Wednesday, November 27, 2013

Cognitive ontology based upon observability

Given a system an intelligent agent can either completely observe the current state of the system or it can only only observe certain parts of the system. Based upon this idea we can categorize cognition in terms of observability as described by the Hasse diagram below:

The category of reasoning includes all cognitive processes that only involve the mind of the agent and not the external world. One fundamental reasoning process is categorization which can be used for example to produce an ontology of abstract structures such as lists, sets, relations, and numbers. Numeric reasoning can involve not just the set of real numbers but surreal numbers as well.

A fundamental part of abstract reasoning is optimization which is the process of finding an optimal item amongst a set of possibilities based upon some criterion such as in linear programming. This also applies to planning problems in games in which the agent must find the best course of action from the current state to achieve a certain goal.

Often times we cannot simply search the entire game tree which means that we need to use reinforcement learning instead to arrive at the optimal solution to the problem. Unlike with logical deduction the reinforcement learning process only produces approximate results. Another learning process is clustering which allows the agent to create new categories rather then simply working with the established ones.

All of these abstract reasoning processes are fundamentally divorced from the problems of incomplete and inconsistent information that arise from perception. It may be the case that incomplete information is a fundamental property of agents in the physical world as agents can only ever perceive things in their past light cone. Perception include vision, hearing, smell, taste, and touch among other senses. Intelligent agents should be able to learn from all these different sorts of perceptual signals.

Sunday, November 24, 2013

Basic cognitive ontology

A basic ontology of cognitive processes is presented in the hasse diagram below. This defines cognition in terms of perception, optimization, learning, and categorization.

I have been saying for a while now that perceptual learning is fundamental to cognition and I have written about this so it shouldn't be too surprising that perception and learning are both listed above. What is perhaps more interesting is that I now list categorization as a core cognitive process.

After creating so many ontologies like this one it is my contention that the categorization and characterization of entities plays a fundamental role in cognition. Simply having a vast ontology like the one provided by Cyc is not enough however as an intelligent agent should also be able to form new categories on the fly using clustering.

In a way essentially every cognitive process can be described as optimization, learning for example can be described as finding an optimal model that fits an experiential data set. Optimization and learning are deeply connected through the process of reinforcement learning which teaches an intelligent agent to perform optimally in its environment. Optimization and learning are so closely tied to one another it is questionable rather they should even be described as separate processes.

Wednesday, November 20, 2013

Mereology and causality

The basis of the ontological category of concretia is spacetime. We perceive the parts of spacetime (events) and relate those to one another through cause and effect relations (causality). In this ontology spacetime is itself an event: it is the largest possible event.

In this ontology causality relations do not occur between the atoms of causal set theory but rather between parts of spacetime which may be composed of such atoms. This implies that we are actually comparing different subsets of a partial order to one another.

Precedence occurs when all the elements of one part of spacetime cause another part. Causual independence occurs when the light cone associated with one of the events doesn't intersect the other events spatiotemporal presence. On possible example is that everything that happens in a given year on Earth is causually independent of everything that happens in a given year on Alpha Centauri as there is 4.367 light years of separation between us.

Monday, November 18, 2013

Causual set theory

Just as mereology partially orders physical structures based upon parthood causality partially orders physical processes based upon precedence. Both mereology and causality should play a fundamental role in any ontology dealing with physical objects. The Standard Upper Merged Ontology (SUMO) seems well suited enough to deal with both sorts of reasoning as it explicitly distinguishes between physical objects and physical processes in its ontology.

Causal reasoning plays a fundamental role in AI as any intelligent agent should be able to perform causal recognition by taking an effect and then determining what its causes may be and prediction by taking a cause and then determining what its effects may be. In either case probabilistic methods may play a role in the reasoning process as a system may use the frequencies of certain effects produced by a cause in order to help predict what the effect will be.

One interesting theory is that of causal sets which unifies quantum gravity and relativity by positing that the physical universe is fundamentally structured by causality. Time dilation indicates that we cannot place an absolute total ordering on events so the only way to effectively understand time is with a causal partial order like the one provided by causal set theory.

Sunday, November 10, 2013

On the ontological nature of incidence structures

Lets consider the following functional dependencies structure #{(#{} #{}) (#{0} #{0}) (#{0} #{1}) (#{1} #{0}) (#{1} #{1}) (#{0 1} #{0 1}) (#{0 1} #{0}) (#{0 1} #{1}) (#{0 1} #{}) (#{0} #{}) (#{1} #{})}. Should this structure be classified as a set, a relation over a set, or a relation over the power set of a set? Depending upon how this structure is classified it could have a size of 2, 4, or 11.

My recently line of thinking is that the underlying set of an incidence structure should be metadata and most ontological questions will be answered by asking about the overlying set itself which means that this set would still qualify as a transitive binary relation regardless of what it is defined over. I am not sure that this is the right approach but this seems to make sense for now.

For non-incidence structures like measure spaces, metric spaces, and rings there is no need to deal with the problem of different underlying sets. With this approach I believe I am on track to classifying most mathematical structures though I haven't exactly worked out exactly how to classify state spaces yet.

Saturday, November 9, 2013

Closure of binary relations

The three main classes of binary relations that exhibit closure operations are reflexive, transitive, and symmetric classes. From the intersection of these classes we get some other classes of relations that exhbit closure operations:

Preorder: reflexive and transitive
Dependency relation: reflexive and symmetric
Equivalence relation: reflexive, symmetric, and transitive

Another class of relation that exhibits a closure operation is the set of complete relations. Its corresponding closure operation is complete closure.