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According to Kant's (1724 -- 1804) philosophical aesthetics, laid down in his Critique of the Power of Judgement (1790), beauty is "subjective purposefulness", reflected by the "harmony of the cognitive faculties", which are "understanding" and "imagination". On the one hand, understanding refers to the mental capability to find regularities in sensory manifolds, while imagination refers to intuition, fantasy, and creativity of the mind, on the other hand. Inspired by the reinforcement learning theory of Schmidhuber, I present a neural network analogy for the harmony of the faculties in terms of generative adversarial networks (GAN) - also often employed for artificial music composition -- by identifying the generator module with the faculty of imagination and the discriminator module with the faculty of understanding. According to the GAN algorithm, both modules are engaged in an adversarial game, thereby optimizing a particular objective function. In my reconstruction, the convergence of the GAN algorithm during the reception of art, either music or fine, entails the harmony of the faculties and thereby a neural network analogue of subjective purposefulness, i.e., beauty., Comment: 33 pages, 3 figures

We adopt some basic ideas on quantum-theoretical modeling of tonal attraction and develop them further in an alternative direction. Fitting Gaussian Mixture Models (GMM) to the Krumhansl-Kessler (KK) probe tone profiles for static attraction opens the possibility to investigate the underlying wave function as the stationary ground state of an anharmonic quantum oscillator with a schematic Hamiltonian involving a perturbation potential. We numerically verify the fulfilment of the associated stationary Schr\"odinger equation and also inspect its excited states as a solution basis for the corresponding time-dependent Schr\"odinger equation. With their help we calculate the temporal evolution of any initial state. As an example, we study the dynamics of transpositions of the stationary KK wave function across Regener's line of fifths. This offers potential models for dynamic tonal attraction and also for the behavior of deflected key profiles within the Hamiltonian dynamics of a given tonality., Comment: 12 pages, 6 figures, submitted to MCM Conference 2024

Computational modeling of neurodynamical systems often deploys neural networks and symbolic dynamics. A particular way for combining these approaches within a framework called vector symbolic architectures leads to neural automata. An interesting research direction we have pursued under this framework has been to consider mapping symbolic dynamics onto neurodynamics, represented as neural automata. This representation theory, enables us to ask questions, such as, how does the brain implement Turing computations. Specifically, in this representation theory, neural automata result from the assignment of symbols and symbol strings to numbers, known as G\"odel encoding. Under this assignment symbolic computation becomes represented by trajectories of state vectors in a real phase space, that allows for statistical correlation analyses with real-world measurements and experimental data. However, these assignments are usually completely arbitrary. Hence, it makes sense to address the problem question of, which aspects of the dynamics observed under such a representation is intrinsic to the dynamics and which are not. In this study, we develop a formally rigorous mathematical framework for the investigation of symmetries and invariants of neural automata under different encodings. As a central concept we define patterns of equality for such systems. We consider different macroscopic observables, such as the mean activation level of the neural network, and ask for their invariance properties. Our main result shows that only step functions that are defined over those patterns of equality are invariant under recodings, while the mean activation is not. Our work could be of substantial importance for related regression studies of real-world measurements with neurosymbolic processors for avoiding confounding results that are dependant on a particular encoding and not intrinsic to the dynamics., Comment: 28 pages, 8 figures

Recognizing a basic difference between the semiotics of humans and machines presents a possibility to overcome the shortcomings of current speech assistive devices. For the machine, the meaning of a (human) utterance is defined by its own scope of actions. Machines, thus, do not need to understand the conventional meaning of an utterance. Rather, they draw conversational implicatures in the sense of (neo-)Gricean pragmatics. For speech assistive devices, the learning of machine-specific meanings of human utterances, i.e. the fossilization of conversational implicatures into conventionalized ones by trial and error through lexicalization appears to be sufficient. Using the quite trivial example of a cognitive heating device, we show that - based on dynamic semantics - this process can be formalized as the reinforcement learning of utterance-meaning pairs (UMP)., Comment: 48 pages, 4 tables

Speech-controlled user interfaces facilitate the operation of devices and household functions to laymen. State-of-the-art language technology scans the acoustically analyzed speech signal for relevant keywords that are subsequently inserted into semantic slots to interpret the user's intent. In order to develop proper cognitive information and communication technologies, simple slot-filling should be replaced by utterance meaning transducers (UMT) that are based on semantic parsers and a mental lexicon, comprising syntactic, phonetic and semantic features of the language under consideration. This lexicon must be acquired by a cognitive agent during interaction with its users. We outline a reinforcement learning algorithm for the acquisition of syntax and semantics of English utterances, based on minimalist grammar (MG), a recent computational implementation of generative linguistics. English declarative sentences are presented to the agent by a teacher in form of utterance meaning pairs (UMP) where the meanings are encoded as formulas of predicate logic. Since MG codifies universal linguistic competence through inference rules, thereby separating innate linguistic knowledge from the contingently acquired lexicon, our approach unifies generative grammar and reinforcement learning, hence potentially resolving the still pending Chomsky-Skinner controversy., Comment: 25 pages, 2 figures, 9 tables. arXiv admin note: text overlap with arXiv:1906.04447

Background / introduction. Vector symbolic architectures (VSA) are a viable approach for the hyperdimensional representation of symbolic data, such as documents, syntactic structures, or semantic frames. Methods. We present a rigorous mathematical framework for the representation of phrase structure trees and parse trees of context-free grammars (CFG) in Fock space, i.e. infinite-dimensional Hilbert space as being used in quantum field theory. We define a novel normal form for CFG by means of term algebras. Using a recently developed software toolbox, called FockBox, we construct Fock space representations for the trees built up by a CFG left-corner (LC) parser. Results. We prove a universal representation theorem for CFG term algebras in Fock space and illustrate our findings through a low-dimensional principal component projection of the LC parser states. Conclusions. Our approach could leverage the development of VSA for explainable artificial intelligence (XAI) by means of hyperdimensional deep neural computation. It could be of significance for the improvement of cognitive user interfaces and other applications of VSA in machine learning., Comment: 36 pages, 3 figures

Speech-controlled user interfaces facilitate the operation of devices and household functions to laymen. State-of-the-art language technology scans the acoustically analyzed speech signal for relevant keywords that are subsequently inserted into semantic slots to interpret the user's intent. In order to develop proper cognitive information and communication technologies, simple slot-filling should be replaced by utterance meaning transducers (UMT) that are based on semantic parsers and a \emph{mental lexicon}, comprising syntactic, phonetic and semantic features of the language under consideration. This lexicon must be acquired by a cognitive agent during interaction with its users. We outline a reinforcement learning algorithm for the acquisition of the syntactic morphology and arithmetic semantics of English numerals, based on minimalist grammar (MG), a recent computational implementation of generative linguistics. Number words are presented to the agent by a teacher in form of utterance meaning pairs (UMP) where the meanings are encoded as arithmetic terms from a suitable term algebra. Since MG encodes universal linguistic competence through inference rules, thereby separating innate linguistic knowledge from the contingently acquired lexicon, our approach unifies generative grammar and reinforcement learning, hence potentially resolving the still pending Chomsky-Skinner controversy., Comment: 13 pages, 1 figure

Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research sofar. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception., Comment: Submitted ms, 38 pages, 7 figures

Computation is classically studied in terms of automata, formal languages and algorithms; yet, the relation between neural dynamics and symbolic representations and operations is still unclear in traditional eliminative connectionism. Therefore, we suggest a unique perspective on this central issue, to which we would like to refer as to transparent connectionism, by proposing accounts of how symbolic computation can be implemented in neural substrates. In this study we first introduce a new model of dynamics on a symbolic space, the versatile shift, showing that it supports the real-time simulation of a range of automata. We then show that the Goedelization of versatile shifts defines nonlinear dynamical automata, dynamical systems evolving on a vectorial space. Finally, we present a mapping between nonlinear dynamical automata and recurrent artificial neural networks. The mapping defines an architecture characterized by its granular modularity, where data, symbolic operations and their control are not only distinguishable in activation space, but also spatially localizable in the network itself, while maintaining a distributed encoding of symbolic representations. The resulting networks simulate automata in real-time and are programmed directly, in absence of network training. To discuss the unique characteristics of the architecture and their consequences, we present two examples: i) the design of a Central Pattern Generator from a finite-state locomotive controller, and ii) the creation of a network simulating a system of interactive automata that supports the parsing of garden-path sentences as investigated in psycholinguistics experiments.

We improve the results by Siegelmann & Sontag (1995) by providing a novel and parsimonious constructive mapping between Turing Machines and Recurrent Artificial Neural Networks, based on recent developments of Nonlinear Dynamical Automata. The architecture of the resulting R-ANNs is simple and elegant, stemming from its transparent relation with the underlying NDAs. These characteristics yield promise for developments in machine learning methods and symbolic computation with continuous time dynamical systems. A framework is provided to directly program the R-ANNs from Turing Machine descriptions, in absence of network training. At the same time, the network can potentially be trained to perform algorithmic tasks, with exciting possibilities in the integration of approaches akin to Google DeepMind's Neural Turing Machines., Comment: 11 pages, 3 figures

The concept of complementarity in combination with a non-Boolean calculus of propositions refers to a pivotal feature of quantum systems which has long been regarded as a key to their distinction from classical systems. But a non-Boolean logic of complementary features may also apply to classical systems, if their states and observables are defined by partitions of a classical state space. If these partitions do not satisfy certain stability criteria, complementary observables and non-Boolean propositional lattices may be the consequence. This is especially the case for non-generating partitions of nonlinear dynamical systems. We show how this can be understood in more detail and indicate some challenging consequences for systems outside quantum physics, including mental processes., Comment: 14 pages, 6 figures

Recurrent temporal dynamics is a phenomenon observed frequently in high-dimensional complex systems and its detection is a challenging task. Recurrence quantification analysis utilizing recurrence plots may extract such dynamics, however it still encounters an unsolved pertinent problem: the optimal selection of distance thresholds for estimating the recurrence structure of dynamical systems. The present work proposes a stochastic Markov model for the recurrent dynamics that allows to derive analytically a criterion for the optimal distance threshold. The goodness of fit is assessed by a utility function which assumes a local maximum for that threshold reflecting the optimal estimate of the system's recurrence structure. We validate our approach by means of the nonlinear Lorenz system and its linearized stochastic surrogates. The final application to neurophysiological time series obtained from anesthetized animals illustrates the method and reveals novel dynamic features of the underlying system. As a conclusion, we propose the number of optimal recurrence domains as a statistic for classifying an animals' state of consciousness., Comment: 7 pages, 3 figures, submitted to EPL. External Supplement available at https://www.researchgate.net/publication/301525222_GrabenEA_EPLsupp

Quasistationarity is ubiquitous in complex dynamical systems. In brain dynamics there is ample evidence that event-related potentials reflect such quasistationary states. In order to detect them from time series, several segmentation techniques have been proposed. In this study we elaborate a recent approach for detecting quasistationary states as recurrence domains by means of recurrence analysis and subsequent symbolisation methods. As a result, recurrence domains are obtained as partition cells that can be further aligned and unified for different realisations. We address two pertinent problems of contemporary recurrence analysis and present possible solutions for them., Comment: 24 pages, 6 figures. Draft version to appear in Proc Royal Soc A

Quantum mechanics emerged as the result of a successful resolution of stringent empirical and profound conceptual conflicts within the development of atomic physics at the beginning of the last century. At first glance, it seems to be bizarre and even ridiculous to apply ideas of quantum physics in order to improve current psychological and linguistic or semantic ideas. However, a closer look shows that there are some parallels in the development of quantum physics and advanced theories of cognitive science. In psychology, geometric models of meaning have a long tradition. However, they suffer from many shortcomings which are illustrated by discussing several puzzles of bounded rationality. In the first part of this article, we argue that the present account of quantum cognition - taking quantum probabilities rather than classical probabilities - can give a more systematic description of these puzzles than the alternate and rather eclectic treatments in the traditional framework of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always and does not automatically provide a deeper understanding and a true explanation of these puzzles. In the second part of this article, we explain the foundational issue from the perspective of Piron, Foulis, Randall, and others, and we apply it to the foundation of quantum cognition. In this connection, we show that quantum probabilities are of (virtual) conceptual necessity if grounded in an abstract algebraic framework of orthomodular lattices. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives) that are combined into a uniform system while considering certain capacity restrictions. It is at this point that one important aspect of the whole idea of bounded rationality directly enters the theoretical scenery of quantum cognition: resource limitation., Comment: 54 pages, 10 figures

By means of an intriguing physical example, magnetic surface swimmers, that can be described in terms of Dennett's intentional stance, I reconstruct a hierarchy of necessary and sufficient conditions for the applicability of the intentional strategy. It turns out that the different levels of the intentional hierarchy are contextually emergent from their respective subjacent levels by imposing stability constraints upon them. At the lowest level of the hierarchy, phenomenal physical laws emerge for the coarse-grained description of open, nonlinear, and dissipative nonequilibrium systems in critical states. One level higher, dynamic patterns, such as, e.g., magnetic surface swimmers, are contextually emergent as they are invariant under certain symmetry operations. Again one level up, these patterns behave apparently rational by selecting optimal pathways for the dissipation of energy that is delivered by external gradients. This is in accordance with the restated Second Law of thermodynamics as a stability criterion. At the highest level, true believers are intentional systems that are stable under exchanging their observation conditions., Comment: 27 pages; 4 figures (Fig 1. Copyright by American Physical Society); submitted to Journal of Consciousness Studies

Turing machines and G\"odel numbers are important pillars of the theory of computation. Thus, any computational architecture needs to show how it could relate to Turing machines and how stable implementations of Turing computation are possible. In this chapter, we implement universal Turing computation in a neural field environment. To this end, we employ the canonical symbologram representation of a Turing machine obtained from a G\"odel encoding of its symbolic repertoire and generalized shifts. The resulting nonlinear dynamical automaton (NDA) is a piecewise affine-linear map acting on the unit square that is partitioned into rectangular domains. Instead of looking at point dynamics in phase space, we then consider functional dynamics of probability distributions functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space of a dynamic field theory (DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. We call the resulting representation \emph{dynamic field automaton}., Comment: 21 pages; 6 figures. arXiv admin note: text overlap with arXiv:1204.5462

We present a microscopic approach for the coupling of cortical activity, as resulting from proper dipole currents of pyramidal neurons, to the electromagnetic field in extracellular fluid in presence of diffusion and Ohmic conduction. Starting from a full-fledged three-compartment model of a single pyramidal neuron, including shunting and dendritic propagation, we derive an observation model for dendritic dipole currents in extracellular space and thereby for the dendritic field potential that contributes to the local field potential of a neural population. Under reasonable simplifications, we then derive a leaky integrate-and-fire model for the dynamics of a neural network, which facilitates comparison with existing neural network and observation models. In particular, we compare our results with a related model by means of numerical simulations. Performing a continuum limit, neural activity becomes represented by a neural field equation, while an observation model for electric field potentials is obtained from the interaction of cortical dipole currents with charge density in non-resistive extracellular space as described by the Nernst-Planck equation. Our work consistently satisfies the widespread dipole assumption discussed in the neuroscientific literature., Comment: 31 pages; 3 figures; to appear in "Neural Fields: Theory and Applications", edited by S. Coombes, P. beim Graben, R. Potthast, and J. J. Wright; Springer Verlag Berlin (2014)

In their target article, \citet{WangBusemeyer13} [A quantum question order model supported by empirical tests of an a priori and precise prediction. \emph{Topics in Cognitive Science}] discuss question order effects in terms of incompatible projectors on a Hilbert space. In a similar vein, Blutner recently presented an orthoalgebraic query language essentially relying on dynamic update semantics. Here, I shall comment on some interesting analogies between the different variants of dynamic semantics and generalized quantum theory to illustrate other kinds of order effects in human cognition, such as belief revision, the resolution of anaphors, and default reasoning that result from the crucial non-commutativity of mental operations upon the belief state of a cognitive agent., Comment: Comment on a target article "A quantum question order model supported by empirical tests of an a priori and precise prediction", Topics in Cognitive Science, by Wang and Busemeyer (2013)

We propose an algorithm for the detection of recurrence domains of complex dynamical systems from time series. Our approach exploits the characteristic checkerboard texture of recurrence domains exhibited in recurrence plots (RP). In phase space, RPs yield intersecting balls around sampling points that could be merged into cells of a phase space partition. We construct this partition by a rewriting grammar applied to the symbolic dynamics of time indices. A maximum entropy principle defines the optimal size of intersecting balls. The final application to high-dimensional brain signals yields an optimal symbolic recurrence plot revealing functional components of the signal., Comment: Accepted for Physical Review Letters; 10 pages, 2 figures

We present a biophysical approach for the coupling of neural network activity as resulting from proper dipole currents of cortical pyramidal neurons to the electric field in extracellular fluid. Starting from a reduced threecompartment model of a single pyramidal neuron, we derive an observation model for dendritic dipole currents in extracellular space and thereby for the dendritic field potential that contributes to the local field potential of a neural population. This work aligns and satisfies the widespread dipole assumption that is motivated by the "open-field" configuration of the dendritic field potential around cortical pyramidal cells. Our reduced three-compartment scheme allows to derive networks of leaky integrate-and-fire models, which facilitates comparison with existing neural network and observation models. In particular, by means of numerical simulations we compare our approach with an ad hoc model by Mazzoni et al. [Mazzoni, A., S. Panzeri, N. K. Logothetis, and N. Brunel (2008). Encoding of naturalistic stimuli by local field potential spectra in networks of excitatory and inhibitory neurons. PLoS Computational Biology 4 (12), e1000239], and conclude that our biophysically motivated approach yields substantial improvement., Comment: 31 pages, 4 figures

Cognitive computation such as e.g. language processing, is conventionally regarded as Turing computation, and Turing machines can be uniquely implemented as nonlinear dynamical systems using generalized shifts and subsequent G\"odel encoding of the symbolic repertoire. The resulting nonlinear dynamical automata (NDA) are piecewise affine-linear maps acting on the unit square that is partitioned into rectangular domains. Iterating a single point, i.e. a microstate, by the dynamics yields a trajectory of, in principle, infinitely many points scattered through phase space. Therefore, the NDAs microstate dynamics does not necessarily terminate in contrast to its counterpart, the symbolic dynamics obtained from the rectangular partition. In order to regain the proper symbolic interpretation, one has to prepare ensembles of randomly distributed microstates with rectangular supports. Only the resulting macrostate evolution corresponds then to the original Turing machine computation. However, the introduction of random initial conditions into a deterministic dynamics is not really satisfactory. As a possible solution for this problem we suggest a change of perspective. Instead of looking at point dynamics in phase space, we consider functional dynamics of probability distributions functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space of a dynamic field theory (DFT). Solving the Frobenius-Perron equation, yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. Thus, the symbolically meaningful NDA macrostate dynamics becomes represented by iterated function dynamics in DFT; hence we call the resulting representation dynamic field automata., Comment: 5 pages, no figure

Quantum entanglement relies on the fact that pure quantum states are dispersive and often inseparable. Since pure classical states are dispersion-free they are always separable and cannot be entangled. However, entanglement is possible for epistemic, dispersive classical states. We show how such epistemic entanglement arises for epistemic states of classical dynamical systems based on phase space partitions that are not generating. We compute epistemically entangled states for two coupled harmonic oscillators., Comment: 13 pages, no figures; International Journal of Theoretical Physics, 2012

I pinpoint an interesting similarity between a recent account to rational parsing and the treatment of sequential decisions problems in a dynamical systems approach. I argue that expectation-driven search heuristics aiming at fast computation resembles a high-risk decision strategy in favor of large transition velocities. Hale's rational parser, combining generalized left-corner parsing with informed $\mathrm{A}^*$ search to resolve processing conflicts, explains gardenpath effects in natural sentence processing by misleading estimates of future processing costs that are to be minimized. On the other hand, minimizing the duration of cognitive computations in time-continuous dynamical systems can be described by combining vector space representations of cognitive states by means of filler/role decompositions and subsequent tensor product representations with the paradigm of stable heteroclinic sequences. Maximizing transition velocities according to a high-risk decision strategy could account for a fast race even between states that are apparently remote in representation space., Comment: The paper has been withdrawn by the author due to some inconsistencies in the argumentation

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations., Comment: 43 pages, 4 figures

We present a microscopic approach for the coupling of cortical activity, as resulting from proper dipole currents of pyramidal neurons, to the electromagnetic field in extracellular fluid in presence of diffusion and Ohmic conduction. As a result, neural activity becomes represented by a continuous neural field equation, while an observation model for electric field potentials is obtained from the interaction of cortical dipole currents with charge density in non-resistive extracellular space as described by the Nernst-Planck equation., Comment: that paper has been redrawn due to a more recent submission