74.406 natural language processing - formal logic - propositional calculus/logic (proplog)...

74.406 Natural Language Processing- Formal Logic -

Propositional Calculus/Logic (PropLog)

First-Order Predicate Logic/Calculus (FOL or FOPL)• Formal Language (Syntax of formulae; wff)• Inference System • Semantics through Interpretation Function

Formal Language

A Formal Language is specified as L = (NT, T, P, S)

NT Set of Non-Terminal Symbols

T Set of Terminal Symbols

P Set of Production or Grammar Rules

S Start Symbol (top-level node in syntax tree / parse tree)

A formal language specifies the syntactically correct or well-formed expressions of a language.

Propositional Calculus

Propositional Logic (PL)

Propositional Logic:• symbols for facts, statements (propositions)• logical connectives AND, OR, NOT, ,

"Rules" – condition, consequence; implications

Example: “Dog Mood” tongue_out thirstygrowl OR bark angryears_back AND tail_in anxious

Propositional Logic - Syntax

Propositional Logic (Symbols, Terminals): propositional symbols P, p, Q, q, r, ... logical connectives , , , , brackets ( , )

Inductive Definition of well-formed formulae (wff):1.Propositional symbols P, Q, ... are wffs.2.If P is a wff, then also (P).3.If P and Q are formulae then also

(P Q); (P Q); (P Q); (P Q)

Propositional Logic - Semantics• assign truth values to atomic formulae

(propositions)• determine truth values for complex formulae

(composed from basic propositions using connectives)

p q p q p q p p q p q

F F F F T T T

F T F T T T T

T F F T F F F

T T T T F T T

Truth table

Propositional Logic – Example

Example: “Dog Moods” tongue_out thirstygrowl OR bark angryears_back AND tail_in anxious

Exercise: Set-up a truth-table for “Dog Moods” Write in the left-side columns the observable propositional symbols (growl, bark,...) and in the right columns the derived propositions (anxious,...).

Example, Exercise – Truth table

Example: “Dog Moods” tongue_out thirstygrowl OR bark angryears_back AND tail_in anxious

tong thirst growl bark growl bark

angry ears tail ears tail

anx

F F F F F F F F F F

F F T F T T T F F F

T T F T T T F T F F

T T T T T T T T T T

Example – Truth table for

Example:

If I win the lottery, every CS420 student gets $1.000.I win the lottery every CS420 student gets $1.000

p q p q p qF F T T

F T T T

T F F F

T T T T

First-Order Predicate Logic

Syntax and Semantics

Syntax of FOPL - Example

Predicate Symbols P, Q, married, ...

Function Symbols f, g, father-of, ...

Variables x, y, z, ...

Constants Sally, John, block-1, c, ...

Connectives , , , ,

Quantifiers ,

Terms x, Sally, father-of (Sally)

Sentences married (Sally, John), P (c)

(atomic, complex) x: married (Sally, x),

x y: P (x, y) Q (x) R (y)

FOPL as Formal Language: Symbols

NT Non-TerminalsFormula, atomic- Formula, complex- Formula, Term, Connective, Quantifier, Predicate, Function, Variable, Constant

T TerminalsPredicate Symbols P, Q, married, ..., T, FFunction Symbols f, g, father-of, ...Variables x, y, z, ...Constants Sally, block-1, c

(Binary) Connectives , , , Negation Symbol

(Unary Connective)Quantifiers , Equality Symbol =Other Symbols ( , ) , :

Notes on FOPL SyntaxThe term well-formed formula (wff) is often used.

equivalent to the term ‘sentence’.

wffs are sentences if all their variables are bound by quantifiers.

bound variable x: married (Sally, x)

open formula: a variable in the formula is not bound, it is free x: married (Sally, x) happy (y)

closed formula: all variables in the formula are bound xy: married (x, y) happy (x) happy

(y)

scope of a quantifier: all occurrences of quantified variables in formulae until over-ruled by new quantifier

Equivalence of Formulae

x: (x) x: (x)

x: (x) x: (x)

x: (x) y: (y)

Predicate Logic - Semantics An Interpretation function determines the semantics of Predicate Logic formulae.

Based on a “Domain” or “Universe” which models “the world”, consists of a set of Individuals (Objects, Constants) with Relations (Roles, Relations, Predicates) among them and Functions (Features, Attributes, Functions).

An Interpretation assigns values to terms and formulae:

Terms constants, variables, function-expressions

Formulae predicate expressions, formulae connected logical connectives, quantified formulae

FOPL: Semantics 1

Define the Semantics of FOPL: 1. Interpretation – Mapping of symbols of the formal

language (predicates, functions, variables, constants) onto the modeled domain (formal: Domain, relational Structure, or Universe)

2. Valuation - Determine the bindings of variables3. Constructive Semantics – Determine the semantics of

complex expressions inductively based on the definition of the semantics of basic expressions

Note: Simpler definitions of semantics exist without explicit Valuation function or explicit notation of the interpretation of predicates, functions, constants, and variables in the domain.

FOPL: Semantics 2

Interpretation

constants I(c) D (0-ary function)

predicates I(P) Dn for P n-ary predicate

functions I(f) Dn →D for f n-ary function

variables I(x) D (see valuation)

-------------------------------------------------------------------

determine constructively based on syntax and above Interpretation:

term I(t) D

sentence I(α) {T,F}

FOPL: Semantics 3Interpretation

term I(f(t1,...,tn)) = I(f)(I(t1),...,I(tn))D

atomic sentence I(P(t1,...,tn)) = T if (I(t1),...,I(tn))I(P)

complex sentence I(α) = T if I(α)=F

| | I(α β) = T if I(α)=T and I(β)=T

I(α β) = T if I(α)=T or I(β)=T

I(α β) = T if I(α)=F or I(β)=T

| I(x: α(x)) = T if I(α(x))=T for at least one

dI(x)

I(x: α(x)) = T if I(α(x))=T for all dI(x)

FOPL: Semantics 3b

Formulae with multiple / nested quantifiers:

Evaluate / Interpret formula from left to right / from outside to inside.

I(x: α(x)) = T if I(α(x))=T for at least one dI(x)

I(x: α(x)) = T if I(α(x))=T for all dI(x)

Easier: Substitute x with constant cC, and later use I(c) instead of I(x).

Task: Interpret the following formulae:

x y: P(x,y)

y x: P(x,y)

What is the difference between them?

FOPL: Semantics 4

Interpretation of open formulae and Satisfiability

Regard complex sentence α with (free) variable x:

α(x)

choose valuation function and determine satisfiability:

valuation function V: V(x) = d D

α(x) satisfiable if there is a valuation V with wrt I,V V(x)=d such that

I(α(d))=T

α(x) true / valid if for every valuation V with wrt I V(x)=d, dD

I(α(d))=T

FOPL: Semantics 5

Model:

Given a set of formulae and a structure D with an Interpretation I, and a valuation V, then D is a model of iff

I() = T for all

FOPL: Semantics 6Semantic-based consequence:

Given a set of formulae and a formula α, and an Interpretation I into a Structure D, we say that α is a logical consequence of iff

if I() = T for all

then I(α) = T

Notation: |= α

FOPL: Inference SystemInference in FOPL:

Derive new formulae by syntactic manipulation of existing formulae (through applying inference rules): • given a set of formulae • apply inference rule (based on )• new formula α is derived; α is a Theorem of . • add new formula to . • The set of valid formulae is now α.

Notation: |– α

α is inferred or derived from .

FOPL: AxiomsThe start-set for inferences in FOPL are the axioms of FOPL.

Axioms describe the general features of a logic, and are always assumed to be valid formulae in this logic.

FOPL Axioms

A1 A2

A3

A4 ( ) (( ) ( ))

A5 x: (x) (y)

A6 (x) y: (y) based on Frost (1986)

Inference Rules – Modus Ponens

Modus Ponens: ,

States that can be concluded provided we know that the formulae and are true in our knowledge base.

Inference Rule UGUniversal Generalization

Universal Generalization:

(x) x: (x)

where (x) is a formula containing the free variable x.

Inference Rules - Universal Quantifier Introduction

Introducing the Universal Quantifier:

(x) x: (x)

(x) is a formula containing the free variable x, which is bound in the conclusion by the universal quantifier.

Inference Rules - Existential Quantifier Introduction

Introducing the Existential Quantifier:

(x) x: (x)

(x) is a formula containing the free variable x, which is bound in the conclusion by the existential quantifier.

Inference Rules - UI

Universal Instantiation:

x: (x) (c)

where (x) is any formula containing the quantified variable x, and (c) is the same as formula (x) but every occurrence of the variable x is substituted with the arbitrary constant c.

Inference Rules - EG

Existential Generalization:

(c) x: (x)

where (c) is a formula containing the arbitrary constant c but not an unbound occurrence of x, and (x) is the same formula as (c) but with every occurrence of the constant c replaced by a variable x. (If x occurs unbound in , use other variable-name.)

Replacement Rules

( ) ( )

IR Replacement Rules

FOPL: Semantics and Inference

In First-Order Predicate Logic, there is a correspondence (regarding the truth status) between formulae derived through logical Inference and their semantic Interpretation.

In other words: Any formula derived by inference* is true if and only if it is true in the semantic interpretation.

Notation:

|– α iff |= α

* in a sound and complete inference system

Inference Systems - Soundness and Completeness

SoundnessAn Inference System is sound iff

if |– α then |= α

Every formula which is derived by formal inference, is semantically true.CompletenessAn Inference System is complete iff

if |= α then |– α

Every formula which is semantically true can be derived by formal inference.

Semantics - Example Predicate Logic Languageconstants Bill-1, John-3, Sally-1, Mary-1, Mary-2predicates happy-together, hate-each-other

Structure Dobjects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Maryrelations: Married, Divorced(Uncle-Bill, Aunt-Sally) Married, (Uncle-John, Mary) Married(or: {(Uncle-Bill, Aunt-Sally), (Uncle-John, Mary)}=Married (Uncle-John, The-woman-I-don't-like) Divorced

InterpretationI(Bill-1)=Uncle-Bill, I(John-3)=Uncle-John, I(Sally-1)=Aunt-Sally, I(Mary-1)=The-woman-I-don't-like, I(Mary-2)=MaryI(happy-together)=Married, I(hate-each-other)=Divorced

True or false? hate-each-other (Bill-1, John-3) happy-together(Bill-1, Sally-1)hate-each-other(John-3, Mary-1)happy-together(John-3, Mary-2)

Semantics and Inference -ExampleStructure Dobjects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Maryrelations: Married, Divorced(Uncle-John, The-woman-I-don't-like) Divorced (Uncle-Bill, Aunt-Sally) Married, (Uncle-John, Mary) Married(or: {(Uncle-Bill, Aunt-Sally), (Uncle-John, Mary)} = Married)

Interpretation II(Bill-1) = Uncle-Bill, I(John-3) = Uncle-John, I(Sally-1) = Aunt-Sally, I(Mary-1) = The-woman-I-don't-like, I(Mary-2) = MaryI(happy-together) = Married, I(hate-each-other) = Divorced

True or false? hate-each-other (Bill-1, John-3) hate-each-other (John-3, Mary-1)happy-together (Bill-1, Sally-1) happy-together (John-3, Mary-2)x: happy-together(Uncle-Bill, x)) x,y,z: happy-together(x,y) hate-each-other (x,z)

What if you want to add a formula? x,y: happy-together(x,y) happy-together(y,x)

Additional References

• Frost, Richard: Introduction to Knowledge Base Systems. Collins Professional and Technical Books, William Collins Sons & Co. Ltd, London, 1986.

• Nilsson, Nils J.: Artificial Intelligence - A new synthesis. Morgan Kaufmann Publishers, San Francisco, CA, 1998.

74.406 natural language processing - formal logic - propositional calculus/logic (proplog)...

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