Download Book Question Set #1: Ertel, Chapter 2: Propositional Logic

Harris Gold
Book Question Set #1: Ertel, Chapter 2: Propositional Logic
1.) What is propositional logic? With respect to AI, what is it good for?
Propositional logic entails connecting propositions with logical operators. In
regards to AI, propositional logic can grant a machine an extremely basic and
limited form of reasoning.
2.) Using BNF, define the language of all “fully parenthesized” propositional
logic formulas, where a propositional logical formula is said to be fully
parenthesized only if it contains one set of parenthesis for each logical
operator. (Hint: look up BNF – Backus-Naur Form)
( formula ) ::= ( a truth value )
( a propositional variable )
( formula )
( formula ˄ formula )
( formula ˅ formula )
( formula → formula )
( formula ↔ formula )
3.) With respect to propositional calculus, what is an interpretation?
An interpretation is an assignment of truth values to the propositional values.
4.) How many interpretations are there for a propositional formula involving 3
variables? 4 variables? 5 variables? N variables?
There are 8 interpretations for a propositional formula involving 3 variables, 16
involving 4, and 32 involving 5.
5.) In words, define…
a. ( NOT A ), where A is a propositional variable
A negation of A
b. ( A AND B ), where A and B are a propositional variable
A conjunction of A and B
c. ( A OR B ), where A and B are a propositional variable
A disjunction of A and B
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d. ( A IMPLIES B ), where A and B are a propositional variable
An implication that, ‘if A then B’ (also known as material implication)
e. ( A EQUIVALENT-TO B ), where A and B are a propositional value
A statement of equivalence where, ‘A if and only if B’
6.) What does it mean for two propositional formulas to be logically equivalent?
If two propositional formulas are logically equivalent, they must evaluate to the
same truth values for all interpretations.
7.) What does it mean for a logical formula to be satisfiable? Give an example of
a logical formula having 3 variables that is satisfiable – and provide a simple
argument that it is, indeed, satisfiable.
For a formula to be called satisfiable, it must be true for at least one interpretation.
((P ^ Q) → ¬R), where P = t, Q = t, and R = f
This will always be valid as long as one variable is false.
8.) What does it mean for a logical formula to be valid? Give an example of a
logical formula having 3 variables that is satisfiable – and provide a simple
argument that it is, indeed valid.
For a formula to be called valid, it must be true for all of its interpretations. True
formulas are also called tautologies.
((A -> B) ^ (B -> C)) -> (A -> C)
9.) What does it mean for a logical formula to be invalid? Give an example of a
logical formula having 3 variables that are satisfiable – and provide a simple
argument that is, indeed, invalid.
For a formula to be called invalid, it evaluates to false for all of its interpretations.
(A^B^C)
If all variables are false, it will evaluate to false.
10.)
What does it mean for a logical formula to be unsatisfiable? Give an
example of a logical formula having 3 variables that are satisfiable – and
provide a simple argument that is, indeed, unsatisfiable.
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11.)
With respect to a logical formula, what is a model? Give an example
of a model for some logical formula having three variables.
A model is every interpretation that satisfies a formula.
An example could be: ( ( A ^ B ) → C )
12.)
What is a proof system in the context of propositional calculus?
A proof system in the context of propositional calculus is the system that allows
us to compute equivalence and put together tautologies.
13.)
What does it mean for a logical formula to follow from, or be entailed
by a set of logical formulas?
In every interpretation where KB (a knowledge base) is true, Q (a query) is also
true. More over, whenever KB is true, Q is also true.
14.)
Convert each of the formulas in Exercise 2.2 to a fully parenthesized
logical formula.
a. ( A ˄ B ) ↔ A ˄ B
(((A˄B))↔(( A)˄( B)))
b. A → B ↔ B →A
((A→B)↔(( B)→( A)))
c. ( ( A → B ) ˄ ( B → A ) ) ↔ ( A ↔ B )
(((A→B)˄(B→A))↔(A↔B))
d. ( A ˅ B ) ˄ ( B ˅ C ) → ( A ˅ C )
(((A˅B)˄((
B)˅C))→(A˅C))
Harris Gold
15.)
Show that each formula of the previous problem is a tautology.
a.) ¬ ( A ^ B ) ↔ ¬ A ^ B
A
B
T
T
F
F
T
F
T
F
¬( A ^ B )
T
F
F
T
¬A^¬B
¬(A^B)↔¬A^B
T
F
F
T
T
T
T
T
b.) A → B ↔ ¬ B → ¬ A
A
B
A→B
¬ B → ¬ A
A→B↔ ¬B
→ ¬ A
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
T
T
T
c.) ( ( A → B ) ˄ ( B → A ) ) ↔ ( A ↔ B )
A
B
A→B
B→A
A↔B
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
F
F
T
((A→B
)˄(B→
A))↔(
A↔B)
T
T
T
T
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d.) ( A ˅ B ) ˄ ( ¬ B ˅ C ) → ( A ˅ C )
A
B
C
A˅B
¬B˅C
A˅C
(A˅B)
˄(¬B˅
C)→(
A˅C)
T
T
T
T
F
F
F
F
T
T
F
F
F
F
T
T
T
F
T
F
F
T
F
T
T
T
T
T
F
F
T
T
T
F
T
T
F
T
F
T
T
T
T
T
F
T
F
T
T
F
T
T
T
F
T
T