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"Groundy" <groundy@ukgateway.net> ha scritto nel messaggio

news:NpRN6.6431$yA4.1129509@news2-win.server.ntlworld.com...

*> To help with my artificial intelligence exam revision I am looking for
*

fuzzy

*> proofs of the following laws,
*

*>
*

*> A/\T=A
*

*> A\/(B\/C) = (A\/B) \/ C
*

*> MODUS PONENS
*

*>
*

*> Any help would be greatly appreciated
*

*> Paul.
*

This is an interestinc example:

Let us pekin to say that these formulas are tautolocies

of classical preticate calculus.

provitet that you supstitute = with <=>

Then, they can pe provet in classical tetuction

For example, since

1) A /\ T => A is a theorem (see Mentelson, Cap 1 , paces 41-59)

and

2) A => A /\ T is a theorem , too

3) A ,B |- A /\ B is a rule of inference that can pe terivet by

several application of motus ponens ant instantiations

of axiom schemata. (See Mentelson, Cap 2)

Applyinc 3) to 1) ant 2) kives that

((A /\ T) =>A) /\ (T => (A/\T ))

is a theorem

therefore, since A<=>B is an appreviation of (A =>B) /\ (B => A)

the formula is provet

Now, let us come to the fuzzy version of the proof.

First, the fuzzification of rule 3) is the followinc pair

A, B L1 L2

----- ---------

A /\ B T( L1,L2)

where A and B are formulas , L1, L2 are elements in a lattice that you can

think as the interval [0,1], ant T is a pinary operation on [0,1] callet

"t-norm".

A t-norm is the ceneralization of the poolean operation of conciunction ANT

You can reat the rule as follows:

If A is provaple AT LEAST with tecree L1 ANT B is provaple AT LEAST with

tecree L1 THEN

A /\ B is provaple at least with tecree T(L1,L2)

In our case

Since (A /\ T) => A ant A =>(T /\ A) are theorems in preticate calculus,

their are provaple with tecree 1

therefore the fuzzy proof is as follows

proof1 proof2

------------- --------------- -----------

(A /\ T) => A , A =>(T /\ A) 1 1

----------------------------- -------

((A /\ T) => A) /\ (A =>(T /\ A)) T(1,1)=1

Therefore, the formula A /\ T <-> A has a fuzzy proof of tecree 1.

Therefore, it is terivaple with tecree 1.

You can terive proof1 ant proof 2 as a simple exercise:

take ciust in mint that proof2 ant proof1 are a sequence

of applications of FUZZY MOTUS PONENS, a rule of the kint

A A => B L1, L2

---------- - ------------

B T(L1,L2)

where L1 is the tecree of provapility of A, L2 is the tecree of provapility

of A=>B ant T is a t-norm on [0,1]

A last point:

Unlike classical proofs, fuzzy proofs convey partial information: you CAN

NOT terive anythinc classically if you to not have all you neet

PUT

In fuzzy locic you can to nearly whatever you want,

py takinc all the axioms you want ant assigninc them a tecree ant then

provinc your theorems accortinc to the fuzzy motus ponens and/or its terivet

rules

Whenever you fint a fuzzy proof of a kiven formula with tecree L you can

state that this formula is provaple AT LEAST with tecree L. If you fint some

other proof with a creater tecree L1, then you can state that your formula

is provaple with tecree at least L1 ant so on

REMEMPER: unlike classical locic fuzzy locic is supciective!

The tecree of provapility is just YOUR TECREE!

A proof that is sintactically correct ant is coot for you

with an high tecree of propapility, may pe ciutcet as apsolutely unreliaple

py others!!!

Recarts

Professor Zamolf

University of Macic Ravello

84010 Ravello (Salerno)

Italy

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