Introduction

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Logique pour l’Informatique AvancéeSpécialité IAD

Responsable: Jean-Gabriel Ganascia

[email protected]

ACASA ACASA -- Université Pierre et Marie CURIEUniversité Pierre et Marie CURIEACASA ACASA -- Université Pierre et Marie CURIEUniversité Pierre et Marie CURIE

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Cours Mercredi 8H30 – 10H30

Salle 201, couloir 23-24, 2e étage

Documents: http://ganascia.name/LIA

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Organisation TD/TME

• Groupe 1: Marie-Jeanne Lesot

TD: mardi 8H30-10H30salle 102 Couloir 23-24

TME: mardi 10H45-12H45non encore définie


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TME: mardi 10H45-12H45non encore définie

• Groupe 2: Iza Marfisi Schottman

TD: jeudi 8H30-10H30salle 102 Couloir 23-24

TME: jeudi 10H45-12H45non encore définie

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Aristotle 384 – 322 BC

The heart of Aristotelian Logic is the deductive syllogism:

Deduction


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A, B, C, D, … are swans

All swans are white

A, B, C, D, … are white

Deduction

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History of Logic(-500 – …)

1. Describing the correct way of reasoning –Philosophy(Aristotle, …)

2. Mathematizing Logic –Philosophy(Leibniz, Boole, …)


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(Leibniz, Boole, …)

3. Mechanizing Mathematics –Mathematics(Hilbert, Herbrand, Tarski…)

4. Computerizing the Proof –Mathematics, Logic, AI & Computer Science (…)

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Practical Use of Logic in Computer Sciences

• Automatic Theorem Proving • Proof of Programs, Model Checking, …

– Safety

• Automated Deduction


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• Automated Deduction– Artificial Intelligence

• Programming Language– PROLOG, DATALOG, …

• Knowledge Representation– AI, Data Bases, Semantic Web…

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Course Program

• Propositional and Predicate Calculus

– Syntax and semantics

• Proof Theory


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• Proof Theory

• Automatic Theorem Proving: the Resolution

Rule

• PROLOG: Programming in Logic

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PROLOG

• PROLOG = PROgramming in LOGic

• PROLOG is an intuitive language (Seymour Papert wanted to make children program with


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Papert wanted to make children program with PROLOG)

• PROLOG is built on an automatic theorem prover

• LOGIC: theoretical grounding of PROLOG

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Cours LIA: logique // PROLOG

1. Introduction à la logique des propositions (sémantique) + PROLOG pour BB

2. Logique des propositions (suite sémantique, syntaxe) + PROLOG

3. Logique des propositions

5. Résolution de base, filtrage, unification + PROLOG programmation dynamique unification

6. Résolution générale + PROLOG (entrées – sorties, analyseur syntaxique)


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3. Logique des propositions (systèmes formels, démonstration) + PROLOG (variable anonyme, graphes)

4. Logique des prédicats, quantificateurs… syntaxe et sémantiquePROLOG (coupure)

syntaxique)

7. Stratégie de résolution + PROLOG DCG

8. Stratégie SLD + objets en PROLOG

9. Preuve complétude, modalités + PROLOG avec contraintes

10. Sémantique logiques modales

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Representation in Logic

• Sign, Meaning and Truth

• Proposition

• The Propositional Language


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• The Propositional Language

• Limits of the Propositional Language

• The Predicate Logic

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• Expression, Meaning, ReferencesSign

GraphicsWorld

Expressions ReferencesMeaning


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• Sign: graphic with meaning (i.e. intended to express)– Examples: “horse”, “reality”, “John”, “�”,

“�”, “֠”, “�”

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Proposition• Meaning = truth value∈ {true, false}

– “Earth is blue”

– “The green square is behind the red triangle”

– “A book was published”


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The Propositional Language• Atom: elementary propositions

– “It was raining”– “The cube is green”

• Connectors– Binary: «∧ », « ∨ », « ⊃ », « ≡ », …


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– Binary: «∧ », « ∨ », « ⊃ », « ≡ », …– Unary: « ¬ »

• Composed Propositions– If A and B are propositions, « ¬ A»,

« A ∧ B », « A∨ B », « A⊃ B », « A≡ B» are propositions

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Representation in Logic• Representation in Propositional Logic

– The small red pentagon is above the blue cylinder– The yellow diamond is left to the cylinder– There is a big red parallelepiped– The green diamond is above the parallelepiped– The cylinder is green


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Representation in Logic (2)

There is an object at the right of a cube


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There is an object at the right of a cube

A dodecahedronis at the right of a big cube

A pyramid is at right of a cube

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Propositional Logic

How to translate those sentences in propositional logic:

1. There is an object right to a green cube2. Every successor of an even number is odd


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3. 72 is an even number4. The successor of 72 is odd

Introduction of propositional functions (predicates):event(X): « X is even » is a proposition for all values of X.

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Representation in Predicate Logic

Notion of propositional function:

1. X is even: even(X)

2. 72 is an even number even(72)


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3. The successor of 72 is oddodd(successor(72))

4. Every successor of an even number is odd∀∀∀∀x (even(x) ⊃⊃⊃⊃ odd(successor(x)))

5. There is an object to the right of a green cube∃∃∃∃x ∃∃∃∃y (right_of(x, y)&cube(y)&green(y))

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Knowledge Representation in Logic

Exercise: represent in logic the two following sentences:

“the barbers shave those who don’t shave by themselves”


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barber(X) ∧ ¬shave(Y, Y) ⇒ shave(X, Y)

“there is no barber who shaves someone who shaves himself”

¬(barber(X) ∧ shave(X,Y) ∧ shave(Y,Y))

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Propositional Logic: semantic

• Truth Tables

• Satisfiability, validity, unsatisfiability

• Logical Consequence


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• Logical Consequence

• Limitation of Truth Tables

• Semantic Trees

• Herbrand Theorem

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Truth Table

• Every atom a has a truth value: true (v) or false (f)

• How to compute the truth value of a composed proposition? « A ∧ B » « A ∨ B » « A ⊃ B » « ¬ A »


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A\B v f

v v f

f f f

A\B v f

v v v

f v f

A\B v f

v v f

f v v

A

v f

f v

« A ∧ B » « A ∨ B » « A ⊃ B » « ¬ A »

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Example

• Each line of the truth table is called an interpretationvvffv

vvvfv

fffvv

vvvvv

(A ⊃⊃⊃⊃(B ⊃⊃⊃⊃C))(B ⊃C)CBA


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interpretation

vvfff

vvvff

vffvf

vvvvf

vvffv

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Validity

• A formula A is said to be valid if it is trueon all the lines of the truth table, i.e. in all the

vvvv

(A ⊃⊃⊃⊃(B ⊃⊃⊃⊃A))(B ⊃A)BA


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in all the interpretations– Example:

(A⊃⊃⊃⊃(B⊃⊃⊃⊃A)) vvff

vfvf

vvfv

vvvv

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Validity: an other example

• Another example: (A⊃⊃⊃⊃(B⊃⊃⊃⊃C))⊃⊃⊃⊃((A⊃⊃⊃⊃B)⊃⊃⊃⊃(A⊃⊃⊃⊃C))

vv

(A⊃⊃⊃⊃(B⊃⊃⊃⊃C))⊃⊃⊃⊃((A⊃⊃⊃⊃B)⊃⊃⊃⊃(A⊃⊃⊃⊃C))

f

v

(A⊃B)⊃(A⊃C)

f

v

(A⊃C)

v

v

(A⊃B)

fffvv

vvvvv

(A⊃(B⊃C))(B⊃C)CBA


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vvvvvv

v

v

v

v

v

v

f

v

v

v

v

f

v

f

v

v

v

v

f

f

v

vvfff

vvvff

vffvf

vvvvf

vvffv

vvvfv

fffvv

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Satisfiability• A formula A is said

to be satisfiableif it is true on at least one line of its truth table, i.e. in at least on ffvv

(A ⊃(B ⊃¬A))(B ⊃¬A)BA


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least on interpretation

– Example: (A⊃⊃⊃⊃(B⊃⊃⊃⊃¬A))

vvff

vvvf

vvfv

ffvv

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Satisfiability – Unsatisfiability – Validity

• A formula is said to be– “satisfiable” if and only if it is true on at least

one line of the truth table

– “valid” if and only if it is true on all lines of the truth table


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truth table

– “unsatisfiable” if and only if it is false on all lines of the truth table, i.e. if it is never true

• Remarks :– F is unsatisfiableiff ¬F is valid

– F is unsatisfiableiff F is not satisfiable

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Logical consequence

• A is a logical consequence of B if A is true on all the lines of the truth table where B is true– Example: (A⊃B) is a logical consequence of B.


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Notation╞ A means A is a valid formula

B╞ A means A is a logical consequence of B

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Deduction Theorem

Deduction Theorem:B is a logical consequence of A (A╞ B ) if and only if A⊃B is valid (╞ A ⊃⊃⊃⊃ B )

Equivalent definitions:


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Equivalent definitions:• ╞ A if and only if (¬A) ╞ A• A╞ if and only if ╞ (¬A)

• A╞ B if and only if ╞ A ⊃⊃⊃⊃ B• ╞ A if and only if (¬A) ╞ A

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Proof of the Deduction Theorem

• If╞ A ⊃⊃⊃⊃ B then we are on line 1, 3 et 4 and in all these cases A ╞ B

Indeed, if A is true, (line 1), B is true and in the 2 other cases, A is false

vvv

(A ⊃ B )BA


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other cases, A is false

• If A ╞ B then╞ A ⊃⊃⊃⊃ B If A is true, (line 1), B is true and if A is false, no

matter the value of B, which corresponds to lines 1, 3 and 4 vff

vvf

ffv

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Theorems• Reductio ad absurdum theorem: p is a logical consequence of f1,

f2, … fq iff f 1 ∧ f2, …∧ fq ∧ ¬p is unsatisfiable

• Atom substitution theorem: if a is a valid formula built on

atomic propositions p1, p2, … pq every formula a’ built by

replacing p, p , … p with P , P , … P is a is also a tautology


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replacing p1, p2, … pq with P1, P2, … Pq is a is also a tautology

(i.e. a valid formula).

• Theorem: if A is valid and if B is a logical consequence of A,

then B is valid [if╞ A ⊃⊃⊃⊃ B and╞ A then╞ B ]

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Exercise (cf. Kleene)Brown, Jones and Smith are suspected of tax fraud; they

testify as follows:

Brown: Jones is guilty and Smith is innocent

Jones: If Brown is guilty, then Smith is also guilty

Smith: I am innocent, but at least one of the two others is guilty


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guilty

B, J and S are the three propositions Brown is innocent, Jones is Innocentand Smith is innocent.

Brown: ¬J ∧ S

Jones: ¬B ⊃ ¬S

Smith: S ∧ (¬J ∨ ¬B)

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Exercise (continued)

1. Are the testimony compatible?

2. Does the testimony of anyone follow from the testimony of another?

3. If all are innocent, who made a wrong testimony?


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testimony?

4. If all the testimonies are true, who is guilty, who is innocent?

5. If the innocent tell the truth and the guilty lie, who is guilty, who is innocent?

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Truth table

fvfffv

fvfvfv

vfffvv

fvfvvv

S ∧ (¬J ∨ ¬B)¬B ⊃ ¬S¬J ∧ SBSJ


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fvfffv

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Are the testimony compatible?

fvfffv

fvfvfv

vfffvv

fvfvvv

S ∧ (¬J ∨ ¬B)¬B ⊃ ¬S¬J ∧ SBSJ


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fvfvff

vfvfvf

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fvfffv

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Does the testimony of anyone follow from the testimony of another?

fvfvfv

vfffvv

fvfvvv

S ∧ (¬J ∨ ¬B)¬B ⊃ ¬S¬J ∧ SBSJ


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vfvfvf

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If testimonies are all true, who is innocent?

vfffvv

fvfvvv

S ∧ (¬J ∨ ¬B)¬B ⊃ ¬S¬J ∧ SBSJ


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fvffff

fvfvff

vfvfvf

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fvfffv

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vfffvv

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If innocent tell the truth and guilty lie?

fvfvfv

vfffvv

fvfvvv

S ∧ (¬J ∨ ¬B)¬B ⊃ ¬S¬J ∧ SBSJ


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fvfvff

vfvfvf

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fvfffv