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Lambdas, Vectors, and Dynamic Logic

Develop: a vector semantics and a dynamic logic for lambda calculus models of language

Mehrnoosh Sadrzadeh Queen Mary University of London

Joint work with Reinhard Muskens (Tilburg) Supported by Royal Society Int. Exchange Award

Why putting lambdas and vector together?

Why putting lambdas and vector together? Because we want to develop a compositional distributional semantics for natural language.

Why putting lambdas and vector together? Because we want to develop a compositional distributional semantics for natural language.

What is that?

Distributional Semantics

Words that occur in similar contexts have similar meanings. “oculist and eye-doctor . . . occur in almost the same environments” “If A and B have almost identical environments. . . we say that they are synonyms.” Harris (1954) “You shall know a word by the company it keeps!” Firth (1957)

Imagine you had never seen the word tesguino, but given the following text: A bottle of tesguino is on the table. Everybody likes tesguino . Tesguino makes you drunk. We make tesguino out of corn. the company it keeps!”. The meaning of a word is thus related to the distribution of words around it. Imagine you had never seen the word tesg¨ uino, but I gave you the following 4 sen-

Speech and Language Processing. Daniel Jurafsky & James H. Martin.

Imagine you had never seen the word tesguino, but given the following text: A bottle of tesguino is on the table. Everybody likes tesguino . Tesguino makes you drunk. We make tesguino out of corn. “an alcoholic drink made of corn”

Co-Occurrence Matrix

• sugar, a sliced lemon, a tablespoonful of apricot

preserve or jam, a pinch each of, their enjoyment. Cautiously she sampled her first pineapple and another fruit whose taste she likened well suited to programming on the digital computer. In finding the optimal R-stage policy from for the purpose of gathering data and information necessary for the study authorized in the

aardvark ... computer data pinch result sugar ... apricot ... 1 1 pineapple ... 1 1 digital ... 2 1 1 information ... 1 6 4 Figure 15.4 Co-occurrence vectors for four words, computed from the Brown corpus,

[| digital |] := (0, …, 2, 1, 0, 1, 0)

Algorithm for Word Meaning Acquisition

Co-Occurrence Matrix

• sugar, a sliced lemon, a tablespoonful of apricot

preserve or jam, a pinch each of, their enjoyment. Cautiously she sampled her first pineapple and another fruit whose taste she likened well suited to programming on the digital computer. In finding the optimal R-stage policy from for the purpose of gathering data and information necessary for the study authorized in the

aardvark ... computer data pinch result sugar ... apricot ... 1 1 pineapple ... 1 1 digital ... 2 1 1 information ... 1 6 4 Figure 15.4 Co-occurrence vectors for four words, computed from the Brown corpus,

[| digital |] := (0, …, 0.25, 0.12, 0, 0.15, 0)

Algorithm for Word Meaning Acquisition

Normalised

Word-Context Matrix

• sugar, a sliced lemon, a tablespoonful of apricot

preserve or jam, a pinch each of, their enjoyment. Cautiously she sampled her first pineapple and another fruit whose taste she likened well suited to programming on the digital computer. In finding the optimal R-stage policy from for the purpose of gathering data and information necessary for the study authorized in the

aardvark ... computer data pinch result sugar ... apricot ... 1 1 pineapple ... 1 1 digital ... 2 1 1 information ... 1 6 4 Figure 15.4 Co-occurrence vectors for four words, computed from the Brown corpus,

Intuition: a word is represented by embedding its representation into a vector space, rather than just treated as an “atom”:

Q: What’s the meaning of life? A: LIFE

Automatic Meaning Acquisition

Curran, 2006

Introduction: launch, implementation, advent, addition arrival, creation, inclusion Evaluation: assessment, examination, appraisal, review audit, analysis, consultation, test, verification Methods: technique, procedure, means, approach, tool, concept, practice, formula

Stronghold

Semantic Similarity

1 2 3 4 5 6 7 1 2 3 digital apricot information Dimension 1: ‘large’ Dimension 2: ‘data’

Evaluations

WordSim-353: noun pairs (cup, coffee) SimLex-999: adjective, noun, and verb pairs (cup, drink) TOEFL: 80 questions “Levied” is closest in meaning to imposed, believed, requested, correlated SCWS: 2003 words in sentences Analogy: a to b is like c to do Athens to Greece is like Oslo to Norway

Applications

Drawing inferences about meanings of words. Named entity recognition, parsing, semantic role labelling., summarisation, essay marking, …

Extending distributional semantics from words to sentences Compositional Distributional Semantics

Vectors for Sentences

1 2 3 4 5 6 1 2 digital [1,1]

result data

information [6,4] 3 4

base 1 base 2

dogs bark mice squeak

At childhood we spent the summers in countryside. Khaterey daram sharbat-e albalooye anjast ast. My mother preferred the jams, however. My worst memory is being forced to take a nap in the afternoons.

A direct distributional approach…

At childhood we spent the summers in countryside. Khaterey daram sharbat-e albalooye anjast ast. My mother preferred the jams, however. My worst memory is being forced to take a nap in the afternoons. “ A memory from then is their sour cherry drink”

A direct distributional approach…

An approach that forgets about grammar

• !

vampires kill men =

• !

vampires + ! kill +

• !

men =

• !

vampires ! kill

• !

men

• !

vampires kill men =

• !

men kill vampires

So what should we do?

Formal Semantics

Syntax Semantics

structure preserving map

Formal Semantics with Vectors

Syntax Semantics

structure preserving map

Formal Grammar Vectors

Compositional Distributional Semantics

Syntax Semantics

structure preserving map

Pregroup Grammar

Coecke, Sadrzadeh, Clark, (Lambek’s 90th Festschrift), 2010 Preller, Sadrzadeh (JoLLI), 2011

Vectors Spaces

strongly monoidal functor

Compositional Distributional Semantics

Syntax Semantics

structure preserving map

Lambek Calculus

Coecke, Grefenstette,Sadrzadeh, (APAL), 2013

Vectors Spaces

homomorphism

Compositional Distributional Semantics

Syntax Semantics

structure preserving map

Lambek Grishin

• G. Wijnholds, (MSc Thesis, ILLC), 2015

Vectors Spaces

homomorphism

Syntax Semantics

structure preserving map

CCG

Maillard, Clark, Grefenstette, (Type Theory and NL, EACL workshop), 2014. Krishnamurty and Mitchell, (CVSC, ACL workshop), 2013. Baroni, Bernardini, Zamparelli, (LILT), 2014.

Vector Spaces

Syntax Semantics

structure preserving map

ACG

Muskens and Sadrzadeh, DSALT , LACL 2016. journal version is to apear in JLM 2018

Vectors Spaces

homomorphism

Royal Society International Exchange Award

CCG Semantics

structure preserving map T ypes

X/Y X \ Y

A

X/Y Y =) X Y X \ Y =) X

Rules

Syntax Semantics

structure preserving map T ypes

3 Vi

A 7! U

Syntax Semantics

structure preserving map T ypes

3 Vi

A 7! U

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

Syntax Semantics

structure preserving map T ypes

3 Vi

3 3 Mij, Tijk, Cijkl, · · ·

A 7! U

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

Ti j···k

2

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

Ti j···k

2

Tkl···w

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

Ti j···k

2

Tkl···w

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

Ti j···k

2

Tkl···w = Ti j···l···w

2

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

2

3 Ti,

Ti j

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

2

3 Ti,

Ti j

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

2

3 Ti,

Ti j

T j

2

Syntax Semantics

structure preserving map T ypes Rules

3 Vi

A 7! U 2 3 Ti, Ti j, Ti jk, Ti jkl, · · ·

X/Y 7! X ⌦ Y⇤ X \ Y 7! Y⇤ ⌦ X

X/Y Y =) X 7! X ⌦ Y⇤ ⌦ Y =) X Y X \ Y =) X 7! Y ⌦ Y⇤ ⌦ X =) X

2

3 Ti,

Ti j

T j

2

x =

7!

Cats chase dogs noisily.

NP ((S \ NP) / NP) NP S \ S

7!

Cats chase dogs noisily.

NP ((S \ NP) / NP) NP S \ S

⌦ ⌦

⌦ S ⌦ N⇤ ⌦ N⇤

N ⌦ S ⌦ S⇤

N N

7!

Cats chase dogs noisily.

NP ((S \ NP) / NP) NP S \ S

⌦ ⌦

⌦ S ⌦ N⇤ ⌦ N⇤

N ⌦ S ⌦ S⇤

N N

Ti

Ti jk

Tk

T jl

2 2 2 2

7!

cats chase dogs noisily

NP ((S/NP) \ NP) NP (S/S)

N ⌦ S ⌦ N⇤ ⌦ N⇤ ⌦ N

N N

N ⌦ S ⌦ S⇤

TiTi jkTkT jl

7!

cats chase dogs noisily

NP ((S/NP) \ NP) NP (S/S)

N ⌦ S ⌦ N⇤ ⌦ N⇤ ⌦ N

N N

N ⌦ S ⌦ S⇤

TiTi jkTkT jl =) TiTijT jl

7!

cats chase dogs noisily

NP ((S/NP) \ NP) NP (S/S)

N ⌦ S ⌦ N⇤ ⌦ N⇤ ⌦ N

N N

N ⌦ S ⌦ S⇤

=) T jT jl

TiTi jkTkT jl =) TiTijT jl

7!

cats chase dogs noisily

NP ((S/NP) \ NP) NP (S/S)

N ⌦ S ⌦ N⇤ ⌦ N⇤ ⌦ N

N N

N ⌦ S ⌦ S⇤

=) Tl

=) T jT jl

TiTi jkTkT jl =) TiTijT jl

7!

cats chase dogs noisily

NP ((S/NP) \ NP) NP (S/S)

N ⌦ S ⌦ N⇤ ⌦ N⇤ ⌦ N

N N

N ⌦ S ⌦ S⇤

=) Tl

=) T jT jl

TiTi jkTkT jl =) TiTijT jl

2 S

Applications

Building the T ensors

2- Degree of correlation between subject and object

X

i

(

• !

S bj ⌦

• !

Obj)i

1- Context vector of the verb 3- Populating a cube tensor from a matrix by copying the

• bject dimension

4- Populating a cube tensor from a matrix by copying the subject dimension

V Mij = Tijk

7! X ⌦ Y 3

• !

Verb ⌦

• !

Verb ⌦

• !

Verb

Relational Frobenius 5- Baroni and Zamparelli, Polajnar et al: Linear and multi-step linear regression for learning these from holistic vectors. Least Sq Kronecker

x y z annual report draw attention

• ld man pulled ceremonial sword

annual report attracted attention

• ld man draw ceremonial sword

Disambiguation

Grefenstette-Sadrzadeh, EMNP 2012, Journal of Computational Linguistics 2015 annual report pulled attention

Model ρ Verb Baseline 0.20 Bigram Baseline 0.14 Trigram Baseline 0.16 Additive 0.10 Multiplicative AdjMult 0.20 AdjNoun 0.05 CategoricalAdj 0.20 Categorical AdjMult 0.14 AdjNoun 0.16 CategoricalAdj 0.19 Kronecker AdjMult 0.26 AdjNoun 0.17 CategoricalAdj 0.27 Upperbound 0.48

• sentence 1

sentence 2

• ld man draw ceremonial sword
• ld man attracted ceremonial sword

annual report draw huge attention annual report attracted huge attention

Disambiguation

Grefenstette-Sadrzadeh, EMNP 2012, Journal of Computational Linguistics 2015

x y z project present problem man shut door programme face difficulty gentleman close eye

Paraphrasing

Kartsaklis-Sadrzadeh, EMNLP 2013

Sentence 1 Sentence 2 man shut door gentleman close eye survey collect information page provide datum project present problem programme face difficulty

Model Ambig. Disamb. BL Verbs only 0.310 ⌧ 0.341 M1 Multiplicative 0.325 ⌧ 0.404 M2 Additive 0.368 ⌧ 0.410 T1 Relational 0.368 ⌧ 0.397 T2 Kronecker 0.404 < 0.412 T3 Copy-subject 0.310 ⌧ 0.337 T4 Copy-object 0.321 ⌧ 0.368 Human agreement 0.550 Difference between T2 and M2 is not s.s.

Similarity

Kartsaklis-Sadrzadeh, EMNLP 2013

Sentence 1 Sentence 2 man shut door gentleman close eye survey collect information page provide datum project present problem programme face difficulty

Model Ambig. Disamb. BL Verbs only 0.310 ⌧ 0.341 M1 Multiplicative 0.325 ⌧ 0.404 M2 Additive 0.368 ⌧ 0.410 T1 Relational 0.368 ⌧ 0.397 T2 Kronecker 0.404 < 0.412 T3 Copy-subject 0.310 ⌧ 0.337 T4 Copy-object 0.321 ⌧ 0.368 Human agreement 0.550 Difference between T2 and M2 is not s.s.

Similarity

Kartsaklis-Sadrzadeh, ACL 2014

ACL 2014

Subject-verb-object report describe result ` document explain process report outline progress ` document describe change value suit budget ` number meet standard book present account ` work show evidence woman marry man ` female join male author retain house ` person hold property report highlight lack document stress need

Entailment

Entailment

Model Inclusion KL-div αSkew WeedsPrec ClarkeDE APinc balAPinc SAPinc SBalAPinc Verb 0.61 0.61 0.66 0.69 0.58 0.74 0.67 0.59 0.63

• 0.55

0.65 0.74 0.79 0.67 0.76 0.71 0.80 0.80 min 0.55 0.71 0.74 0.78 0.63 0.77 0.71 0.73 0.76 + 0.58 0.54 0.71 0.59 0.60 0.65 0.64 0.67 0.67 max 0.58 0.55 0.68 0.58 0.58 0.63 0.61 0.60 0.61 Least-Sqr – – – – – – – – – ⌦rel 0.51 0.64 0.78 0.79 0.69 0.79 0.72 0.84 0.83 ⌦proj 0.64 0.60 0.70 0.69 0.61 0.74 0.70 0.75 0.76 ⌦CpSbj 0.57 0.65 0.73 0.77 0.63 0.73 0.68 0.79 0.78 ⌦CpObj 0.54 0.62 0.73 0.72 0.64 0.76 0.71 0.81 0.79 ⌦FrAdd 0.60 0.60 0.75 0.72 0.67 0.77 0.75 0.84 0.82 ⌦FrMul 0.55 0.62 0.76 0.81 0.68 0.78 0.73 0.86 0.83

Balkir, Kartsaklis, Sadrzadeh , ISAIM 2016, COLING 2016, LACL 2016 Annals of Maths and AI

Lambdas and Dynamic Logic

Montague Semantics

constant c type τ H0(c) h0(τ) woman N woman est man N man est tall NN tall (est)est smokes DS smoke est loves DDS love eest knows SDS λpλxλw.∀w0(Kxww0 → pw0) (st)est every N(DS)S λP 0λPλw.∀x(P 0xw → Pxw) (est)(est)st a N(DS)S λP 0λPλw.∃x(P 0xw ∧ Pxw) (est)(est)st

term homs type homs terms types

A Vector Semantics

c τ H(c) h(τ) woman N woman V tall NN λv.(tall ×1 v) V V smokes DS λv.(smoke ×1 v) V V loves DDS λuv.(love ×2 u) ×1 v V V V knows SDS λuv.(know ×2 u) ×1 v V V V every N(DS)S λvZ.Z(every ×1 v) V (V V )V a N(DS)S λvZ.Z(a ×1 v) V (V V )V Some abstract constants typed with abstract types and their term Matrix Mult Cube Contr Vector

Simpler Option (1)

c τ H(c) h(τ) woman N woman V tall NN λv.(tall + v) V smokes DS λv.(smoke + v) V loves DDS λuv.(love + u) + v V knows SDS λuv.(know + u) + v V every N(DS)S λvZ.Z(every + v) V a N(DS)S λvZ.Z(a + v) V

Simpler Option (2)

c τ H(c) h(τ) woman N woman V tall NN λv.(tall v) V smokes DS λv.(smoke v) V loves DDS λuv.(love u) v V knows SDS λuv.(know u) v V every N(DS)S λvZ.Z(every v) V a N(DS)S λvZ.Z(a v) V

Making things dynamic

U

Update Function Contexts Contexts

:

U

Update Function Contexts Contexts

:

Intransitive Verbs T ransitive Verbs

VU VVU U

Sentences

Making things dynamic

Dynamic Vector Semantics

a τ H(a) ρ(τ) Anna (DS)S λZ.Z(anna) (V U)U woman N λZ.Z(woman) (V U)U tall NN λQZ.Q(λvc.ZvF(tall, v, c)) ((V U)U)(V U)U smokes DS λvc.G(smoke, v, c) V U loves DDS λuvc.I(love, u, v, c) V V U knows SDS λpvc.pJ(know, v, c) UV U every N(DS)S λQ.Q ((V U)U)(V U)U who (DS)NN λZ0QZ.Q(λvc.Zv(QZ0c)) (V U)((V U)U)(V U)U and (αS)(αS)(αS) λR0λRλXλc.R0X(RXc) (ρ(α)U)(ρ(α)U)(ρ(α)U)

N-> (VU)U D -> V S -> U

Examples of Contexts

U

Update Function Contexts Contexts

:

Co-Occurrence Matrices Entity-Relation Cubes

Co-occurrence Matrices

     m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk      + ha, u, v, · · · i =      m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

    

Co-occurrence Matrices

     m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk      + ha, u, v, · · · i =      m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

    

m0

ij := mij + 1

Co-occurrence Matrices

     m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk      + ha, u, v, · · · i =      m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

    

i: tall

m0

ij := mij + 1

j: modifee of tall v

1, for i, j, k indices of by λvc.F(tall, v, c) tall and its modified

Co-occurrence Matrices

     m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk      + ha, u, v, · · · i =      m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

    

by λvc.G(smoke, v, c) increases indices of smoke and its subject i: smoke

m0

ij := mij + 1

j: subject of smoke v

Co-occurrence Matrices

     m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk      + ha, u, v, · · · i =      m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

    

i: love j: subject of love u

by λuv.λc.I(love, u, v, c) 1, for indices of loves

m0

ij := mij + 1

m0

jk := mjk + 1

m0

ik := mik + 1

k: object of love v

Example

1 2 3 4 5 man cat loves fears sleeps 1 Anna 100 700 800 500 400 2 woman 500 650 750 750 600 3 tall 300 50 500 400 400 4 smokes 400 50 600 600 200 5 loves 350 250 ✏ 600 500 6 knows 300 50 200 250 270

update by

= ⇒

G, I, F, J 1 2 3 4 5 man cat loves fears sleeps 1 Anna 100 700 800 500 400 2 woman 500 650 750 750 600 3 tall 650 50 500 400 400 4 smokes 700 50 600 600 200 5 loves 550 750 ✏ 600 500 6 knows 600 250 450 510 700

Entity-Relation Cubes

cijk entity relation entity + (a, u, v) = c0

ijk

entity relation entity where c0

ijk := cijk + 1

Entity-Relation Cube Example

100 Anna is-a smoker

update by G

= ⇒ 400 Anna is-a smoker 50 Anna is tall

update by F

= ⇒ 280 Anna is tall 200 Anna loves cat

update by I

= ⇒ Anna 350 loves cat

Dynamic Logic

φ ::= p | ¬φ | φ ∧ ψ |

Dynamic Logic

φ ::= p | ¬φ | φ ∧ ψ |

||p||(c) := c + ||p|| ||¬φ||(c) := c − ||φ||(c) ||φ ∧ ψ|| := ||ψ||(||φ||(c))

c: context || - ||: update

Dynamic Logic

||p||(c) := c + ||p|| ||¬φ||(c) := c − ||φ||(c) ||φ ∧ ψ|| := ||ψ||(||φ||(c))

+ = c ∩ ||p||

Heim, Karttunen c, ||p||: sets of valuations

Dynamic Logic

||p||(c) := c + ||p|| ||¬φ||(c) := c − ||φ||(c) ||φ ∧ ψ|| := ||ψ||(||φ||(c))

? for us Contexts = Matrices or Cubes

Dynamic Logic with Vector Semantics

||S||(c) := c +0 H(S) c − H(S) := (c +0 H(S))1

S: a sentence H(S): its term hom image

Dynamic Logic with Vector Semantics

B B B @ m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk 1 C C C A +0 ha, u, v, · · · i = B B B @ m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

1 C C C A for m0

ij :=

( 1 mij = 1 1 mij = 0 B B B @ m11 · · · m1k m21 · · · m2k . . . mn1 · · · mnk 1 C C C A 0 ha, u, v, · · · i = B B B @ m0

11 · · · m0 1k

m0

21 · · · m0 2k

. . . m0

n1 · · · m0 nk

1 C C C A for m0

ij :=

( mij = 1 mij = 0

For co-occurence matrices

Dynamic Logic with Vector Semantics

Binary versions of the contexts.

1 2 3 4 5 man cat loves fears sleeps 1 Anna 100 700 500 400 2 woman 650 750 750 600 3 tall 300 50 500 400 400

Dynamic Logic with Vector Semantics

1 2 3 4 5 man cat loves fears sleeps 1 Anna 1 1 1 1 2 woman 1 1 1 1 3 tall 1 1 1 1 1

Binary versions of the contexts.

Dynamic Logic with Vector Semantics

Basically working with the graph of the relation of the matrix.

✓1 0 1 1 ◆ 7! 7! 1 2 1 1 0 2 1 1 {(1, 1), (2, 1), (2, 2)}

+’ = c ∩ ||p||

||S1, · · · , Sn||(c) := ||S1|| · · · ||Sn||(c)

Dynamic Logic with Vector Semantics

This semantics is indeed “dynamic”.

• Observation. The update obtained by a sequence of sentences

is the same as the update obtained by their composition.

Dynamic Logic with Vector Semantics

This semantics is indeed “dynamic”.

• Proposition. The context corresponding to a sequence of

sentences is the empty context (0 matrix or cube) updated by them.

c = ||S1, · · · , Sn||(0)

Application

L context c admits proposition φ ⇐ ⇒ ||φ||(c) = c

Admittance of a proposition by a context (Karttunen&Heim):

Application

L context c admits proposition φ ⇐ ⇒ ||φ||(c) = c

Admittance of a proposition by a context (Karttunen&Heim):

• Definition. A corpus admits a sentence if the context

corresponding to it admits it.

Application

1 2 3 4 5 man cat loves fears sleeps 1 Anna 100 700 500 400 2 woman 650 750 750 600 3 tall 300 50 500 400 400 1 2 3 4 5 man cat loves fears sleeps 1 Anna 1 1 1 1 2 woman 1 1 1 1 3 tall 1 1 1 1 1 1 2 3 4 5 man cat loves fears sleeps 1 Anna 1 1 1 1 2 woman 1 1 1 1 3 tall 1 1 1 1 1

update

Example

‘Cats and dogs are animals that sleep. Cats chase cats and mice. Dogs chase all

• animals. Cats like mice, but mice fear cats, since cats eat mice. Cats smell mice and

mice run from cats.’ It admits the following sentences:

Cats are animals. Dogs are animals. Cats chase cats. Cats chase mice. Dogs chase cats and dogs.

(*) Cats like dogs. (*) Cats eat dogs. (*) Dogs run from cats. (*) Dogs like mice. (*) Mice fear dogs. (*) Dogs eat mice.

(*) Cats are not animals. (*) Dogs do not sleep.

Experimental Evaluation

fracas-013 answer: yes P1 Both leading tenors are excellent. P2 Leading tenors who are excellent are indispensable. Q Are both leading tenors indispensable? H Both leading tenors are indispensable.

Check whether the context P1, P2 admits the sentence H.

Parents have a great influence on the career development of their children. Parents have a powerful influence on the career development of their children.

Experimental Evaluation

Zeichner S1 S2 Check whether the context S1 admits the sentence S2.

Conclusions and Future Work

Developed vector semantics for lambda calculus of NL. Extend to reference resolution: dynamic modality. Experiment with the notion of admittance Developed a dynamic logic for vector semantics. It does not matter where do the lambdas come from, it can be from ACG, CCG, LC, etc.