LLogics for DData and KKnowledgeRRepresentation

Exercise 2: PL, ClassL, Ground ClassL

Outline Reasoning

Truth Table Deduction

Logics PL ClassL Ground ClassL

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Summary of Logics Mentioned by Now (1) Language / Syntax

Choose proper logics for the following symbols

⊓ ⊤ ⊢ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨

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PL ClassLGround ClassL

Summary of Logics Mentioned by Now (2) Semantics

PL pI={True, False}

ClassL pI={set1, …, setn}

Ground ClassL pI={set1, …, setn}, qI={element1, …, elementm}

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Truth Table What is a Truth valuation?

Def. A truth valuation on a propositional language L is a mapping ν assigning to each formula P of L a truth value ν(P).

What is a Truth Table? A truth table is composed of one column for each input

variable and one final column for all of the possible results of the logical operation that the table is meant to represent. Each row of the truth table therefore contains one possible assignment of the input variables, and the result of the operation for those values.

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Example Calculate the Truth Table of the following formulas

A∧B P∨Q X↔Y

What about this? (A∨B→C∨D∨E)∧(¬F↔A)∧(¬F∨G∧¬H∨F)∧(¬I→

¬(D∧J))∧(¬J∨¬D∨E)∧F

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A B A∧B A∨B A↔B

T T T T T

T F F T F

F T F T F

F F F F T

Deduction Deduction

Double negative elimination  P⊢P

Conjunction introduction / elimination  {P, Q}⊢P∧Q; P∧Q⊢P, P∧Q⊢Q

Disjunction introduction / elimination  P⊢P∨Q, Q⊢P∨Q; {P∨Q, P→R, Q→R}⊢R

Bi-conditional introduction / elimination (P→Q)∧(P←Q)⊢(P↔Q) 

De Morgan (P∧Q)⊢P∨Q, (P∨Q)⊢P∧Q

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Proofs of Deduction Rules

P Q ¬¬P P→Q Q→P ¬(P∧Q) ¬P∨¬Q ¬(P∨Q) ¬P∧¬Q

T T T T F F F F F

T F T F T T T F F

F T F T F T T F F

F F F T T T T T T

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P Q P∧Q P∨Q P↔Q

T T T T T

T F F T F

F T F T F

F F F F T

Soundness and Completeness A deduction system is sound if any sentence P that

is derivable from a set Г of sentences is also a logical consequence of that set Г.

A deductive system is complete if every sentence P that is a semantic consequence of a set of sentences Γ can be derived in the deduction system from that set.

A soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable.

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Outline Reasoning

Truth Table Deduction

Logics PL ClassL Ground ClassL

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Warming up List the number of models for the following formulas

1. (A∧B)∨(B∧C)

2. A∨B→D

3.¬A↔B↔C

4. (A↔¬B)∧(A ↔B)

5. Something has to be mentioned:

Suppose the formulas are represented by a PL with 4 propositions: A, B, C and D.

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Wait!

PL Exercises 1 Let A, B, C be propositional sentences, if A⊨B∧C,

then A⊨B or A⊨C or both? What if A⊨B∨C?

The model of ‘B∧C’ is I={B=T, C=T}; If ‘A⊨B∧C’, I should be also a model of ‘A’; Because I assigns True to ‘B’, I is a model of ‘B’; Similarly, I is also a model of ‘C’; So, the proposition is true.

If ‘A⊨B∨C’, a model of ‘B∨C’ can be I’={B=T, C=F}; I’ is not a model of either ‘A⊨B ’ and ‘A⊨C ’.

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PL Exercises 2 Given that P=(A∨B)∧(C∨D∨E), Q1=A∨B,

Q2=(A∨B∨C)∧(B∧C∧D→E), Q3=(A∨B)∧(D∨E), list i that P⊨Qi.

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A B C D E P Q1 Q2 Q3

T T T T T T T T T

F T T T T …

T F T T T

T T F T T

T T T F T

… …

F F F F F …

PL Exercises 2 cont. Given that P=(A∨B)∧(C∨D∨E), Q1=A∨B,

Q2=(A∨B∨C)∧(B∧C∧D→E), Q3=(A∨B)∧(D∨E), list i that P⊨Qi.

Notice that let X= A∨B, Y= D∨E, then we rewrite P=X∧(¬C∨Y), Q1=X

Q2=(X∨C)∧(¬ B∨¬C∨Y),

Q3=X∧Y

So, P⊨Q1;

X ⊨X∨C, (¬C∨Y) ⊨(¬ B∨¬C∨Y) thus P ⊨ Q2,

Y ⊨(¬C∨Y) thus Q3 ⊨ P.

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PL Exercises 3 Suppose p, q, r, s are four

propositional sentences, is the following sentence valid? (p→r) (q→s) (∧ ∧ r∨s)→(p∨q)

A=(p→r) (q→s) (∧ ∧ ¬r∨¬s)→(¬p∨¬q)

=¬((p→r) (q→s) (∧ ∧ ¬r∨¬s))∨ ¬p∨¬q

=(p∧¬r)∨(q∧¬s)∨(r∧s) ∨ ¬p∨¬q

=⊤15

p Q r s A

T T T T T

T T T F T

T T F T T

T T F F T

T F T T T

T F T F T

T F F T T

T F F F T

F T T T T

F T T F T

F T F T T

F T F F T

F F T T T

F F T F T

F F F T T

F F F F T

Are you ‘Sherlock Holmes’?

There was a robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that : (1) Nobody else could have been involved other than A, B

and C.

(2) C never commits a crime without A's participation.

(3) B does not know how to drive. So, is A innocent or guilty?

A∨B∨C C→AB

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A

Knights and Knaves A very special island is inhabited only by knights and

knaves. Knights always tell the truth, and knaves always lie.

You meet two inhabitants: Zoey and Mel.

Zoey tells you that Mel is a knave.

Mel says, ‘Neither Zoey nor I are knaves.’

Can you determine what are they? (who is a knight and who is a knave? )

Z: M M: Z∧M

proof: If M, then Z∧M, then Z, then M.

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Contradictory!

ClassL Exercises 1 T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL?

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ABA

B

ClassL Exercises 2 T={C⊑A, C⊑B} is (A⊓B) satisfiable in ClassL?

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A BcA

BC

A

BC

AB

C

A B

C

ClassL Exercises 3 Suppose a ⊨A, ⊨B, is the following sentence A∧B

satisfiable in PL, what about A⊓B in ClassL?

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A B A∧B

T T T

T F F

F T F

F F F

A B

AB

A B

PL vs. ClassLFor any PL sentence P, suppose a corresponding ClassL P’ is built by changing , ,→, into , , , ∧ ∨ ⊓ ⊔ ⊑(not wff’s are not considered). Then do we have that

⊨P iff ⊨P’?The transmission of truth VS. coverage.

For all wff’s P of PL, there exists a truth valuation ν that ν(P) = True iff there exists a class valuations σ, σ(P’) ≠ ∅ with P’ as the corresponding proposition in ClassL.

NOTE: although there are more possible assignments in ClassL for the ‘same’ formula, some of them may collapse into one case in PL.

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Example Suppose a PL wff ¬A B and a corresponding ⋀

ClassL wff ¬A⊓B, the possible assignments (interpretations) are as the following:

More than one model in ClassL collapse into one model in PL. (All non-empty are mapped to T.)

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A B ¬A B⋀

T T F

T F F

F T T

F F F

A B

BA

…

Ground ClassL What distincts Ground ClassL from ClassL?

The expressiveness to represent ‘instance’.

What for?

To capture the instance/individual in the world.

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Black Hair

White Hair

Exercise: LDKR ClassID Name Nationality Hair

1 Fausto Italian White

2 Enzo Italian Black

3 Rui Chinese Black

4 Bisu Indian Black

… … … …

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TBox={Italian⊑LDKR, BlackHair⊑LDKR, Indian ⊑LDKR}ABox={Italian(Fausto), Italian(Enzo), Chinese(Rui), Indian(Bisu), BlackHair(Enzo), BlackHair(Rui), BlackHair(Bisu), WhiteHair(Fausto)}

LDKR

Italian

Chinese

People

Indian

LDKRItalianChinese

Indian

White

H

air

Black Hair