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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

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

Semantics and Pragmatics of NLP Lambda Terms, Quantifiers, Satisfaction

Alex Lascarides & Ewan Klein

School of Informatics University of Edinburgh

10 January 2008

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

1

Typed Lambda Calculus

2

First Order Logic

3

Truth and Satisfaction

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

Transitive Verbs as Functions

We looked at replacing n-ary relations with functions. How does this work with transitive verbs? Version 1: chase of type <IND, IND >→ BOOL Version 2: chase of type IND → (IND → BOOL) Advantages of Version 2 (called a curryed function): Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

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

Transitive Verbs as Functions

We looked at replacing n-ary relations with functions. How does this work with transitive verbs? Version 1: chase of type <IND, IND >→ BOOL Version 2: chase of type IND → (IND → BOOL) Advantages of Version 2 (called a curryed function): Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

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

Transitive Verbs as Functions

We looked at replacing n-ary relations with functions. How does this work with transitive verbs? Version 1: chase of type <IND, IND >→ BOOL Version 2: chase of type IND → (IND → BOOL) Advantages of Version 2 (called a curryed function): Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

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

Transitive Verbs as Functions

We looked at replacing n-ary relations with functions. How does this work with transitive verbs? Version 1: chase of type <IND, IND >→ BOOL Version 2: chase of type IND → (IND → BOOL) Advantages of Version 2 (called a curryed function): Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

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

Transitive Verbs as Functions

We looked at replacing n-ary relations with functions. How does this work with transitive verbs? Version 1: chase of type <IND, IND >→ BOOL Version 2: chase of type IND → (IND → BOOL) Advantages of Version 2 (called a curryed function): Makes the syntax more uniform. Fits better with compositional semantics (discussed later)

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

Lambda

Lambdas talk about missing information, and where it is. The λ binds a variable. The positions of a λ-bound variable in the formula mark where information is ‘missing’. Replacing these variables with values fills in the missing information. Example: λx.(man x) λ-abstract (λx.(man x) john) application (man john) β-reduction/function application.

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

Types

IND and BOOL are basic types.

If σ, τ are types, then so is (σ → τ). Brackets are

mitted if no ambiguity.

For types τ, we have variables Var(τ), constants Con(τ). Since we are doing first order logic, we will later restrict variables to Var(IND), but allow constants of any type.

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

Types

IND and BOOL are basic types.

If σ, τ are types, then so is (σ → τ). Brackets are

mitted if no ambiguity.

For types τ, we have variables Var(τ), constants Con(τ). Since we are doing first order logic, we will later restrict variables to Var(IND), but allow constants of any type.

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

Types

IND and BOOL are basic types.

If σ, τ are types, then so is (σ → τ). Brackets are

mitted if no ambiguity.

For types τ, we have variables Var(τ), constants Con(τ). Since we are doing first order logic, we will later restrict variables to Var(IND), but allow constants of any type.

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

Types

IND and BOOL are basic types.

If σ, τ are types, then so is (σ → τ). Brackets are

mitted if no ambiguity.

For types τ, we have variables Var(τ), constants Con(τ). Since we are doing first order logic, we will later restrict variables to Var(IND), but allow constants of any type.

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

Terms in Typed Lambda Calculus

We define terms Term(τ) of type τ: Var(τ) ⊆ Term(τ). Con(τ) ⊆ Term(τ). If α ∈ Term(σ → τ) and β ∈ Term(σ) then (α β) ∈ Term(τ) (function application). If x ∈ Var(σ) and α ∈ Term(ρ), then λx.α ∈ Term(τ), where τ = (σ → ρ)

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

Terms in Typed Lambda Calculus

We define terms Term(τ) of type τ: Var(τ) ⊆ Term(τ). Con(τ) ⊆ Term(τ). If α ∈ Term(σ → τ) and β ∈ Term(σ) then (α β) ∈ Term(τ) (function application). If x ∈ Var(σ) and α ∈ Term(ρ), then λx.α ∈ Term(τ), where τ = (σ → ρ)

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

Terms in Typed Lambda Calculus

We define terms Term(τ) of type τ: Var(τ) ⊆ Term(τ). Con(τ) ⊆ Term(τ). If α ∈ Term(σ → τ) and β ∈ Term(σ) then (α β) ∈ Term(τ) (function application). If x ∈ Var(σ) and α ∈ Term(ρ), then λx.α ∈ Term(τ), where τ = (σ → ρ)

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

Terms in Typed Lambda Calculus

We define terms Term(τ) of type τ: Var(τ) ⊆ Term(τ). Con(τ) ⊆ Term(τ). If α ∈ Term(σ → τ) and β ∈ Term(σ) then (α β) ∈ Term(τ) (function application). If x ∈ Var(σ) and α ∈ Term(ρ), then λx.α ∈ Term(τ), where τ = (σ → ρ)

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

Extending to a First Order Language

1 Variables i.e., Var(IND): x, y, z, . . . , x0, x1, x2, . . . 2 Boolean connectives:

¬

BOOL → BOOL

(negation) ∧

BOOL → (BOOL → BOOL)

(and) ∨

BOOL → (BOOL → BOOL)

(or) →

BOOL → (BOOL → BOOL)

(if. . . then)

3 Quantifiers:

∀ (all) ∃ (some)

4 Equality:

= τ → (τ → BOOL)

5 Punctuation: brackets and period

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

Quantifier Syntax

If φ ∈ Term(BOOL), and x ∈ Var(IND), then ∀x.φ and ∃x.φ ∈ Term(BOOL). x ∈ Var(IND) is called an individual variable. Syntactic conventions: Instead of writing ((= α)β), ((∧φ)ψ), etc., we write (= α = β), (φ ∧ ψ), etc. 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.

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

Quantifier Syntax

If φ ∈ Term(BOOL), and x ∈ Var(IND), then ∀x.φ and ∃x.φ ∈ Term(BOOL). x ∈ Var(IND) is called an individual variable. Syntactic conventions: Instead of writing ((= α)β), ((∧φ)ψ), etc., we write (= α = β), (φ ∧ ψ), etc. 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.

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

Quantifier Syntax

If φ ∈ Term(BOOL), and x ∈ Var(IND), then ∀x.φ and ∃x.φ ∈ Term(BOOL). x ∈ Var(IND) is called an individual variable. Syntactic conventions: Instead of writing ((= α)β), ((∧φ)ψ), etc., we write (= α = β), (φ ∧ ψ), etc. 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.

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

Quantifier Syntax

If φ ∈ Term(BOOL), and x ∈ Var(IND), then ∀x.φ and ∃x.φ ∈ Term(BOOL). x ∈ Var(IND) is called an individual variable. Syntactic conventions: Instead of writing ((= α)β), ((∧φ)ψ), etc., we write (= α = β), (φ ∧ ψ), etc. 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.

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

Some Examples

1 ∃x.(love x kim)

Kim loves someone

2 (¬∃x.(love x kim))

Kim doesn’t love anyone

3 ∀x.((robber x) → ∃y.((customer y) ∧ (love y x)))

All robbers love a (perhaps different) customer

4 ∃y.((customer y) ∧ ∀x.((robber x) → (love y x)))

All robbers love the same customer

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

Some Examples

1 ∃x.(love x kim)

Kim loves someone

2 (¬∃x.(love x kim))

Kim doesn’t love anyone

3 ∀x.((robber x) → ∃y.((customer y) ∧ (love y x)))

All robbers love a (perhaps different) customer

4 ∃y.((customer y) ∧ ∀x.((robber x) → (love y x)))

All robbers love the same customer

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

Some Examples

1 ∃x.(love x kim)

Kim loves someone

2 (¬∃x.(love x kim))

Kim doesn’t love anyone

3 ∀x.((robber x) → ∃y.((customer y) ∧ (love y x)))

All robbers love a (perhaps different) customer

4 ∃y.((customer y) ∧ ∀x.((robber x) → (love y x)))

All robbers love the same customer

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

Some Examples

1 ∃x.(love x kim)

Kim loves someone

2 (¬∃x.(love x kim))

Kim doesn’t love anyone

3 ∀x.((robber x) → ∃y.((customer y) ∧ (love y x)))

All robbers love a (perhaps different) customer

4 ∃y.((customer y) ∧ ∀x.((robber x) → (love y x)))

All robbers love the same customer

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Free and Bound Variables

((customer x) ∨ ∀x.((robber x) → ∃y.(person y))) First occurrence of x is free; Second occurrence of x is bound; Occurrence of y is bound. Free variable ≈ pronouns.

She loves Fido

Context needed to interpret she; Something in addition to models so far needed to interpret free variables. A WFF (Term(BOOL)) with no free variables is a (closed)

sentence. FOL sentences ⊂ WFFs.

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

Interpreting FOL Sentences

Task: Compute whether a sentence is true or false with respect to a model.

Is the sentence an accurate description of the situation?

Strategy: Compositionality! Use recursion, but: Subformula of ∀x.(robber x) is (robber x) and this is not a sentence! So. . .

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

Satisfaction: [ [φ] ]M,g = 1

Model M and variable assignment g satisfy the WFF φ. g defined for all individual variables, i.e., x ∈ Var(IND); g(x) ∈ D. If α is an atomic term (∈ Con(τ) ∪ Var(τ)), then ig

V(α) =

g(α) if α is a variable V(α) if α is a constant [ [∃x.φ] ]M,g = 1 iff [ [φ] ]M,g[u/x] = 1 for some u ∈ D g[u/x](x) = u ‘Try to find some value u for x that makes φ true’

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

Value of a term in a model

Where M = D, V: [ [α] ]M,g = ig

V(α)

if α is atomic [ [(α β)] ]M,g = [ [α] ]M,g([ [β] ]M,g) [ [(λx.α)] ]M,g = that function h such that for any u ∈ D, h(u) = [ [α] ]M,g[u/x] [ [α1 = α2] ]M,g = 1 iff [ [α1] ]M,g = [ [α2] ]M,g [ [¬φ] ]M,g = 1 iff [ [φ] ]M,g = 0 [ [φ ∧ ψ] ]M,g = 1 iff [ [φ] ]M,g = 1 and [ [ψ] ]M,g = 1 [ [φ ∨ ψ] ]M,g iff [ [φ] ]M,g = 1 or [ [ψ] ]M,g = 1 [ [φ → ψ] ]M,g = 1 iff [ [φ] ]M,g = 0 or [ [ψ] ]M,g = 1 [ [∃x.φ] ]M,g = 1 iff [ [φ] ]M,g[u/x] = 1 for some u ∈ D [ [∀x.φ] ]M,g = 1 iff [ [φ] ]M,g[u/x] = 1 for every u ∈ D.

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

Truth (in terms of Satisfaction)

If φ ∈ Term(BOOL), we often write M, g | = φ instead of [ [φ] ]M,g′ = 1. It doesn’t matter which g you use for sentences, so: Truth: A sentence φ is true in a model M (written M | = φ) iff for any g, M, g | = φ Validity: A sentence φ is valid (written | = φ) iff for any M, M | = φ Entailment: φ1, . . . , φn | = ψ iff if M, g | = φi for all i, 1 ≤ i ≤ n, then M, g | = ψ