🗊Презентация Quantum Semantics

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Quantum Semantics, слайд №1Quantum Semantics, слайд №2Quantum Semantics, слайд №3Quantum Semantics, слайд №4Quantum Semantics, слайд №5Quantum Semantics, слайд №6Quantum Semantics, слайд №7Quantum Semantics, слайд №8Quantum Semantics, слайд №9Quantum Semantics, слайд №10Quantum Semantics, слайд №11Quantum Semantics, слайд №12Quantum Semantics, слайд №13Quantum Semantics, слайд №14Quantum Semantics, слайд №15Quantum Semantics, слайд №16Quantum Semantics, слайд №17Quantum Semantics, слайд №18Quantum Semantics, слайд №19Quantum Semantics, слайд №20Quantum Semantics, слайд №21Quantum Semantics, слайд №22Quantum Semantics, слайд №23Quantum Semantics, слайд №24Quantum Semantics, слайд №25Quantum Semantics, слайд №26Quantum Semantics, слайд №27Quantum Semantics, слайд №28Quantum Semantics, слайд №29Quantum Semantics, слайд №30
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Quantum Semantics, слайд №1
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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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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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Outer product Outer product for is a operator: Arbitrary can be written in a basis for and for : , where
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Outer product Outer product for is a operator: Arbitrary can be written in a basis for and for : , where

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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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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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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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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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Trace inner product and are density matrices same dimension and
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Trace inner product and are density matrices same dimension and

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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
Описание 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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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 => .
Описание 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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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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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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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 .
Описание 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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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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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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Graphical calculus
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Graphical calculus

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Graphical calculus
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Graphical calculus

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

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

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Examples “John likes Mary” “John does not like Mary”
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Examples “John likes Mary” “John does not like Mary”

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- 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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- 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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– categorical representation of meaning space
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– categorical representation of meaning space

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“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 .
Описание 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 .

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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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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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Example: “John likes Mary”
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Example: “John likes Mary”

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Example
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Example

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Example
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Example

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Example
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Example

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Example
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Example

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Example
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Example

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Readings Esma Balkir. Using Density Matrices in a Compositional Distributional Model of Meaning. // Master thesis. University of Oxford. 2014 Joachim Lambek. Type grammars as pregroups. Grammars, 4(1):21{39, 2001.
Описание слайда:
Readings Esma Balkir. Using Density Matrices in a Compositional Distributional Model of Meaning. // Master thesis. University of Oxford. 2014 Joachim Lambek. Type grammars as pregroups. Grammars, 4(1):21{39, 2001.

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