The Problem of the Jealous Husbands

Solving the Jealous Husband Problem with Answer Set Programming (ASP)
ASP
Project
Author

Mahmut Osmanovic

Published

December 9, 2024

1.0 | Introduction

The Jealous Husband Problem is a classic river-crossing puzzle that presents the challenge of safely transporting three married couples across a river using a boat, while adhering to specific constraints. This problem first appeared in the medieval text Propositiones ad Acuendos Juvenes. An elegant solution can be found using Answer Set Programming (ASP), a declarative programming paradigm that is particularly effective for tackling complex combinatorial problems.

2.0 | Problem Description

We need to transport three husbands (m1, m2, m3) and their respective wives (w1, w2, w3) from the left bank of a river to the right bank using a boat. However, we face the following constraints:

1. The boat can carry at most two people at a time.
2. A man and a woman cannot be in the boat together unless they are married.
3. A woman cannot remain on the same side of the river with a man unless her husband is also present.
4. The boat must always return to the other side to pick up the remaining passengers.
5. The goal is to ensure all six individuals and the boat reach the right bank.

3.0 | Approach to the Solution

ASP is a natural choice for this problem because of its ability to model discrete states and transitions efficiently. I will subsequently describe how I modeled the solution.

3.1 | Entities

1. Persons: Represented as m1, m2, m3 (men) and w1, w2, w3 (women).
2. Vehicle: A boat capable of carrying up to two people.
3. Locations: The left and right banks of the river

3.2 | Rules and Constraints

I incorporated the following ASP rules: 1. Initial State: All individuals and the boat start at the left bank (at_bank(left, Person, 0))..
2. Transition Rules: The boat can move between banks, carrying at most two people (1 { cross(P, T) : person(P), ... } 2).. The location of each person and the boat is updated at each time step.
3. Jealousy Constraints: A man and a woman can only share the boat if they are married (:- cross(P1, T), cross(P2, T), man(P1), woman(P2), not married(P1, P2)).. A woman cannot remain with another man on the same bank without her husband being present :- at_bank(Current_Bank, Female, T), woman(Female), at_bank(Current_Bank, M1, T), man(M1), not married(M1, Female), married(Husband, Female), not at_bank(Current_Bank, Husband, T)..
4. Goal State: The solution terminates successfully when all persons and the boat are on the right bank.
5. Inertia: Entities retain their position unless acted upon.

3.3 | Output

The program outputs a sequence of actions (cross/2) indicating who should cross the river and when.

4.0 | ASP Code (Clingo)

Here is the full implementation of the problem in ASP:

% Entities and Relationships
time(0..10). 
person(m1; m2; m3; w1; w2; w3).
man(m1; m2; m3). woman(w1; w2; w3).
married(m1, w1). married(m2, w2). married(m3, w3).
vehicle(boat).
side(left; right). 

% Initial State
at_bank(left, m1, 0). at_bank(left, w1, 0).
at_bank(left, m2, 0). at_bank(left, w2, 0).
at_bank(left, m3, 0). at_bank(left, w3, 0).
at_bank(left, boat, 0).

% Goal State
goal_reached :- at_bank(right, m1, T), at_bank(right, w1, T),
                at_bank(right, m2, T), at_bank(right, w2, T),
                at_bank(right, m3, T), at_bank(right, w3, T).

% Crossing Rules
1 { cross(P, T) : person(P), at_bank(Current_Bank, P, T), at_bank(Current_Bank, boat, T)} 2 :- time(T).

% Inertia Conditions
at_bank(Current_Bank, P, T+1) :- time(T), not -at_bank(Current_Bank, P, T+1), at_bank(Current_Bank, P, T).

% Effects
at_bank(Other_Bank, P, T+1) :- 
    cross(P, T), 
    at_bank(Current_Bank, P, T), 
    side(Other_Bank), 
    Current_Bank != Other_Bank.
-at_bank(Current_Bank, P, T+1) :- 
    cross(P, T),
    at_bank(Current_Bank, P, T).
at_bank(Other_Bank, boat, T+1) :-
    cross(P, T),
    at_bank(Current_Bank, boat, T),
    side(Other_Bank),
    Current_Bank != Other_Bank.
-at_bank(Current_Bank, boat, T+1) :-
    cross(P, T),
    at_bank(Current_Bank, boat, T).

% Constraints
:- cross(P1, T), cross(P2, T), man(P1), woman(P2), not married(P1, P2). 
:- at_bank(Current_Bank, Female, T), woman(Female), at_bank(Current_Bank, M1, T), man(M1), not married(M1, Female), married(Husband, Female), not at_bank(Current_Bank, Husband, T).

% Termination
:- not goal_reached.

% Output
#show cross/2.

There are a total of 486 unique solutions. Note that the boat constraint is superfluous. All answersets that include invalid boat combiniations will anyhow be deleted. That is due to that they only occur when there is an invalid combination of men and woman at either bank. However, for the

5.0 | Insights

5.1 | Advantages of ASP

  • Declarative Nature: ASP allowed us to focus on “what” the solution should look like, rather than “how” to compute it.
  • Constraints Handling: Expressing the jealousy constraints and managing state transitions was straightforward with ASP’s expressive syntax.

5.2 | Key Takeaways

  • Proper modeling of relationships (e.g., married couples) and logical constraints ensures correctness.
  • Constraints Handling: The use of time steps (time/1) enables the representation of sequential actions.

5.3 | Challenges and Solutions

One of the main challenges was ensuring that constraints (e.g., jealousy) were enforced in both crossing and staying scenarios. This was addressed using integrity constraints in ASP.

6.0 | Final Words

This example demonstrates the power and flexibility of ASP in solving classic logic puzzles. It highlights how ASP can be applied to real-world scheduling, planning, and constraint satisfaction problems.