(after G. Sandu). Consider a game played by Builder and Critic over two sets of propositional formulas called Yes and No. The following rules are automatic without player action: a) if is in one set, it is replaced by in the other set b) if o A is in Yes, it is replaced there by the two formulas and c) if V is in No, it is replaced there by the two formulas and The players move in turns as follows: Critic selects a formula, either in Yes or in No. Builder replaces the selected formula (V in Yes or ^ in No) with either or . The game ends when both Yes and No contain only atomic formulas. Critic wins if some atomic formula is in both Yes and No. Otherwise, Builder wins. Prove that Builder has a winning strategy in the game (Yes={1,...,n}, No={1,...,m}) if and only if the set of formulas {1,..., n, 1,...,m} is satisfiable.
(after G. Sandu). Consider a game played by Builder and Critic over two sets of propositional formulas called Yes and No. The following rules are automatic without player action: a) if is in one set, it is replaced by in the other set b) if o A is in Yes, it is replaced there by the two formulas and c) if V is in No, it is replaced there by the two formulas and The players move in turns as follows: Critic selects a formula, either in Yes or in No. Builder replaces the selected formula (V in Yes or ^ in No) with either or . The game ends when both Yes and No contain only atomic formulas. Critic wins if some atomic formula is in both Yes and No. Otherwise, Builder wins. Prove that Builder has a winning strategy in the game (Yes={1,...,n}, No={1,...,m}) if and only if the set of formulas {1,..., n, 1,...,m} is satisfiable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Game of Builder and Critic with Propositional Formulas
**Overview:**
Consider a strategic game played between two entities, Builder and Critic, involving two sets of propositional formulas labeled "Yes" and "No." The game is governed by specific rules that operate automatically without direct player intervention:
#### Automatic Rules:
1. **Negation Handling (¬φ):**
- If ¬φ exists in one set, it is swapped with φ in the opposite set.
2. **Conjunction Handling (φ ∧ ψ):**
- If φ ∧ ψ is present in the "Yes" set, it is replaced there by the formulas φ and ψ.
3. **Disjunction Handling (φ ∨ ψ):**
- If φ ∨ ψ is contained in the "No" set, it is substituted there with the formulas φ and ψ.
#### Player Turns:
- **Critic's Move:**
- Critic chooses a formula (φ ∨ ψ in "Yes" or φ ∧ ψ in "No").
- **Builder's Move:**
- Builder replaces the chosen formula with either φ or ψ.
#### Game Termination and Winning Conditions:
- The game concludes once both "Yes" and "No" consist solely of atomic formulas.
- **Critic's Victory:** Occurs if any atomic formula is shared by both sets.
- **Builder's Victory:** Happens if no atomic formula is common.
#### Objective:
Prove that Builder possesses a winning strategy if, and only if, the combination of formulas:
\[
\{φ_1, \ldots, φ_n, ¬ψ_1, \ldots, ¬ψ_m\}
\]
is satisfiable. This implies that it is possible to assign truth values to the variables that make the set consistent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6419945a-88d3-48ad-a566-88e8836c5974%2F64df7edc-fed6-428b-9e3c-28d04bf7bd0d%2Fci0vs6r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Game of Builder and Critic with Propositional Formulas
**Overview:**
Consider a strategic game played between two entities, Builder and Critic, involving two sets of propositional formulas labeled "Yes" and "No." The game is governed by specific rules that operate automatically without direct player intervention:
#### Automatic Rules:
1. **Negation Handling (¬φ):**
- If ¬φ exists in one set, it is swapped with φ in the opposite set.
2. **Conjunction Handling (φ ∧ ψ):**
- If φ ∧ ψ is present in the "Yes" set, it is replaced there by the formulas φ and ψ.
3. **Disjunction Handling (φ ∨ ψ):**
- If φ ∨ ψ is contained in the "No" set, it is substituted there with the formulas φ and ψ.
#### Player Turns:
- **Critic's Move:**
- Critic chooses a formula (φ ∨ ψ in "Yes" or φ ∧ ψ in "No").
- **Builder's Move:**
- Builder replaces the chosen formula with either φ or ψ.
#### Game Termination and Winning Conditions:
- The game concludes once both "Yes" and "No" consist solely of atomic formulas.
- **Critic's Victory:** Occurs if any atomic formula is shared by both sets.
- **Builder's Victory:** Happens if no atomic formula is common.
#### Objective:
Prove that Builder possesses a winning strategy if, and only if, the combination of formulas:
\[
\{φ_1, \ldots, φ_n, ¬ψ_1, \ldots, ¬ψ_m\}
\]
is satisfiable. This implies that it is possible to assign truth values to the variables that make the set consistent.
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