Презентация Quantum Semantics онлайн

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30 слайдов
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1,2,3,4,5,6,7,8,9,10,11
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1.93 MB
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Слайды и текст к этой презентации:

№1 слайд

Содержание слайда:

№2 слайд

Содержание слайда: Bra-ket notation (Dirac, 1939) – vector (Hilbert) space, - field - pure state (vector, or operator ) – ket - effect of state (dual vector, dual operator Hermitian conjugate) – bra Inner product of and is

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Содержание слайда: Outer product Outer product for is a operator: Arbitrary can be written in a basis for and for : , where

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Содержание слайда: Eigenvectors and eigenvalues , if is an orthogonal basis in which is diagonal. - are eigen vectors - are eigen values Easy to check:

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Содержание слайда: Density operator (matrix) If - are pure states, - are probabilities over them, then is a dense operator Positive operator: for all Theorem: is a density operator iff it’s a positive Hermitian operator with trace =1.

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Содержание слайда: Trace inner product and are density matrices same dimension and

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Содержание слайда: Distributional Semantics “You shall know the word by the company it keeps” (Firth) Obtain meaning high dimensional vector representations from large corpora automatically Compositionality DS can not be applied for entire sentence (lack of frequency) Entailment entails if the meaning of a word is included in the meaning of a word (is-a ) - subsumption relation non symmetric

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Содержание слайда: Distributional Inclusion Hypothesis If is semantically narrower than , then a significant number of salient distributional features of are also included in the feature vector of : Hypothesis 1: If => then all the characteristic features of is expected to appear in . Hypothesis 2: If all the characteristic features of appear in , then => .

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Содержание слайда: Category Theory A monoidal category C is a category consisting of the following: a functor called the tensor product an object called the unit object a natural isomorphism whose components are called the associators a natural isomorphism whose components are called the left unitors a natural isomorphism whose components are called the right unitors

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Содержание слайда: Category Theory The objects of the category are thought to be types of systems A morphism is a process that takes a system of type to a system of type . for and , is the composite morphism that takes a system of type into a system of type by applying the process after . Morphisms of type are called elements of .

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Содержание слайда: Compact closed categories A monoidal category is compact closed if for each object , there are also left and right dual objects and , and morphisms that satisfies The maps of compact categories are used to represent correlations, and in categorical quantum mechanics they model maximally entangled states.

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Содержание слайда: Graphical calculus

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Содержание слайда: Graphical calculus

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Содержание слайда: Compositional Distributional Model Pregroup grammars (Lambek) A partially ordered monoid consists of: a set a monoid multiplication operator satisfying the condition for all and thу monoidal unit where for all a partial order on

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Содержание слайда: Pregroup (Lambek, 2001) A pregroup is a partially ordered monoid in which each element has both a left adjoint and a right adjoint such that and Adjoints have properties: Uniqueness: Adjoints are unique Order reversal: If then and The unit is self adjoint: Multiplication operation is self adjoint: and Opposite adjoints annihilate: Same adjoints iterate:

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Содержание слайда: Pregroup grammar means ( reduces to ) “John likes Mary” “John” and “Mary” assigned to type (noun) “likes” is assigned to compound type “likes” takes a noun from the left and from the right, and returns a sentence

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Содержание слайда: Basic types : noun : declarative statement : infinitive of the verb : glueing type

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Содержание слайда: Pregroups as compact closed categories is a concrete instance of a compact closed category Test snake identities: …

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Содержание слайда: Examples “John likes Mary” “John does not like Mary”

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Содержание слайда: - finite dimensional vector space - finite dimensional vector spaces over the base field R together with linear maps, form a monoidal category FVect as a compact closed category.

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Содержание слайда: – categorical representation of meaning space

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Содержание слайда: “From-the-meanings-of-words-to-the-meanings-of-the-sentence” map Let be a string of words, each with a meaning space representation . Let be a pregroup type such that . Then the meaning vector for the string is: , where is defined to be the application of the compact closure maps obtained from the reduction to the composite vector space .

№23 слайд

Содержание слайда: Example: “John likes Mary” It has the pregroup type vector representations and The morphism in corresponding to the map is of type: From the pregroup reduction we obtain the compact closure maps . In this translates into:

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