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Logics for Data and Knowledge Representation. Exercises: Propositional Logic. Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Syntax Symbols and formation rules Modeling a problem Semantics Assignments, models Logical implication Reasoning Satisfiability and validity - PowerPoint PPT Presentation
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Logics for Data and KnowledgeRepresentation
Exercises: Propositional Logic
Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
Outline Syntax
Symbols and formation rules Modeling a problem Semantics
Assignments, models Logical implication Reasoning
Satisfiability and validity Some nice problems
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Symbols in PL
1. Which of the following symbols are used in PL?
⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
2. Which of the following symbols are in well formed formulas?
⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Symbols in PL (solution)
1. Which of the following symbols are used in PL?
⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
2. Which of the following symbols are in well formed formulas?
⊓ ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨
Remember the BNF grammar:
<Atomic Formula> ::= A | B | ... | P | Q | ... | | ⊥ ⊤
<wff> ::= <Atomic Formula> | ¬<wff> | <wff> ∧ <wff> | <wff> ∨ <wff> | <wff> → <wff> | <wff> ↔ <wff>
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Formation rules Which of the following is not a wff?
1. MonkeyLow ∨ BananaHigh
2. MonkeyLow BananaHigh
3. MonkeyLow BananaHigh
4. MonkeyLow → GetBanana
NUM. 3 IS WRONG!
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Modeling: Bananas
Bananas may differ in many ways. However, there are red and yellow bananas. I like bananas, but I eat only yellow bananas. If I do not eat at least a banana I get crazy.
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Highlight relevant words, define a PL language and a theory for the problem below.
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Modeling: Bananas (possible solution)
Bananas may differ in many ways. However, there are red and yellow bananas. I like bananas, but I eat only yellow bananas. If I do not eat at least a banana I get crazy.
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L = {RedBanana, YellowBanana, EatBanana, GetCrazy}
T = {EatBanana → YellowBanana, EatBanana → GetCrazy}
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Truth valuations and Truth Tables A truth valuation on a PL language L is a mapping ν that assigns to
each formula P of L a truth value ν(P). A truth table is composed of one column for each input variable and
one (or more) 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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A B A∧B
T T T
T F F
F T F
F F F
POSSIBLEASSIGNEMENTS
VARIABLESLOGICAL
OPERATION
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Example Calculate the Truth Table of the following formulas:
(1) A ∧ B; (2) P ∨ Q; (3) X ↔ Y.
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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
POSSIBLEASSIGNEMENTS
VARIABLES (1)
(2)
(3)
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Exercise: Truth Valuations Compute truth valuations for the formulas:
(A∨B)∧¬C
(A∨B)→C
(A∨B→C∨D∨E)∧(¬F↔A)∧(¬F∨G∧¬H∨F)∧(¬I→¬(D∧J))∧(¬J∨¬D∨E)∧F
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Provide the models for the propositions RECALL: a truth valuation ν is a model for a proposition P iff
ν(P) = true
List the models for the following formulas:1. A ∧ ¬B
2. (A ∧ B) ∨ (B ∧ C)
3. A ∨ B → C4. ¬A ↔ B ↔ C
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A B A ¬ ∧ B
T T F
T F T
F T F
F F F
MODEL
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Entailment RECALL: ⊨ ψ iff all models satisfying the formulas in also
satisfy ψ
Let A, B, C be propositional sentences.
(a) If A ⊨ B ∧ C, then A ⊨ B or A ⊨ C or both?
(b) What if A ⊨ B ∨ C?
Proof: The only model satisfying ‘B ∧ C’ is ν = {B=T, C=T}.
If ‘A ⊨ B ∧ C’ then ν should be also a model of ‘A’. However, since ν assigns true to B, ν is a model of ‘B’. Similarly, ν is also a model of ‘C’. So, (a) above is true for both.
Assume now that ‘A ⊨ B ∨ C’. A model of ‘B ∨ C’ is ν = {B=T, C=F}.
ν is not a model of ‘C’, therefore A ⊭ C. The other cases are similar.
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Exercise: prove entailment Given that:
P = (A ∨ B ) ∧ (C ∨ D ∨ E)
Q1 = A ∨ B
Q2 = (A ∨ B ∨ C) ∧ ((B ∧ C ∧ D) → E)
Q3 = (A ∨ B) ∧ (D ∨ E)
Does P ⊨ Qi ?
Proof: Let X = A ∨ B, Y = D ∨ E, then we can rewrite:
P = X (¬∧ C ∨ Y); Q1 = X; Q2 = (X ∨ C) ∧ (¬B ¬∨ C ∨ Y); Q3 = X Y∧
P ⊨ Q1 is obvious.
Since ‘X X ⊨ ∨ C’ and ‘(¬ C ∨ Y) ⊨ (¬B ¬∨ C ∨ Y)’, then P ⊨ Q2.
Since ‘Y ⊨ (¬C ∨ Y)’, then Q3 P (and not vice versa).⊨
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Exercise: prove entailment using truth tables Given that:
P = (A ∨ B ) ∧ (C ∨ D ∨ E)
Q1 = A ∨ B
Q2 = (A ∨ B ∨ C) ∧ (B ∧ C ∧ D → E)
Q3 = (A ∨ B) ∧ (D ∨ E)
(1) List all truth assignments such that P ⊨ Qi
(2) Is there any assignment such that P ⊨ Qi for all i?
Solution to (1): First compute the truth tables for all the propositions above. Then, list all rows for which both P and Q i are true.
Solution to (2): Check whether there is any assignment for which all the sentences above are true.
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Logical implication and deduction Deduction (⊢) is the process of inferring new knowledge from
known facts.
Logical implication (→) is a way to include deduction directly in the language. It is an alternative way to implement deduction.
For instance: ¬(A B) → (¬A ¬B) iff ¬(A B) ∨ ∧ ∨ ⊢ (¬A ¬B)∧
We provided some well known tautologies in the theoretical part (e.g. the De Morgan Law above)
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Prove using truth tables, the following deductions
1. Double negative elimination
P ⊢ P
2. Conjunction introduction / elimination
(a) {P, Q} ⊢ P∧Q; (b) P ∧ Q ⊢ P; (c) P ∧ Q ⊢ Q. Disjunction introduction / elimination
(a) P ⊢P ∨ Q; (a) Q ⊢ P ∨ Q; (c) {P ∨ Q, P → R, Q → R} ⊢ R
2. Bi-conditional introduction / elimination
(P → Q) ∧ (Q → P) ⊢ (P ↔ Q)
1. De Morgan
(a) (P ∧ Q) ⊢ P ∨ Q; (b) (P ∨ Q) ⊢ P ∧ Q
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Proofs of the Deduction Rules
P Q ¬¬P P→Q Q→P ¬(P∧Q) ¬P∨¬Q ¬(P∨Q) ¬P∧¬Q
T T T T T 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
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Soundness and Completeness A deductive 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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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Prove validity (I) Suppose p, q, r, s are four propositional sentences, is the following
sentence valid?
A = (p → r) (q → s) (∧ ∧ r ∨ s) →(p ∨ q)
A way to prove validity is to show that the proposition entails ⊤. This can be done by applying well known tautologies (e.g. De Morgan). In alternative we can show using truth tables that all the assignments are true.
(p → r) (q → s) (∧ ∧ ¬r ∨ ¬s) → (¬p ∨ ¬q) =
¬((p → r) (q → s) (∧ ∧ ¬r ∨ ¬s)) ∨ ¬p ∨ ¬q =
(p ∧ ¬r) ∨ (q ∧ ¬s) ∨ (r ∧ s) ∨ ¬p ∨ ¬q =
= ⊤
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Prove validity (II)
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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
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Calculus with Tableaux
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Γ = {A B, B, A}
B
A B
A
closed
B
A
closed
Γ = {B A, B}
B A
A
closed
B
B
Γ = {(A B), A}
A B
A
closed
B
A
Calculus with Tableaux
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Γ = {(A B) (B A), B}
(A B) ( B A)
(A B)
closed
B A
B
A B
A B
B A
closed
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.
Is A innocent or guilty?
Proof: The 3 points above can be translated in PL as follows:(1) θ1 = A B∨ ∨ C;
(2) θ2 = C → A;
(3) θ3 = B → (B ∧ A) ∨ (B ∧ C)
Does {θ1 , θ2 , θ3} ⊨ A ? Yes!
We can prove it by showing that θ1 ∧ θ2 ∧ θ3 → A is a tautology.
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
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?)
Proof: The two sentences above can be translated in PL as follows:
(1) Z → M; (2) M → Z ∧ M.
We can use truth tables to prove that there are two possible situations:
- Both lie (they are both knaves)
- Zoey tells the true (is a Knight) and Mel lies (is a knave)
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SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Recall the DPLL algorithm Input: a proposition P in CNF Output: true if "P satisfiable" or false if "P unsatisfiable"
boolean function DPLL(P) {
if consistent(P) then return true;
if hasEmptyClause(P) then return false;
foreach unit clause C in P do
P = unit-propagate(C, P);
foreach pure-literal L in P do
P = pure-literal-assign(L, P);
L = choose-literal(P);
return DPLL(P L) OR DPLL(P L);
}
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REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVTIONS :: CONCLUSIONSSYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL
Consistency Check
Empty Close Test
Unit Propagation
Pure Literal Elimination
Splitting rule
Use DPLL to prove satisfiability
B ¬A (¬C A) (B C)
1. Is it consistent? YES, there are no contradictions
2. Are there any empty clauses? No, go ahead.
3. Assign the right truth value to all literals and simplify the formula:Assign B = T, A = F and then the formula simplifies to ¬C
4. Assign the right value to pure literals:Assign C = F. Done.
5. No need for the splitting rule
The formula is satisfiable at least for the assignment (a model for it): {A = F, B = T, C = F}
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS :: DPLL