Geometric Representations for Minimalist Grammars

Graben, Peter beim; Gerth, Sabrina

doi:10.1007/s10849-012-9164-2

Cited by 13 publications

(12 citation statements)

References 33 publications

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“…Otherwise rectangular states would remain stabilized as described by Eq. (10). All these problems provide promising prospects for future research.…”

Section: Discussionmentioning

confidence: 99%

“…Accordingly, the value of ϕ is retrieved through an unbinding ψ + , u where ψ + is the adjoint of the address ψ where ϕ is bound to. These operations require further underlying structure of the employed function spaces that are therefore chosen as Banach or Hilbert spaces where either adjoint or bi-orthogonal basis functions are available (see [10,[14][15][16]38] for examples).…”

Section: Variables and Data Typesmentioning

confidence: 99%

“…From a mathematical point of view, an algorithm is an element of an abstract algebra that has a representation as an operator on the space of variable allocations, which is well-known as denotational semantics in computer science [50]. The algebra product is the concatenation of instructions being preserved in the representation which is thereby an algebra homomorphism [10,15]. Concatenating instructions or composing operators takes place step-by-step in discrete time.…”

Section: Algorithms and Sequential Processesmentioning

confidence: 99%

“…When the solid curve displaying |A(t)|wf (u) is above u (the dotted curve), then the dynamics (10) leads to an increase of u, indicated by the arrows pointing to the right. In the case where |A(t)|wf (u) < u, a decrease of u is obtained from (10). This is indicated by the arrows pointing to the left.…”

Section: Dynamic Field Automatamentioning

confidence: 99%

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Universal Neural Field Computation

Graben

Potthast

2014

Neural Fields

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Turing machines and Gödel 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 Gödel 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 dynamic field automaton.

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“…Otherwise rectangular states would remain stabilized as described by Eq. (10). All these problems provide promising prospects for future research.…”

Section: Discussionmentioning

confidence: 99%

Section: Variables and Data Typesmentioning

confidence: 99%

Section: Algorithms and Sequential Processesmentioning

confidence: 99%

Section: Dynamic Field Automatamentioning

confidence: 99%

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Universal Neural Field Computation

Graben

Potthast

2014

Neural Fields

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“…However, it should be noticed that beimGraben and Gerth (2012) presented a geometric representation theory for minimalist grammars in a mathematical sense.21 Similarly, in a letter to Stapp, Werner Heisenberg mentions "that Niels Bohr was very interested in the ideas of William James" (Stapp 1972…”

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confidence: 99%

Quantum cognition and bounded rationality

Blutner

Graben

2015

Synthese

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We consider several puzzles of bounded rationality. These include the Allais-and Ellsberg paradox, the disjunction effect, and related puzzles. 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 treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives). They are combined into a uniform system (i.e. orthomodular lattice) via a mechanism preventing the simultaneous realization of perspectives. Gleason's theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic B Reinhard Blutner

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