first-order logic (and beyond) johan bos
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First-Order Logic (and beyond) Johan Bos. Overview of this lecture. Introduction to first-order logic Discourse Representation Theory Using the Lambda-Calculus. Logical languages. propositional logic modal logic description logic first-order logic ( predicate logic) second-order logic - PowerPoint PPT PresentationTRANSCRIPT
First-Order Logic
(and beyond)
Johan Bos
Overview of this lecture
• Introduction to first-order logic
• Discourse Representation Theory
• Using the Lambda-Calculus
Logical languages
• propositional logic
• modal logic
• description logic
• first-order logic (predicate logic)
• second-order logic
• higher-order logic
expressive power
This lecture
• In this lecture we will try to map English to First-Order Logic
• First-order logic extends propositional logic with variables and quantifiers
• As we will see it is capable for modelling sub-sentential semantics
First-order logic
• First-order logic is a language
• So we will look at its ingredients
• We will define the syntax, or in other words, the “grammar”
• We will look at the semantics only from an informal point of view
Ingredients of first-order logic
• Terms (variables or contants)– Variables: x, y, z, …– Constants: m’, j’, …
• Predicate Symbols– One-place predicate symbols: walk, smoke, …– Two-place predicate symbols: see, love, …
• Connectives: , ,, , • Punctuation: brackets ( ) and the comma ,• The quantifiers
– Universal quantifier: – Existential quantifier:
Syntax of first-order logic
• If P is a one-place relation symbol, and t a term, then P(t) is a first-order formula
• If R is a two-place relation symbol, and t1 and t2 are terms, then R(t1,t2) is a first-order formula
• If is a first-order formula, then so is • If and are first-order formulas, then so are
(), (), () and ()• If is a first-order formula, and x a variable,
then x and x are first-order formulas• Nothing else is a first-order formula
Examples of first-order formulas
• Mia walks. walk(mia’)
• A dog barks. x(dog(x) bark(x))
• Vincent likes every dog. x(dog(x) like(vincent’,x))
Semantics of the quantifiers
x true if and only if there is an x such that is true
x true if and only if for all x it is the case that is true
Truth and Models
• Truth in first-order logic is often defined with the help of models
• A model M is usually taken to consist of two parts (M = <D,F>): (1) a domain of entities (D)(2) an interpretation function (F) for all non-logical symbols
• The truth-definition with models was introduced by the famous logician Alfred Tarski
Example model
• M = <D,F>• D = {d1,d2,d3}• F(mia’) = d1
F(vincent’) = d2F(person) = {d1,d2}F(dog) = {d3}F(love) = {(d1,d2),(d2,d2),(d2,d1),(d2,d3)}F(hate) = {(d1,d3)}
Semantics of the quantifiers
x true in M if and only if we can map x to at least one member of D such that is true in M
x true if and only if for all members of D, if we map x it, it is the case that is true in M
Free variables
• The quantifiers bind variables• For instance, x binds all
occurrences of x in the formula • Variables that are not bound are
called free• For instance, the following two formulas
contain free variables:– walk(x)– smoke(y) y person(y)
Closed formulas
• Formulas that have no free variables are called closed
• Usually we’re only interested in closed formulas --- translating a natural language sentence to first-order logic should produce a closed formula
• Free variables can be thought of as “pronouns”.
What’s wrong with these translations?
• A dog barks. (x dog(x) bark(x))
• A dog barks. x(dog(x) bark(x))
• Every dog barks. x(dog(x) bark(x))
Lambdas and Higher-order Logic
• Fine, we have seen how we can represent English (or Italian) sentences into logic, but what about – noun phrases, – verb phrases, – nouns, – determiners, – adjectives, – prepositions, and so on?
Montague Grammar
• Richard Montague usedhigher order logic to translatesub-sentence fragments intologic
• Basically we add to two new constructs to first-order logic:– the lambda operator λ – function application ()
Examples with lambdas
• The lambda binds variables and can be seen as a “place-holder” for missing information
• Examples:
Mia mia’
man λz.man(z)
love λx. λy. love(y,x)
every λp. λq. x(p(x) q(x))
Example derivation
loves Mia λx. λy. love(y,x) (mia’) = λy. love(y,mia’)
Every man λp. λq. x(p(x) q(x))(λz.man(z)) = λq. x(λz.man(z)(x) q(x)) = λq. x(man(x) q(x))
Every man loves Mia λq. x(man(x) q(x)) (λy. love(y,mia’)) = x(man(x) λy. love(y,mia’)(x)) = x(man(x) love(x,mia’))
Discourse Representation Theory
• Nice so far, but what about translating pronouns that have antecedents across sentences?– Mia dances. She is happy.– A man smokes. He likes Mia.
• Hans Kamp introduced DRT(Discourse Representation Theory) to deal with a lotof anaphoric phenomena.
Problematic cases for FOL
• A woman dances. She is happy. x(woman(x) dance(x)) happy(x)
• Every farmer who owns a donkey beats it.x((farmer(x) y(donkey(y) own(x,y))) beat(x,y))
Problematic cases for FOL
• A woman dances. She is happy. x(woman(x) dance(x)) happy(x) x(woman(x) dance(x) happy(x))
• Every farmer who owns a donkey beats it.x((farmer(x) y(donkey(y) own(x,y))) beat(x,y))xy((farmer(x) donkey(y) own(x,y)) beat(x,y))
Discourse Representation Theory
• DRT is a theory of natural language semantics using DRSs to represent texts (discourse)
• A DRS encapsulates both content and context– Content: the meaning of the text so far– Context: information to interpret anaphoric
expressions in subsequent sentences
DRT examples
• Discourse Representation Structures (DRS)– Discourse referents (first-order variables)– Structure plays role in pronoun resolution
A dog barked.
x
dog(x)
bark(x)
DRT examples
• Discourse Representation Structures (DRS)– Discourse referents (first-order variables)– Structure plays role in pronoun resolution
A dog barked.
x
dog(x)
bark(x)
Every dog barked.
x
dog(x) bark(x)
Accessibility (1)
• Discourse referents are accessible if they are in the same DRS
A dog barked. It was happy.
x y
dog(x)
bark(x)
happy(y)y = ???
Accessibility (1)
• Discourse referents are accessible if they are in the same DRS
A dog barked. It was happy.
x y
dog(x)
bark(x)
happy(y)y = x
Discourse referent x is accessible
Accessibility (2)
• Discourse referents are not accessible if they are part of a nested DRS
Every dog barked. ?It was happy.
y
happy(y)
y = ??
x
dog(x) bark(x)
Discourse referent x is not accessible
Donkey Sentences
• DRT solves the donkey sentence problem
Every farmer that owns a donkey beats it.
x y
farmer(x)
donkey(y)
own(x,y)
beat(x,y)
Further Reading
• Gamut, Volume 2(Montague Grammar)
• Kamp & Reyle (Discourse Representation Theory)