜂 ⎧𝜔 ← 𝜔 ⎪ 𝜽𝒗 ⎪ 𝑏 ←𝑏 𝜂 ⎪ 𝜽 𝒗 , (43) 𝑐 ←𝑐 𝜂 ⎨ 𝜽𝒗 ⎪𝐸 ←𝐸 𝜂 ⎪ ⎪ 𝜽𝒗 ⎩ 𝜆←𝜆 𝜂 where 𝜂 denotes the learning rate. In this paper, we propose an energy conservation algorithm (ECA) for the SRBM, which makes the SRBM somewhat biologically plausible. To this end, we need to remove equation (41). Thus, 𝐸 is a given number. If the running time of the SRBM follows 𝑡
𝑡
𝑡
⋯ , we assume that 𝐸
SRBM at the time 𝑡 . We further assume that 𝐸
𝐸 is the initial energy of the 𝒗, 𝒉 stands for the energy that
the SRBM absorbs or consumes by activating neurons 𝒗 and 𝒉 at the time 𝑡 . Thus, by energy conservation, the energy of the SRBM should obey: 11
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𝐸
𝐸
𝐸
,
(44)
where 𝑘 0,1,2, … ∞. From the perspective of a biological organism, the SRBM halts (dies) when 𝐸
𝟎, 𝟎
0, where 𝟎 denotes that all neurons stay at inactive state. This yields
𝜽𝒗 𝜽𝒗
;
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⎨ 𝑐 ←𝑐 𝜂 ⎪ ⎪ ⎩ 𝜆←𝜆 𝜂 𝐸 ←𝐸 𝐸; end
p ro
positive number; 𝐸 𝜂 0.1; g 0; 0 while 𝐸 g g 1; do 𝜽𝒗 𝜂 ⎧𝜔 ← 𝜔 ⎪ ⎪ 𝑏 ←𝑏 𝜂 𝜽𝒗
Pr e-
Energy Conservation Algorithm (ECA):
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a definitely biological meaning: If a SRBM wants to absorb or consume energy, it should activate its neurons by itself; otherwise, it will halt (die). In this sense, the run of the SRBM is a self-motivated activity. Thus, the ECA is designed as below:
Like the RBM, seeking maximum likelihood parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ in the SRBM
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obeys a process of unsupervised-learning. More importantly, by the Theorem 1, we have 𝐸 𝟎, 𝟎 0. This means, by the ECA, that the SRBM cannot halt. Because each run of the SRBM arouses an unsupervised learning, non-halt implies that the SRBM is a self-motivated learning system in which the learning is a kind of self-loop activity. Therefore, from the perspective of the ECA, the SRBM resembles a biological organism, where 𝒗 stands for the sensory nervous system and 𝒉 stands for the reflective nervous system. Self-motive is a universal biological phenomenon. For example, based on an unknown self-motive, earthworms crawl out of the ground. From a psychological perspective, self-motive can be regarded as the most primitive self-motivation. There is a large body of psychological literature [21-24] arguing that self-awareness is a highstage self-motivation. Based on the ECA, we can propose a definition for self-motive in the theoretical framework of Boltzmann machine. If we regard a biological organism as a Boltzmann machine, then the organism dies as it has zero energy with all neurons 12
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being inactive. This corresponds to the halt of the Boltzmann machine. Thus, we have the following definition: Definition 1 (Halt): If one denotes the energy of a Boltzmann machine by 𝐸 𝒗, 𝒉 , the Boltzmann machine halts when 𝐸 𝟎, 𝟎 0, where 𝟎 denotes that all neurons stay in an inactive state.
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of
By equation (2) and Definition 1, we know that the traditional Boltzmann machines can halt. Because each run of the SRBM arouses an unsupervised learning, non-halt implies a series of self-motivated learning behaviors. Therefore, for Boltzmann machines, we can propose a definition for self-motive as below: Definition 2 (Self-motive): If a Boltzmann machine cannot halt, then it is a selfmotivated system.
Pr e-
The ECA can be thought of as a potential algorithm for the SRBM. Nevertheless, it is hard to run the ECA because there is no a practical way of computing equation (42) and the second terms in equations (38)-(40). In this paper, we use the contrastive divergence algorithm [26] to approximately approach the ECA. To implement the contrastive divergence algorithm, equation (41) must hold, that is, 𝜆 1. However, even if we employ the contrastive divergence algorithm, it is hard to compute 𝐸 . Fortunately, the following Theorem 2 will avoid this difficulty. Theorem 2: For the SRBM, if 𝐸
Proof. By Theorem 1, one has 𝐸 𝒗, 𝒉 𝐸 𝐸 𝐸 . By 𝐸 0, we complete the proof. □
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𝐸
0, then one has 𝐸
0.
0, which can be rewritten as
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By Theorem 2, if 𝐸 0 can be always justified, then the SRBM runs forever. Therefore, we use 𝐸 0 to replace 𝐸 0 in the ECA. Next, we give the contrastive divergence approximation to the ECA. The approximation algorithm is to test if 𝐸 0 always holds. The programming language is based on the MATLAB. For a given visible input 𝝂 , we carry out the following programming to learn it: Programming 1:
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1; 𝐸 𝜂 0.1; g 0; 0.001 while 𝐸 g g 1; do
13
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⎧𝜔 ← 𝜔 ⎪ ⎨ ⎪ ⎩ 𝐸
𝜂∙ 𝑃
𝑏 ←𝑏
𝑐 ←𝑐 ∑
1𝝂
ℎ
∑
𝜂∙ 𝑃
ℎ ∑
𝜔 ℎ𝑣
𝑃
∙𝜈
𝜂∙ 𝜈
𝜈
1𝝂
𝑃 ∑
𝑏𝑣
ℎ
1𝝂
∙𝜈 ;
ℎ
1𝝂
𝑐ℎ;
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end
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p ro
𝝂 is the output vector, which is the learning result of the SRBM, where 𝑘 stands for the step number of Gibbs sampling [26]. Here, we consider 𝑘 1, 𝑛 1 and 𝑚 25 . 𝝂 denotes an image consisting of 5 5 blocks, and it is randomly extracted from the image gallery, which includes 2 different images. Because of the infinite self-motive of the SRBM, the vector 𝝂 should reproduce all 2 images. The detailed Programming 1 can be found in Supplemental Materials, by which 𝝂 indeed examines all 2 images. Here, for simplicity, we only check 𝑚 25 . By using the Programming 1, the readers can check that our conclusion holds for any 𝑚. This verifies that the SRBM cannot halt. By Definition 2, this means that the SRBM is a self-motivated system. Boltzmann machines are well known because of their excellent learning function in pattern recognition [26]. In the next section, we apply the selfmotive of the SRBM to perform the task of searching images.
7. Search task
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By Programming 1, we know that the SRBM cannot halt. During the running process, it learns images continuously, and records each image. This means that the SRBM can constantly browse an image gallery with a huge number of images by itself, unless the target image has been found. Here, we consider that an image consists of 3 3 blocks, that is, 𝑚 9. Learning image 𝝂 is randomly extracted from the image gallery, which includes 2 different images. Once 𝝂 is learned, then it can be recorded in the form of 𝝂 . In this process, we hope to seek the target image 𝝂∗ . If the SRBM finds 𝝂∗ , we should have 𝝂 𝝂∗ . Thus, we design the following programming. Programming 2:
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𝑟𝑎𝑛𝑑 3,3 ; 𝝂 𝐸 1; 𝑒 1; 𝜂 0.1; g 0; 0.001 & 𝑒 while 𝐸 g g 1;
0.01
14
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do
∑
𝑒
𝝂
∑
𝜂∙ 𝑃
∙𝜈
1𝝂
𝑏 ←𝑏
𝑐 ←𝑐
𝐸
ℎ
ℎ ∑
𝜔 ℎ𝑣
ℎ
𝑃
𝜂∙ 𝜈
𝜈
1𝝂
𝑃 ∑
𝑏𝑣
1𝝂
; ℎ
1𝝂
𝑐ℎ;
𝝂∗ ;
end
∙𝜈
p ro
⎨ ⎪ ⎩
𝜂∙ 𝑃
of
⎧𝜔 ← 𝜔 ⎪
Image Gallery
□■□ □□□ □□□
□□■ □□□ □□□
Pr e-
■□□ □□□ □□□
𝝂
……
■■■ ■■■ ■■■
is randomly extracted
■■■ □□□ ■■■
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Unsupervised-learning
Learning result 𝝂
■■■ □□□ ■■■ Matching
■■■ Target Image 𝝂∗ □□□ ■■■
Figure 3. The target image 𝝂∗ is hidden in the image gallery, which includes 2 is randomly extracted from the image gallery. different images. Learning image 𝝂 is learned, then the SRBM records it in the form of 𝝂 . The SRBM finds Once 𝝂 ∗ 𝝂 if and only if 𝝂 𝝂∗ . 15
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8. Conclusion
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p ro
of
The detailed Programming 2 can be found in Supplementary Materials. We have plotted the result of search task in Figure 3, from which one can find that, due to the will examine all 2 images, and 𝝂∗ is finally infinite self-motive, the vector 𝝂 found. To perform the search task, we use the contrastive divergence algorithm to approximately approach the ECA. This requires that equation (41) holds, which leads to 𝜆 1. However, equation (41) will break the energy conservation equation (44). Therefore, the contrastive divergence algorithm only provides a quite rough approximation for the ECA. To rigidly perform the ECA, we need to find a way of computing equation (42) and the second terms in equations (38)-(40). This will be explored in the future. Here, we need to clarify that traditional Boltzmann machines cannot finish the self0 , by motive task above. Since traditional Boltzmann machines require 𝐸 Programming 2 it is easy to check that each run of the search task will produce 𝐸 0. 𝐸 0, which, by Definition 1, indicates halt. However, This means 𝐸 𝒗, 𝒉 𝐸 we highlight that the self-motive endowed by the SRBM is only a kind of self-loop with a biological sense; that is, if an organism wants to perform learning motivation, it have to take a positive energy to activate its neurons. Such a self-loop makes sense if and only if the organism has a positive (zero-point) energy even though all neurons are inactive. By contrast, traditional Boltzmann machines fail to capture such a biological feature.
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Different from lifeless-physical systems, a well-functioning free-market society will obey a Boltzmann-like distribution that has a self-referential entropy. We show that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a self-referential Boltzmann machine (SRBM), where each person plays the role of a neuron. Because of the self-reference of the entropy, the SRBM always has a positive energy, even if all neurons are inactive. This implies the presence of a kind of positive zero-point energy. We show that such a positive zero-point energy leads to the SRBM being a self-motivated learning system if the energy conservation is taken into account. This makes the SRBM somewhat biologically plausible. Because the existing empirical investigations have confirmed that a well-functioning freemarket society will obey a Boltzmann-like income distribution [19-20], we conclude that such a society is actually a SRBM. This implies that a human society functions like a kind of complex system (or organism) with potential self-motivations, just as motivated by an “invisible hand” coined by Adam Smith. This finding contradicts the long-standing reductionism that is popular in the neoclassical economics. On the other hand, there has been a large body of psychological literature proposing that selfawareness consists of different self-motives. Thus, the SRBM provides a potential perspective for exploring self-awareness. As a simple application, we also employ the self-motive of the SRBM to perform the task of searching images.
16
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Appendix A Derivation for equation (7): Because 𝜀 is a function of 𝑋, we conclude 𝑁 𝑑𝑁
|
,
, ,
𝑑𝛼
|
𝑑𝛽
,
𝑁 𝛼, 𝛽, 𝑋 . Thus, we have: , ,
|
,
, ,
|
,
𝑁.
, ,
|
,
𝐸.
, ,
|
,
𝛽∑
(A.1)
(A.2)
p ro
By equations (5) and (6), one has:
𝑑𝑋.
of
, ,
.
(A.3)
(A.4)
By substituting equation (A.1) into equation (A.3), it is easy to obtain:
On the other hand, we have: 𝑑𝛼
𝑑𝑁
𝑑 𝛽
𝑑
𝛼𝑑
𝑑 𝛼
𝑑𝛼
𝑑𝑋 .
Pr e-
𝑑𝐸
.
(A.5)
(A.6)
Substituting equation (A.6) into equation (A.5) yields: 𝑑𝐸
𝑑 𝛽
𝑑𝑁
𝑑 𝛼
𝑑
𝑑𝑋.
(A.7)
Finally, by substituting equations (A.2), (A.3) and (A.4) into equation (A.7), we get: 𝑑𝑁
𝑑𝐸
𝑑 𝑁
𝛼𝑁
∑
𝛽𝐸
𝑑𝑋.
(A.8)
al
□
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Proof of Lemma 1: The characteristic equation of the partial differential equation (9) is .
By
(A.9)
, one can get a first integral:
𝐶,
(A.10)
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where 𝐶 is a constant. On the other hand, it is easy to calculate: 𝑆𝑑𝑁
𝑁𝑑𝑆
𝑁 𝑑
.
(A.11)
Substituting equation (A.11) into
𝑑
yields:
𝑑 𝑙𝑛𝑁 .
(A.12) 17
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Thus, we get another first integral: 𝑙𝑛𝑁
𝐶,
(A.13)
where 𝐶 is a constant. Combining equations (A.10) and (A.13), we obtain the general solution (10). □
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Appendix B
0 is not self-referential, we introduce the
p ro
To verify that the entropy for following two propositions. 0 for 𝑘
Proposition 2: If
1,2, … , 𝑙 and if 𝑁 , 𝑆 , and 𝑋 are independent
with each other, then we have:
|
𝑆
, ,
𝑁
∑
,
𝐸 𝑁, 𝑆, 𝑋 ,
Pr e-
, ,
𝑁
.
(B.1) (B.2)
Proof. Because 𝑁, 𝑆, and 𝑋 are independent with each other, by equations (7) and (8) it is easy to obtain equation (B.1). Next we verify equation (B.2). As 𝜀 is a function of 𝑋, equation (6) indicates: 𝑑𝐸
|
𝑑𝛼
,
|
𝑑𝛽
,
|
𝑑𝑋.
,
(B.3)
𝑑𝐸
|
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On the other hand, as 𝐸 is a function of 𝑁, 𝑆, and 𝑋 as well, one should have: 𝑑𝑁
,
|
,
𝑑𝑆
|
,
𝑑𝑋.
(B.4)
𝑑𝑋.
(B.5)
By equations (5) and (6) we have:
𝑑𝑆
𝑑𝛼
|
𝑑𝛽
urn
|
𝑑𝑁
|
,
,
𝑑𝛼
|
,
|
𝑑𝛽
,
|
, ,
𝑑𝑋.
(B.6)
Substituting equations (B.5) and (B.6) into equation (B.4) yields: 𝑑𝐸
|
∙
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|
∙
,
|
,
,
∙
|
|
|
,
|
,
|
,
,
|
,
,
∙
∙
|
∙
| |
𝑑𝛼
,
𝑑𝛽
,
|
,
,
.
(B.7)
𝑑𝑋
Comparing equations (B.3) and (B.7), we have: |
,
|
,
|
,
∙
|
,
|
,
∙
|
,
.
The partial derivative of equation (8) with respect to 𝑋 yields: 18
(B.8)
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|
|
,
𝛼
,
|
Substituting
|
𝛽
,
|
,
,
|
,
.
(B.9) and equation (B.9) into equation (B.8)
,
yields: |
|
,
,
.
(B.10)
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Substituting equation (A.4) into equation (B.10), we obtain equation (B.2). □
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By Proposition 2, equation (B.2) ensures the validity of the complete differential (7). By observing equation (8), equation (B.1) is actually the definition of the entropy. However, the following proposition further shows that equation (B.1) may contradict equation (7)
Pr e-
Proposition 3: If the complete differential (7) holds for any 𝐸 𝑁, 𝑆, 𝑋 satisfying equation (B.1), then one has 𝑆 𝑁 𝛼𝑁 𝛽𝐸 𝑁, 𝑆, 𝑋 . Proof. The complete differential (7) indicates: |
,
|
,
, .
(B.11) (B.12)
Substituting equations (B.11) and (B.12) into equation (8) yields: 𝑁
, ,
𝑆
, ,
𝑁
𝐸 𝑁, 𝑆, 𝑋 .
(B.13)
𝑁
, ,
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Here, we can construct a partial differential equation: 𝑆
, ,
𝑁
, ,
𝜖
𝐸 𝑁, 𝑆, 𝑋 ,
(B.14)
where 𝜖 is a parameter, which is independent of 𝑁, 𝑆 and 𝑋. , ,
urn
Obviously, if 𝑙𝑖𝑚 𝜖 ∙ →
obtain: 𝐸
𝑁, 𝑆, 𝑋
0 , by equations (B.13) and (B.14) it is easy to
𝐸 𝑁, 𝑆, 𝑋 .
Later, we will verify 𝑙𝑖𝑚 𝜖 ∙
(B.15) , ,
0.
→
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The characteristic equation of equation (B.14) yields: .
(B.16)
Using equation (B.16) we can obtain the following first integrals: 𝐷,
𝑙𝑛𝑁
(B.17)
𝐷,
(B.18) 19
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𝑁𝑒
𝐷,
(B.19)
where 𝐷 , 𝐷 and 𝐷 are constants. Based on equations (B.17)-(B.19), the general solution of equation (B.14) equals: 𝐹
,
0,
𝑙𝑛𝑁, 𝑁𝑒
(B.20)
∙
,
|
|
,
,
∙
|
,
, |
,
∙
|
0
,
which can be rewritten as: , ,
|
| ,
𝑁
,
.
| ,
p ro
|
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where 𝐹 𝑥, 𝑦, 𝑧 is a smooth function of 𝑥, 𝑦 and 𝑧. The partial derivative of equation (B.20) with respect to 𝑋 equals:
(B.21)
(B.22)
, ,
𝑙𝑖𝑚 𝜖 ∙
Pr e-
As 𝐹 𝑥, 𝑦, 𝑧 is a smooth function of 𝑥, 𝑦 and 𝑧, equation (B.22) implies that one can obtain: 0,
→
which means that equation (B.15) holds. Equation (B.15) indicates that, for every 𝛿 integer
so that when
, ,
𝛿.
, one has:
𝑚𝑎𝑥
Let us order positive integer
𝐸 𝑁, 𝑆, 𝑋
𝜂∙
, ,
(B.25)
. Thus, for every 𝛿
,
so that when
Jo
𝐸 𝑁, 𝑆, 𝑋
(B.24) 0 , there always exists a positive
al
so that when
urn
𝜖∙
0 , there always exists a positive
, one has:
|𝐸 𝑁, 𝑆, 𝑋 𝐸 𝑁, 𝑆, 𝑋 | 𝛿. Equation (B.23) indicates that, for every 𝛿 integer
(B.23)
0 , there always exists a
, one has: 𝛿,
(B.26)
𝛿.
(B.27)
Equations (B.26) and (B.27) imply that 𝐸 𝑁, 𝑆, 𝑋
is a series of the
solution of equation (B.13). Therefore, by the condition of Proposition 3, we conclude that
𝑑𝐸 𝑁, 𝑆, 𝑋
is a series of complete differential. However, equation 20
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(B.22) indicates that 𝑙𝑖𝑚 𝑑𝐸 𝑁, 𝑆, 𝑋 does not exist. This implies 𝑆 →
𝑁
𝛼𝑁
𝛽𝐸 𝑁, 𝑆, 𝑋 . □ Obviously, Proposition 3 contradicts equation (8), which is the definition of the 0 for 𝑘
entropy. This means that when
1,2, … , 𝑙, the definition of the entropy
0 for 𝑘
differs from the case
of
does not always ensure the validity of the complete differential (7). This significantly 1,2, … , 𝑙, where equation (9) does guarantee the
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validity of the complete differential (11). In this sense, the so-called complete differential (7) in the thermodynamics is only a physical result rather than a rigid mathematical result.
Appendix C
𝑃∗ 1, … , 𝑁
∗
𝑎∗
∏
,
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As the income structure of an 𝑁-agent society obeys the Boltzmann-like income distribution (15), the joint probability distribution among 𝑁 agents, 𝑃∗ 1, … , 𝑁 , can be written as: (C.1)
where 𝑎 and 𝑃∗ 1, … , 𝑁 have been regarded as the unnormalized probabilities. To derive equation (C.1), we assume that the income probability of each agent is independent. As the probability of obtaining 𝜀 units of income is 𝑎∗ , by the assumption of independence, the joint probability among 𝑎∗ agents, each of which ∗
∗
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obtains 𝜀 units of income, equals 𝑎∗ 𝑎∗ ∙ 𝑎∗ ⋯ 𝑎∗ . Equation (C.1) is the result 𝜀 ⋯ 𝜀. of taking into account all income levels 𝜀 Substituting equation (15) into equation (C.1) yields: ∗
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∏
𝑃∗ 1, … , 𝑁
∗
∗
,
∙
(C.2)
∏
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where we have used equations (5) and (6). Obviously, the GDP 𝐸 should be a function with respect to the state variables of agents. Let us denote the state variables of 𝑁 agents by the vector 1 (or 𝑣 𝑣 , … , 𝑣 , ℎ , … , ℎ , where 𝑚 𝑛 𝑁. For example, we can use ℎ 0 (or 𝑣 0) to denote 1) to denote that the agent 𝑖 is active in markets, and ℎ inactivity. Thus, without loss of generality, the GDP 𝐸 𝑣 , … , 𝑣 , ℎ , … , ℎ can be expanded as the Taylor’s series: 𝐸 𝑣 ,…,𝑣 ,ℎ ,…,ℎ
∑
∑
𝜎 ℎℎ
∑
∑
𝜌 𝑣𝑣
∑ 21
∑
𝐸 0, … ,0,0, … ,0 𝑏𝑣
∑ ∑
𝜔 ℎ𝑣 𝑐ℎ,
(C.3)
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where 𝜔 , 𝜎 and 𝜌 represent weights, and 𝑏 and 𝑐 represent biases. Here we only expand the series up to the two-order terms. Using equation (C.3), 𝑃∗ 1, … , 𝑁 can be rewritten in the form of normalized probability: 𝒗,𝒉
𝑃
𝑒
𝒗, 𝒉
where 𝑍
∑𝒗,𝒉 𝑒
𝒗,𝒉
∙ 𝒗,𝒉 ∙ 𝒗,𝒉
𝒗,𝒉
,
(C.4)
𝒗,𝒉
denotes the partition function, 𝒗
𝑣 ,…,𝑣
and
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References:
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𝒉 ℎ , … , ℎ . ∑𝒗∙ and ∑𝒉∙ denote the sum over all possible 𝒗 and 𝒉 , respectively. Obviously, if we regard each agent as a neuron, then the GDP 𝐸 𝒗, 𝒉 denotes energy. Without loss of generality, we use 𝒗 to denote the visible neurons and 𝒉 the hidden neurons.
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[1]. G. A. Mashour, A. G. Hudetz1. Neural Correlates of Unconsciousness in LargeScale Brain Networks. Trends in neurosciences 41, 150-160 (2018) [2]. Wissner-Gross A. D., Freer, C. E. Causal entropic forces. Phys. Rev. Lett. 110, 168702 (2013) [3]. J. L. Perez Velazquez. Finding simplicity in complexity: general principles of biological and nonbiological organization. Journal of Biological Physics 35, 209–221 (2009) [4]. R. Guevara Erra, D. M. Mateos, R. Wennberg, J. L. Perez Velazquez. Statistical mechanics of consciousness: Maximization of information content of network is associated with conscious awareness. Phys. Rev. E 94, 052402 (2016) [5]. D. M. Mateos, R. Wennberg, R. Guevara, and J. L. Perez Velazquez. Consciousness as a global property of brain dynamic activity. Phys. Rev. E 96, 062410 (2017) [6]. I. D. Vincenzo, I. Giannoccaro, G. Carbone, P. Grigolini. Criticality triggers the emergence of collective intelligence in groups. Phys. Rev. E 96, 022309 (2017) [7]. D. M. Mateos, R. Guevara Erra, R. Wennberg, J. L. Perez Velazquez. Measures of entropy and complexity in altered states of consciousness. Cognitive Neurodynamics 12, 73–84 (2018) [8]. J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. PNAS 79, 2554-2558 (1982) [9]. D. H. Ackley, G. E. Hinton, T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science 9, 147-169 (1985) [10]. G. E. Hinton, R. R. Salakhutdinov. Reducing the Dimensionality of Data with Neural Networks. Science 313, 504-507 (2006) [11]. Y. LeCun, Y. Bengio, G. Hinton. Deep learning. Nature 521, 436–444 (2015) [12]. D. Silver et al. Mastering the game of Go with deep neural networks and tree search. Nature 529, 484–489 (2016) 22
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[13]. J. V. Neumann. Theory of Self-Reproducing Automata. University of Illinois Press (1966) [14]. D. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books (1979) [15]. Y. Tao. Competitive market for multiple firms and economic crisis. Phys. Rev. E 82, 036118 (2010) [16]. Y. Tao, X. Chen. Statistical Physics of Economic Systems: a Survey for Open Economies. Chinese Physics Letters 29, 058901 (2012) [17]. Y. Tao. Universal Laws of Human Society’s Income Distribution. Physica A 435, 89-94 (2015) [18]. Y. Tao. Spontaneous economic order. Journal of Evolutionary Economics 26, 467500 (2016) [19]. Y. Tao, X. Wu, T. Zhou, W. Yan, Y. Huang, H. Yu, B. Mondal, V. M. Yakovenko. Exponential structure of income inequality: evidence from 67 countries. Journal of Economic Interaction and Coordination 14, 345-376 (2019) [20]. Y. Tao. Swarm intelligence in humans: A perspective of emergent evolution. Physica A 502, 436-446 (2018) [21]. T. S. Duval, R. A. Wicklund. A theory of objective self-awareness. New York: Academic Press (1972) [22]. R. A. Wicklund. Objective self-awareness. Advances in Experimental Social Psychology 8, 233-275 (1975) [23]. T. S. Duval, P. J. Silvia. Self-awareness and causal attribution: A dual systems theory. Boston: Kluwer Academic (2001) [24]. P. J. Silvia, T. S. Duval. Objective self-awareness theory: Recent progress and enduring problems. Personality and Social Psychology Review 5, 230-340 (2001) [25]. H.E. Stanley, et al., Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics. Physica A 224, 302-321 (1996). [26]. A. Fischer, C. Igel. Training restricted Boltzmann machines: An introduction. Pattern Recognition 47, 25-39 (2014) [27]. B. Linder. Thermodynamics and Introductory Statistical Mechanics. WileyInterscience, New York (2004)
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Acknowledgements: Yong Tao was supported by the Fundamental Research Funds for the Central Universities (Grant No. SWU1409444) and the State Scholarship Fund granted by the China Scholarship Council.
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Highlights
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· Self‐referential Boltzmann machine is a Boltzmann machine with self‐referential entropy · Self‐reference of entropy leads to a positive energy even if all neurons are inactive · Self‐referential Boltzmann machine is a self‐motivated system · Self‐motive can be regarded as the most primitive self‐motivation · Self‐referential Boltzmann machine runs as if a biological organism runs
24
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Declaration of Interest Statement
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The author declares no competing interests.
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PII: DOI: Reference:
S0378-4371(19)32102-8 https://doi.org/10.1016/j.physa.2019.123775 PHYSA 123775
To appear in:
Physica A
Received date : 1 March 2019 Revised date : 31 October 2019 Please cite this article as: Y. Tao, Self-referential Boltzmann machine, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123775. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Self-referential Boltzmann Machine Yong Tao†
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College of Economics and Management, Southwest University, Chongqing, China Department of Management, Technology and Economics, ETH Zurich, Switzerland
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Abstract: We recently reported that the income structure of the low and middle class (about 95% of populations) in a well-functioning free-market country would follow a Boltzmann-like distribution that has a self-referential entropy [Physica A 502, 436-446 (2018)]. The empirical evidences cover 66 free-market countries and the Hong Kong SAR. By contrast, the entropy of a physical system is not self-referential. This finding implies that the self-reference may be a potential difference between biological and lifeless-physical systems. In this paper, we argue that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a self-referential Boltzmann machine (SRBM), where each person plays the role of a neuron. Because of the self-reference of the entropy, we show that the SRBM always has a positive energy even if all neurons are inactive. This implies the presence of a kind of positive zero-point energy. Based on such a positive zero-point energy, we further show that the SRBM may be a self-motivated system with a biological sense. Our finding supports that a human society functions like a kind of complex system (or organism) with potential self-motivations, just as motivated by an “invisible hand” coined by Adam Smith. As a simple application, we apply the self-motive of the SRBM to perform the task of searching images.
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Keywords: Boltzmann machine; Self-reference; Self-motive; Self-awareness; Gibbs term PACS numbers: 87.85.dq; 87.19.lv; 89.75.Fb
†
Correspondence to: [email protected] or [email protected] 1
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1. Introduction
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Understanding the origin of self-awareness is one of the most challenging and fundamental questions in 21st century science [1]. Recently, some scholars have observed the potential connections between consciousness, intelligence, and entropy. [2-7]. On the one hand, Wissner-Gross and Freer [2] proposed a deep connection between individual intelligence and entropy maximization. On the other hand, Perez Velazquez et al [3-7] suggested that consciousness can be considered as a consequence of the most probable distribution that maximizes entropy of brain functional networks. In the equilibrium statistical physics, the most probable distribution related to the entropy maximization is the Boltzmann distribution. Interestingly, the Boltzmann distribution has been employed to construct Boltzmann machines, which are stochastic neural networks [8], assuming that the energy distribution of neurons obeys the Boltzmann distribution [9]. For years, because of the endeavor of Hinton [10], Boltzmann machines have contributed to the opening of the field of deep learning architectures [11]. As an application, deep learning architectures were recently used to construct AlphaGo [12], a computer program, which beat the world’s No.1 ranked player of the game Go. A remarkable merit of Boltzmann machines is due to the unsupervised-learning function, which captures the feature of human-like learning. Despite these advances, we still do not know how to construct a neural network that exhibits human-like thinking. The most significant feature of human thinking is the emergence of self-awareness. Motivated by Godel’s Incompleteness Theorem, Neumann [13] and Hofstadter [14] conjectured that the emergence of self-awareness might be due to the “self-reference” of neural networks in the brain. Unfortunately, no one has clarified to date how to construct a neural network that exhibits self-reference. Recently, Tao theoretically showed [15-18] that a well-functioning free-market society evolves according to the rule of entropy maximization and the household income structure will obey a Boltzmann-like distribution. By analyzing datasets of household income from 66 countries and Hong Kong SAR, Tao et al confirmed that [19], for all the countries, the income structure for lower and middle classes (about 95% of populations) exactly follows the Boltzmann-like distribution. More recently, by doing an in-depth investigation for the Boltzmann-like distribution, Tao showed that [20], different from the entropy in a physical system, the entropy in a human society is self-referential. As a human society is obviously a biological system, Tao identified the “self-reference” of entropy as a central characteristic for distinguishing between biological and lifelessphysical systems [20]. The main purpose of this paper is to show that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a kind of Boltzmann machine, which we call the “self-referential Boltzmann machine” (SRBM). Because of the self-reference of the entropy, we will find that the SRBM always has a positive energy, even if all neurons are inactive. This singular feature leads to the possibility that the SRBM may be a self-motivated system with a biological sense. There is a large body of psychological literature arguing that self-awareness consists of 2
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2. Boltzmann machines
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different self-motives [21-24]. Thus, the SRBM provides a potential perspective for exploring the origin of self-awareness, which bridges the literature between neural networks and psychology. In particular, due to the self-motivation feature of the SRBM, we argue that a human society should be a kind of complex system (or organism) [25]. This finding contradicts the long-standing reductionism that is popular in the neoclassical economics. The remainder of this paper is organized as follows. Section 2 briefly introduces Boltzmann machines. In section 3, we show that the income structure of a wellfunctioning free-market society will obey a Boltzmann-like distribution, and that, different from Boltzmann’s physical systems, the entropy in a human society exhibits self-reference. In section 4, we show that if a human society obeys such a Boltzmannlike income distribution, it will spontaneously form a SRBM, where each person plays the role of a neuron. In section 5, we show that the SRBM always has a positive energy even if all neurons are inactive. In section 6, we show that this singular feature leads to the possibility that the SRBM may be a self-motivated system with a biological sense. Furthermore, we design an algorithm for the SRBM. In section 7, we apply the selfmotive of the SRBM to perform the task of searching images. In section 8, our conclusion follows.
𝒗,𝒉
𝑃 𝒗, 𝒉
𝒗,𝒉
,
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∑𝒗,𝒉
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For simplicity, we only introduce the Restricted Boltzmann Machine (RBM), which is a special Boltzmann machine [10, 26]. A RBM consists of 𝑚 visible units 𝒗 𝑣 ⋯ 𝑣 , representing observable data, and 𝑛 hidden units 𝒉 ℎ ⋯ ℎ , to capture the dependencies between observed variables [26]. The random variables and the joint probability distribution 𝒗 𝒉 take values belonging to 0,1 under the model is given by the Boltzmann distribution [26]: (1)
with the energy function: 𝐻 𝒗, 𝒉
∑
∑
𝜔 ℎ𝑣
∑
𝑏𝑣
∑
𝑐ℎ,
(2)
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where 𝑇 denotes the temperature. For all 𝑖 ∈ 1, … , 𝑛 and 𝑗 ∈ 1, … , 𝑚 , 𝜔 is a real valued weight associated with the edge between units 𝑣 and ℎ , and 𝑏 and 𝑐 are real valued bias terms associated with the 𝑗 th visible and the 𝑖 th hidden variable, respectively [26]. The summation ∑𝒗,𝒉 in equation (1) runs over all 2 states, where 𝑚 𝑛 𝑁 . The RBM is shown in Figure 1. The main purpose of the RBM is to maximize the log-likelihood [26]: ∗
, ∗, ∗
: 𝑙𝑛 ∑𝒉 𝑃 𝒗, 𝒉 ,
where the summation ∑𝒉 runs over 2
(3) states. 3
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ℎ ℎ ℎ 𝑐 𝑐 ⋯ 𝑐
𝜔
𝑏 𝑏 ⋯ 𝑏 𝑣 𝑣 𝑣
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Figure 1. The RBM with 𝑛 hidden and 𝑚 visible neurons
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The optimal parameters 𝜔∗ , 𝑏 ∗ , 𝑐 ∗ determine the “learning memory” of neural networks [10, 26]. In general, it is impossible to find the analytical solution to the optimal problem (3). Therefore, the existing literature focused on seeking the numerical approximation of the solution 𝜔∗ , 𝑏∗ , 𝑐 ∗ [10, 26]. Because the optimal problem (3)
Pr e-
is to find the maximum likelihood parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ , the learning obeys a kind
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of unsupervised process. This feature makes the RBM approach the human-like learning. However, there is no evidence showing that the RBM exhibits human-like thinking. The most significant feature of the human thinking is the emergence of selfawareness. The existing psychological literature [21-24] argued that self-awareness might consist of different self-motives. In this paper, we will show, different from the RBM, that the SRBM may be a self-motivated system with a biological sense. To this end, let us first introduce the Boltzmann-like distribution that has been found in human societies [15-20].
3. Self-referential entropy
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Different from physical systems, Tao showed that [20] the entropy of a human society would exhibit self-reference. To confirm this, let us consider a human society consisting of 𝑁 self-interested agents [15-20], where the production and consumption behaviors among 𝑁 agents obey the Arrow-Debreu general equilibrium model (ADGEM) [18-20], which describes the ideally free markets. If one employs the symbol to denote a potential income distribution in which there are 𝑎 agents, each 𝑎 of which obtains 𝜀 units of income and 𝜀 𝜀 ⋯ 𝜀 , then Tao showed that, due to the constraint of ADGEM, the most probable income distribution 𝑎∗ obeys a Boltzmann-like pattern [18-20]: 𝑎∗
,
(4)
𝑘 1,2, … , 𝑙, where 𝑔 denotes the number of industries, and 𝛼 and 𝛽 are Lagrange multipliers [15-20]. 4
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By using equation (4), the total number of agents, 𝑁, and gross domestic product (GDP), 𝐸, can be written in the form [18, 20]: 𝑁
∑
,
(5)
𝐸
∑
.
(6)
sense in the economics1 [15-20]. Therefore, we employ
of
Without loss of generality, we assume that 𝜀 is a function of 𝑋 ; that is, 𝜀 𝜀 𝑋 . In the statistical physics, 𝑋 denotes the volume [27]; however, 𝑋 makes no 0 to describe a human
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0 to describe a physical system [27].
society [15-20] and employ
By using equations (5) and (6), one can obtain: 𝑑𝐸
𝑑𝑁
𝑑 𝑁
𝛼𝑁
∑
𝛽𝐸
𝑑𝑋.
(7)
The derivation for equation (7) can be found in Appendix A. We discuss equation 0 and
0, where 𝑘
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(7) in terms of two cases; that is,
1,2, … , 𝑙.
Let us define the entropy as: 𝑆 𝑁 𝛼𝑁 𝛽𝐸. In section 4, we will verify that 𝑆 indeed stands for the entropy. In this paper, we mainly explore the case
(8)
0, where we will show that the
entropy in a human society is self-referential. This differs from the entropy in a physical system. To verify this, let us introduce the following proposition.
,
𝑁
𝑆
0 for 𝑘
1,2, … , 𝑙 and if
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Proposition 1: If
𝑁
,
𝐸 𝑁, 𝑆
(9)
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is solvable, then 𝑆 is independent of 𝑁.
Before verifying Proposition 1, we prove a lemma: Lemma 1: The partial differential equation (9) has a general solution: Φ
,
𝑙𝑛𝑁
0,
(10)
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where Φ 𝑎, 𝑏 is a smooth function of 𝑎 and 𝑏. Proof. See Appendix A.
We cannot find a variable with an economic meaning in the neoclassical economics that corresponds to 𝑋.
1
5
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𝑬 𝜶
𝒅𝑬
𝜷
𝒅𝑵
𝟏 𝜷
𝒅 𝑵
𝜶𝑵
𝑬 𝑵, 𝑺 𝜷𝑬 .
𝑵
𝜶𝑵
𝜷𝑬 𝑵, 𝑺 .
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Definition of Entropy
Self-reference
Figure 2. The entropy of a human society, 𝑺, is defined by itself.
Proof of Proposition 1: Obviously, if 𝑑𝐸
𝑑𝑁
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Now we start to prove Proposition 1.
0 for 𝑘 𝑑𝑆.
1,2, … , 𝑙, equation (7) yields: (11)
, .
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Thus, 𝑆 is independent of 𝑁 if and only if:
(12) (13)
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Substituting equations (12) and (13) into equation (8) we obtain equation (9). Therefore, 𝑆 is independent of 𝑁 if and only if equation (9) is solvable. By Lemma 1, we complete the proof. □
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By the Proposition 1, it is easy to verify that the entropy 𝑆 is defined by itself. Let us show it. When Proposition 1 holds, 𝑆 is independent of 𝑁; thus, 𝐸 is a function of 𝑆 and 𝑁, i.e., 𝐸 𝐸 𝑁, 𝑆 . Substituting 𝐸 𝐸 𝑁, 𝑆 into equation (8), which is the definition of entropy, we obtain: 𝑆 𝑁 𝛼𝑁 𝛽𝐸 𝑁, 𝑆 , (14) where 𝑆 is defined by 𝑆, indicating a self-reference. We also plot Figure 2 to show such a self-reference. However, the self-reference of the entropy does not occur in the case
0,
which corresponds to Boltzmann’s physical systems [27], where 𝑋 denotes the volume.
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0 in Appendix B, where we explain why the
We carefully discuss the case
entropy in a physical system is not self-referential. 0, which ensures the self-reference of
Henceforth, we only consider the case
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the entropy. In this case, equations (4) and (9) constitute a Boltzmann-like distribution that has a self-referential entropy. In next section, we employ it to construct a SRBM.
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4. Self-referential Boltzmann machine
Substituting equations (12) and (13) into equation (4), we get Tao’s Boltzmann-like distribution [20]: 𝑎∗ ,
𝑁𝐸 𝑆 𝑁 𝐸 𝑘 1,2, … , 𝑙 , 𝐸
where 𝐸
(15)
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𝐸
∑
and 𝐸
𝑎∗ 𝜀 . Here, we have considered equation
(9), which ensures the self-reference of the entropy. Now we show that if a human society obeys the Boltzmann-like income distribution (15), it will spontaneously form a kind of Boltzmann machine, which we call the SRBM. To this end, we assume that the income distribution among 𝑁 agents obeys equation (15); thus, the joint probability distribution among 𝑁 agents, 𝑃 𝒗, 𝒉 , can be written as: 𝒗,𝒉
𝒗, 𝒉
𝒗,𝒉
∑𝒗,𝒉
𝒗,𝒉
∙ 𝒗,𝒉
𝒗,𝒉
,
(16)
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𝑃
∙ 𝒗,𝒉
𝜽
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where each agent plays the role of a neuron. Therefore, we do not distinguish between agents and neurons2 in this paper. Without loss of generality, we use 𝒗 to denote the visible neurons and 𝒉 the hidden neurons. The derivation for equation (16) can be found in Appendix C. Like the RBM (3), the probability that the neural network assigns to the visible vector 𝒗 is given by summing over all possible hidden vectors: ∑𝒉 𝑃 𝒉, 𝒗 ; therefore, the SRBM is determined by the maximum likelihood estimate: : 𝑙𝑛 ∑𝒉 𝑃
𝒗, 𝒉 ,
(17)
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𝑆 𝑁 𝐸 𝒗, 𝒉 , (18) 𝑠. 𝑡. 𝐸 𝒗, 𝒉 𝑁𝐸 𝒗, 𝒉 where 𝜽 denotes the maximum likelihood parameters. Different from the traditional Boltzmann machines, there is a constraint equation (18) in the SRBM. Furthermore, the maximum likelihood formulation of the optimal problem (17) indicates that the Boltzmann-like distribution (15) “spontaneously” induces the human society to form a SRBM. From this meaning, if a human society obeys the Boltzmann-like income 2
Likewise, we do not distinguish between income and energy. 7
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distribution (15), it can be regarded as a SRBM. Now we show that the SRBM (17) can be simplified to a refined form. To this end, let us write down the average values of 𝑆 𝑁, 𝐸 𝒗, 𝒉 ∑𝒗,𝒉 𝑆 𝑁, 𝐸 𝒗, 𝒉
∙𝑃
𝐸 ∑𝒗,𝒉 𝐸 𝒗, 𝒉 ∙ 𝑃 If we order: 𝛺
𝒗,𝒉
∑𝒗,𝒉 𝑒
𝒗, 𝒉 ,
(19)
𝒗, 𝒉 .
∙ 𝒗,𝒉
(20)
of
𝑆̅
𝒗,𝒉
,
𝑙𝑛𝛺,
𝐸
𝑙𝑛𝛺,
where 𝐸 𝒗, 𝒉
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then one has: 𝑁
and 𝐸 𝒗, 𝒉 :
and 𝐸 𝒗, 𝒉
(21) (22) (23)
, see equations (12) and (13).
𝑑𝑁
𝑑𝐸
Pr e-
Using equations (22) and (23), it is easy to obtain: 𝑑 𝑙𝑛𝛺
𝛼
𝑙𝑛𝛺
𝛽
𝑙𝑛𝛺 ∙ 𝑑𝛼
where we have used d𝑙𝑛𝛺
𝑙𝑛𝛺 ,
(24)
𝑙𝑛𝛺 ∙ 𝑑𝛽.
On the other hand, equation (11) can be written in the form: 𝑑𝑆̅.
𝑑𝑁
𝑑𝐸
(25)
Comparing equations (24) and (25), we have: 𝑙𝑛𝛺
𝛼
𝑙𝑛𝛺
𝛽
𝑙𝑛𝛺.
(26)
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𝑆̅
𝑃
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The consistency between equations (19) and (26) will be justified immediately. Substituting equation (21) into equation (26) one has: ∑𝒗,𝒉 𝑃 𝒗, 𝒉 ∙ 𝑙𝑛𝑃 𝒗, 𝒉 . (27) 𝑆̅ Now, we start to simplify equation (16). Substituting equation (18) into equation (16) one has: ,
𝒗, 𝒉
𝒗,𝒉
,
∑𝒗,𝒉
𝒗,𝒉
.
(28)
Combining equations (19) and (27) we have: ∑𝒗,𝒉 𝑆 𝑁, 𝐸 𝒗, 𝒉
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∙𝑃
𝒗, 𝒉
∑𝒗,𝒉 𝑃
𝒗, 𝒉 ∙ 𝑙𝑛𝑃
𝒗, 𝒉 .
(29)
Substituting equation (28) into equation (29) yields:
∑𝒗,𝒉 𝑒
,
𝒗,𝒉
1,
(30)
which implies that equation (28) can be simplified to: 𝑃
𝒗, 𝒉
𝑒
,
𝒗,𝒉
.
(31) 8
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From equations (27) and (31), we verify that 𝑆 indeed stands for the entropy3. The consistency between equations (19) and (26) is ensured by equation (30). Using equations (30) and (31), the SRBM (17) can be rewritten in the form: ,
: 𝑙𝑛 ∑𝒉 𝑒 ,
𝑠. 𝑡. ∑𝒗,𝒉 𝑒 where 𝑆 𝑁, 𝐸 𝒗, 𝒉
𝒗,𝒉
𝒗,𝒉
1
,
(32)
is determined by equation (18). Equation (32) is the standard
of
𝜽
form of the SRBM.
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5. Central feature of the SRBM
The general solution of equation (18) has been found to be equation (10). Thus, any function 𝑆 𝑁, 𝐸 𝒗, 𝒉
satisfying equation (10) can be used to construct the SRBM. 𝒗,𝒉
𝑆 𝑁, 𝐸 𝒗, 𝒉
Pr e-
Here, we only consider the solution: 𝑁 ∙ 𝑙𝑛𝑁,
(33)
which satisfies the well-known thermodynamic relation 𝑑𝑆
as 𝑁 is a constant.
Generally, 𝐸 𝒗, 𝒉 can be expanded as the Taylor’s series: 𝐸 𝒗, 𝒉 ∑
∑
𝐸 𝟎, 𝟎 ∑
𝜌 𝑣𝑣
∑ 𝑏𝑣
∑
𝜔 ℎ𝑣
∑
∑
∑
𝜎 ℎℎ
𝑐ℎ,
(34)
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where we only expand the series up to the two-order terms. For the SRBM, we have the following theorem.
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Theorem 1: Assume that equation (9) holds. If 𝑁 ≫ 1 and 𝐸 𝒗, 𝒉 𝐸 𝒗, 𝒉 0. Proof. We assume 𝐸 𝒗, 𝒉 Φ 0,
𝑙𝑛𝑁
0 . Substituting it into equation (10) yields
0, which can be rewritten as
𝑙𝑛𝑁
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. Since 𝑁 ≫ 1,
𝑁𝑙𝑛𝑁 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ∙ 𝑁 0. Therefore, by equation (31) we have 1, contradicting 0 𝑃 𝒗, 𝒉 1. □
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we conclude 𝑆 𝑃 𝒗, 𝒉 𝑒
0, one has
Theorem 1 implies that the SRBM should always have a positive energy. Here, we further explain that 𝐸 𝒗, 𝒉 0 indeed holds for the SRBM. To this end, substituting equations (33) and (34) into equation (32) one has: Strictly speaking, 𝑆 stands for the amount of information, and the average value 𝑆̅ stands for the entropy, i.e., the average amount of information. However, we do not distinguish the amount of information and entropy in this paper. 3
9
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,
,
, ,
𝐸
∑
𝜌 𝑣𝑣
∑
,
∙
𝑠. 𝑡. ∑𝒗,𝒉 𝑒 where
∙
: 𝑙𝑛 ∑𝒉 𝑒
𝐸 𝟎, 𝟎 ∑
∑
𝐸
and ∑
𝑏𝑣 ∙
By using ∑𝒗,𝒉 𝑒
(35)
1 ∑
∑
𝜔 ℎ𝑣
𝑐ℎ.
∑
𝜎 ℎℎ
of
,
1, it is easy to verify 𝐸 𝒗, 𝒉
𝐸
𝐸
0. In
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particular, for 𝑁 1 , we have 𝐸 𝒗, 𝒉 0 , which implies a positive zero-point energy 𝐸 𝟎, 𝟎 0 . This central feature significantly differs from the traditional Boltzmann machines. For example, by equation (2), it is easy to check that the RBM leads to 𝐻 𝟎, 𝟎 0. Here, we also attempt to explain the possible underlying origin why the SRBM has a positive zero-point energy. To this end, we observe that if 𝑁 ≫ 𝒗,𝒉
1, equation (33) can be rewritten as 𝑆
𝒗,𝒉
𝑁 ∙ 𝑙𝑛𝑁
ln𝑁!, where an
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al
Pr e-
additional Gibbs term ln𝑁! emerges. Such a Gibbs term is usually related to quantum effects in physics, which requires the presence of a kind of zero-point energy, that is, 𝐸 𝟎, 𝟎 0. Because equation (33) is the solution of equation (9) that guarantees the self-reference of the entropy 𝑆 , we think that the presence of a positive zero-point energy is actually due to self-reference. From the perspective of Neumann [13] and Hofstadter [14], self-reference may be a central difference of distinguishing between a biological organism and a lifeless-physical system. For example, both authors [13-14] conjectured that a self-loop or self-replication of a biological organism makes sense if and only if a kind of self-reference occurs. Now, we show that a kind of self-loop with a biological sense may occur in the SRBM because it has a positive zero-point energy. To verify this, we can simply observe that a SRBM can activate its neurons by consuming its zero-point energy; in other words, a SRBM can activate its neurons by itself. This will lead to a kind of self-loop with a biological sense, which we call the “self-motive”. To assess this, we design the algorithm for the SRBM (35).
6. Algorithm
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0 and 𝜌 0 . This corresponds to the For simplicity, we order 𝑇 1 , 𝜎 case of the RBM. Thus, equation (35) can be simplified to: ,
, ,
𝑠. 𝑡. 𝑙𝑛 ∑𝒗,𝒉 𝑒
where 𝐸
∙
: 𝑙𝑛 ∑𝒉 𝑒
∑
∑
,
∙
0
𝜔 ℎ𝑣
∑
(36) 𝑏𝑣
∑
𝑐ℎ.
For the optimal problem (36), we can construct a Lagrange function: 𝐿 𝜽|𝒗
𝑙𝑛 ∑𝒉 𝑒
∙
𝜆 ∙ 𝑙𝑛 ∑𝒗,𝒉 𝑒 10
∙
,
(37)
Journal Pre-proof
where 𝜆 denotes the Lagrange multiplier and 𝜽
𝜔 ,𝑏 ,𝑐 ,𝐸 .
By applying the same technique developed by [26] into equation (37), it is easy to obtain the optimal conditions as below: 𝜽𝒗 𝑃 ℎ 1|𝝂 ∙ 𝜈 𝜆 ∙ ∑𝒗 𝑃 𝒗 ∙ 𝑃 ℎ 1|𝝂 ∙ 𝜈 , (38)
𝜽𝒗 𝜽𝒗
𝑃
ℎ
𝜆
1,
𝜽𝒗 where
𝒗 ∙𝜈,
1|𝝂
𝜆 ∙ ∑𝒗 𝑃
∙
𝑙𝑛 ∑𝒗,𝒉 𝑒
𝑃
𝜎 𝑡
𝜆 ∙ ∑𝒗 𝑃
𝒗
∑𝒉 𝑃
(40) (41)
and
𝑃
ℎ
ℎ
1|𝝂
𝜎 ∑
𝜔 𝜈
(42) 𝑐
with
Pr e-
.
𝒗, 𝒉
(39)
1|𝝂 ,
𝒗 ∙𝑃
,
of
𝜈
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𝜽𝒗
Equation (42) is due to equation (30), which ensures the self-reference of the entropy, see the derivation for equation (30) in section 4. Computing equation (42) requires examining all potential combinations 𝒗, 𝒉 . This leads to a huge amount of computation, which implies an unavoidable cost of producing self-reference. By using equations (38)-(42), the optimal parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ , 𝐸 ∗ can be found
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al
according to the following iterative algorithm: 𝜽𝒗 𝜂 ⎧𝜔 ← 𝜔 ⎪ 𝜽𝒗 ⎪ 𝑏 ←𝑏 𝜂 ⎪ 𝜽 𝒗 , (43) 𝑐 ←𝑐 𝜂 ⎨ 𝜽𝒗 ⎪𝐸 ←𝐸 𝜂 ⎪ ⎪ 𝜽𝒗 ⎩ 𝜆←𝜆 𝜂 where 𝜂 denotes the learning rate. In this paper, we propose an energy conservation algorithm (ECA) for the SRBM, which makes the SRBM somewhat biologically plausible. To this end, we need to remove equation (41). Thus, 𝐸 is a given number. If the running time of the SRBM follows 𝑡
𝑡
𝑡
⋯ , we assume that 𝐸
SRBM at the time 𝑡 . We further assume that 𝐸
𝐸 is the initial energy of the 𝒗, 𝒉 stands for the energy that
the SRBM absorbs or consumes by activating neurons 𝒗 and 𝒉 at the time 𝑡 . Thus, by energy conservation, the energy of the SRBM should obey: 11
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𝐸
𝐸
𝐸
,
(44)
where 𝑘 0,1,2, … ∞. From the perspective of a biological organism, the SRBM halts (dies) when 𝐸
𝟎, 𝟎
0, where 𝟎 denotes that all neurons stay at inactive state. This yields
𝜽𝒗 𝜽𝒗
;
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⎨ 𝑐 ←𝑐 𝜂 ⎪ ⎪ ⎩ 𝜆←𝜆 𝜂 𝐸 ←𝐸 𝐸; end
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positive number; 𝐸 𝜂 0.1; g 0; 0 while 𝐸 g g 1; do 𝜽𝒗 𝜂 ⎧𝜔 ← 𝜔 ⎪ ⎪ 𝑏 ←𝑏 𝜂 𝜽𝒗
Pr e-
Energy Conservation Algorithm (ECA):
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a definitely biological meaning: If a SRBM wants to absorb or consume energy, it should activate its neurons by itself; otherwise, it will halt (die). In this sense, the run of the SRBM is a self-motivated activity. Thus, the ECA is designed as below:
Like the RBM, seeking maximum likelihood parameters 𝜔∗ , 𝑏∗ , 𝑐 ∗ in the SRBM
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obeys a process of unsupervised-learning. More importantly, by the Theorem 1, we have 𝐸 𝟎, 𝟎 0. This means, by the ECA, that the SRBM cannot halt. Because each run of the SRBM arouses an unsupervised learning, non-halt implies that the SRBM is a self-motivated learning system in which the learning is a kind of self-loop activity. Therefore, from the perspective of the ECA, the SRBM resembles a biological organism, where 𝒗 stands for the sensory nervous system and 𝒉 stands for the reflective nervous system. Self-motive is a universal biological phenomenon. For example, based on an unknown self-motive, earthworms crawl out of the ground. From a psychological perspective, self-motive can be regarded as the most primitive self-motivation. There is a large body of psychological literature [21-24] arguing that self-awareness is a highstage self-motivation. Based on the ECA, we can propose a definition for self-motive in the theoretical framework of Boltzmann machine. If we regard a biological organism as a Boltzmann machine, then the organism dies as it has zero energy with all neurons 12
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being inactive. This corresponds to the halt of the Boltzmann machine. Thus, we have the following definition: Definition 1 (Halt): If one denotes the energy of a Boltzmann machine by 𝐸 𝒗, 𝒉 , the Boltzmann machine halts when 𝐸 𝟎, 𝟎 0, where 𝟎 denotes that all neurons stay in an inactive state.
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of
By equation (2) and Definition 1, we know that the traditional Boltzmann machines can halt. Because each run of the SRBM arouses an unsupervised learning, non-halt implies a series of self-motivated learning behaviors. Therefore, for Boltzmann machines, we can propose a definition for self-motive as below: Definition 2 (Self-motive): If a Boltzmann machine cannot halt, then it is a selfmotivated system.
Pr e-
The ECA can be thought of as a potential algorithm for the SRBM. Nevertheless, it is hard to run the ECA because there is no a practical way of computing equation (42) and the second terms in equations (38)-(40). In this paper, we use the contrastive divergence algorithm [26] to approximately approach the ECA. To implement the contrastive divergence algorithm, equation (41) must hold, that is, 𝜆 1. However, even if we employ the contrastive divergence algorithm, it is hard to compute 𝐸 . Fortunately, the following Theorem 2 will avoid this difficulty. Theorem 2: For the SRBM, if 𝐸
Proof. By Theorem 1, one has 𝐸 𝒗, 𝒉 𝐸 𝐸 𝐸 . By 𝐸 0, we complete the proof. □
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𝐸
0, then one has 𝐸
0.
0, which can be rewritten as
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By Theorem 2, if 𝐸 0 can be always justified, then the SRBM runs forever. Therefore, we use 𝐸 0 to replace 𝐸 0 in the ECA. Next, we give the contrastive divergence approximation to the ECA. The approximation algorithm is to test if 𝐸 0 always holds. The programming language is based on the MATLAB. For a given visible input 𝝂 , we carry out the following programming to learn it: Programming 1:
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1; 𝐸 𝜂 0.1; g 0; 0.001 while 𝐸 g g 1; do
13
Journal Pre-proof
⎧𝜔 ← 𝜔 ⎪ ⎨ ⎪ ⎩ 𝐸
𝜂∙ 𝑃
𝑏 ←𝑏
𝑐 ←𝑐 ∑
1𝝂
ℎ
∑
𝜂∙ 𝑃
ℎ ∑
𝜔 ℎ𝑣
𝑃
∙𝜈
𝜂∙ 𝜈
𝜈
1𝝂
𝑃 ∑
𝑏𝑣
ℎ
1𝝂
∙𝜈 ;
ℎ
1𝝂
𝑐ℎ;
of
end
Pr e-
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𝝂 is the output vector, which is the learning result of the SRBM, where 𝑘 stands for the step number of Gibbs sampling [26]. Here, we consider 𝑘 1, 𝑛 1 and 𝑚 25 . 𝝂 denotes an image consisting of 5 5 blocks, and it is randomly extracted from the image gallery, which includes 2 different images. Because of the infinite self-motive of the SRBM, the vector 𝝂 should reproduce all 2 images. The detailed Programming 1 can be found in Supplemental Materials, by which 𝝂 indeed examines all 2 images. Here, for simplicity, we only check 𝑚 25 . By using the Programming 1, the readers can check that our conclusion holds for any 𝑚. This verifies that the SRBM cannot halt. By Definition 2, this means that the SRBM is a self-motivated system. Boltzmann machines are well known because of their excellent learning function in pattern recognition [26]. In the next section, we apply the selfmotive of the SRBM to perform the task of searching images.
7. Search task
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By Programming 1, we know that the SRBM cannot halt. During the running process, it learns images continuously, and records each image. This means that the SRBM can constantly browse an image gallery with a huge number of images by itself, unless the target image has been found. Here, we consider that an image consists of 3 3 blocks, that is, 𝑚 9. Learning image 𝝂 is randomly extracted from the image gallery, which includes 2 different images. Once 𝝂 is learned, then it can be recorded in the form of 𝝂 . In this process, we hope to seek the target image 𝝂∗ . If the SRBM finds 𝝂∗ , we should have 𝝂 𝝂∗ . Thus, we design the following programming. Programming 2:
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𝑟𝑎𝑛𝑑 3,3 ; 𝝂 𝐸 1; 𝑒 1; 𝜂 0.1; g 0; 0.001 & 𝑒 while 𝐸 g g 1;
0.01
14
Journal Pre-proof
do
∑
𝑒
𝝂
∑
𝜂∙ 𝑃
∙𝜈
1𝝂
𝑏 ←𝑏
𝑐 ←𝑐
𝐸
ℎ
ℎ ∑
𝜔 ℎ𝑣
ℎ
𝑃
𝜂∙ 𝜈
𝜈
1𝝂
𝑃 ∑
𝑏𝑣
1𝝂
; ℎ
1𝝂
𝑐ℎ;
𝝂∗ ;
end
∙𝜈
p ro
⎨ ⎪ ⎩
𝜂∙ 𝑃
of
⎧𝜔 ← 𝜔 ⎪
Image Gallery
□■□ □□□ □□□
□□■ □□□ □□□
Pr e-
■□□ □□□ □□□
𝝂
……
■■■ ■■■ ■■■
is randomly extracted
■■■ □□□ ■■■
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al
Unsupervised-learning
Learning result 𝝂
■■■ □□□ ■■■ Matching
■■■ Target Image 𝝂∗ □□□ ■■■
Figure 3. The target image 𝝂∗ is hidden in the image gallery, which includes 2 is randomly extracted from the image gallery. different images. Learning image 𝝂 is learned, then the SRBM records it in the form of 𝝂 . The SRBM finds Once 𝝂 ∗ 𝝂 if and only if 𝝂 𝝂∗ . 15
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8. Conclusion
Pr e-
p ro
of
The detailed Programming 2 can be found in Supplementary Materials. We have plotted the result of search task in Figure 3, from which one can find that, due to the will examine all 2 images, and 𝝂∗ is finally infinite self-motive, the vector 𝝂 found. To perform the search task, we use the contrastive divergence algorithm to approximately approach the ECA. This requires that equation (41) holds, which leads to 𝜆 1. However, equation (41) will break the energy conservation equation (44). Therefore, the contrastive divergence algorithm only provides a quite rough approximation for the ECA. To rigidly perform the ECA, we need to find a way of computing equation (42) and the second terms in equations (38)-(40). This will be explored in the future. Here, we need to clarify that traditional Boltzmann machines cannot finish the self0 , by motive task above. Since traditional Boltzmann machines require 𝐸 Programming 2 it is easy to check that each run of the search task will produce 𝐸 0. 𝐸 0, which, by Definition 1, indicates halt. However, This means 𝐸 𝒗, 𝒉 𝐸 we highlight that the self-motive endowed by the SRBM is only a kind of self-loop with a biological sense; that is, if an organism wants to perform learning motivation, it have to take a positive energy to activate its neurons. Such a self-loop makes sense if and only if the organism has a positive (zero-point) energy even though all neurons are inactive. By contrast, traditional Boltzmann machines fail to capture such a biological feature.
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Different from lifeless-physical systems, a well-functioning free-market society will obey a Boltzmann-like distribution that has a self-referential entropy. We show that if a human society obeys such a Boltzmann-like income distribution, it will spontaneously form a self-referential Boltzmann machine (SRBM), where each person plays the role of a neuron. Because of the self-reference of the entropy, the SRBM always has a positive energy, even if all neurons are inactive. This implies the presence of a kind of positive zero-point energy. We show that such a positive zero-point energy leads to the SRBM being a self-motivated learning system if the energy conservation is taken into account. This makes the SRBM somewhat biologically plausible. Because the existing empirical investigations have confirmed that a well-functioning freemarket society will obey a Boltzmann-like income distribution [19-20], we conclude that such a society is actually a SRBM. This implies that a human society functions like a kind of complex system (or organism) with potential self-motivations, just as motivated by an “invisible hand” coined by Adam Smith. This finding contradicts the long-standing reductionism that is popular in the neoclassical economics. On the other hand, there has been a large body of psychological literature proposing that selfawareness consists of different self-motives. Thus, the SRBM provides a potential perspective for exploring self-awareness. As a simple application, we also employ the self-motive of the SRBM to perform the task of searching images.
16
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Appendix A Derivation for equation (7): Because 𝜀 is a function of 𝑋, we conclude 𝑁 𝑑𝑁
|
,
, ,
𝑑𝛼
|
𝑑𝛽
,
𝑁 𝛼, 𝛽, 𝑋 . Thus, we have: , ,
|
,
, ,
|
,
𝑁.
, ,
|
,
𝐸.
, ,
|
,
𝛽∑
(A.1)
(A.2)
p ro
By equations (5) and (6), one has:
𝑑𝑋.
of
, ,
.
(A.3)
(A.4)
By substituting equation (A.1) into equation (A.3), it is easy to obtain:
On the other hand, we have: 𝑑𝛼
𝑑𝑁
𝑑 𝛽
𝑑
𝛼𝑑
𝑑 𝛼
𝑑𝛼
𝑑𝑋 .
Pr e-
𝑑𝐸
.
(A.5)
(A.6)
Substituting equation (A.6) into equation (A.5) yields: 𝑑𝐸
𝑑 𝛽
𝑑𝑁
𝑑 𝛼
𝑑
𝑑𝑋.
(A.7)
Finally, by substituting equations (A.2), (A.3) and (A.4) into equation (A.7), we get: 𝑑𝑁
𝑑𝐸
𝑑 𝑁
𝛼𝑁
∑
𝛽𝐸
𝑑𝑋.
(A.8)
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□
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Proof of Lemma 1: The characteristic equation of the partial differential equation (9) is .
By
(A.9)
, one can get a first integral:
𝐶,
(A.10)
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where 𝐶 is a constant. On the other hand, it is easy to calculate: 𝑆𝑑𝑁
𝑁𝑑𝑆
𝑁 𝑑
.
(A.11)
Substituting equation (A.11) into
𝑑
yields:
𝑑 𝑙𝑛𝑁 .
(A.12) 17
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Thus, we get another first integral: 𝑙𝑛𝑁
𝐶,
(A.13)
where 𝐶 is a constant. Combining equations (A.10) and (A.13), we obtain the general solution (10). □
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Appendix B
0 is not self-referential, we introduce the
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To verify that the entropy for following two propositions. 0 for 𝑘
Proposition 2: If
1,2, … , 𝑙 and if 𝑁 , 𝑆 , and 𝑋 are independent
with each other, then we have:
|
𝑆
, ,
𝑁
∑
,
𝐸 𝑁, 𝑆, 𝑋 ,
Pr e-
, ,
𝑁
.
(B.1) (B.2)
Proof. Because 𝑁, 𝑆, and 𝑋 are independent with each other, by equations (7) and (8) it is easy to obtain equation (B.1). Next we verify equation (B.2). As 𝜀 is a function of 𝑋, equation (6) indicates: 𝑑𝐸
|
𝑑𝛼
,
|
𝑑𝛽
,
|
𝑑𝑋.
,
(B.3)
𝑑𝐸
|
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On the other hand, as 𝐸 is a function of 𝑁, 𝑆, and 𝑋 as well, one should have: 𝑑𝑁
,
|
,
𝑑𝑆
|
,
𝑑𝑋.
(B.4)
𝑑𝑋.
(B.5)
By equations (5) and (6) we have:
𝑑𝑆
𝑑𝛼
|
𝑑𝛽
urn
|
𝑑𝑁
|
,
,
𝑑𝛼
|
,
|
𝑑𝛽
,
|
, ,
𝑑𝑋.
(B.6)
Substituting equations (B.5) and (B.6) into equation (B.4) yields: 𝑑𝐸
|
∙
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|
∙
,
|
,
,
∙
|
|
|
,
|
,
|
,
,
|
,
,
∙
∙
|
∙
| |
𝑑𝛼
,
𝑑𝛽
,
|
,
,
.
(B.7)
𝑑𝑋
Comparing equations (B.3) and (B.7), we have: |
,
|
,
|
,
∙
|
,
|
,
∙
|
,
.
The partial derivative of equation (8) with respect to 𝑋 yields: 18
(B.8)
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|
|
,
𝛼
,
|
Substituting
|
𝛽
,
|
,
,
|
,
.
(B.9) and equation (B.9) into equation (B.8)
,
yields: |
|
,
,
.
(B.10)
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Substituting equation (A.4) into equation (B.10), we obtain equation (B.2). □
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By Proposition 2, equation (B.2) ensures the validity of the complete differential (7). By observing equation (8), equation (B.1) is actually the definition of the entropy. However, the following proposition further shows that equation (B.1) may contradict equation (7)
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Proposition 3: If the complete differential (7) holds for any 𝐸 𝑁, 𝑆, 𝑋 satisfying equation (B.1), then one has 𝑆 𝑁 𝛼𝑁 𝛽𝐸 𝑁, 𝑆, 𝑋 . Proof. The complete differential (7) indicates: |
,
|
,
, .
(B.11) (B.12)
Substituting equations (B.11) and (B.12) into equation (8) yields: 𝑁
, ,
𝑆
, ,
𝑁
𝐸 𝑁, 𝑆, 𝑋 .
(B.13)
𝑁
, ,
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Here, we can construct a partial differential equation: 𝑆
, ,
𝑁
, ,
𝜖
𝐸 𝑁, 𝑆, 𝑋 ,
(B.14)
where 𝜖 is a parameter, which is independent of 𝑁, 𝑆 and 𝑋. , ,
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Obviously, if 𝑙𝑖𝑚 𝜖 ∙ →
obtain: 𝐸
𝑁, 𝑆, 𝑋
0 , by equations (B.13) and (B.14) it is easy to
𝐸 𝑁, 𝑆, 𝑋 .
Later, we will verify 𝑙𝑖𝑚 𝜖 ∙
(B.15) , ,
0.
→
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The characteristic equation of equation (B.14) yields: .
(B.16)
Using equation (B.16) we can obtain the following first integrals: 𝐷,
𝑙𝑛𝑁
(B.17)
𝐷,
(B.18) 19
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𝑁𝑒
𝐷,
(B.19)
where 𝐷 , 𝐷 and 𝐷 are constants. Based on equations (B.17)-(B.19), the general solution of equation (B.14) equals: 𝐹
,
0,
𝑙𝑛𝑁, 𝑁𝑒
(B.20)
∙
,
|
|
,
,
∙
|
,
, |
,
∙
|
0
,
which can be rewritten as: , ,
|
| ,
𝑁
,
.
| ,
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|
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where 𝐹 𝑥, 𝑦, 𝑧 is a smooth function of 𝑥, 𝑦 and 𝑧. The partial derivative of equation (B.20) with respect to 𝑋 equals:
(B.21)
(B.22)
, ,
𝑙𝑖𝑚 𝜖 ∙
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As 𝐹 𝑥, 𝑦, 𝑧 is a smooth function of 𝑥, 𝑦 and 𝑧, equation (B.22) implies that one can obtain: 0,
→
which means that equation (B.15) holds. Equation (B.15) indicates that, for every 𝛿 integer
so that when
, ,
𝛿.
, one has:
𝑚𝑎𝑥
Let us order positive integer
𝐸 𝑁, 𝑆, 𝑋
𝜂∙
, ,
(B.25)
. Thus, for every 𝛿
,
so that when
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𝐸 𝑁, 𝑆, 𝑋
(B.24) 0 , there always exists a positive
al
so that when
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𝜖∙
0 , there always exists a positive
, one has:
|𝐸 𝑁, 𝑆, 𝑋 𝐸 𝑁, 𝑆, 𝑋 | 𝛿. Equation (B.23) indicates that, for every 𝛿 integer
(B.23)
0 , there always exists a
, one has: 𝛿,
(B.26)
𝛿.
(B.27)
Equations (B.26) and (B.27) imply that 𝐸 𝑁, 𝑆, 𝑋
is a series of the
solution of equation (B.13). Therefore, by the condition of Proposition 3, we conclude that
𝑑𝐸 𝑁, 𝑆, 𝑋
is a series of complete differential. However, equation 20
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(B.22) indicates that 𝑙𝑖𝑚 𝑑𝐸 𝑁, 𝑆, 𝑋 does not exist. This implies 𝑆 →
𝑁
𝛼𝑁
𝛽𝐸 𝑁, 𝑆, 𝑋 . □ Obviously, Proposition 3 contradicts equation (8), which is the definition of the 0 for 𝑘
entropy. This means that when
1,2, … , 𝑙, the definition of the entropy
0 for 𝑘
differs from the case
of
does not always ensure the validity of the complete differential (7). This significantly 1,2, … , 𝑙, where equation (9) does guarantee the
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validity of the complete differential (11). In this sense, the so-called complete differential (7) in the thermodynamics is only a physical result rather than a rigid mathematical result.
Appendix C
𝑃∗ 1, … , 𝑁
∗
𝑎∗
∏
,
Pr e-
As the income structure of an 𝑁-agent society obeys the Boltzmann-like income distribution (15), the joint probability distribution among 𝑁 agents, 𝑃∗ 1, … , 𝑁 , can be written as: (C.1)
where 𝑎 and 𝑃∗ 1, … , 𝑁 have been regarded as the unnormalized probabilities. To derive equation (C.1), we assume that the income probability of each agent is independent. As the probability of obtaining 𝜀 units of income is 𝑎∗ , by the assumption of independence, the joint probability among 𝑎∗ agents, each of which ∗
∗
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obtains 𝜀 units of income, equals 𝑎∗ 𝑎∗ ∙ 𝑎∗ ⋯ 𝑎∗ . Equation (C.1) is the result 𝜀 ⋯ 𝜀. of taking into account all income levels 𝜀 Substituting equation (15) into equation (C.1) yields: ∗
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∏
𝑃∗ 1, … , 𝑁
∗
∗
,
∙
(C.2)
∏
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where we have used equations (5) and (6). Obviously, the GDP 𝐸 should be a function with respect to the state variables of agents. Let us denote the state variables of 𝑁 agents by the vector 1 (or 𝑣 𝑣 , … , 𝑣 , ℎ , … , ℎ , where 𝑚 𝑛 𝑁. For example, we can use ℎ 0 (or 𝑣 0) to denote 1) to denote that the agent 𝑖 is active in markets, and ℎ inactivity. Thus, without loss of generality, the GDP 𝐸 𝑣 , … , 𝑣 , ℎ , … , ℎ can be expanded as the Taylor’s series: 𝐸 𝑣 ,…,𝑣 ,ℎ ,…,ℎ
∑
∑
𝜎 ℎℎ
∑
∑
𝜌 𝑣𝑣
∑ 21
∑
𝐸 0, … ,0,0, … ,0 𝑏𝑣
∑ ∑
𝜔 ℎ𝑣 𝑐ℎ,
(C.3)
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where 𝜔 , 𝜎 and 𝜌 represent weights, and 𝑏 and 𝑐 represent biases. Here we only expand the series up to the two-order terms. Using equation (C.3), 𝑃∗ 1, … , 𝑁 can be rewritten in the form of normalized probability: 𝒗,𝒉
𝑃
𝑒
𝒗, 𝒉
where 𝑍
∑𝒗,𝒉 𝑒
𝒗,𝒉
∙ 𝒗,𝒉 ∙ 𝒗,𝒉
𝒗,𝒉
,
(C.4)
𝒗,𝒉
denotes the partition function, 𝒗
𝑣 ,…,𝑣
and
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References:
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𝒉 ℎ , … , ℎ . ∑𝒗∙ and ∑𝒉∙ denote the sum over all possible 𝒗 and 𝒉 , respectively. Obviously, if we regard each agent as a neuron, then the GDP 𝐸 𝒗, 𝒉 denotes energy. Without loss of generality, we use 𝒗 to denote the visible neurons and 𝒉 the hidden neurons.
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[1]. G. A. Mashour, A. G. Hudetz1. Neural Correlates of Unconsciousness in LargeScale Brain Networks. Trends in neurosciences 41, 150-160 (2018) [2]. Wissner-Gross A. D., Freer, C. E. Causal entropic forces. Phys. Rev. Lett. 110, 168702 (2013) [3]. J. L. Perez Velazquez. Finding simplicity in complexity: general principles of biological and nonbiological organization. Journal of Biological Physics 35, 209–221 (2009) [4]. R. Guevara Erra, D. M. Mateos, R. Wennberg, J. L. Perez Velazquez. Statistical mechanics of consciousness: Maximization of information content of network is associated with conscious awareness. Phys. Rev. E 94, 052402 (2016) [5]. D. M. Mateos, R. Wennberg, R. Guevara, and J. L. Perez Velazquez. Consciousness as a global property of brain dynamic activity. Phys. Rev. E 96, 062410 (2017) [6]. I. D. Vincenzo, I. Giannoccaro, G. Carbone, P. Grigolini. Criticality triggers the emergence of collective intelligence in groups. Phys. Rev. E 96, 022309 (2017) [7]. D. M. Mateos, R. Guevara Erra, R. Wennberg, J. L. Perez Velazquez. Measures of entropy and complexity in altered states of consciousness. Cognitive Neurodynamics 12, 73–84 (2018) [8]. J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. PNAS 79, 2554-2558 (1982) [9]. D. H. Ackley, G. E. Hinton, T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science 9, 147-169 (1985) [10]. G. E. Hinton, R. R. Salakhutdinov. Reducing the Dimensionality of Data with Neural Networks. Science 313, 504-507 (2006) [11]. Y. LeCun, Y. Bengio, G. Hinton. Deep learning. Nature 521, 436–444 (2015) [12]. D. Silver et al. Mastering the game of Go with deep neural networks and tree search. Nature 529, 484–489 (2016) 22
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al
Pr e-
p ro
of
[13]. J. V. Neumann. Theory of Self-Reproducing Automata. University of Illinois Press (1966) [14]. D. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books (1979) [15]. Y. Tao. Competitive market for multiple firms and economic crisis. Phys. Rev. E 82, 036118 (2010) [16]. Y. Tao, X. Chen. Statistical Physics of Economic Systems: a Survey for Open Economies. Chinese Physics Letters 29, 058901 (2012) [17]. Y. Tao. Universal Laws of Human Society’s Income Distribution. Physica A 435, 89-94 (2015) [18]. Y. Tao. Spontaneous economic order. Journal of Evolutionary Economics 26, 467500 (2016) [19]. Y. Tao, X. Wu, T. Zhou, W. Yan, Y. Huang, H. Yu, B. Mondal, V. M. Yakovenko. Exponential structure of income inequality: evidence from 67 countries. Journal of Economic Interaction and Coordination 14, 345-376 (2019) [20]. Y. Tao. Swarm intelligence in humans: A perspective of emergent evolution. Physica A 502, 436-446 (2018) [21]. T. S. Duval, R. A. Wicklund. A theory of objective self-awareness. New York: Academic Press (1972) [22]. R. A. Wicklund. Objective self-awareness. Advances in Experimental Social Psychology 8, 233-275 (1975) [23]. T. S. Duval, P. J. Silvia. Self-awareness and causal attribution: A dual systems theory. Boston: Kluwer Academic (2001) [24]. P. J. Silvia, T. S. Duval. Objective self-awareness theory: Recent progress and enduring problems. Personality and Social Psychology Review 5, 230-340 (2001) [25]. H.E. Stanley, et al., Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics. Physica A 224, 302-321 (1996). [26]. A. Fischer, C. Igel. Training restricted Boltzmann machines: An introduction. Pattern Recognition 47, 25-39 (2014) [27]. B. Linder. Thermodynamics and Introductory Statistical Mechanics. WileyInterscience, New York (2004)
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Acknowledgements: Yong Tao was supported by the Fundamental Research Funds for the Central Universities (Grant No. SWU1409444) and the State Scholarship Fund granted by the China Scholarship Council.
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Highlights
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· Self‐referential Boltzmann machine is a Boltzmann machine with self‐referential entropy · Self‐reference of entropy leads to a positive energy even if all neurons are inactive · Self‐referential Boltzmann machine is a self‐motivated system · Self‐motive can be regarded as the most primitive self‐motivation · Self‐referential Boltzmann machine runs as if a biological organism runs
24
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Declaration of Interest Statement
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The author declares no competing interests.
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