Classical Logic and Quantum Logic with Multiple and Common Lattice Models

Pavičić, Mladen

doi:10.1155/2016/6830685

Cited by 8 publications

(7 citation statements)

References 39 publications

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“…Such logic employs models which evaluate particular combinations of propositions by mapping from a set of propositions to an algebra (lattice), through which a correspondence with measurement values indirectly emerges. Since the algebra must be an orthomodular lattice and cannot be a Boolean algebra then this non-classical logic which has an orthomodular lattice as one of its models is "empirical" whenever describing non-classical measurements, simply because it can be linked to its algebraic model which serves for such a description: an orthomodular Hilbert lattice, that is, the lattice of closed subspaces of a complex Hilbert space ( Pavičić , 2016). 5 The terminology in the neuroscience literature includes at least columns, bundles, groups, COUs, etc.…”

Section: Part 2: the Hardware Devicementioning

confidence: 99%

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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A model of computation describes how units of information and an organizational architecture transform input to output. In the cerebral cortex the unit of information is a normalized collection of neuronal membrane potentials representing a vector of complex amplitudes. The organizational architecture of the cortex allows for programming operator matrices in dendritic arborizations on vectors of complex amplitudes. Three well-documented facts are sufficient to define the cortical model of computation: first, cortical dendritic inputs to compute membrane potentials are, in general, complex numbers (amplitudes) that may be configured as state vectors in complex Hilbert spaces; second, normalization is a canonical neural computation, specifically, for vectors of cortical membrane amplitudes representing orthogonal states; and, third, dendritic arborizations are programmable devices in a universal sense for mathematical and logical functions. These three facts, well studied and accepted for years, are all that is needed for a simple, well-defined, and well-known model of computation. A great deal of research is presently devoted to understanding particular neural algorithms, circuits, and programming. Understanding the algorithm that a particular circuit implements is not to be confused with identifying the much more basic underlying model of computation. The analogy for classical computing is that the classical model of computing stores and manipulates information fundamentally in the form of bits – essentially a finite set of possible states, typically, {0,1}. The primitive classical container of information is analogous to a two-state switch. Nevertheless, sophisticated computational circuitry and algorithms are developed on that simple classical realization. In contrast to the finite set of states found in the fundamental unit of classical information, the fundamental units of information in the cerebral cortex form a model of computation over the complex field in normalized sets of mutually inhibitory amplitudes. This means that the cortex represents information in a fundamental container of information (a vector of amplitudes in membrane potentials) that can encode discrete states over a continuum of values (no doubt down to some constant.) A well-known result from probability theory and standard neuroimaging analysis argue that these mutually inhibitory fundamental units of information may normalize complex amplitudes under the 2-norm. Sufficiently sophisticated computational means are available in the cortex to program unitary operators, state reduction and expansion operators and tensor products that operate on vectors of normalized membrane amplitudes. This model of computation is typically viewed as a probabilistic or predictive model of computation. Again, the model of computation is not a particular circuit or implementation of any one algorithm. The model of computation is the arena in which cortical circuits and algorithms are implemented. The fundamental arena then for cortical computations is a field of complex amplitudes normalized for probabilistic computation.

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Section: Part 2: the Hardware Devicementioning

confidence: 99%

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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“…Recall that closed linear subspaces, say, H A and H B , of a Hilbert space H are called commutable if the following condition holds [4]:…”

Section: Admissibility and The Indefiniteness Of Valuationmentioning

confidence: 99%

Probabilities in the logic of quantum propositions

Bolotin

2018

Preprint

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In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.

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“…is applicable to them [12]. It is readily to see that any pair of the subspaces in the Boolean blocks, i.e., H ′ , H ′′ ∈ L(Σ q ), adheres to this condition, but it is not applicable to a pair H ′ , H ′′ ∈ L(H) where H ′ = ran( P ), H ′′ = ran( Q) and P Q = Q P .…”

Section: The Classical Limit Of Quantum Logicmentioning

confidence: 99%

Classical limit of quantum propositions

Bolotin¹

2018

Preprint

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Contrary to classical semantics, the disjunction of two experimental propositions relating to pure states of a quantum system ("quantum propositions" for short) can be true even in the case where neither disjunct is true. This suggests that in such case either both disjuncts are false and so the distributive laws are not applicable to quantum propositions (this inference is accepted in quantum logic) or the disjuncts are not bivalent, i.e., neither true nor false, therefore the principle of bivalence is not applicable to quantum propositions. But, to accept the latter inference, one must explain how quantum propositions become bivalent in the classical limit. This paper shows the emergence of bivalence through the interaction between a quantum system and its environment and compares the environmentally induced bivalence with the classical limit of quantum logic.

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Classical Logic and Quantum Logic with Multiple and Common Lattice Models

Cited by 8 publications

References 39 publications

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Probabilities in the logic of quantum propositions

Classical limit of quantum propositions

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