Let's break down the tasks from the image you provided. The questions seem to cover logic, fuzzy logic systems, and goal stack planning. Here’s an outline of how we can approach these questions:

Question 2: Prove the invalidity of a marriage using First-Order Logic

This question involves using First-Order Logic (FOL) to demonstrate that Clara and Albert's marriage is invalid based on kinship. The steps would include:

1. Defining relationships in FOL:

   • Let $P(x, y)$ represent "x is a parent of y."
   • Let $S(x, y)$ represent "x is a sibling of y."
   • Let $C(x, y)$ represent "x is the child of y."
   • Let $M(x, y)$ represent "x is married to y."
   • The rule given: "A marriage is invalid if the two are related as uncle/aunt and niece/nephew."

2. Establish the facts:

   • Clara is the child of Peter.
   • Peter is the father of Albert, making Clara and Albert siblings.

3. Prove the invalidity:

   • Use the rule that marriage between close relatives (uncle/aunt and niece/nephew or siblings) is invalid.

Question 3: Fuzzy Logic System for Promotion

This problem involves designing a fuzzy logic system to model promotion increases based on different inputs.

1. Membership Functions:

   • Define membership functions for the mandatory variable (current level) and one of the optional variables (loyalty, dedication, or work experience).
   • The membership functions could be triangular, trapezoidal, or Gaussian, based on specific thresholds for low, medium, and high values.

2. Fuzzy Rules:

   • Define rules like:
     • "If loyalty is high and current level is low, then increase is high."
     • "If dedication is low, increase is low."

3. Model:

   • Choose either Mamdani or Sugeno model. Mamdani is common for more intuitive fuzzy logic systems with fuzzy output, while Sugeno is useful for systems with numerical outputs.

4. Fuzzification and Inference:

   • Convert the crisp input values into fuzzy values, apply the rules, and then defuzzify the output to get the promotion increase.

Question 4: Goal Stack Planning

This involves transforming an initial block-world configuration into a goal configuration using a planner.

1. Initial State:

   • Blocks B on A, A on the table, C on the table, D on the table, and the arm is empty.

2. Goal State:

   • Blocks C on A, B on D, A and D on the table, and the arm is empty.

3. Steps:

   • Identify the actions (like unstack, stack, pickup) to move from the initial state to the goal state, pushing and popping from the goal stack as needed.

Let me know which question you'd like to start solving in more detail, or if you'd like clarification on any part of these outlines!