# Knowledge Representation and Reasoning

Knowledge can be represented within the machine environment using several different approaches. The most common approach is with propositional calculus or with first order predicate calculus. This approach is common to mathematics, philosophy, linguistics, and conventional computer science. Another approach is through Theory-based Semantics, which states that knowledge is dynamically expressed through associative relationships of concepts, ideas and thought patterns that constrain meaning according to the theory that justifies their relationships. Logic-based representations assumes that meaning is represented through language-based propositions that adhere to universal truth-conditions, whereas theory-based semantics take the quantum relativist view, that meaning exists as a condition under which it can be verified and certified as acceptable without regard to universal truth-conditions. In either case, the test of veracity of any knowledge representation is in its capacity to reduce question uncertainty. Those who adhere to the theory-based semantics approach, accept logic-based representations as valid for mechanical, process oriented behaviors, but question the fidelity of this system of thought when reasoning with behaviors common to the human experience such as beliefs, values, and social behavior. From their perspective, axiological, value-based knowledge representations transcend the limitations of logic.

## Logic-based Representations

Propositional calculus deals with the truth or falsehood of logical statements (propositions). Predicate calculus deals with statements involving the properties of objects (these properties and relationships are called predicates like verbs are in grammar).

In propositional calculus variables such as p, q and r, are used to represent truth or falsehood. These variables are modified and joined with operators such as and, or, and not. All of these combine two statements into a single statement except for not, which only modifies one statement. There is a standard way to represent these logical operators in symbolic logic. OR is represented with a sans-serif, capital V (from the Latin vel). AND is represented with the same symbol as OR, but upside-down (resembling the A in and). NOT is represented with a tilde (~), but is also written another way. A statement involving several variables can be called a compound statement. A compound statement's value (true or false) is determined from the values of the variable inside it. A statement that is the negation (the not operation) of another has the opposite truth value. If p is true, ~p is false, and vice versa. A statement formed with the and operation is true only if both of the statements joined are true, it is false otherwise. A statement formed with the or operation is true if at least one the statements joined together are true (one or the other, or both), and false if both statements are false.

Reasoning in propositional calculus consists of determining whether statements are true or false in the presence of incomplete information. This can be done using logical equivalences such as the associativity and commutativity of AND and OR, and the double negative law.

First order predicate calculus allows more expressiveness than propositional calculus. Predicate calculus allows properties of arbitrary objects to be described. Predicate calculus includes two quantifiers. The existential quantifier is used to state that an object with a given property exists and can be read as "there exist". The universal quantifier is used to state that all objects have a property and can be read as "for all". The existential quantifier is written as a backwards capital sans-serif E. The universal quantifier is written as an upside-down sans-serif capital A. The universal quantifier and the existential quanifier are related to each other; if all objects have a property, then no object exists that doesn't have that property. Peano's axioms define numbers using first order predicate calculus, except for the law of induction which requires second order predicate logic, where properties can have properties (e.g. equality being transitive). Peano's axioms say that 1 is a number, every number has a successor (which is also a number), that no number has 1 as a successor (1 is the first number), that no two numbers have the same successor (the number doesn't have forks), and states the law of induction (if 1 has a property, and any number having that property means the next number has it too, then all numbers have that property). First order predicate calculus is used instead of higher order logics in knowledge representation because it is more difficult to do reasoning in higher order logics, and first order logic usually is expressive enough.

## Theory-based Semantics

Theory-based semantics as defined by Dr. Richard L. Ballard, describes knowledge representations that are based on the premise that the binding element of human thought is "theory," and that theory constrains the meaning of concepts, ideas and thought patterns according to their associative relationships. For this reason, knowledge stores of theory-based semantic representations do not just represent meaning, they precisely embody the very knowledge they are intended to represent - they KNOW. Ballard's Knowledge Science says that knowledge (Knowledge = Theory + Information), is any input of theory or facts that reduces question uncertainty. From this perspective, theory represents 85% or more of knowledge with information (data, facts of situations and circumstances), representing 15% or less. The built-in, a`priori intelligence of theory also defines concept values and purpose, which in turn, determines each concepts influence on every other concept, idea or thought pattern with which it is associated. Theory-based semantics holds that this state of intelligence is valid whether a concept is held in the mind, or is represented within the machine environment. Once learned by people or machines, theory endures with great tenacity, changing only when new paradigms of thought subsumes or replaces the well-justified theories that we use to understand our world. The endurance of data and facts, however, are quite different. They are in a constant state of flux as situations and circumstances dynamically change one moment to the next. From the standpoint of theory-based semantics, most appearances of change are not new - they are only new to us. The facts may be different, but most often, the theory that defines situations and circumstances remains the same.