INTRODUCTION TO ARTIFICIAL INTELLIGENCE

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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

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### INTRODUCTION TO ARTIFICIAL INTELLIGENCE

Massimo PoesioLECTURE 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
• 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
• 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
• 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
• 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
• TERMS
• Constants
• Variables
• PREDICATES: 1 argument ( HAPPY), two arguments (LIKES), etc
• CONNECTIVES (from the propositional calculus): ~, &, ∨, →, ↔
• QUANTIFIERS: ∀ ∃
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
• 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
• 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

Fred

HAPPY PEOPLE

John

Matilda

Massimo

Lucy

HAPPY(m) = T

HAPPY(f) = F

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

AIRPLANES

BIRDS

SWALLOWS

Tweety

Lou

Airplane1

Roger

Loreto

FLYING THINGS

Swallows are birds

Birds fly

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
• 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
• 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
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

An example of BARBARA:

A Birds fly

A Swallows are birds

A Swallows fly

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

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
• Modern logics make it possibile to represent every type of knowledge
• Different types of knowledge have different EXPRESSIVE POWER
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
• 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

(∀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

UNIVERSAL INSTANTIATION

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