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Optimal Homing Sequences for Machines with Timing Constraints

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A timed-IUT is a machine with timing constraints on its state transfer function. ... An optimal timed homing sequence is found using a breadth-first search algorithm ... – PowerPoint PPT presentation

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Title: Optimal Homing Sequences for Machines with Timing Constraints


1
Optimal Homing Sequences for Machines with Timing
Constraints
  • Ariel Stulman stulman_at_jct.ac.il
  • Simon Bloch simon.bloch_at_univ-reims.fr
  • H.G. Mendelbaum mendel_at_jct.ac.il

2
Introduction
  • Homing sequences (HS) are input sequences that
    induce specific output so as to allow a tester to
    infer the final state of an IUT (Implementation
    Under Test) regardless of its initial state.
  • An optimal homing sequence is a HS containing the
    minimal number of input symbols that still allow
    for a solution.

3
Introduction
  • A timed-IUT is a machine with timing constraints
    on its state transfer function.
  • A timed homing sequence is a HS that includes the
    timing for the input literals.
  • An optimal timed homing sequence is minimal timed
    homing sequence (fewest input literals).

4
Description of Problem
  • Finding an optimal homing sequence is a well
    known problem, fully solved.
  • (see Kohavis book Switching and Finite
    Automata Theory)
  • We wish to find an optimal timed homing sequence,
    taking into account the timing constraints of the
    t-IUT.

5
Proposed Solution
  • Define a current state uncertainty (CSU) as a
    vector containing groups of states, grouped based
    on similar output to some specific input
    sub-sequence.
  • A homogeneous state uncertainty vector is a CSU
    vector where each sub-group contains only one
    element (one or more times).

6
Proposed Solution
  • Let us build a tree that with every edge we
    associate an input symbol and a timing
    constraint, and with every node we associate a
    current state uncertainty vector based on the
    concatenation of the input and constraints on the
    edges on the path to the specific node.

7
Proposed Solution
  • An optimal timed homing sequence is found using a
    breadth-first search algorithm looking for a node
    that is associated with a homogenous state
    uncertainty vector.
  • The optimal timed homing sequence is constructed
    by concatenating the labeling (input and time
    constraints) on the edges of the tree on the path
    to the above node.

8
Example t-IUT
9
Example timed homing tree
10
Example timed homing tree
  • Time regions were chosen based on possible
    relevant changes to the t-IUT.
  • All terminal nodes were marked by a dashed line.
  • The two non-relevant (NR) nodes were inserted for
    the sake of completeness.

11
Example optimal timed HS
  • Node 9 is the first node found during a breadth
    first search that contains a homogenous
    uncertainty vector (each sub-group contains only
    one element one or more times).
  • The optimal timed HS is constructed by
    concatenating the labels on the path to node 9
    namely, 0c3 1cgt2.

12
Example optimal timed HS
  • Normally, a table is constructed to determine the
    final state based on the t-IUTs output sequence.
  • This is a special case where no table is needed
    because the t-IUT will reach the same final state
    (A) regardless of the initial state.

13
Example optimal timed HS
  • Optimal timed homing sequence 0c3 1cgt2

14
Proof of optimality
  • Nodes at higher levels of the tree, necessarily
    represent shorter HSs than nodes at lower levels
    (there are fewer edges in its path).
  • Any node that is not associated with a homogenous
    CSU vector cannot represent a complete solution
    for uniquely identifying the final state.

15
Proof of optimality
  • By the definition of a breadth first search,
    there cannot be a node at a higher level of the
    tree that is also associated with a homogenous
    CSU vector (otherwise it will be found by the
    search first).
  • Hence, the path to the node found must represent
    the shortest sequence that uniquely identifies
    every final state.

16
Closing Remarks
  • We proposed a method for generation of optimal
    homing sequences for machines with timing
    constraints.
  • Comparison to another possible method is
    presented in the written article.
  • The relevance of our algorithm is mainly for the
    field of testing real-time machines.
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