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Computer Optics

Modeling of quantum-like cognitive phenomena by the Fourier-holography technique under the choice of alternatives

Alexander V. Pavlov

Published Aug 1, 2021• 2 citations

Abstract

The article is dedicated to the search for a biologically motivated mechanism of the cognitive phenomenon of violation of the classical formula of total probability for the disjunction of incompatible events, which is considered by a number of researchers as a quantum-like phenomenon. A classical mechanism implemented by the 6f Fourier holography scheme of the resonant architecture that does not require reference to quantum mechanics either in its physical nature or at the level of formalism is demonstrated. In the analysis, the decision-making is interpreted as a choice of alternatives by using the non-cooperative game "Prisoner's Dilemma". The approach to the task is based on the search for a mechanism for forming a conditional estimate under a condition that contradicts the rule of monotonous decision logic. It is demonstrated that this estimate, in contrast to the unconditional and conditional one with a non-contradictory condition, is formed by logic with exception. The ring architecture of the holographic setup corresponds to the biologically inspired neural network concept of the excitation ring and implements cognitive dissonance on logic with exception. Conditions and ranges of violation of the classical formula of total probability in relation to the correlation radius of the reference image recorded in a hologram storing the monotone logic inference rule are analytically determined. The analytical model is confirmed by a quantitative coincidence of the results of numerical modeling with the published results of natural experiments.

DOI: 10.18287/2412-6179-co-830

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The article is dedicated to the search for a biologically motivated mechanism of the cognitive phenomenon of violation of the classical formula of total probability for the disjunction of incompatible events, which is considered by a number of researchers as a quantum-like phenomenon. A classical mechanism implemented by the 6f Fourier holography scheme of the resonant architecture that does not require reference to quantum mechanics either in its physical nature or at the level of formalism is demonstrated. In the analysis, the decision-making is interpreted as a choice of alternatives by using the non-cooperative game "Prisoner's Dilemma". The approach to the task is based on the search for a mechanism for forming a conditional estimate under a condition that contradicts the rule of monotonous decision logic. It is demonstrated that this estimate, in contrast to the unconditional and conditional one with a non-contradictory condition, is formed by logic with exception. The ring architecture of the holographic setup corresponds to the biologically inspired neural network concept of the excitation ring and implements cognitive dissonance on logic with exception. Conditions and ranges of violation of the classical formula of total probability in relation to the correlation radius of the reference image recorded in a hologram storing the monotone logic inference rule are analytically determined. The analytical model is confirmed by a quantitative coincidence of the results of numerical modeling with the published results of natural experiments.

Subject of study. A 4 f Fourier-holography scheme with phase conjugation in the correlation plane and the corresponding Grossberg “Instar” neural-network model were examined to assess the feasibility of implementing quantum-like models of information processing. Aim of study. The aim was to develop a Fourier-holography scheme model consistent with the mathematical formalism of quantum mechanics used in quantum-like cognitive models. Method. Analytical modeling and numerical simulations were performed by representing the processed information with a homogeneous isotropic random field model. Main results. The results showed that the Fourier-holography setup incorporating nonlinear phase conjugation in the correlation plane could serve as a limited analog of a quantum system in terms of parallel processing and choice from a superposition. In this case, the choice was not random; rather, it was determined by the nonlinear transfer function of the phase-conjugating mirror. The shift-invariance, ensured by a thin Fourier hologram, imparted an additional capability—making decisions in the form of a superposition described by a linear prediction model; some researchers consider this approach as a possible mechanism underlying the phenomenon of intuition. Practical significance. An analytical description, consistent with the information-processing model, is required for developing holographic simulators of quantum information-processing systems and computing devices.

A Fourier-holography scheme with superimposed holograms and phase conjugation is considered. It is shown that the sub-holograms, arising from the recording of superimposed holograms due to quadratic nonlinearity of the exposure characteristics of holographic recording media, form the connections necessary for the realisation of the quantum-logic mechanism of sequential projection of the input vector onto the subspaces of the recorded images. A classical mechanism of the 'Linda' cognitive phenomenon belonging to the category of quantum-like ones is described. An experimental illustration of the self-sustained formation of sub-holograms due to the nonlinearity of the exposure characteristics of holographic recording media is given.

The disjunction effect in human decision making is often taken to show that the classical law of total probability is violated, motivating quantum-like models. We re-examine this claim for the Prisoner's Dilemma disjunction effect. Under the mental-event reading of the opponent-choice events, the conventional classical decision-process model implicitly builds in a certainty-only premise: its standard partition assumptions leave no room for ambiguity, forcing every participant to be certain that the opponent will defect or will cooperate. We relax this by introducing a new classical model in which each participant carries a continuous expectation parameter representing the anticipated likelihood of opponent defection, and the participant pool is partitioned by expectation level; the resulting ambiguity set is precisely the union of the interior expectation bins. In contrast, under the quantum-like event semantics, ambiguous pure states are generic (dense and of full unitarily invariant measure on the unit sphere), so "certainty states" are mathematically exceptional. We prove that an instance of our classical model can realize any empirically observed triple of defection rates across the three information conditions, including strong disjunction-effect patterns, while strictly obeying the classical law of total probability. We further prove that for any such triple produced by a standard quantum-like model of the same experiment, there exists a classical instance reproducing it exactly. In this sense, classical and quantum-like approaches have the same observable-rate expressiveness; their substantive difference lies in how ambiguity is represented and in their respective event semantics, not in a breakdown of classical probability.

In cognitive psychology, some experiments of games were reported [1, 2, 3, 4], and these demonstrated that real players did not use the "rational strategy" provided by classical game theory. To discuss probabilities of such "irrational choice", recently, we proposed a decision-making model which is based on the formalism of quantum mechanics [5, 6, 7, 8]. In this paper, we briefly explain the above model and calculate the probability of irrational choice in several prisoner's dilemma (PD) games.

We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.

We proceed towards an application of the mathematical formalism of quantum mechanics to cognitive psychology — the problem of decision-making in games of the Prisoners Dilemma type. These games were used as tests of rationality of players. Experiments performed in cognitive psychology by Shafir and Tversky [1, 2], Croson [3], Hofstader [4, 5] demonstrated that in general real players do not use "rational strategy" provided by classical game theory; this psychological phenomenon was called the disjunction effect. We elaborate a model of quantum-like decision making which can explain this effect ("irrationality" of plays). Our model is based on quantum information theory. The main result of this paper is the derivation of Gorini-Kossakowski-Sudarshan-Lindblad equation whose equilibrium solution gives the quantum state used for decision making. It is the first application of this equation in cognitive psychology.