README's for A1 & A2

Author: pspanoudakis

HopfieldNetworkTraining/README.md

@@ -0,0 +1,89 @@
+## Group A - Exercise 2
+***
+
+### Context
+A [Hopfield Network](https://en.wikipedia.org/wiki/Hopfield_network) operates recurrently, as following:
+
+- It consists of $N$ self-fed nodes-neurons, with 2 possible states: $+1$ & $-1$.
+- We can store a number of $N$-dimensional vectors, with possible element values $+1$ & $-1$.
+- When the network is fed with an $N$-dimensional vector, its state will eventually balance itself in one of the stored vectors, which is the closest to the given vector.
+
+
+A Hopfield Network is trained by storing a number of $N$-dimensional vectors.
+This storage is implemented by calculating the $N \times N$ weight vector of the network.
+
+Each weight of the network is attached to a connection between 2 nodes. The weight matrix $W$ is calculated as following:
+
+$$ W = \sum_{i=1}^M (Y^3_{i} \cdot Y_i) - M \cdot I $$
+
+Where:
+ - $M$ is the number of $N$-dimensional vectors
+ - $Y_i$ is one of the $1 \times N$ vectors to be stored
+ - $Y_i^T$ is the $N \times 1$ column vector (the reversed $Y_i$)
+ - $I_n$ is the $N \times N$ [identity matrix](https://en.wikipedia.org/wiki/Identity_matrix)
+
+For example, to store $M = 3$ vectors: $(+1, -1, -1, +1)$, $(−1, −1, +1, −1)$, $(+1, +1, +1, +1)$ to a network with $N = 4$ nodes:
+
+$$ W = \begin{pmatrix}
+        +1 \\
+        -1 \\
+        -1 \\
+        +1 \\
+        \end{pmatrix}
+        \cdot
+        \begin{pmatrix}
+        +1 & -1 & -1 & +1
+        \end{pmatrix}
+        +
+        \begin{pmatrix}
+        -1 \\
+        -1 \\
+        +1 \\
+        -1 \\
+        \end{pmatrix}
+        \cdot
+        \begin{pmatrix}
+        -1 & -1 & +1 & -1
+        \end{pmatrix}
+        +
+        \begin{pmatrix}
+        +1 \\
+        +1 \\
+        +1 \\
+        +1 \\
+        \end{pmatrix}
+        \cdot
+        \begin{pmatrix}
+        +1 & +1 & +1 & +1
+        \end{pmatrix} $$
+
+Finally,
+
+$$ W = \begin{pmatrix}
+        0 & 1 & 1 & 3   \\
+        1 & 0 & 1 & 1   \\
+        -1 & 1 & 0 & -1 \\
+        3 & 1 & -1 & 0  \\
+        \end{pmatrix} $$
+
+### Task
+Implement a `hopfield/2` predicate:
+- The first argument is a given list of vectors to be stored in a Hopfield Network. Every vector is also a list with the corresponding vector elements.
+- The second argument will return the matrix containing the network weights, as a list of lists (one list per row).
+
+### Execution Example
+Input:
+
+    ?- hopfield([
+                    [+1,-1,-1,+1],
+                    [-1,-1,+1,-1],
+                    [+1,+1,+1,+1]
+                ], W).
+
+
+Output:
+
+    W = [[0,1,-1,3], [1,0,1,1], [-1,1,0,-1], [3,1,-1,0]]
+
+### Implementation
+The `hopfield` predicate is implemented in `hopfield.pl`, along with several helper predicates.

MatrixDiagonals/README.md

@@ -0,0 +1,21 @@
+## Group A - Exercise 1
+***
+
+### Task
+Implement a `diags(Matrix, DiagsDown, DiagsUp)` predicate:
+- `Matrix` is a given 2-D matrix (a list of lists)
+- `DiagsDown` will return a list of the descending diagonals of `Matrix`
+- `DiagsUp` will return a list of the ascending diagonals of `Matrix`
+
+### Execution Example
+Input:
+
+    ?- diags([[a,b,c,d],[e,f,g,h],[i,j,k,l]],DiagsDown,DiagsUp).
+
+Output:
+
+    DiagsDown = [[i],[e,j],[a,f,k],[b,g,l],[c,h],[d]]
+    DiagsUp = [[a],[b,e],[c,f,i],[d,g,j],[h,k],[l]]
+
+### Implementation
+The `diags` predicate is implemented in `diags.pl`, along with several helper predicates.