EE290N - Specification and Modeling of Reactive Real-Time Systems

Lecture 3 - September 3, 1996, Scribe: Peggy Laramie.

Terms Defined :

These terms can be found in the glossary

Inputs/Outputs of Processes

(s^m, s^n) is a paritition of s^N if N=n+m (the space can be divided)

A key question: If P1 and P2 are indivually determinate is the intersection of P1 and P2 determinate?

What constraints on processes are needed in order to make it determinate?

Partially Ordering of Tags and Events

There exists a reflective, antisymmetric, transitive relation "<" between tags.

Odering of Tags => Ordering of Events

  • Given two events: e1=(t1,v1) and e2=(t2,v2), e2>e1 <=> t2>t1
  • poset : (T,R), R is a partially order
  • timed system : (T,R), T is totally ordered

    Timed Systems

    Metric Time - (T, d) is a set T and a metric d:TxT -> R .

    Metric

    Must satisfy the following conditions :
    1. d(t1, t2) = d(t2, t1)
    2. d(t1, t2) = 0 <=> t1=t2
    3. d(t1, t2) >= 0
    4. d(t1, t2) + d(t2, t3) >= d(t1, t3)

    Ultra Metric

    Must satisfy the following coniditions :
    1. d(t1, t2) = d(t2, t1)
    2. d(t1, t2) = 0 <=> t1=t2
    3. d(t1, t2) >= 0
    4. max(d(t1, t2), d(t2, t3)) >= d(t1, t3)

    Time and Events

    A continuum (a closed connected set) is a set T such that there does not exist two disjoint open sets O1 and O2 such that O1 union O2 = T is the entire set.

    A continuous-time system is a metric timed system where T is a continuum and T(s)=T for each signal s in any tuple s that satisfies the system.

    Discrete Time Systems

    A two sided discrete-event system Q is a timed system where for each s, there exists an order-preserving bijection from the integers to T(s).

    A one sided discrete-event system Q is a timed system where for each s, there exists an order-preserving bijection from the natural numbers to T(s).

    Homework Assignment : Due 9-10-96

    Let (x,d) be a metric space. Prove that :
    1. Both x and Null are closed
    2. Intersection of any number of closed sets is closed
    3. Union of any two closed sets is closed