Optimal Homing Sequences for Machines with Timing Constraints

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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

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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.

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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).

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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.

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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).

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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.

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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.

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Example t-IUT
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Example timed homing tree
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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.

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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.

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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.

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Example optimal timed HS

• Optimal timed homing sequence 0c3 1cgt2

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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.

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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.

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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.