Programming Assignment 3

Assignment 3: CSPs

Table of Contents

Introduction

Autograder

Q1: Table Constraint

Q2: Forward Checking

Q3: GacEnforce and GAC

Q4: AllDiff for Sudoku

Q5: NValues Constraint

Q6: Plane Scheduling Problem

Introduction

In this project, you will implement some new constraints and backtracking search algorithms.

Note that this code base is unrelated to the Berkeley pacman code base. So you will not need any of

the files from A1 nor A2. (If anyone has a good idea as to how to use CSPs within the pacman

framework please let us know).

The code for this project contains the following files, available as a zip archive.

Files you’ll edit:

backtracking.py

Where all of the code related to backtracking search is located. You will

implement forward checking and gac search in this file.

csp_problems.py

Where all of the code related implementing different CSP problems is

located. You will implement a new version of the nQueens CSP and a CSP

to solve the plane scheduling problem in this file.

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constraints.py

Where all of the code related implementing various constraints is located.

You will implement the NValues constraint in this file.

Files you can ignore:

csp.py File containing the definitions of Variables, Constraints, and CSP classes.

util.py Some basic utility functions.

nqueens.py Solve nQueens problems.

sudoku.py Solve sudoku problems.

plane_scheduling.py Solve plane scheduling problems.

autograder.py

Program for evaluating your solutions. As always your solution might also

be evaluated with additional tests besides those performed by the

autograder.

Files to Edit and Submit: You will fill in portions of backtraking.py , csp.py , and csp_problems.py

during the assignment. You may also add other functions and code to these file so as to create a

modular implementation. Do not add code to other files and import them. These files cannot be

submitted so your solution relying on these files will fail our testing You will submit these file with your

modifications. Please do not change the other files in this distribution.

Evaluation: Your code will be autograded for technical correctness. The tests in autograder.py will

be run. Please do not change the names of any provided functions or classes within the code, or you

will wreak havoc on the autograder.

Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for

help. There will be scheduled help sessions (to be announced), the piazza discussion forum will be

monitored and questions answered, and you can also ask questions about the assignment during

office hours. These things are for your support; please take advantage of them. If you can’t make our

office hours, let us know and we will arrange a different appointment. We want the assignment to be

rewarding and instructional, not frustrating and demoralizing. But, we don’t know when or how to help

unless you ask.

Piazza Discussion: Please be careful not to post spoilers.

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What to Submit

You will be using MarkUs to submit your assignment. You will submit three files:

1. Your modified backtracking.py

2. Your modified csp_problems.py

3. Your modified constraints.py

Note: In the various parts below we ask a number of questions. You do not have to hand in answers

to these questions, rather these questions are designed to help you understand the material.

AutoGrader

autograder.py is not the same as the Berkeley autograder. You can only run the command

python autograder.py -q qn

where qn is one of q1 , q2 , q3 , q4 , or q5 .

Or you can run the grader on all questions together with the command

python autograder.py

The autograder does not generate any graphical output.

CSPs

Question 1 (4 points): Implementing a Table Constraint

backtracking.py already contains an implementation of BT (plain backtracking search) while

csp_problems.py contains an implementation of the nQueens problem. Try running

python nqueens.py 8

to solve the 8 queens problem using BT. If you run

python nqueens.py -c 8

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the program will find all solutions to the 8-Queens problem. Try

python nqueens.py –help

to see the other arguments you can use. (However, you haven’t implemented FC nor GAC yet, so you

can’t use these algorithms yet.) Try some different small numbers with the ‘-c’ option, to see how the

number of solutions grows with the number of Queens. Also observe that even numbered queens are

generally faster to solve, and the time to find a single solution for ‘BT’ grows quite quickly. Observe

the number of nodes explored. Later once you have FC and GAC implemented you will see that they

explore fewer nodes.

For this question look at constraint.py . There you will find the class QueensTableConstraint that you

have to implement for this question. This class creates a table constraint to capture the nQueens

constraint. Once you have that implemented you can run

python nqueens.py -t 8

to solve the nQueens CSP using your table constraint implementation. Check a number of sizes and

‘-c’ options: you should get the same solutions returned irrespective of whether or not you use ‘-t’.

That is, your table constraint should yield the same behavior as the original QueensConstraint

Question 2 (5 points): Forward Checking

In backtracking.py you will find the unfinished function FC . You have to complete this function. Note

that the essential subroutine FCCheck has already been implemented for you. Note that your

implementation must deal correctly with finding one or all solutions. Check how this is done in the

already implemented BT algorithm…just be sure that you restore all pruned values even if FC is

terminating after one solution.

After implementing FC you will be able to run

python nqueens.py -a FC 8

to solve 8-Queens with forward checking. Solve some different sizes and check how the number of

nodes explored differs from when BT is used.

Also try solving sudoku using the command

python sudoku.py 1

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Which will solve board #1 using Forward Checking. Try other boards (1 to 7). Use

python sudoku.py –help

to see the other arguments you can use.

Also try

python sudoku.py -a ‘BT’ 1

to see how BT performs compared to FC. Finally try

python sudoku.py -a ‘FC’ -c 1

To find all solutions using FC. Check if any of the boards 1-7 have more than one solution.

Note also that if you have a sudoku board you would like to solve, you can easily add it into

sudoku.py and solve it. Look at the code in this file to see how input boards are formatted and placed

in the list boards . Once a new board is added to the list boards it can be solved with the command

python sudoku.py -a ‘FC’ k where k is the position of the new board in the list boards

Question 3 (7 points): GacEnforce and GAC

In backtracking.py you will find unfinished GacEnforce and GAC routines. Complete these functions.

After finishing these routines you will be able to run

python nqueens.py -a GAC 8

Try different numbers of Queens and see how the number of nodes explored differs from when you

run FC .

Does GAC also take less time than FC on sudoku ? What about on nqueens ?

Now try running

python sudoku.py -e 1

which will not do any backtracking search, it will only run GacEnforce at the root.

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Try running only GacEnforce on each board to see which ones are solved by only doing GacEnforce .

Question 4 (2 points): AllDiff for Sudoku

In csp_problems.py you will find the function sudokuCSP . This function takes a model parameter that is

either ‘neq’ or ‘alldiff’ . When model == ‘neq’ the returned CSP contains many binary not-equals

constraints. But when model == ‘alldiff’ the model should contain 27 allDifferent constraints.

Complete the implementation of sudokuCSP so it properly handles the case when model == ‘alldiff’

using allDifferent constraints instead of binary not-equals.

Note that this question is very easy as you can use the class AllDiffConstraint(Constraint) that is

already implemented in constraints.py . However, you must successfully complete Question 3 to get

any marks on this question.

Question 5 (4 points): NValues Constraint

The NValues Constraint is a constraint over a set of variables that places a lower and an upper bound

on the number of those variables taking on value from a specified set of values.

In constraints.py you will find an incomplete implementation of class NValuesConstraint . In

particular, the function hasSupport has not yet been implemented. Complete this implementation.

Question 6 (10 points): Plane Scheduling

Implement a solver for the following plane scheduling problem by encoding the problem as a CSP

and using your already developed code to find solutions.

You have a set of planes, and a set of flights each of which needs to be flown by some plane. The

task is to assign to each plane a sequence of flights so that:

1. Each plane is only assigned flights that it is capable of flying (e.g., small planes cannot fly transAtlantic flights).

2. Each plane’s initial flight can only be a flight departing from that plane’s initial location.

3. The sequence of flights flown by every plane must be feasible. That is, if F2 follows F1 in a

plane’s sequence of flights then it must be that F2 can follow F1 (normally this would mean that

F1 arrives at the same location that F2 departs).

4. Certain flights terminate at a maintenance location. All planes must be serviced with a certain

minimum frequency. If this minimum frequency is K then in the sequence of flights assigned to a

plane at least one out every subsequence of K flights must be a flight terminating at a

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maintenance location. Note that if a plane is only assigned J flights with J < K, then it satisfies this

constraint.

5. Each flight must be scheduled (be part of some plane’s sequence of flights). And no flight can be

scheduled more than once.

Example

Say we have two planes AC-1 and AC-2, and 5 flights AC001, AC002, AC003, AC004, and AC005.

Further:

1. AC-1 can fly any of these flights while AC-2 cannot fly AC003 (but can fly the others).

2. AC-1 can start with flight AC001, while AC-2 can start with AC004 or AC005.

3. AC002 can follow AC001, AC003 can follow AC002, and AC001 can follow AC003 (these form a

one-way circuit since we can’t, e.g., fly AC003 first then AC002). In addition, AC004 can follow

AC003, AC005 can follow AC004, and AC004 can follow AC005.

4. AC003 and AC005 end at a maintenance location.

5. The minimum maintenance frequency is 3.

In this case a legal solution would be for AC-1’s schedule to be the sequence of flights [AC001,

AC002, AC003, AC004] while AC-2’s schedule is [AC005] (notice that for AC-1 every subsequence of

size 3 at least one flight ends at a maintenance location). Another legal schedule would be for AC-1

to fly [AC001, AC002, AC003, AC004, AC005] and AC-2 to fly [] (i.e., AC-2 does do any flights).

Your task is to take a problem instance, where information like that given in the above example is

specified, and build a CSP representation of the problem. You then solve the CSP using any of the

search algorithms, and from the solution extract a legal schedule for each plane. Note that the set of

constraints you have (and have built in the previous questions) are sufficient to model this problem

(but feel free to implement further constraints if you need them for the CSP model you develop).

See plane_scheduling.py for the details of how problem instances are specified; csp_problems.py

contains the class PlaneProblem for holding a specific problem.

You are to complete the implementation of solve_planes in the file csp_problems.py . This function

takes a PlaneProblem , constructs a CSP, solves that CSP with backtracking search, converts the

solutions of the CSP into the required format (see the solve_planes starter code for a specification of

the output format) and then returns the solutions.

You can also test your code with plane_scheduling.py . The command

python plane_scheduling.py -a GAC -c K

where K is the problem number, will invoke your code (from csp_problems.py ) on the specified

problem. (Use

python plane_scheduling.py –help

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for further information). It can be particularly useful to test your code on problems 1-4 as these

problems only test one of the constraints you have to satisfy.

A Few Hints:

First you should decide on the variables and the domain of values for these variables that you want to

use in your CSP model. You should design your variables so that it makes it easy to check the

constraints. Avoid variables that require an exponential number of values: performing GAC on such

constraints will be too expensive. A number of values equal to the number of flights times number of

planes values would be ok.

Try not to use table constraints over large numbers of variables. Table constraints over two or three

variables are fine: performing GAC on table constraints with large numbers of variables becomes very

expensive.

In some models it is useful to observe that if plane P can fly up to K different flights, then the length of

its sequence of flights is at most K. For example, in the example above, AC-1 can fly 5 different flights

while AC-2 can fly 4 different flights. So clearly, the sequence of flights flown by AC-1 can’t be more

than 5 long, and for AC-2 in sequence can’t be more than 4 long.

As an example of a set of variables and values that would be inadequate consider having a variable

for every flight with values being the set of planes that can fly that flight. This a reasonable number of

variables, and it makes the last constraint, that every flight is scheduled only once, automatically

satisfied (since every variable can only have one value). However, these variables by themselves will

not be sufficient, as we won’t be able to determine the sequencing of the set of flights assigned to a

particular plane. Potentially, such variables might be useful, but other variables would have to be

added to model the sequencing part of the CSP.

Submission

You’re not done yet! You will also need to submit your code to MarkUs.

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