Introduction to artificial intelligence
This presentation is the property of its rightful owner.
Sponsored Links
1 / 32

INTRODUCTION TO ARTIFICIAL INTELLIGENCE PowerPoint PPT Presentation


  • 79 Views
  • Uploaded on
  • Presentation posted in: General

INTRODUCTION TO ARTIFICIAL INTELLIGENCE. Massimo Poesio LECTURE 3: Logic: predicate calculus, psychological evidence. PREDICATE CALCULUS. The propositional calculus is only concerned with connectives – statements not containing connectives are left unanalyzed Massimo is happy: p

Download Presentation

INTRODUCTION TO ARTIFICIAL INTELLIGENCE

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Introduction to artificial intelligence

INTRODUCTION TO ARTIFICIAL INTELLIGENCE

Massimo PoesioLECTURE 3: Logic: predicate calculus, psychological evidence


Predicate calculus

PREDICATE CALCULUS

  • The propositional calculus is only concerned with connectives – statements not containing connectives are left unanalyzed

    • Massimo is happy: p

  • In predicate calculus, or predicate logic, atomic statements are decomposed into TERMS and PREDICATES

    • Massimo is happy: HAPPY(m)

    • Students like AI: LIKE(students,AI)

  • In this way it is possible to state general properties about predicates: for instance, every professor at the University of Trento is happy, etc.


First order logic

FIRST-ORDER LOGIC

  • Predicate calculus becomes FIRST ORDER LOGIC when we add QUANTIFIERS – logical symbols that make it possible to make universal and existential statements (i.e., to translate statements A, E, I and O of syllogisms)


The existential quantifier

THE EXISTENTIAL QUANTIFIER

  • Used to traduce statements like

    • Some birds are swallows

  • Notation:

    • ∃(backwards E, for Exist – Peano, 1890)

    • ‘Some birds are swallows’ 

    • There exists an x, such that x is a bird, and x is a swallow

    • (∃ x) (BIRD(x) & SWALLOW(x))


The universal quantifier

THE UNIVERSAL QUANTIFIER

  • To represent

    • All men are mortal

    • But also: Swallows are birds

  • Notation:

    • ∀for inverted A (alle)

  • Conversion of universal statements requires conditional:

    • For every x, is x is a man, then x is mortal

    • (∀ x) (MAN(x) → MORTAL(x))


The syntax of fol vocabulary

THE SYNTAX OF FOL: VOCABULARY

  • TERMS

    • Constants

    • Variables

  • PREDICATES: 1 argument ( HAPPY), two arguments (LIKES), etc

  • CONNECTIVES (from the propositional calculus): ~, &, ∨, →, ↔

  • QUANTIFIERS: ∀ ∃


The syntax of fol phrases

THE SYNTAX OF FOL: PHRASES

  • If P is an n-ary predicate and t1, … tn are terms, then P(t1,…,tn) is a formula.

  • If φ and ϕ are formulas, then ~φ, φ & ϕ , φ ∨ϕ , φ →ϕ and φ ↔ ϕ are formulas

  • If ϕ is a formula and x is a variable, then (∀ x) ϕ and (∃ x) ϕ are formulas.


Scope and binding

SCOPE AND BINDING

  • Let x be a variable and ϕ a formula, and let (∀ x) ϕ and (∃ x) ϕ be formulas. then ϕ is the SCOPE of x in these formulas.

  • An occurrence of x is BOUND if it occurs in the scope of (∀ x) or (∃ x)

  • Examples (PMW p. 141)


The semantics of fol

THE SEMANTICS OF FOL

  • As in the case of propositional calculus, statements (formulas) can be either true or false

  • But the other phrases of the language have set-theoretic meanings:

    • Terms denote set elements

    • Unary predicates denote sets

    • N-ary predicates denote n-ary relations

    • Quantifiers denote relations between sets


Set theory recap

SET THEORY RECAP

Fred

HAPPY PEOPLE

John

Matilda

Massimo

Lucy

HAPPY(m) = T

HAPPY(f) = F


Set theory recap relations

SET THEORY RECAP: RELATIONS

PEOPLE

SUBJECTS

John

AI

Matilda

Logic

Fred

Maths

Massimo

LIKES(j,AI) = T

LIKES(m,Maths) = F


Set theory recap quantifiers

SET THEORY RECAP: QUANTIFIERS

AIRPLANES

BIRDS

SWALLOWS

Tweety

Lou

Airplane1

Roger

Loreto

FLYING THINGS

Swallows are birds

Birds fly


The semantics of fol1

THE SEMANTICS OF FOL

  • If t is a term and P a unary predicate, then [P(t)] = TRUE iff [t] ∈[P]

  • If φ and ϕ are formulas, then

    • [~φ] = TRUE iff [φ] = FALSE

    • [φ & ϕ] = TRUE iff [φ] = TRUE and [ϕ] = TRUE

  • [(∀ x) ϕ] = TRUE iff for every value a for x in model M, [ϕ(a/x)] = TRUE

  • [(∃ x) ϕ] = TRUE iff there is at least one object a in model M such that [ϕ(a/x)] = TRUE


Some tautologies of fol

SOME TAUTOLOGIES OF FOL

  • Laws of Quantifier Distribution:

    • (∀x) (φ(x) & ϕ(x)) ≡ (∀x) φ(x) & (∀x) ϕ(x)

    • “Every object is formed of elementary particles and has a spin” iff “Every object is formed of elementary particles” and “Every object has a spin”

  • Law of Quantifier Negation:

    • ~ (∀x) (φ(x)) ≡ (∃y) (~ φ(y))

    • “It is not the case that every object is made of cheese” iff “there is an object which is not made of cheese”


From syllogisms to fol

FROM SYLLOGISMS TO FOL

  • Four types of syllogism:

    • Universal affirmative: All Ps are Qs

    • Universal negative: All Ps are not Qs (No P is a Q)

    • Particular affirmative: Some P is a Q

    • Particular negative: Some P is not a Q


The square of opposition

THE SQUARE OF OPPOSITION


The square of opposition1

THE SQUARE OF OPPOSITION


From syllogisms to fol1

FROM SYLLOGISMS TO FOL

  • Syllogism in FOL:

    • Universal affirmative: (∀ x) (P(x) → Q(x))

    • Universal negative: (∀y) (P(y) → ~ Q(y))

    • Particular affirmative: (∃z) (P(z) & Q(z))

    • Particular negative: (∃ w) (P(w) & ~ Q(w))


From syllogysm to fol

FROM SYLLOGYSM TO FOL

An example of BARBARA:

A Birds fly

A Swallows are birds

A Swallows fly


Barbara in predicate calculus

BARBARA IN PREDICATE CALCULUS

(∀x) (BIRD(x) → FLY(x))

(∀y) ( SWALLOW(y) → BIRD(y))

(∀z) ( SWALLOW(z) → FLY(z))


Set theoretic demonstrations of validity of syllogisms

SET THEORETIC DEMONSTRATIONS OF VALIDITY OF SYLLOGISMS

R

Q

Q

P

A: All Ps are Qs

R

A: All Qs are Rs

P

A: All Ps are Rs

(A more general method exists)


Representing knowledge in logic 2

REPRESENTING KNOWLEDGE IN LOGIC, 2

  • Modern logics make it possibile to represent every type of knowledge

  • Different types of knowledge have different EXPRESSIVE POWER


Representing knowledge in logic 21

REPRESENTING KNOWLEDGE IN LOGIC, 2

  • “Tutte le biciclette hanno due ruote”

  • Propositional calculus: p

  • Predicate logic + quantifiers:

    • (∀ x) (BICYCLE(x) → HAS_TWO_WHEELS(x))

    • Can be used to represent DARII

  • Explicit representation of the number 2:

    • (∀ x) (BICYCLE(x) → HAS_WHEELS(x,2))

  • Set of wheels:


Deduction in fol

DEDUCTION IN FOL

  • The system of inference rules for FOL includes all the inference rules from the propositional calculus, together with four new rules for quantifier introduction and elimination

  • The tableaus system has also been extended


Natural deduction for fol 1

NATURAL DEDUCTION FOR FOL, 1

(∀y) P(y)

UNIVERSAL INSTANTIATION

∴P(c) (for any constant c)

P(c) (for any constant c)

UNIVERSAL GENERALIZATION

∴ (∀y) P(y)


Ui and ug examples

UI AND UG EXAMPLES

(∀y) MADE-OF-ATOMS(y)

UNIVERSAL INSTANTIATION

∴ MADE-OF-ATOMS(c) (for any c)


Natural deduction for fol 2

NATURAL DEDUCTION FOR FOL, 2

(∃y) P(y)

EXISTENTIAL INSTANTIATION

∴ P(k) (for a new k)

P(c) (for a constant c)

EXISTENTIAL GENERALIZATION

∴ (∃ y) P(y)


Beyond first order logic

BEYOND FIRST ORDER LOGIC

  • Artificial Intelligence research moved beyond first order logic in several directions:

    • Beyond using logic as a formalization of valid inference only, developing logics for non-valid (or NONMONOTONIC / UNCERTAIN) reasoning

    • Developing simpler logics in which inference can be done more efficiently (description logics, discussed in later lectures)


Psychological evidence on reasoning

PSYCHOLOGICAL EVIDENCE ON REASONING

  • First order logic and the propositional calculus are good formalizations of ‘sound’ reasoning, and are therefore the basis for work on proving mathematical truths

  • But are they a good formalization of the way people reason?

  • Evidence suggests that this is not the case

    • The WASON SELECTION TASK perhaps the best known example of this evidence


The wason selection task

THE WASON SELECTION TASK

  • Subjects are asked to verify the truth of a statement (typically, a conditional statement) by turning over cards


Wason test example

WASON TEST: EXAMPLE

If A CARD SHOWS AN EVEN NUMBER ON ONE SIDE, then THE OPPOSITE FACE IS RED

Answer: the second and fourth card


Readings

READINGS

  • Basics:

    • B. Partee, A. ter Meulen, R. Wall, Mathematical Methods in Linguistics, Springer, ch. 5, 6, 7

    • (in Italian): D. Palladino, Corso di Logica, Carocci

  • To know more:

    • History of logic: P. Odifreddi, Le menzogne di Ulisse, Tea, ch. 1-7

    • Inference: P. Blackburn, J. Bos, Representation and Inference for Natural Language, CSLI

    • K. Stenning and M. van Lambalgen, Human Reasoning and Cognitive Science, MIT Press

  • Logic on the Web:

    • http://www.thelogiccourse.com/

    • Do the Wason selection task: http://coglab.wadsworth.com/experiments/WasonSelection.shtml


  • Login