EE290N - Specification and Modeling of Reactive Real-Time Systems

Lecture 7 - September 17, 1996, Scribe: Roberto Passerone (roby@ic.eecs).

Terms Defined :

  • No new terms have been defined.

    Lecture:

    The lecture starts with a clarification on the set of all possible discrete events signals.

    A signal s in a timed system is a discrete event signal if there exists an order preserving bijection from the natural numbers to T(s), where T(s) is the set of all tags in s. Let Sd be the set of all possible discrete event signals.

    A tuple of signal ss in a timed system is a discrete event tuple of signals if there exists an order preserving bijection from the natural numbers to T(ss), where T(ss) is the set of all tags in ss. Let Sd^(N) be the set of all possible discrete event N-tuple of signals.

    Then clearly Sd X Sd = Sd^2 != Sd^(2). In fact, suppose:

  • s1 = { ( t, v ) ; t = 0, 1, 2, ... ; v = 1 }
  • s2 = { ( t, v ) ; t = i / i + 1; i = 0, 1, 2, ... ; v = 1 }

    then s1 (- Sd and s2 (- Sd, but ( s1, s2 ) (/- Sd^(2), although ( s1, s2 ) (- Sd^2.

    A summary of the math is then presented:

    Then an example of the use of the Banach theorem is presented to show how to obtain the fixed point as the limit of a sequence. The theorem proves that if the function is D-causal and the metric space is complete, then the limit exists for a sequence of this kind:

  • s1 = f( s0 );
  • s2 = f( s1 );
  • s3 = f( s2 );
  • ...

    and the limit is the fixed point. Using the metric:

  • lim(i->oo) si = s iff for all e > 0, there exists M > 0 such that for all n > M => e > d( s, sn ).

    The Banach fixed point theorem applies only if the metric space is complete. To define completeness we also need the notion of a Cauchy sequence:

    Given a metric space ( X, d ): ( in our context X = Sd^(N) and d is the Cantor metric )

  • Def: a sequence { a1, a2, a3, ... } C= X is a Cauchy sequence iff d( am, an ) -> 0 as m, n -> oo (no matter how m and n -> oo)
  • Def: a metric space A C= X is complete if every Cauchy sequence has a limit in A.

    To prove that a metric space is complete we need to show that every Cauchy sequence has a limit in the space. Once completeness is proved, then the Banach fixed point theorem applies.

    A similar theorem applies in the case that the metric space be compact.

    Given a metric space ( X, d ):

  • Def: A C= X is compact if every infinite sequence in A has an infinte subsequence that converges to a point in A.

    Intuitively, the notion of compactness is a more general notion of finiteness.

    It is then informally stated that:

  • Compactness => Completeness => Closeness

    For example, the set of the real numbers is complete but not compact. In fact, the sequence of the integer numbers has no infinite subsequence that converges.

    The set of discrete event signal is not compact if |V| = oo. In fact, define a sequence like:

  • sn = { ( 0, vn ) } n = 1, 2, 3, ...

    where all the vn are distinct. Then, d( sn, sm ) = 1 for all m and n. Thus, no infinite subsequence of sn converges.

    An example of a compact metric space is [ 0, 1 ] C R, with the usual metric d( x, y ) = | x - y |.