Monday, March 31, 2014

Logical foundations of mathematics

The object of mathematics in general is to reason under conditions of certainty. This involves the consideration of logical propositions with truth values such as true and false. With truth values such as true and false we can form predicates that return such truth values and an ontology of such predicates.

With predicates such as sets we can describe every object encountered in mathematics. And with an ontology we can get a clear picture of all these different types of objects. Then we can use logic and our ontology to reason about mathematical objects. By combining certain mathematical knowledge with logical models of uncertainty domains we can use logic to describe all declarative knowledge.

Wednesday, March 26, 2014

Logical models of uncertain domains

Well the set theory and logical predicate calculus provide a good foundation for understanding certain abstract domains we need to extend our logic to be able to deal with uncertainty. Given a system with unknown characteristics then we can create a model of that domain. Models classify uncertain domains by allowing us to apply predicates. Here are two of the simplest kinds of models:
  • Simple models: the simplest possible model of an uncertain domain is a set of values the domain may take. Applying a predicate to such a model yields true if the predicate is a subclass of the set of values of model, false it if it is independent of the set, and unknown otherwise.
  • Probabilistic models: a more advanced type of model is one in which applications of a predicate return probability values. Probabilities should be higher the more general a predicate is so that the probability that an object is an entity is always one.
Given any system with uncertain characteristics we can create a logical model of the system. If our agent is embodied within some environment then it can create a world model of the entire environment including it. If that environment is our physical reality then a world model containing all physical entities and their mereological and causal properties can be produced. In this way logic can be used to organize all declarative knowledge about real or imagined entities.

Sunday, February 23, 2014

Logical organization of declarative knowledge

Mathematical logic provides the foundation for the organization of all declarative knowledge stored by an intelligent agent. By creating an ontology we can organize our knowledge base in terms of the logical classification of entities. The main categories in a logical ontology of mathematical entities are ordered collections, unordered collections, algebraic structures, and elements of algebraic structures. Relations are specified as unordered collections of ordered collections and algebraic structures are specified as a system of relations over a underlying set.

Well Von Neumann ordinals can be specified as unordered collections numbers including naturals, integers, rationals, reals, surreals, complex numbers, numbers, surcomplex numbers, quaternions, octonions, among others can be specified as elements of algebraic structures. In this way the four categories of the upper ontology described previously are sufficiently powerful to classify most mathematical entities.

The only limitation of this approach to classification is that partially observable systems cannot be described as mathematical structures in the traditional sense. Nonetheless, it is my contention that we can represent the set of states a partially observable system may take and reason about them using logic. We can enrich the system with a mereology and then form a logical ontology of parts of the system.

Well it is certainly true that logic provides an ideal foundation for the formalization of all declarative knowledge this doesn't mean that we don't need more then this for general intelligence such as a system of procedural knowledge for approximate optimization, a module system for loading domain specific knowledge such as interaction, and an advanced learning system for acquiring new knowledge and transforming modules.

Tuesday, February 18, 2014

Classification of partially observable systems

Our logical ontology provides us with the means to classify abstract fully observable systems such as combinatorial games. This approach places mathematical entities into an advanced ontology that includes lists, sets, relations, and algebraic structures among others. Partially observable systems such as spacetime provide us with the conundrum of determining how to classify and identify entities based upon uncertain and limited perceptual information on the external environment.

It is my contention that the methods of mathematical logic should be extended to deal with partially observable systems by providing the means to classify entities based upon a limited set of observations about them. Given a partially observable system the process of classification involves taking the environment and classifying it as an entity and then receiving additional and using that to further specify the class of our environment.

Given our observations we deduce that physical reality is composed of galaxies, star systems, planets, molecules, atoms, subatomic particles, and various other classes of concrete entities. All of these different concrete entities are parts of the universe in the logical mereology. The universe is the top level entity in the mereology and all other instances of concrete entities are parts of it.

Friday, February 14, 2014

Signatures of algebraic structures

Algebraic structures have signatures associated with them. These signatures specify the set of relations defined over the underlying set of the algebraic structure. Here are a few signatures:
  • Semiring: (S + *)
  • Ordered group: (S <= +)
  • Ordered ring: (S <= + *)
  • Ordered differential composition ring: (S <= + * c d)
The internal relations of algebraic structures are used to define mathematical objects such as numbers and games. Games are algebraic structures with a binary relation move between positions and a unary relation winning that tells rather or not a given position is winning or not.

Tuesday, February 4, 2014

Subalgebras and reducts

Given an algebraic structure with a variety of symbols such as addition and multiplication associated with it we can take subalgebras of the structure and reducts of the structure. Subalgebras are produced by taking a subset of the underlying set of the structure. Reducts are produced by taking a subset of the set of symbols associated with the structure.

For example an additive group (S,+) may be a reduct of a field (S,+,*) and an ordered group (S,<=,+) both of which are reducts of an ordered field (S,<=,+,*). A subreduct of a structure is a substructure of the algebraic structure that is both a subalgebra and a reduct because it uses a subset of the underlying set and a subset of the set of symbols used by the structure. The ring of integers is a subreduct of the ordered field of rational numbers.

Saturday, February 1, 2014

Algebraic structures and symbols

When I first constructed my system for dealing with algebraic structures I only provide support for structures with a pair of operations like rings and fields which were numbered so there were no symbols involved.
  • <= comparison
  • + addition
  • * multiplication
  • c composition
  • d differentiation
  • m metric
  • w weight
The first five of the above symbols are used for ordered differential composition rings like those of transseries. The second two describing metric and weighting functions subsume what was previously provided by the weighted hypergraphs system. Vector spaces will also be provided with the scalar multiplication operation being denoted by the symbol s for now.