## Tuesday, November 30, 2010

### Fuzzy Inferencing

Fuzzy Inferencing

The process of fuzzy reasoning is incorporated into what is called a Fuzzy Inferencing System. It is comprised of three steps that process the system inputs to the appropriate system outputs. These steps are 1) Fuzzification, 2) Rule Evaluation, and 3) Defuzzification. The system is illustrated in the following figure.

1Fuzzification

is the first step in the fuzzy inferencing process. This involves a domain formation where crisp inputs are transformed into fuzzy inputs. Crisp inputs are exact inputs measured by sensors and passed into the control system for processing, such as temperature, pressure, rpm's, etc.. Each crisp input that is to be processed by the FIU has its own group of membership functions or sets to which they are transformed. This group of membership functions exists within a universe of discourse that holds all relevant values that the crisp input can possess. The following shows the structure of membership functions within a universe of discourse for a crisp input.

2 Degree of membership:

degree to which a crisp value is compatible to a membership function, value from 0 to 1, also known as truth value or fuzzy input.membership function, MF: defines a fuzzy set by mapping crisp values from its domain to the sets associated degree of membership.

3.crisp inputs: distinct or exact inputs to a certain system variable, usually measured

4.parameters external from the control system, e.g. 6 Volts.

5.label: descriptive name used to identify a membership function.

6.scope: or domain, the width of the membership function, the range of concepts, usually numbers, over which a membership function is mapped.

7.universe of discourse: range of all possible values, or concepts, applicable to a system variable. When designing the number of membership functions for an input variable, labels must initially be determined for the membership functions. The number of labels correspond to the number of regions that the universe should be divided, such that each label describes a region of behavior. A scope must be assigned to each membership function that numerically identifies the range of input values that correspond to a label. The shape of the membership function should be representative of the variable. However this shape is also restricted by the computing resources available. Complicated shapes require more complex descriptive equations or large lookup tables. The next figure shows examples of possible shapes for membership functions.
Classical Planners use the STRIPS (Stanford Research Institute Problem Solver)

language to describe states and operators. It is an efficient way to represent

planning algorithms.

### Fuzzy Concepts. Hedges

Another important feature of fuzzy systems is the ability to define "hedges," or

modifier of fuzzy values. These operations are provided in an effort to maintain

close ties to natural language, and to allow for the generation of fuzzy statements

through mathematical calculations. As such, the initial definition of hedges and

operations upon them will be quite a subjective process and may vary from one

project to another. Nonetheless, the system ultimately derived operates with the

same formality as classic logic. The simplest example is in which one transforms

the statement "Jane is old" to "Jane is very old." The hedge "very" is usually

defined as follows:

m"very"A(x) = mA(x)^2 Thus, if mOLD(Jane) = 0.8, then mVERYOLD(Jane) = 0.64.

Other common hedges are "more or less" [typically SQRT(mA(x))], "somewhat,"

"rather," "sort of," and so on. Again, their definition is entirely subjective, but

their operation is consistent: they serve to transform membership/truth values in a

systematic manner according to standard mathematical functions.

A more involved approach to hedges is best shown through the work of Wenstop

in his attempt to model organizational behavior. For his study, he constructed

arrays of values for various terms, either as vectors or matrices. Each term and

hedge was represented as a 7-element vector or 7x7 matrix. He ten intuitively

assigned each element of every vector and matrix a value between 0.0 and 1.0,

inclusive, in what he hoped was intuitively a consistent manner. For example, the

term "high" was assigned the vector 0.0 0.0 0.1 0.3 0.7 1.0 1.0

and "low" was set equal to the reverse of "high," or 1.0 1.0 0.7 0.3 0.1 0.0 0.0

Wenstop was then able to combine groupings of fuzzy statements to create new

fuzzy statements, using the APL function of Max-Min matrix multiplication.

These values were then translated back into natural language statements, so as to

allow fuzzy statements as both input to and output from his simulator. For

example, when the program was asked to generate a label "lower than sortof

low," it returned "very low;" "(slightly higher) than low" yielded "rather low,"

etc. The point of this example is to note that algorithmic procedures can be

devised  which translate "fuzzy" terminology into numeric values, perform

reliable operations upon those values, and then return natural language statements

in a  reliable manner.

### Fuzzy Concepts.

The notion central to fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. For example, let us take the statement: "Jane is old."

If Jane's age was 75, we might assign the statement the truth value of 0.80. The statement could be translated into set terminology as follows:

"Jane is a member of the set of old people."

This statement would be rendered symbolically with fuzzy sets as:

mOLD(Jane) = 0.80

where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 1.0.

At this juncture it is important to point out the distinction between fuzzy systems and probability. Both operate over the same numeric range, and at first glance both have similar values: 0.0 representing False (or non- membership), and 1.0 representing True (or membership). However, there is a distinction to be made between the two statements:

The probabilistic approach yields the natural-language statement, "There is an 80% chance that Jane is old," while the fuzzy terminology corresponds to "Jane's degree of membership within the set of old people is 0.80." The semantic difference is significant:

the first view supposes that Jane is or is not old (still caught in the Law of the Excluded Middle); it is just that we only have an 80% chance of knowing which set she is in. By contrast, fuzzy terminology supposes that Jane is "more or less" old, or some other term corresponding to the value of 0.80. Further distinctions arising out of the operations will be noted below.

The next step in establishing a complete system of fuzzy logic is to define the operations of EMPTY, EQUAL, COMPLEMENT (NOT), CONTAINMENT, UNION (OR), and INTERSECTION (AND). Before we can do this rigorously, we must state some formal definitions:

Definition 1

: Let X be some set of objects, with elements noted as x. Thus,

X = {x}.

Definition 2

: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases.

Definition 3

: A is EMPTY iff for all x, mA(x) = 0.0.

Definition 4

: A = B iff for all x: mA(x) = mB(x) [or, mA = mB].

Definition 5

: mA' = 1 - mA.

Definition 6

: A is CONTAINED in B iff mA <= mB.

Definition 7

: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)).

Definition 8:

C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

It is important to note the last two operations, UNION (OR) and INTERSECTION

(AND), which represent the clearest point of departure from a probabilistic theory for sets to fuzzy sets. Operationally, the differences are as follows:

For independent events, the probabilistic operation for AND is multiplication,

which (it can be argued) is counterintuitive for fuzzy systems. For example, let us

presume that x = Bob, S is the fuzzy set of smart people, and T is the fuzzy set of

tall people. Then, if mS(x) = 0.90 and uT(x) = 0.90, the probabilistic result would

be: mS(x) * mT(x) = 0.81 whereas the fuzzy result would be:

MIN(uS(x), uT(x)) = 0.90

The probabilistic calculation yields a result that is lower than either of the two

initial values, which when viewed as "the chance of knowing" makes good sense.

However, in fuzzy terms the two membership functions would read something

like "Bob is very smart" and "Bob is very tall." If we presume for the sake of

argument that "very" is a stronger term than "quite," and that we would correlate

"quite" with the value 0.81, then the semantic difference becomes obvious. The

probabilistic calculation would yield the statement If Bob is very smart, and Bob

is very tall, then Bob is a quite tall, smart person.The fuzzy calculation, however,

would yield If Bob is very smart, and Bob is very tall, then Bob is a very tall,

smart person.Another problem arises as we incorporate more factors into our

equations (such as the fuzzy set of heavy people, etc.). We find that the ultimate

result of a series of AND's approaches 0.0, even if all factors are initially high.

Fuzzy theorists argue that this is wrong: that five factors of the value 0.90 (let us

say, "very") AND'ed together, should yield a value of 0.90 (again, "very"), not

0.59 (perhaps equivalent to "somewhat").

Similarly, the probabilistic version of A OR B is (A+B - A*B), which approaches

1.0 as additional factors are considered. Fuzzy theorists argue that a sting of low

of the resulting membership grade should be the strongest membership value in

the collection. The skeptical observer will note that the assignment of values to

linguistic meanings (such as 0.90 to "very") and vice versa, is a most imprecise

operation. Fuzzy systems, it should be noted, lay no claim to establishing a formal

procedure for assignments at this level; in fact, the only argument for a particular

assignment is its intuitive strength. What fuzzy logic does propose is to establish a

formal method of operating on these values, once the primitives have been

established.

### Planning as Search

Planning as Search:

There are two main approaches to solving planning problems, depending on the kind of search space that is explored:

1. Situation-space search

2. Planning-space searchIn situation space search

In Situation-Space search

• the search space is the space of all possible states or situations of the world

• initial state defines one node

• a goal node is a state where all goals in the goal state are satisfied

• a solution plan is the sequence of actions (e.g. operator instances) in the path
from the start node to a goal node.

In Plan-Space Search

• the search space is the space of all possible plans

• a node corresponds to a partial plan

• initially we will specify an "initial plan" which is one node in this space

• a goal node is a node containing a plan which is complete, satisfying all of the
goals in the goal state

• the node itself contains all of the information for determining a solution plan
(e.g.sequence of actions)

Goal Interaction

Most planning algorithms assume that the goals to be achieved are independent or nearly independent in the sense that each can be solved separately and then the solutions concatenated together. If the order of solving a set of goals (either the original goals or a  set of sub-goals which are the preconditions of an operator) fails because solving a latter goal undoes an earlier goal, then this version of the STRIPS algorithm fails. Hence, situation-space planners do not allow for interleaving of steps in any solution it finds.

Principle of Least Commitment:The principle of least commitment is the idea of never making a choice unless required to do so. The advantage of using this principle is you won't have to backtrack later! In planning, one application of this principle is to never order plan steps unless it's necessary for some reason. So, partial-order planners exhibit this property because constraint ordering steps will only be inserted when necessary. On the other hand, situation-space progression planners make commitments about the order of steps as they try to find a solution and therefore may make mistakes from poor guesses about the right order of steps.

## Monday, November 29, 2010

### Situation Calculus

Situation Calculus

Situation calculus is a version of first-order-logic (FOL) that is augmented so that it can reason about actions in time.

• Add situation variables to specify time. A situation  is a snapshot of the world at

an interval of time when nothing changes

• Add a special predicate holds(f,s) that means "f is true in situation s"

• Add a function result(a,s) that maps the current situations into a new situation as

a result of performing action a.