Programming Assignment: Planning

Goal: gaining familiarity with theorem-proving technology

Goals for this homework

1. Learn how to use Prover9, a state-of-the-art theorem prover for FOL

2. Brief introduction to a different style of programming: logic programming

3. Practice converting a planning problem into FOL using situation calculus conventions

Step 1: Install Prover9/Mace4• Download and install Prover9/Mace4 on some machine that you will be able to use for a

while. Lab machines will not have this, but if you don’t have access to another machine, let me know, and I will try to get this installed for you on a lab machine.

http://www.cs.unm.edu/~mccune/prover9/gui/v05.html

• Also, check out the documentation and examples. Manual:http://www.cs.unm.edu/~mccune/prover9/manual/2009-11A/

Examples:http://www.cs.unm.edu/~mccune/prover9/examples/2009-11A/

I especially recommend the “Schubert Steamroller” example, which is fairly easy to understand. Look for it under “Prover9 Miscellany”. Another one that is fairly easy to understand is the “Trivial set theory problem”.

The Prover9/Mace4 GUI

Step 2: Run Prover9/Mace4 on an Example Problem

1. Load the Assumptions and Goals from the Schubert Steamroller problemhttp://www.cs.unm.edu/~mccune/prover9/examples/2009-11A/misc/steam.in

into Prover9/Mace4, either using the GUI or command line. I’ll explain things using the GUI, so if you want to use the command line, you’ll need to look it up in the manual.

2. Check to make sure the assumptions and goals are entered correctly by clicking “Well Formed?” in the GUI.

3. In the GUI, on the right panel, you should see two Start buttons, one for Prover9 and one for Mace4. Prover9 tries to prove that the goals must logically follow from the assumptions. Mace4 tries to find a counter-example.

Press each of the Start buttons, and record what happens in a text file called “steamroller-output.txt”. Note that at least one of these will not succeed, so it may take up to the given time limit (60 seconds by default).

Step 3: Run Prover9/Mace4 on a Real Problem from Industry

One common use of theorem provers in software engineering is to verify that some critical piece of code works correctly, or to search for a piece of code that satisfies the input/output properties being looked for.

Your next task is to test Prover9 on this kind of industry task. We will take the problem from the TPTP (Thousands of Problems for Theorem Provers):http://www.cs.miami.edu/~tptp/

TPTP format isn’t directly compatible with Prover9, so we’ll use this converted version of a TPTP problem:http://www.cs.unm.edu/~mccune/prover9/examples/2009-11A/fof/SWC258+1.in

Copy and paste these axioms into the assumptions of Prover9, and run Prover9. Verify that it can prove this. (Note: the “goals” should be empty; they have already been included as “negative” assumptions.)

Copy the output proof from Prover9 into a text file called “swc258-output.txt”.

Step 4: Formulate your own problem

For this assignment, I will ask you to prove that there is a solution to two games/puzzles, or find a counter-example.

The main task is to formulate the puzzles in situation calculus.

We’ll start with a simple one (next slide), and then work up to a harder one.

Monkey and BananasPuzzle:A monkey is in a room. Suspended from the ceiling is a bunch of bananas, beyond the monkey's reach. However, in the room there are also a crate and a stick. The ceiling is just the right height so that a monkey standing on a crate could knock the bananas down with the stick, if the crate is directly below the bananas. The monkey knows how to move left and right, climb up and down, pick up things, pull things left and right, and wave a stick in the air. How can the monkey knock the bananas down?

Note: the problem is trivial for humans, but actually somewhat hard for monkeys and other animals.

Left Mid Right

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Monkey and BananasYour Job:1. Formulate this domain in Situation

Calculus. Store your set of axioms and goals in a file called “monkey-bananas.txt”.

2. Copy your axioms and goals into Prover9, and have it prove that a plan exists. (You may find that you need to correct your axioms to make this work.) Save the output of Prover9 to “monkey-bananas-output.txt”.

3. Extra credit (optional, 1 point): Examine the output of Prover9 to determine the exact sequence of steps that Prover9 comes up with for the plan. Save this plan to a file called “monkey-bananas-plan.txt”.

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Step 5: Peg Solitaire: Triangle VersionPeg Solitaire consists of a board with some pegs in it, where one of the pegs is missing.

The only action is a Jump action, which moves one peg over top an adjacent peg, and lands the first peg in an empty slot.

The peg that was jumped over is removed from the game.

The goal of the game is to remove all pegs except one, and to have the final peg land in the same spot as the initial hole.

Initial Board for Triangle Peg Solitaire

A jump, with resulting boardThe goal state:

Your task for Step 51. Write a set of assumption and goal formulas that describe the Triangle Peg Solitaire Game.

Store them in a text file called “peg-solitaire.txt”.

Hint: To help with debugging your formulation of the problem, you may want to start with a smaller board, and not require that the final peg land in the original open spot. If you can get that to work, then work toward the full problem.

2. Load these into Prover9/Mace4, and determine whether Triangle Peg Solver can be solved. Save the results (outputs of Prover9 and Mace4) in a text file called “peg-solitaire-output.txt”.

Note: Peg Solitaire is, in fact, solvable. However, the shortest solution takes 14 steps, with a fairly big branching factor. I don’t know for sure whether Prover9 can actually find a solution. It’s fine to determine that Prover9 can’t solve this, so long as you’re sure that you have formulated the problem correctly. Just record the output saying that Prover9 couldn’t find a solution.

3. Extra credit (optional, 1 point): inspect the output of Prover9 to determine the exact sequence of steps that gets from the initial state to the goal state. Save the steps you found in a file called “peg-solitaire-plan.txt”.

To turn in

You should turn in a single zip archive called <your-name>.zip. It should contain: • steamroller-output.txt• swc258-output.txt• monkey-bananas.txt and monkey-bananas-output.txt• peg-solitaire.txt and peg-solitaire-output.txt• If you did the extra credit, monkey-bananas-plan.txt

and/or peg-solitaire-plan.txtEmail or otherwise transfer this zip file to your TA.