Abstract

We explore the relationship of group generators where each element of the group is assigned a vector. Each successive element that is being generated by the generator is then set up to be orthogonal at step sizes of the generator. We then would like to find some structure in the space spanned by such vectors in a way that we can optimize the information content of moving in a disjoint manner between vectors in this space. Inspiration comes from the musical scale of 12 notes arranged by their frequencies, with a generator of 5. We hope such structures such as keys and scales can be imported from it into our system with the hope of increasing the information content of such progressions using algebraic information theory. It can be seen that permutations on an agents input and actions vector my be induced to have more information (be less random) this way.

. INTRODUCTION

The problem we hope to solve is one of , given a collection of objects where n can be selected at any one time. And we change the objects at each iteration or time step. How would we maximize the information content of such a progression?

The reason that this may be an interesting question is that we may optimize the input vector of an agent in reinforcement learning this way. Or, with little modification, generate new tactics for players in live games, or make for more efficient operations research at a mine or a shop or even a government. There has been such a problem that has been âsolvedâ by artists all over the world. That is music theory. A scale of 12 notes is defined that form a group with a generator of 1 when describing the raw notes and 5 when describing the frequencies. This leads to a chromatic scale of 12 notes with each 7Th note being a 5Th apart in terms of frequencies. As you know this generator 5 induces a lot of structure in this system with the goal of maximizing the information content of the arrangements of chords and melodies that composers may come up with. We shall create a scale of 12 notes but rather than define a movement of a fifth in terms of frequencies we shall use vector notation and equate it to orthogonality. So every 5th vector is orthogonal in a cycle of vectors. That gives us a circle of vectors that correspond to the key signatures in a musical scale. Using the equivalence between this system and the circle of fifths we can use this to reorder the notes in ascending order according to their lettering. Automatically we can see that with this system certain masking out and masking in of notes as a preselection of permitted collections correlates with major and minor scales. In fact, it can be shown that adding notes from outside of-these scales in most cases correlates with selecting vectors that share a large component with vectors/notes already in the scale. And possesses less information content. In music, this shows up as disharmony. Of course, the structures we have built have a lot of nuance, it may be good to select a not from the minor of a major scale in a certain key in order to increase information content. In short with our system of vectors we can tackle the problem mentioned above using algorithmic information theory. Also note that when an agent seeks to optimize the information content of the changes in its input vector , it may choose to move in 5Thâs, where its input vector is one of 12. But occasionally more âsurpriseâ can be generated by moving in 4Thâs for example. But the tension created by this must be resolved with a 5ths movement of some sort that is strong. This phenomena is the way we craft the exact nature of the information content found within the agentâs policy

.II. RESEARCH OBJECTIVES

A research objective would be to find out what sort of structures are defined or induced when there are more than one generator to the group.

Another research objective would be to find out is if we can apply a genetic algorithm to the actions vector of an agent in such a way that the structure in the state is maximized including the global reward in the fitness function as well

Another research objective is to find out if the information content found in the firing of neurons can be modeled with a fitness function and extend the input vector such that the extra dimensions represent nodes that can fire or not in harmony with reality.

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