Semantics and Pragmatics of NLP Lascarides & Klein Lambda - PowerPoint PPT Presentation

SPNLP: Lambda Terms, Quantifiers, Satisfaction Semantics and Pragmatics of NLP Lascarides & Klein Lambda Terms, Quantifiers, Satisfaction Outline Typed Lambda Calculus Alex Lascarides & Ewan Klein First Order Logic Truth and School of Informatics Satisfaction University of Edinburgh 10 January 2008

SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & Typed Lambda Calculus 1 Klein Outline Typed Lambda 2 First Order Logic Calculus First Order Logic Truth and Satisfaction Truth and Satisfaction 3

Transitive Verbs as Functions SPNLP: Lambda Terms, Quantifiers, We looked at replacing n -ary relations with functions. How Satisfaction does this work with transitive verbs? Lascarides & Klein Outline Version 1: chase of type < IND , IND > → BOOL Typed Version 2: chase of type IND → ( IND → BOOL ) Lambda Calculus First Order Advantages of Version 2 (called a curryed function ): Logic Truth and Satisfaction Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

Transitive Verbs as Functions SPNLP: Lambda Terms, Quantifiers, We looked at replacing n -ary relations with functions. How Satisfaction does this work with transitive verbs? Lascarides & Klein Outline Version 1: chase of type < IND , IND > → BOOL Typed Version 2: chase of type IND → ( IND → BOOL ) Lambda Calculus First Order Advantages of Version 2 (called a curryed function ): Logic Truth and Satisfaction Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

Transitive Verbs as Functions SPNLP: Lambda Terms, Quantifiers, We looked at replacing n -ary relations with functions. How Satisfaction does this work with transitive verbs? Lascarides & Klein Outline Version 1: chase of type < IND , IND > → BOOL Typed Version 2: chase of type IND → ( IND → BOOL ) Lambda Calculus First Order Advantages of Version 2 (called a curryed function ): Logic Truth and Satisfaction Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

Transitive Verbs as Functions SPNLP: Lambda Terms, Quantifiers, We looked at replacing n -ary relations with functions. How Satisfaction does this work with transitive verbs? Lascarides & Klein Outline Version 1: chase of type < IND , IND > → BOOL Typed Version 2: chase of type IND → ( IND → BOOL ) Lambda Calculus First Order Advantages of Version 2 (called a curryed function ): Logic Truth and Satisfaction Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

Transitive Verbs as Functions SPNLP: Lambda Terms, Quantifiers, We looked at replacing n -ary relations with functions. How Satisfaction does this work with transitive verbs? Lascarides & Klein Outline Version 1: chase of type < IND , IND > → BOOL Typed Version 2: chase of type IND → ( IND → BOOL ) Lambda Calculus First Order Advantages of Version 2 (called a curryed function ): Logic Truth and Satisfaction Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

Lambda SPNLP: Lambda Terms, Lambdas talk about missing information, and where it is. Quantifiers, Satisfaction The λ binds a variable. Lascarides & Klein The positions of a λ -bound variable in the formula mark Outline where information is ‘missing’. Typed Lambda Replacing these variables with values fills in the Calculus missing information. First Order Logic Example: Truth and Satisfaction λ x . ( man x ) λ -abstract ( λ x . ( man x ) john ) application ( man john ) β -reduction/function application.

Types SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & Klein IND and BOOL are basic types. If σ, τ are types, then so is ( σ → τ ) . Brackets are Outline Typed omitted if no ambiguity. Lambda Calculus For types τ , we have variables Var ( τ ), constants First Order Con ( τ ). Logic Truth and Since we are doing first order logic, we will later restrict Satisfaction variables to Var ( IND ), but allow constants of any type.

Types SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & Klein IND and BOOL are basic types. If σ, τ are types, then so is ( σ → τ ) . Brackets are Outline Typed omitted if no ambiguity. Lambda Calculus For types τ , we have variables Var ( τ ), constants First Order Con ( τ ). Logic Truth and Since we are doing first order logic, we will later restrict Satisfaction variables to Var ( IND ), but allow constants of any type.

Types SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & Klein IND and BOOL are basic types. If σ, τ are types, then so is ( σ → τ ) . Brackets are Outline Typed omitted if no ambiguity. Lambda Calculus For types τ , we have variables Var ( τ ), constants First Order Con ( τ ). Logic Truth and Since we are doing first order logic, we will later restrict Satisfaction variables to Var ( IND ), but allow constants of any type.

Types SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & Klein IND and BOOL are basic types. If σ, τ are types, then so is ( σ → τ ) . Brackets are Outline Typed omitted if no ambiguity. Lambda Calculus For types τ , we have variables Var ( τ ), constants First Order Con ( τ ). Logic Truth and Since we are doing first order logic, we will later restrict Satisfaction variables to Var ( IND ), but allow constants of any type.

Terms in Typed Lambda Calculus SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & We define terms Term ( τ ) of type τ : Klein Var ( τ ) ⊆ Term ( τ ). Outline Typed Con ( τ ) ⊆ Term ( τ ). Lambda Calculus If α ∈ Term ( σ → τ ) and β ∈ Term ( σ ) then First Order Logic ( α β ) ∈ Term ( τ ) (function application). Truth and If x ∈ Var ( σ ) and α ∈ Term ( ρ ) , then λ x .α ∈ Term ( τ ) , Satisfaction where τ = ( σ → ρ )

Terms in Typed Lambda Calculus SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & We define terms Term ( τ ) of type τ : Klein Var ( τ ) ⊆ Term ( τ ). Outline Typed Con ( τ ) ⊆ Term ( τ ). Lambda Calculus If α ∈ Term ( σ → τ ) and β ∈ Term ( σ ) then First Order Logic ( α β ) ∈ Term ( τ ) (function application). Truth and If x ∈ Var ( σ ) and α ∈ Term ( ρ ) , then λ x .α ∈ Term ( τ ) , Satisfaction where τ = ( σ → ρ )

Terms in Typed Lambda Calculus SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & We define terms Term ( τ ) of type τ : Klein Var ( τ ) ⊆ Term ( τ ). Outline Typed Con ( τ ) ⊆ Term ( τ ). Lambda Calculus If α ∈ Term ( σ → τ ) and β ∈ Term ( σ ) then First Order Logic ( α β ) ∈ Term ( τ ) (function application). Truth and If x ∈ Var ( σ ) and α ∈ Term ( ρ ) , then λ x .α ∈ Term ( τ ) , Satisfaction where τ = ( σ → ρ )

Terms in Typed Lambda Calculus SPNLP: Lambda Terms, Quantifiers, Satisfaction Lascarides & We define terms Term ( τ ) of type τ : Klein Var ( τ ) ⊆ Term ( τ ). Outline Typed Con ( τ ) ⊆ Term ( τ ). Lambda Calculus If α ∈ Term ( σ → τ ) and β ∈ Term ( σ ) then First Order Logic ( α β ) ∈ Term ( τ ) (function application). Truth and If x ∈ Var ( σ ) and α ∈ Term ( ρ ) , then λ x .α ∈ Term ( τ ) , Satisfaction where τ = ( σ → ρ )

Extending to a First Order Language SPNLP: Lambda Terms, Quantifiers, 1 Variables i.e., Var ( IND ): x , y , z , . . . , x 0 , x 1 , x 2 , . . . Satisfaction Lascarides & 2 Boolean connectives: Klein ¬ BOOL → BOOL (negation) Outline ∧ BOOL → ( BOOL → BOOL ) (and) Typed ∨ BOOL → ( BOOL → BOOL ) (or) Lambda Calculus → BOOL → ( BOOL → BOOL ) (if. . . then) First Order Logic 3 Quantifiers: ∀ (all) Truth and ∃ (some) Satisfaction 4 Equality: = τ → ( τ → BOOL ) 5 Punctuation: brackets and period

Quantifier Syntax SPNLP: Lambda Terms, If φ ∈ Term ( BOOL ), and x ∈ Var ( IND ), then ∀ x .φ and Quantifiers, Satisfaction ∃ x .φ ∈ Term ( BOOL ). Lascarides & Klein x ∈ Var ( IND ) is called an individual variable. Outline Typed Syntactic conventions: Lambda Calculus First Order Instead of writing ((= α ) β ) , (( ∧ φ ) ψ ) , etc., we write Logic (= α = β ) , ( φ ∧ ψ ) , etc. Truth and Satisfaction Instead of writing e.g., ((chase fido) john), we sometimes write (chase fido john). NB this is equivalent to chase(john, fido) on a relational approach.

Quantifier Syntax SPNLP: Lambda Terms, If φ ∈ Term ( BOOL ), and x ∈ Var ( IND ), then ∀ x .φ and Quantifiers, Satisfaction ∃ x .φ ∈ Term ( BOOL ). Lascarides & Klein x ∈ Var ( IND ) is called an individual variable. Outline Typed Syntactic conventions: Lambda Calculus First Order Instead of writing ((= α ) β ) , (( ∧ φ ) ψ ) , etc., we write Logic (= α = β ) , ( φ ∧ ψ ) , etc. Truth and Satisfaction Instead of writing e.g., ((chase fido) john), we sometimes write (chase fido john). NB this is equivalent to chase(john, fido) on a relational approach.