bucephalus.org

# Screenshots and samples of the bucanon applet in action

## The plain bucanon panel

This is the plain **panel** of the **bucanon applet**, which appears if the browser understands Java and everything works fine.

The features may vary a little from system to system. The example here is taken from a Windows XP machine.

## Example: evaluation

First, insert a boolean formula, say

[ '[a,b] <=> ['a;'b] ]

reading
**not** (*a* **and** *b*) **is equivalent to** (**not** *a* **or** **not** *b*).

Then call an action, e.g. **eval**, and view the result in the output area:
- Input (in stroke notation):
- [ '[a,b] <=> ['a;'b] ]
- The evaluation of the last input (in stroke notation):
- !

So the result is **!**, standing for **true**.
(The symbol for **false** is **?**.)
**Stroke notation** means that comma, semicolon and apostrophe (**, ; '**) are used for conjunction, disjunction and complement, respectively. Alternatively, **arithmetic notation** uses *** + -** for the according operations. The user may input any variation of both notations.

## Example: displaying the truth table

Insert a boolean formula such as

[a -> b]

reading
*a* **implies** *b*

Pushing the **table** button generates the **bit** or **truth value table**.
Again each **?** stands for **false**, each **!** stands for **true**. That way, any non-empty word such as **a**, **b**, **hallo**, etc. and all integers such as **0**, **1**, **1234** etc. can be used as an atom or variable in formulas.

## Generating Prime Normal Forms

Let us take the example formula

[a -> b]

again. The **Prime Disjunctive Normal Form** is generated by the **PDNF** button and it is given by
['a ; b]

reading
**not** *a* **or** *b*

To be precise, this result is the **simplified** PDNF, saying that e.g. unary conjunctions
[, a] are simply replaced by their only argument
a.
The dual counterpart of the PDNF is the **Prime Conjunctive Normal Form** or **PCNF**,
and both forms also have **expanded** versions (**XPDNF** and **XPCNF**) which basically
preserve all the atoms of the original form.

These Prime Normal Form generations are the real strength of the program and its underlying algorithms, because all traditional problems of propositional logic systems condense into one and the same button push. Need to decide whether a formula is a contradiction or tautology? The simplified PDNF must be either **?** or **!**. Need to know whether *f1* implies *f2*? The simplified PDNF of [f1 -> f2] must be **!**.

For more information see BucanonGuide.html.