Non-Monotonic Reasoning -...

Université Paris 5 – René DescartesLIPADE

Non-Monotonic ReasoningIntroduction

Jean-Guy Mailly ([email protected])

2017

1

Administration

I Jean-Guy Mailly : [email protected]– Bureau 814 I

I 30h de cours/TD : lundi, 12h15–15h00I Modalités de contrôle de connaissances :

I Un contrôle (' mi-semestre)I Un examen (fin de semestre)I Note finale : max(EX , EX+CC

2 )

I Moodle : Cours Raisonnement Non Monotonehttps:

//moodle.parisdescartes.fr/course/view.php?id=6631

J.-G. Mailly | Intro to NMR

https://moodle.parisdescartes.fr/course/view.php?id=6631

https://moodle.parisdescartes.fr/course/view.php?id=6631

2

Aims

Non-Monotonic ReasoningReasoning with information which is: contradictory, uncertain,imprecise, revisable, with exceptions. . .

Common sense reasoning

Hot research topicI Mainly since the 80sI A lot of publications in major AI conferences (IJCAI, ECAI, AAAI,

KR,. . . )I Dedicated workshop on even years: International Workshop on

Non-Monotonic Reasoning


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Plan

I Introduction to NMR - Background KnowledgeI Non-monotonic inference relations

I Maximal consistent subsetsI Closed World AssumptionI Default logic

I Introduction to Belief change (BC)I Belief RevisionI Links between inference and BC


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Outline

Introduction to NMR

Background on Propositional Logic

Short Introduction to First Order Logic


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Introduction to NMRThe usual example

Does Tweety fly?I Tweety is a bird

(b)

I Birds fly

(b → f )

I Tweety is a penguin

(p)

I Penguins don’t fly

(p → ¬f )

Propositional logicK = {b,b → f ,p,p → ¬f}

K ` fK ` ¬f

K is inconsistent: we can deduce anything from K (e.g. K ` ¬b)


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Does Tweety fly?I Tweety is a bird (b)I Birds fly (b → f )I Tweety is a penguin (p)I Penguins don’t fly (p → ¬f )

Propositional logicK = {b,b → f ,p,p → ¬f}

K ` fK ` ¬f



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Propositional logicK = {b,b → f ,p,p → ¬f}K ` f

K ` ¬f



5



Propositional logicK = {b,b → f ,p,p → ¬f}K ` fK ` ¬f



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Monotonicity

IntuitivelyA formalism is monotonic if we can’t lose consequences when weadd new information

Formally∀K a set of formulas and ∀α, β formulas, if

K ` α

thenK ∪ {β} ` α

I This property is satisfied by classical logicI Not suitable for common sense reasoning


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Solutions

I Define new formalisms which don’t satisfy monotonicity

I Non-monotonic inference relations

I Not « just » add the new piece of information when it bringsinconsistency

I Belief Change


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Solutions

I Define new formalisms which don’t satisfy monotonicityI Non-monotonic inference relations

I Not « just » add the new piece of information when it bringsinconsistency

I Belief Change


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Outline

Introduction to NMR




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Syntax of Propositional Logic

I Boolean variables V = {x1, . . . , xn}I can receive the value 0 (false, F, ⊥) or 1 (true, T, >)I set of connectives, usually {¬,∨,∧}

A (well-formed) formula:I ϕ = xi , for xi ∈ V , is a formula; it is called an atomI if ϕ is a formula, then ¬ϕ is a formula; it is called the negation ofϕ

I if ϕ and ψ are formulas, then ϕ ∧ ψ is a formula; it is called theconjunction of ϕ and ψ

I if ϕ and ψ are formulas, then ϕ ∨ ψ is a formula; it is called thedisjunction of ϕ and ψ


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Semantics of Propositional Logic

Interpretation: mapping ω from formulas to {0,1}. An interpretationsatisfies a formula ϕ if:

I ϕ = xi , and ω(xi) = 1I ϕ = ¬ϕ′, and ω(ϕ′) = 0I ϕ = ϕ′ ∧ ϕ′′, and ω(ϕ′) = ω(ϕ′′) = 1I ϕ = ϕ′ ∨ ϕ′′, and ω(ϕ′) = 1 or ω(ϕ′′) = 1

When ω(ϕ) = 1, ω is a model of ϕ, denoted ω |= ϕ

I Alternative notations: Set of atoms {xi ∈ V | ω(xi) = 1}Vector of bits ω(x1), ω(x2), . . . , ω(xn)

I Set of models of ϕ: mod(ϕ)I ϕ′ is a consequence of ϕ if mod(ϕ) ⊆ mod(ϕ′): ϕ ` ϕ′

I ϕ′ is equivalent to ϕ if mod(ϕ) = mod(ϕ′): ϕ ≡ ϕ′

I A formula ϕ is called consistent when mod(ϕ) 6= ∅


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Additional Connectives

I Implication: ϕ→ ϕ′ ≡ ¬ϕ ∨ ϕ′

I Equivalence: ϕ↔ ϕ′ ≡ (ϕ→ ϕ′) ∧ (ϕ′ → ϕ)

I Exclusive or: ϕ⊕ ϕ′ ≡ (ϕ ∧ ¬ϕ′) ∨ (¬ϕ ∧ ϕ′)


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Truth Tables

To determine the models of a formula:I Split it into simpler formulasI Apply the semantics of the connectives on these simpler

formulasE.g. ϕ = (a ∨ b) ∧ (¬a ∨ c)

a b c a ∨ b ¬a ∨ c ϕ0 0 0 0 1 00 0 1 0 1 00 1 0 1 1 10 1 1 1 1 11 0 0 1 0 01 0 1 1 1 11 1 0 1 0 01 1 1 1 1 1


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Inference Rules

Modus Ponensϕ,ϕ→ ψ

ψ: If ϕ is true and ϕ→ ψ is true, then we deduce ψ

Modus Tollens¬ψ,ϕ→ ψ¬ϕ : If ψ is false and ϕ→ ψ is true, then we deduce ¬ϕ

RefutationProving that ϕ ` ψ is equivalent to prove that ϕ ∧ ¬ψ ` ⊥


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Models of Conjunction/Disjunction

Computing models of a complex formula ϕ can be simplified bysplitting ϕ in sub-formulas:

I if ϕ = ϕ1 ∧ ϕ2, then mod(ϕ) = mod(ϕ1) ∩mod(ϕ2)

I if ϕ = ϕ1 ∨ ϕ2, then mod(ϕ) = mod(ϕ1) ∪mod(ϕ2)


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Distributivity of Connectives

Disjunction and conjunction can be distributed/factorized:I ϕ ∧ (ψ1 ∨ ψ2) ≡ (ϕ ∧ ψ1) ∨ (ϕ ∧ ψ2)

I ϕ ∨ (ψ1 ∧ ψ2) ≡ (ϕ ∨ ψ1) ∧ (ϕ ∨ ψ2)


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De Morgan’s Laws

I ¬(ϕ1 ∨ ϕ2) ≡ ¬ϕ1 ∧ ¬ϕ2

I ¬(ϕ1 ∧ ϕ2) ≡ ¬ϕ1 ∨ ¬ϕ2


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Normal Forms

I A literal is an atom x ∈ V or its negation ¬x , with x ∈ VI A clause is a disjunction of literals l1 ∨ · · · ∨ lnI A cube (or term) is a conjunction of literals l1 ∧ · · · ∧ ln

Disjunctive Normal Form (DNF)A formula ϕ is in DNF if it is a disjunction of cubes

Conjunctive Normal Form (CNF)A formula ϕ is in CNF if it is a conjunction of clauses

TheoremAny propositional formula can be rewritten into an equivalent DNF orCNF formula


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Outline

Introduction to NMR




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Syntax of First Order Logic (1/3)

In Propositional Logic, there is a single kind of “data”: propositions,represented as Boolean variables. First Order Logic generalizes PL:

I P: infinite set of predicatesI F : infinite set of functionsI V : infinite set of variables (any kind of variables, not Boolean)I The usual set of connectives: {¬,∧,∨} (and also→,⇔,⊕, . . . )I Quantifiers : ∀,∃


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PredicateA predicate p is a mapping from a list of variables x1, . . . , xn to {0,1}.When all the variables are instanciated, p(x1, . . . , xn) is “equivalent” to aproposition from PL.

I n = 0, p is a propositionI n = 1, p(x) gives a property of x (e.g. prime(x))I n = 2, p(x , y) defines a binary relation (e.g. smallerThan(x , y))

FunctionA function f is a mapping from a list of variables x1, . . . , xn to aconstant.

I n = 0, f is a constantI n = 1, (e.g. square(x) = x2)I n = 2, (e.g. sum(x , y) = x + y )


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TermI Any variable is a termI For any terms t1, . . . , tn, and f a n-ary function, f (t1, . . . , tn) is a

term (in particular, constants are terms)

FormulasI For any terms t1, . . . , tn, and p a n-ary predicate, p(t1, . . . , tn) is a

formulaI Connectives: usual rules (negation, conjunction, disjunction)I If ϕ is a formula and x ∈ V , then ∀x , ϕ and ∃x , ϕ are formulas


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ExampleSocrates

Socrates is a man. All men are mortals.Can we deduce that Socrates is mortal?

In PLs, s → man,man→ mortal

In FOLman(s) ∧ (∀x ,man(x)→ mortal(x))


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ExampleDistances

Given a set V , a mapping d on V is a distance iff

(∀x ,equals(d(x , x),0)) ∧ (∀x , y ,equals(d(x , y),d(y , x)))∧(∀x , y , z,greaterOrEquals(add(d(x , y),d(y , z)),d(x , z)))


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Semantics of FOL (1)

Interpretation In FOLω = (D, ωc , ωv ) with

I D = {d1, . . . ,dn} is the non-empty domainI ωc is defined as

I ωc(f ) maps f (d1, . . . , dm) to an element of DI ωc(P) maps P(d1, . . . , dm) to an element of {0, 1}

I ωv maps variables x to an element of D


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Truth Value In FOLI If x is a free variable (non quantified), then ω(x) = ωv (x)

I ω(f (t1, . . . , tn)) = (ωc(f ))(ω(t1), . . . , ω(tn))I ω(P(t1, . . . , tn)) = (ωc(P))(ω(t1), . . . , ω(tn))I If ϕ,ψ are formulas, then ¬ϕ,ϕ ∧ ψ,ϕ ∨ ψ are interpreted like in

PLI If ϕ is a formula, then ω(∀x , ϕ) = 1 iff ωx←d (ϕ) = 1 for all d ∈ DI If ϕ is a formula, then ω(∃x , ϕ) = 1 iff ωx←d (ϕ) = 1 for at least

one d ∈ D


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Truth Value In FOLI If x is a free variable (non quantified), then ω(x) = ωv (x)I ω(f (t1, . . . , tn)) = (ωc(f ))(ω(t1), . . . , ω(tn))

I ω(P(t1, . . . , tn)) = (ωc(P))(ω(t1), . . . , ω(tn))I If ϕ,ψ are formulas, then ¬ϕ,ϕ ∧ ψ,ϕ ∨ ψ are interpreted like in


one d ∈ D


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Truth Value In FOLI If x is a free variable (non quantified), then ω(x) = ωv (x)I ω(f (t1, . . . , tn)) = (ωc(f ))(ω(t1), . . . , ω(tn))I ω(P(t1, . . . , tn)) = (ωc(P))(ω(t1), . . . , ω(tn))

I If ϕ,ψ are formulas, then ¬ϕ,ϕ ∧ ψ,ϕ ∨ ψ are interpreted like inPL

I If ϕ is a formula, then ω(∀x , ϕ) = 1 iff ωx←d (ϕ) = 1 for all d ∈ DI If ϕ is a formula, then ω(∃x , ϕ) = 1 iff ωx←d (ϕ) = 1 for at least

one d ∈ D


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Truth Value In FOLI If x is a free variable (non quantified), then ω(x) = ωv (x)I ω(f (t1, . . . , tn)) = (ωc(f ))(ω(t1), . . . , ω(tn))I ω(P(t1, . . . , tn)) = (ωc(P))(ω(t1), . . . , ω(tn))I If ϕ,ψ are formulas, then ¬ϕ,ϕ ∧ ψ,ϕ ∨ ψ are interpreted like in

PL

I If ϕ is a formula, then ω(∀x , ϕ) = 1 iff ωx←d (ϕ) = 1 for all d ∈ DI If ϕ is a formula, then ω(∃x , ϕ) = 1 iff ωx←d (ϕ) = 1 for at least

one d ∈ D


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PLI If ϕ is a formula, then ω(∀x , ϕ) = 1 iff ωx←d (ϕ) = 1 for all d ∈ D

I If ϕ is a formula, then ω(∃x , ϕ) = 1 iff ωx←d (ϕ) = 1 for at leastone d ∈ D


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one d ∈ D


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ExampleSocrates

Socrates is a man. All men are mortals.Can we deduce that Socrates is mortal?

In FOLman(s) ∧ (∀x ,man(x)→ mortal(x))

I D = {Socrates,Plato,MickeyMouse}I ωv (s) = SocratesI ωc(man) = fman s.t. fman(x) = 1 iff x ∈ {Socrates,Plato}I ωc(mortal) = fmortal s.t. fmortal(x) = 1


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ExampleDistances

In FOL


I D = R+ = {x ∈ R | x ≥ 0}I ωv : no free variables so nothing to doI ωc(equals) = f= s.t. f=(x , y) = 1 iff x = yI ωc(greaterOrEquals) = f≥ s.t. f≥(x , y) = 1 iff x ≥ yI ωc(add) = f+ s.t. f+(x , y) = x + yI ωc(d) = fd

What is fd?I fd can be any mapping from x , y to a positive real numberI if for a given fd , the formula is true, then d is a distance


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ExampleDistances

In FOL



What is fd?

I fd can be any mapping from x , y to a positive real numberI if for a given fd , the formula is true, then d is a distance


27

ExampleDistances

In FOL



What is fd?I fd can be any mapping from x , y to a positive real numberI if for a given fd , the formula is true, then d is a distance


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