15 Logic Interview Questions and Answers – CLIMB

Logic forms the backbone of problem-solving and decision-making in various fields, from computer science and mathematics to philosophy and engineering. Mastery of logical principles enables individuals to construct sound arguments, identify fallacies, and develop algorithms that can efficiently tackle complex problems. Understanding logic is essential for anyone looking to enhance their analytical skills and apply them in practical scenarios.

This article offers a curated selection of logic-based questions and answers designed to help you prepare for interviews. By working through these examples, you will sharpen your reasoning abilities and be better equipped to demonstrate your logical thinking skills to potential employers.

Logic Interview Questions and Answers

1. Construct a truth table for the expression (A ∧ B) → ¬C.

To construct a truth table for the expression (A ∧ B) → ¬C, evaluate the expression for all possible combinations of truth values for A, B, and C. The table will have columns for A, B, C, A ∧ B, ¬C, and (A ∧ B) → ¬C. List all combinations of truth values for A, B, and C, and compute the values for A ∧ B, ¬C, and (A ∧ B) → ¬C accordingly.

A	B	C	A ∧ B	¬C	(A ∧ B) → ¬C
T	T	T	T	F	F
T	T	F	T	T	T
T	F	T	F	F	T
T	F	F	F	T	T
F	T	T	F	F	T
F	T	F	F	T	T
F	F	T	F	F	T
F	F	F	F	T	T

2. Simplify the logical expression ¬(A ∨ B) ∧ (¬A ∨ ¬B).

To simplify the expression ¬(A ∨ B) ∧ (¬A ∨ ¬B), use De Morgan’s laws and Boolean algebra:

• Apply De Morgan’s law: ¬(A ∨ B) = ¬A ∧ ¬B
• Substitute: (¬A ∧ ¬B) ∧ (¬A ∨ ¬B)
• Distribute ∧ over ∨: (¬A ∧ ¬B ∧ ¬A) ∨ (¬A ∧ ¬B ∧ ¬B)
• Simplify: ¬A ∧ ¬B

The simplified expression is ¬A ∧ ¬B.

3. Apply Modus Ponens to derive a conclusion from the premises: P → Q and P.

Modus Ponens allows deriving a conclusion from a conditional statement and its antecedent:

• P → Q (If P, then Q)
• P (P is true)

Conclusion: Q (Q is true)

Example:

• If it is raining, then the ground is wet. (P → Q)
• It is raining. (P)

Conclusion: The ground is wet. (Q)

4. Describe the resolution principle and provide an example of its application.

The resolution principle is used in automated theorem proving. It operates on clauses, which are disjunctions of literals. If two clauses contain a literal and its negation, they can be combined to produce a new clause without the literal and its negation.

Example:

Clauses:
1. (A ∨ B)
2. (¬A ∨ C)

Resolution: (B ∨ C)

5. Convert the expression (A ∨ B) ∧ (¬A ∨ C) into Conjunctive Normal Form (CNF).

The expression (A ∨ B) ∧ (¬A ∨ C) is already in Conjunctive Normal Form (CNF) as it is a conjunction of disjunctions.

6. Prove that if n is an even integer, then n^2 is also even.

To prove that if n is an even integer, then n^2 is also even, express n as n = 2k, where k is an integer. Then:

n^2 = (2k)^2 = 4k^2 = 2(2k^2)

Since 2k^2 is an integer, n^2 is even.

7. Explain logical equivalence and provide an example.

Logical equivalence occurs when two statements yield the same truth value in every scenario. For instance, “A and B” and “B and A” are equivalent. De Morgan’s laws also illustrate equivalence:

• ¬(A and B) ≡ (¬A or ¬B)
• ¬(A or B) ≡ (¬A and ¬B)

Example:

A = True
B = False

expression1 = not (A and B)
expression2 = (not A) or (not B)

print(expression1 == expression2)  # Output: True

8. Describe different proof techniques and give an example of each.

Proof techniques include:

1. Direct Proof: Start with known facts to prove a statement.
Example: Prove the sum of two even numbers is even.

   def is_even(n):
       return n % 2 == 0

   a, b = 4, 6
   assert is_even(a) and is_even(b)
   assert is_even(a + b)  # 4 + 6 = 10, which is even

2. Proof by Contradiction: Assume the opposite and show a contradiction.
Example: Prove √2 is irrational.

   import math

   def is_rational(x):
       return x == int(x)

   assert not is_rational(math.sqrt(2))

3. Proof by Induction: Prove a base case and an inductive step.
Example: Prove the sum of the first n natural numbers is (n(n + 1))/2.

   def sum_natural_numbers(n):
       return n * (n + 1) // 2

   assert sum_natural_numbers(5) == 15

4. Proof by Contrapositive: Prove “If not Q, then not P.”
Example: Prove if a number is not divisible by 3, it is not the sum of two numbers each divisible by 3.

   def is_divisible_by_3(n):
       return n % 3 == 0

   a, b = 3, 6
   assert is_divisible_by_3(a) and is_divisible_by_3(b)
   assert is_divisible_by_3(a + b)

9. Define tautologies and contradictions and provide examples.

A tautology is always true, like A or not A. A contradiction is always false, like A and not A.

10. Explain the basics of first-order logic and its differences from propositional logic.

First-order logic (FOL) extends propositional logic with quantifiers and predicates, allowing for more complex statements about objects and their relationships. FOL includes quantifiers like ∀ and ∃, predicates, variables, and functions, making it more expressive than propositional logic.

11. Identify common logical fallacies and provide examples.

Common logical fallacies include:

• Ad Hominem: Attacking the person instead of the argument.

  Example: “You can’t trust John’s opinion on climate change because he’s not a scientist.”

• Straw Man: Misrepresenting an argument to attack it.

  Example: “People who support space exploration want to waste money on useless projects instead of helping the poor.”

• Appeal to Ignorance: Arguing that a lack of evidence proves something.

  Example: “No one has ever proven that aliens don’t exist, so they must be real.”

• False Dilemma: Presenting two options as the only possibilities.

  Example: “We can either ban all cars or accept that pollution will destroy the planet.”

• Slippery Slope: Arguing that a small step will lead to a chain of events.

  Example: “If we allow students to redo assignments, soon they’ll expect to retake entire courses.”

• Circular Reasoning: Repeating what it assumes without proving it.

  Example: “The Bible is true because it says so in the Bible.”

• Hasty Generalization: Making a broad generalization based on limited evidence.

  Example: “My friend got sick after eating at that restaurant, so it must have terrible hygiene.”

• Red Herring: Introducing an irrelevant topic to divert attention.

  Example: “Why worry about the environment when there are so many people unemployed?”

12. Formulate a modal logic expression involving necessity and possibility.

Modal logic extends classical logic with necessity (◻) and possibility (◇). A common expression is:

◻P → ◇P

This states that if P is necessarily true, then it is also possibly true.

13. Outline the steps involved in implementing an automated theorem prover.

Implementing an automated theorem prover involves:

1. Formalization of Logic: Choose a formal system of logic.
2. Representation of Knowledge: Convert statements into formal logical expressions.
3. Inference Engine: Develop an engine to apply logical rules.
4. Search Strategy: Implement a strategy to explore possible proofs.
5. Proof Checking: Verify the correctness of potential proofs.
6. Optimization and Heuristics: Improve efficiency with optimizations.
7. User Interface: Develop an interface for user interaction.

14. Explain how a SAT solver works and provide a simple example.

A SAT solver transforms a problem into a Boolean formula and determines if there is a satisfying assignment. The process involves:

1. Problem Encoding: Encode the problem into a Boolean formula.
2. Search Algorithm: Use an algorithm to explore truth value assignments.
3. Backtracking: Backtrack if a conflict arises.
4. Solution or Unsatisfiability: Find a satisfying assignment or determine unsatisfiability.

Example:

from pysat.solvers import Glucose3

solver = Glucose3()

solver.add_clause([1, -2])  # A ∨ ¬B
solver.add_clause([-1, 2])  # ¬A ∨ B

is_satisfiable = solver.solve()

if is_satisfiable:
    model = solver.get_model()
    print("Satisfiable with assignment:", model)
else:
    print("Unsatisfiable")

solver.delete()

15. Write a Prolog program to solve the problem of finding the ancestor of a person.

In Prolog, define relationships and rules using facts and rules to find an ancestor.

Example:

% Facts
parent(john, mary).
parent(mary, susan).
parent(susan, tom).

% Rule
ancestor(X, Y) :- parent(X, Y).
ancestor(X, Y) :- parent(X, Z), ancestor(Z, Y).

This defines parent-child relationships and a rule for finding an ancestor.