Assignment 2 Marks Uploaded

1. Translate the following statements into first-order-logic statements. Use function symbols as commonly used in mathematics, e.g. +,-,x,% etc for add, substract, multiply, mod, = for equality etc. (e.g. instead of "successor(y,x)", use "y = x+1" etc.).
   • Every number has a unique successor
   • Not every number has a predecessor
   • The sum of any two odd numbers is even
   • x is a prime
   • There is no largest prime
   • x and y are relatively prime
   1. Translate these into FOL statements
   2. Convert these statements into clause form.
2. Choose an Universe such as U = (a,b). Assign appropriate values in a truth table so as to disprove the following, where V = forall and E = exists:
   • (Vx)(Ey) P(x,y) <=> (Ey)(Vx) P(x,y)

     What of the following - can you construct a proof or disproof for it?

   • (Ex)(Vy) P(x,y) ==> (Vx)(Ey) P(x,y)
3. Consider this argument:
   • If all drugs are contaminated then all negligent technicians are scoundrels.
   • If there are any drugs that are contaminated then all drugs are contaminated and unsafe
   • All germicides are drugs
   • Only the negligent are absent-minded
   Therefore
   • If any technician is absent-minded then if some germicides are contaminated he is a scoundrel.

   Using as predicates D(x) = drug(x), C(x) = contaminated(x), N(x) = negligent(x) T(x), S(x), U(x), G(x), A(x), convert these statements in FOL form. For example, the goal (the part after "therefore") may be written as follows:

   (Vx){ (T(x)^A(x)) => {(Ey)[G(y)^C(y)] => S(x)}}
   1. Obtain the FOL form for this argument.
   2. Obtain the proof using standard Quantified FOL
   3. Obtain the Clause form for the premises alone
   4. Prove the goal using forward resolution
   5. Prove the goal using resolution refutation.

DUE DATE

Assignment 1

1. Make FIVE pictures of your face, more or less frontal, with the hair etc cropped out. Convert your images into 50x50 scale. Everyone should mail 5 pics each of the above specifications to mrityu@iitk.ac.in by Sunday night (8/12/2006) 11:59pm. The collection of all the pics of this class will be available to you on the first link given below after Sunday midnight. Please stick with the specifications given above. All images should be in grayscale only.
   
2. Given the corpuses :
   1. CS365 Students Database
   2. Yale Database
   Take each 50 × 50 pixel training image and vectorize into an 2500-dimensional vector. Then perform principal component analysis (PCA) on the entire set of training image vectors, retaining the first d principal components. Note, rather than actually computing the huge covariance matrix, you can use a pseudo-matrix inversion (Singular Value Decomposition) - we will discuss this on Monday. The d eigenvectors (when converted back to images) are the Eigenfaces. Display the d eigenfaces as images in your report.

1. database 1 alone.
2. database 1 and 2 together.

SUBMISSION

1. The first principal components arising out of parts I and II, Comment on any special characteristic that you may notice.
2. Results of tests on your own face (5th image), as well as ONE OTHER 50x50 image that is NOT a face (can be anything - landscape, hostel scene, etc.) Comment on your results.
3. A link to your source code.

DUE DATE