## Saturday, December 28, 2013

### Degree of a transseries

I believe that the construction of surreal numbers and transseries should be done through a process of extensions going in order from natural numbers, to integers, to rationals, to reals, and then to real polynomials. In keeping with the fact that these number systems are extensions of one another we can compute the integer part of a real number or the standard part of a real number.

An analogous notion is computing the degree of a transseries. The way I think we should go about doing this is to compute the standard part of $\frac{ln(f(x))}{ln(x)}$. Since transseries are closed under logs and division this remains a transseries for any transseries so we can still compute its standard part.

If we plug in a constant we get $\frac{ln(c)}{ln(x)}$ which goes to zero for any constant. if we plug in $ln(x)$ we get $\frac{ln(ln(x))}{ln(x)}$ which also goes to zero, If we plug in the square root we get $\frac{ln(\sqrt{x})}{ln(x)}$ which goes to 1/2, if we plug in quadratics we get a degree of two, if we plug in quintics we get a degree of three, and if we go on to plug in exponentials we get a degree of infinity. It is necessary to use the infinity to deal with exponential degrees just as it is necessary to use infinity to deal with transfinite values when computing the standard part.

The next phase to build our transseries system is to make use of exponential and logarithmic values. With exponentials we get growth rates greater then any polynomial degree and with logarithms we get growth rates that are so small they are not distinguishable in the degrees system. In essence exponentials are of infinite degree and logarithms are of infinitesimal degree.

## Thursday, December 26, 2013

### Asymptotic analysis of combinatorial species

Given a collection of classes of entities we can arrange those entities in terms of generality in order to get an ontology. In particular this can be applied to mathematical structures over a set:

Well it is certainly the case that there is no set number of structures we can apply to a set there is a set number of structures in all the other classes described in the above ontology. The number of families of sets over a set is $2^{2^n}$. The number of N-ary relations over a set is $2^{n^k}$ which in the case of unary relations is $2^n$, in the case of binary relations is $2^{n^2}$, and in the case of ternary relations is $2^{n^3}$.

Some species although different from other species have the same growth rates as them. Coreflexive relations have a growth rate of $2^n$ like unary relations because both of them involve picking out subsets of a set. This leads to a natural bijection between unary relations and coreflexive binary relations based upon membership so for example the unary relation $\{(1), (2), (3)\}$ might go to the binary relation $\{(1, \; 1), (2, \; 2), (3, \; 3)\}$.

We can get the growth rates for a variety of other classes of structures. For reflexive relations we get $2^{n^2 -n}$, for symmetric relations we get $2^{\frac{n^2}{2}+\frac{n}{2}}$, for independency relations we get $2^{\frac{n^2}{2} - \frac{n}{2}}$, for antisymmetric relations we get $2^n 3^{\frac{n^2}{2}+\frac{n}{2}}$, for asymmetric relations we get just $3^{\frac{n^2}{2}+\frac{n}{2}}$, for functions we get $(n+1)^n$, and for binary operations we get $(n+1)^{2^{n}}$.

Each of these growth rates can be expressed as transseries, for example, $(n+1)^{2^{n}}$ can be expressed as $e^{\log(x+1)e^{\log(2)x}}$. The ontological order is a suborder of the asymptotic order on the collection of species as every subspecies of some other species has a smaller growth rate then its parent species.

## Thursday, December 19, 2013

### The role of light in perception

One of the most important sources of perception of the real the world is light. Light includes radio waves, microwaves, infrared light, ultraviolet rays, x rays, and gamma rays in addition to the visible light that is most familiar to humans. These other sources of light are essential to the astronomical perception of physical entities across the universe.

Through light we can get images and videos of the environment. With these perceptual stimuli we can begin to create spatiotemporal models of the environment. With the process of perceptual recognition we can give semantic meaning to stimuli by relating them to categories.

## Thursday, December 12, 2013

### Perceptual recognition

Given a set of observations of a partially observable system and a set of categories of parts of the partially observable system it is useful for us to be able to recognize rather or not parts of the environment belong to one of our categories. There are two types of perceptual recognition: non-localized recognition and localized recognition.

The process of non-localized recognition simply compares our observations to the definition of our category in order to make a determination. Localized recognition on the other hand attempts to locate entities within our model of the external environment.

An example of localized recognition is that we given some observations of a star such as its spectra we can determine that said star is a sun-like star. On the other hand with localized recognition we can determine that said star is the sun. Localized recognition generally requires information about the location of objects in the environment relative to one another.

## Friday, December 6, 2013

### Perception action cycle

An agent receives inputs from its environment in the form of perceptions and it outputs things to its environment in the form of actions. Perceptions and actions are combined together in the perception action cycle. The perception action cycle is an inherently causal process as actions precede perceptions which may themselves be preceded by previous actions.

In stochastic systems it is especially important for agents to be able to perform causal reasoning so that they can determine what the most probable effects of their actions will be. The predicted outcomes of an action can be integrated into an agent's plans so that an agent can be prepared to deal with all contingencies.

## Wednesday, December 4, 2013

### Algebraic ontology

I have created an ontology that contains all the most common structured collections used in mathematics including sets, lists, relations, binary relations, partial orders, lattices, distributive lattices, functions, ternary relations, monoids, groups, semirings, rings, fields, residuated lattices, bilattices, etc.

I intend to actively work on improving this ontology so that I can provide a formalization of the most important elements of algebra for my reasoning engine.