Available online at www.sciencedirect.com ScienceDirect Cognitive Systems Research 59 (2020) 231–246 www.elsevier.com/locate/cogsys Consciousness as a logically consistent and prognostic model of reality q Evgenii Vityaev ⇑ Sobolev Institute of Mathematics, Koptuga 4, Novosibirsk, Russia Novosibirsk State University, Pirogova 2, Novosibirsk, Russia Received 31 May 2019; received in revised form 16 September 2019; accepted 18 September 2019 Available online 20 September 2019 Abstract The work demonstrates that brain might reflect the external world causal relationships in the form of a logically consistent and prog- nostic model of reality, which shows up as consciousness. The paper analyses and solves the problem of statistical ambiguity and pro- vides a formal model of causal relationships as probabilistic maximally specific rules. We suppose that brain makes all possible inferences from causal relationships. We prove that the suggested formal model has a property of an unambiguous inference: from consistent pre- mises we infer a consistent conclusion. It enables a set of all inferences to form a consistent model of the perceived world. Causal rela- tionships may create fixed points of cyclic inter-predictable properties. We consider the ‘‘natural” classification introduced by John St. Mill and demonstrate that a variety of fixed points of the objects’ attributes forms a ‘‘natural” classification of the external world. Then we consider notions of ‘‘natural” categories and causal models of categories, introduced by Eleanor Rosch and Bob Rehder and demon- strate that fixed points of causal relationships between objects attributes, which we perceive, formalize these notions. If the ‘‘natural” classification describes the objects of the external world, and ‘‘natural” concepts the perception of these objects, then the theory of inte- grated information, introduced by G. Tononi, describes the information processes of the brain for ‘‘natural” concepts formation that reflects the ‘‘natural” classification. We argue that integrated information provides high accuracy of the objects identification. A computer-based experiment is provided that illustrates fixed points formation for coded digits. Ó 2019 Elsevier B.V. All rights reserved. Keywords: Clustering; Categorization; Natural classification; Natural concepts; Integrated information; Concepts 1. Introduction The work analyses and solves such problem of causal reflection of the external world as a statistical ambiguity The work demonstrates that the human brain may (Section 2.3). The problem is solved in such a way that reflect the external world causality in the form of a logically it is possible to obtain a formal model of causal relation- consistent and prognostic model of reality that shows up as ships, which provides a consistent and prognostic model consciousness. of the external world. To discover these causal relation- ships by the brain, a formal model of neuron that is in line with Hebb rule (Hebb, 1949), is suggested. We sup- q № The work is supported by the Russian Science Foundation grant 17- pose that brain makes all possible inferences/predictions 11-01176 in part concerning the mathematical results in 2.4–2.6 and by Russian Foundation for Basic Research # 19-01-00331-a in other parts. from those causal relationships. We prove (see Section 2.5) ⇑ Address: Sobolev Institute of Mathematics, Koptuga 4, Novosibirsk, that the suggested formal model of causal relationships Russia. has a property of an unambiguous inference/predictions, E-mail address: [email protected]. https://doi.org/10.1016/j.cogsys.2019.09.021 1389-0417/Ó 2019 Elsevier B.V. All rights reserved. 232 E. Vityaev / Cognitive Systems Research 59 (2020) 231–246 namely, consistent implications are drawn out from con- 2. The process of brain reflection of objects by causal rela- sistent premises. It enables a set of all inferences/predic- tions marked by blue lines; tions, which brain makes from causal relationships, to 3. Formation of the systems of interconnected causal rela- form a consistent and predictive model of the perceived tionships, indicated by green ovals. world. What is particularly important is that causal rela- tionships may create fixed points of cyclic inter- In G. Tononi’s theory only the third point of reflection predictable properties that create a certain ‘‘resonance” is considered. The totality of the excited groups of neurons of inter-predictions. In terms of interconnections between form a maximally integrated conceptual structure that neurons, these are cellular assemblies of neurons that defined by G. Tononi as qualia. Integrated information is mutually excite each other and form the systems of highly also considered as a system of cyclic causality. Using inte- integrated information. In the formal model these are log- grated information, the brain is adjusted to perceiving ically consistent fixed points of causal relationships. We ‘‘natural” objects of the external world. argue (Section 2.1) that if attributes of the external world In terms of integrated information, phenomenological objects are taken regardless of how persons perceive them, properties are formulated as follows. In brackets an inter- a complex of fixed points of the objects’ attributes forms a pretation of these properties from the point of view of ‘‘natural” classification of the external world. If the fixed ‘‘natural” classification is given. points of causal relationships of the external world objects, which persons perceive, are taken, they form 1. Composition – elementary mechanisms (causal relation- ‘‘natural” concepts described in cognitive sciences ships) can be combined into the higher-order ones (‘‘nat- (Section 2.2). ural” classes in the form of causal loops produce a If the ‘‘natural” classification describes objects of the hierarchy of ‘‘natural” classes); external world, and ‘‘natural” concepts are the perception 2. Information – only mechanisms that specify ‘‘differences of these objects, then the theory of integrated information that make a difference” within a system shall be taken (Tononi, 2004; Tononi, Boly, Massimini, & Koch, 2016; into account (only a system of ‘‘resonating” causal rela- Ozumi, Albantakis & Tononi, 2014) describes the informa- tionships, forming a class and ‘‘differences that make a tion processes of the brain when these objects are difference” is important. See illustration in the computer perceived. experiment below); G. Tononi defines consciousness as a primary concept, 3. Integration – only information irreducible to non- which has the following phenomenological characteristics: interdependent components shall be taken into account composition, information, integration, exclusion (Ozumi (only system of ‘‘resonating” causal relations, indicating et al., 2014). For a more accurate determination of these an excess of information and perception of highly corre- properties G. Tononi introduces the concept of integrated lated structures of ‘‘natural” object is accounted for); information: ‘‘integrated information characterizing the 4. Exclusion – only maximum of integrated information reduction of uncertainty is the information, generated by counts (only values of attributes that are ‘‘resonating” the system that comes in a certain state after the causal at the fix-point and, thus, mostly interrelated by causal interaction between its parts, which is superior information relationships, form a ‘‘natural” class or ‘‘natural” generated independently by its parts themselves” (Tononi, concept). 2004). The process of reflection of causal relationships of the These properties are defined as the intrinsic properties of external world (Fig. 1) shall be further considered. It the system. We consider these properties as the ability of includes: the system to reflect the complexes of external objects’ cau- sal relations, and consciousness as the ability of a complex 1. The objects of the external world (cars, boats, berths) hierarchical reflection of a ‘‘natural” classification of the which relate to certain ‘‘natural” classes; external world. Fig. 1. Brain reflection of causal relationships between objects attributes. E. Vityaev / Cognitive Systems Research 59 (2020) 231–246 233 Theoretical results on consistency of inference and con- conditional probability and use maximum available infor- sistency of fixed points of our formal model are supposing mation). Work (Vityaev, 2013) shows that the semantic that a probability measure of events is known. However, if probabilistic inference might be considered as a formal we discover causal relationships on the training set, and model of neuron that satisfy the Hebb rule, in which the intend to predict properties of a new object out of the train- semantic probabilistic reasoning discover all most precise ing set and belonging to a wider general population, or to conditional relationships. A set of such neurons might cre- recognize a new object as a member of some ‘‘natural” con- ate a consistent and prognostic model of the external cept, there might be inconsistencies. Here, a certain crite- world. Causal relationships may form fixed points of cyclic rion of maximum consistency is employed (see inter-predictable properties that produce a certain ‘‘reso- Section 2.8), which is based upon information measure, nance” of mutual predictions. Cycles of inferences about close in meaning to an entropic measure of integrated causal relations, are mathematically described by the ‘‘fixed information (Tononi, 2004). The process of recognizing points”. These points are characterized by the property,