Glossary

# Default Logic Simulation - Online-Help

This applet should visualize the method for finding extensions of a given default theory (comprising definite knowledge plus a number of default assumptions) presented in the lecture "Wissensbasierte Systeme" at the TU Vienna. This lecture is obligatory for students of computer science.

### System requirements:

For running this applet all you need is a Java 1.1 capable browser, e.g. Netscape 4.06 or higher. We have tested it under Solaris and Windows95/NT using Netscape.
Description of the applet: After loading you can choose an example from a list (partly examples from the lecture, partly taken from old exams):

By clicking the button  you can open this Help-Site.

After you have selected an example, the sets W (the initial knowledge) and D (a set of Defaults), which represent the chosen default theory, are shown.
The button  remains inactive until you have chosen an example. Clicking this button you get a short description of the selected example.
The button  also remains inacative until you select an example, it opens the simulation window, which shows how candidates for extensions are calculated.

After the candidates for extensions have been calculated, a table showing the list of candidates to be tested is displayed in the applet window, e.g.:

You can select one of the canditates for testing either by selecting it in this list   or by simply clicking on it in the table. The testing of a candidate is again shown in an own simulation window, that visualizes how you can check if a candidate really is an extension or not.

### Simulation window

The calculation of possible candidates and testing of a single candidate is shown in an own window. Here you can see the most important elements of the user interface:

### Glossary

• Consequent: the conclusion of a default.
• DS refer to classical reduct.
• Default: a default is a classical inference rule plus a set of consistency conditions, which allows you to model rules that are "normally" true, that is, as long as the consistency condidtions are not violated.

• A Default consists of three parts,  the prerequisite, a set of justifications (consistency condiditons) and the consequent (conclusion).

Formal definition: Let A(X), B1(X), B2(X), ..., Bn(X), C(X) be first order formulas, with X = (x1, x2, ..., xn) being the set of free variables occuring in these formulas. Then a Default is an expression of the form:

A(X)                                     ...    prerequisite
{ B1(X), B2(X), ..., Bn(X)}   ...    consistency conditions ( = justifications)
C(X)                                     ...    consequent,  conclusion

The Default  for example can be read as follows: "If A is valid and there are no objections to B1, B2, ..., Bn , then C.

•  Default theory:  a default theory T=(W,D) is an ordered pair, where W is a set of closed formulas and D is a set of defaults. W represents the premises of T, that means the initial knowledge ("hard facts"). D represents relations that are normally valid.
• Extension: The extension of a default theory (W,D) is intuitively spoken one possible world which emerges from the initial knowledge when applying a maximum number of defaults, that don't "block" themselves or each other.

• Formal definition (Reiter 1980):
Let T = (W,D) be a (closed) Default Theory
1) We define an operator T as follows:
Let S be a set of closed formulas. Then T(S) is the smallest set that fulfills the following conditions:
DT1(S):    K = Th(K)    ...    that means K is its deductive closure.
DT2(S):    W is a subset of K.
DT3(S):    If D and A  K and B1B2, ..., Bn S, then C  K.
2) A set E of closed formulas is an extension of T iff T(E) = E, that means E is a fixed point of T.
• -Operator (read "Gamma-Operator"): refer to Extension (T)
• Justifications: The justifications represent consistency conditions, that means if you want to apply a default, all justifications have to be satisfied.
• classical Reduct: Let S be a set of closed formulas (in this special case a candidate for an extension) and let D be a set of (closed) defaults. The classical reduct of D with respect to S is the following set:

• DS = {  | D, B1,B2,...,BnS }
The classical reduct is a set of residues.
• Minimality/maximality of extensions: Let E, F be extensions of a default theory T=(W,D), then  E  F  always implies that E = F.
• normal: A default of the form  is called normal; a default theory, where all defaults in D are normal is called normal default theory. Each (closed) normal default theory has at least one extension!
• Prerequisite: the precondition for applying a default. A default with only a prerequisite but no justifications is very similar to a classical inference rule.
• residue: the residue of a default  is the classical inference rule .
• seminormal: A default of the form  is called seminormal; a default theory where all defaults in D are seminormal is called seminormal default theory. With seminormal defaults you can encode priorities.
• Th(W): The deductive closure of a set W, that means all formulas that can be classically derived from W, e.g. using resolution.
• ThDs(W): Let S be a closed set and T=(W,D) a default theory, then ThDs(W) is the set of all formulas, which can be classically derived from W by using in addition the rules in the classical reduct DS.

• The set ThDs(W) equals T(S) !!!! Since each fixed point of T(S) represents an extension, S is an extension of the default theory T = (W,D) iff
ThDs(W) = S
The tests of the respective candidates in our applet are based on this concept, as you will find out when studying the simulations!