COMPUTATIONAL-LOGIC Homework 2 Solved

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Consider the following pigeonhole problem:

  • There are n pigeons and m
  • Each pigeon has to live in a hole.
  • Each hole can have at most one pigeon.

We would like to find a hole for every pigeon. Clearly, the problem is not solvable if n > m.

Let pij be an atom for 1 ≤ i n and 1 ≤ j m. The atom pij is T iff the pigeon i live in the hole j. Consider the following clause:

pi1 ∨ pi2 ∨ ··· ∨ pim.

This clause says that pigeon i lives in a hole. Moreover, consider

^

¬pik ∨ ¬pjk.

1≤i<jn

This formula says that at most one pigeon lives in hole k. Please write a program such that:

  • it accepts two positive numbers n and m as inputs.
  • it outputs a CNF formula in DIMACS SAT format.
  • the generated CNF formula specifies the pigeonhole problem with n pigeons and m

You can use any programming language of your choice. Please send me the following files:

  • your program source code with instructions on how to use it;
  • the output files (in DIMACS SAT format) for
    • n = 3 and m = 3;
    • n = 4 and m = 3.
  • the outputs of MiniSat on the above two input files.