Re: Fuzzy proofs.

From: professor Zamolf (zamolf@yahoo.com)
Date: Mon May 28 2001 - 15:30:13 MET DST

  • Next message: Sidney Thomas: "Re: Fuzzy proofs."

    "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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