Path Delay Fault Classification Based on ENF Analysis

1
Path Delay Fault Classification Based on ENF
Analysis

• Matrosova A., Nikolaeva E.
• Tomsk State University, Russia

2
Abstract

• Path delay fault classification based on using
  ENF and an addition of its product is suggested.
  It allows clarifying a nature of single and
  multiple PDF manifestation and getting test
  patterns. This approach is also used for
  investigation of combinational circuit
  testability properties concerning PDFs.

Key words path delay faults, equivalent normal
form (ENF).
3
Introduction

• In recent years with development of nanometer
  technologies, delay testing has become very
  important problem.
• The objective of delay testing is to detect
  timing defects degrading the performance of a
  circuit.
• Among the proposed delay fault models, the path
  delay fault model (PDF model) is considered the
  most accurate and has received wide attention.

4
Introduction

• In order to observe delay defects, it is
  necessary to generate and propagate transitions
  in the circuit. This requires application of a
  pair of vectors v1, v2.
• The first vector v1 stabilizes all signals in the
  circuit. It means the signal value on any circuit
  pole coincides with the function value
  corresponding to this pole and vector v1. The
  function depends on the circuit input variables.
  The second vector v2 causes the desired
  transition.
• Take into account that delays of falling (1/0)
  transition and rising (0/1) transition along of
  the same path from a primary input to a primary
  output in a circuit may be different. We need a
  pair of vectors v1, v2 for each transition of a
  path.

5
Introduction

• Single and multiple PDFs are distinguished. In
  this paper we first consider single PDFs.
• In accordance with the conditions of fault
  manifestation single PDFs are divided into
  robust, non robust and functional ones.
• Robust fault manifestation does not depend on
  delays of other paths of a circuit.
• Non robust fault shows itself only if all other
  paths of a circuit are free fault.
• Functional fault may manifest itself only if the
  certain other paths are sensitized together with
  the path considered.
• When several paths are fault we say about
  multiple PDF.

6
Introduction

• Several attempts were made to formalize these
  informal notions for gates of circuit, for vector
  pairs (in the case of robust PDF) and so on.
• Here we try to formalize above mentioned notions
  for vector pairs and ENF of arbitrary
  combinational circuit using addition of ENF
  product.
• In this way we hope to clarify a nature of PDF
  manifestation in order to facilitate getting a
  vector pair for different types of PDFs.

7
Equivalent normal form (ENF)

• Consider equivalent normal form (ENF) that
  represents a function implementing with a circuit
  and all circuit paths. Each ENF literal is
  supplied with index sequence enumerating gates of
  the path.
• It should be noted that literal with the same
  index sequence may appear in different ENF
  products.
• Literals of the same product of ENF have
  different index sequences.
• ENF of the circuit (Fig.1) is as follows (1).

Fig.1. The combinational circuit.
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Equivalent normal form (ENF)
Except ENF let consider sum S of products derived
from ENF with removing index sequences of
literals

• This sum may have products with both the same
  literals and inverse literals.
• Product having the same literals or/and inverse
  literals will be called non ordinary product.
• In the example considered ____ is non ordinary
  product.
• Otherwise a product is ordinary one.
• Further we will consider ENF and S together.

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Robust and non robust PDFs. Notions

• A notion of orthogonal pair of products can be
  extended to non ordinary products.
• If one product of the pair has literal xi and
  another product has literal __ then these
  products are orthogonal otherwise they are not
  orthogonal.
• If a non ordinary product has literals __,__ then
  this product is empty.
• Let __ be non empty (possibly non ordinary)
  product from a sum S of products, xi be literal
  of __ and a the path corresponding to this
  literal in ENF.
• Eliminate in __ all repetitive literals. As a
  result we have got ordinary product K comprising
  xi.
• Now change in K a literal xi for __ and obtain
  product __. Product __ is called an addition of K
  relative to literal xi. For example, ___________,
  xi  e, ________.

10
Robust and non robust PDFs. Notions

• First formalize notions of robust PDFs separately
  for falling and rising sequences.
• Let Ka be set of products from S so that each
  product is not orthogonal to K and comprises
  literal xi marked with the index sequence
  representing the path a in the corresponding ENF.
• Ka presents additional possibilities of
  sensitizing the path a.
• In product sum S for ________, and _____ we have
  __________.

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Robust and non robust PDFs. Notions

• Choose a Boolean vector v of input variables of a
  circuit. Subset of Ka in which any product takes
  the 1 value on vector v denote as Ka(v). If
  v  01101, then Ka(v) is empty (in our example).
• Let M be derived from the sum S of products by
  eliminating __ and Ka(v). M and Ka(v) are
  originated by v.
• Consider falling sequence corresponding to a.
• Let v1 be vector that stabilizes all signals in
  the circuit ensuring the 1 value of the function
  and v2 provides the desired (falling)
  transition.
• Denote as u minimal cube covering v1, v2 and as
  k(u) product corresponding to u.

12
Robust and non robust PDFs. Dfinitins

• Definition1. PDF of falling sequence manifests
  itself as robust one under the following
  conditions.
• 1. Product _____ takes the 1 value on a vector
  v1.
• 2. Sum M of products takes the 0 value on a
  vector v1.
• 3. Product __ takes the 1 value on a vector v2.
• 4. Sum S of products takes the 0 value on a
  vector v2.
• 5. Product k(u) is orthogonal to each product
  from M.
• 6. There exists a product from a set _________ in
  which literal xi appears only once.

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Robust and non robust PDFs. Dfinitins

• Now consider rising sequence corresponding to a.
  Let v1 be a vector that stabilizes all signals in
  the circuit ensuring the 0 value of the function
  and v2 is a vector that provides the desired
  (rising) transition.
• Definition2. PDF of rising sequence manifests
  itself as robust one under the following
  conditions.
• 1. Product _ takes the 1 value on a vector v1.
• 2. Sum S of products takes the 0 value on a
  vector v1
• 3. Product K takes the 1 value on a vector v2.
• 4. Sum M of products takes the 0 value on a
  vector v2.
• 5. Product k(u) is orthogonal to each product
  from M.
• 6. There exists a product from set ___________ in
  which literal xi appears only once.
• The conditions of robust PDF manifestation of
  rising sequence are derived from the conditions
  of robust PDF manifestation of falling sequence
  (for the same path a) through changing v1 for v2
  and conversely.

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Robust and non robust PDFs. Dfinitins

• Now regard conditions of non robust PDF
  manifestation for a.
• Definition3. PDF of falling sequence manifests
  itself as non robust one under the following
  conditions.
• 1. Product K(__) takes the 1 value on a vector
  v1.
• 2. Sum M of products takes the 0 value on a
  vector v1.
• 3. Product __ takes the 1 value on a vector v2.
• 4. Sum S of products takes the 0 value on a
  vector v2.
• 5. Product k(u) is not orthogonal to some
  products from M.
• 6. There exists a product from a set __________
  in which literal xi appears only once.
• The condition 5 means that some products from M
  may take the 1 value when changing v1 for v2. If
  other paths in a circuit are not free fault then
  masking PDF of falling sequence of a is possible.
  To exclude masking we have to suppose that all
  other paths are free fault.
• The conditions of non robust PDF manifestation of
  rising sequence are derived from the conditions
  of non robust

15
Robust and non robust PDFs. Dfinitins

• The conditions of non robust PDF manifestation of
  rising sequence are derived from the conditions
  of non robust
• PDF manifestation of falling sequence through
  changing v1 for v2 and conversely and eliminating
  the condition 6.
• Definition4. PDF of rising sequence manifests
  itself as non robust one under the following
  conditions.
• 1. Product __ takes the 1 value on a vector v1.
• 2. Sum S of products takes the 0 value on a
  vector v1.
• 3. Product K takes the 1 value on a vector v2.
• 4. Sum M of products takes the 0 value on a
  vector v2.
• 5. Product k(u) is not orthogonal to some
  products from M.
• Take into consideration that the condition 5 in
  the definitions 1 4 divides path delay faults
  into robust and non robust ones.

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Robust and non robust PDFs. Theorems

• Call ordinary product K expansible relative to
  literal xi if new product obtained with
  elimination of literal xi be implicant of the
  function. (In our case a function is represented
  with sum S of products). Then xi is expanding
  literal.
• Otherwise K is non expansible relative to literal
  xi and xi is not expanding literal. Notice that
  prime implicant is non expansible relative to
  each literal.
• Theorem1. To detect either robust or non robust
  manifestation of PDF originated with non empty
  product __(K) and literal xi it is necessary that
  K be non expansible with respect to literal xi.
• Let K be non expansible with respect to xi and P
  be obtained from S with elimination of product __
  and a set of products Ka.

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Robust and non robust PDFs. Theorems

• Theorem2. To detect either robust or non robust
  PDF (originated with non empty product __(K) and
  not expanding literal xi) it is necessary that
  the cube corresponding to K be not completely
  covered with the cubes corresponding to the
  products of P.
• Let Q be set of Boolean vectors representing
  minterms of K not covered with P.
• Find among a set __, Ka products that dont
  contain repeating literals xi. Join them into a
  set __.
• Let __ be result of intersection of Q and cubes
  corresponding to __.
• Theorem3. To detect PDF of both robust sequences
  and non robust falling sequence it is necessary
  that a set __ be not empty.
• Theorem4. To detect PDF of non robust rising
  sequence it is necessary that a set Q be not
  empty

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Robust and non robust PDFs. Resume

• Taking into consideration Theorems 1-3 and
  definitions 1-4 we conclude the following.
• 1. Test patterns of a pair v1, v2 that detects
  PDF of robust falling sequence may be applied for
  detecting PDF of robust rising sequence and
  inversely.
• 2. Test patterns of a pair v1, v2 that detects
  PDF of non robust falling sequence may be applied
  for detecting PDF of non robust rising sequence.
• 3. Test patterns of a pair v1, v2 detecting PDF
  of non robust rising sequence not always detects
  non robust falling sequence.

19
Robust and non robust PDFs. Vector pair
generation

• Represent __ as sum D of all prime implicants.
  Also represent all minterms of __ product on
  which S takes the 0 value as sum __ of all prime
  implicants.
• Let k be product from D and _ be product from __.
  Let __,__ be derived from __,__ removing literals
  __,__, correspondingly. Consider all pairs
  ___,___ originated by __,__.
• Theorem5. If a pair __,__ consists of not
  orthogonal products then the pair originates
  vectors v1, v2 that detect PDF of robust falling
  and rising sequences. Vector v1(v2) turns into 1
  product ___ and vector v2 (v1) turns into 1
  product ___ for falling (rising) sequence.
• If __,__ are not empty, then checking pairs __,__
  we may find vectors v1, v2 for detecting PDF of
  both robust sequences.
• In this paper we discuss only possibilities of
  getting pair v1, v2 without consideration of
  calculation problems.

20
Robust and non robust PDFs. Vector pair
generation
21
Robust and non robust PDFs. Examples
Fig.1. The combinational circuit.
22
Robust and non robust PDFs. Examples

• Consider next example. We want to find vector
  pair for path ____ using product _______,_______.
  The product is non expansible with respect to d.
  Product ______, _________, __ contains repeated
  literal d, but product ___ from Ka contains the
  only literal, _______________________.
  Representing Q as sum of products we obtain the
  only product __ (Fig.2). Then we derive __ and
  represent it as sum of products, _________.
  ____________________. Consider _______ and
  _______,_______,_______. According to the
  Theorem 5 there exists vector pair detecting PDF
  of robust sequences. Vector v1  10000 turns into
  1 expression ___________ and vector v2  10010
  turns into 1 expression ___________ for falling
  sequence of the path ____. M  P, u  100-0,
  ___________, k(u) is orthogonal to M. It means
  the fault considered manifests itself as robust
  one.

23
Robust and non robust PDFs. Examples

• The pair v2, v1 detects robust PDF of rising
  sequence of the path _____.

Fig. 3. Sensitizing the falling robust sequence
of the path _____
.
Notice that in the paper S. Devadas, K. Keitzer
Synthesis of Robust Delay-Fault-Testable
Circuits Theory. IEEE Transactions on
Computer-Aided Design, vol.11, NO 1. January
1992. p.87-101. when formalizing conditions of
robust path delay manifestation authors dont
consider the condition 6. Keeping this condition
allowed to find vector pair for the path _____
from product ________.The authors suppose that
this product must be ignored.
24
Functional PDFs. Notions

• Consider product __ comprising literal xi
  corresponding to path a.
• Let K be derived from __ with elimination of
  repeated literals. Product K is expansible
  relative to xi. (In the expression S the product
  _________ is expansible relative to d).
• If we want to derive from K new product which is
  not function implicant (S represents function),
  then we must exclude certain subset of literals
  together with xi.
• Let Xj be minimal subset of literals that
  originates the product __ from (___) K so that
  ___ is not expansible relative to xi.
• Let d be subset of paths corresponding to Xj in
  __ and K. Each path from d is free fault.
• Next change in K each literal from xi, Xj for
  inverse literal and obtain product __. __ is
  called an addition of K relative to literals xi,
  Xj.

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Functional PDFs. Notions

• Let Ka be subset of S in which every product
  contains all literals of a set xi, Xj with the
  corresponding index sequences in ENF and every
  product is not orthogonal to K (the same literals
  have the same index sequence).
• Subset Ka represents additional possibilities of
  sensitizing the path a together with a set d of
  paths.
• Subset of Ka in which any product takes the 1
  value on a vector v denote as Ka(v). Subset M is
  derived from S by eliminating __ and Ka(v).

26
Functional PDFs. Notions

• As each path from d is free fault then functional
  falling sequence is not detectable.
• Then consider rising sequence corresponding to a
  and a set d.
• Let v1 be a vector that stabilizes all signals in
  a circuit ensuring the 0 value of the function
  and v2 be a vector that provides the desired
  (rising) transition.
• Definition5. PDF of rising sequence manifests
  itself as functional one under the following
  conditions.
• 1. Product __ takes the 1 value on a vector v1.
• 2. Sum S of products takes the 0 value on a
  vector v1.
• 3. Product K(___) takes the 1 value on a vector
  v2.
• 4. Sum M of products takes the 0 value on a
  vector v2.
• 5. Product k(u) may be not orthogonal to some
  products from M.

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Functional PDFs. Example
Consider a circuit of Fig.4.
Fig4. Manifestation of functional PDF for the
path __

• We have the following ENF _________________.
  ____________, choose _____, K is non expansible
  per a and b. Let ____, ______, _____, kDab,
  _________, v1  00, v2  11. PDF of rising
  sequence of the path a36 manifests itself as
  functional one together with the path b36.

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Functional PDFs. Definitions

• Let K(__) be expansible relative to literal xi
  and non expansible relative to literal xj.
• Elimination of xj originates the product ___ from
  K so that it becomes non expansible relative to
  xi. Let ? be free fault path corresponding to xj
  in K (___).
• Change in K each literal from xi, xj for
  inverse literals and obtain product __ that is an
  addition to K.
• Let Ka be subset of S in which every product
  contains all literals of a set xi, xj with the
  corresponding index sequences in ENF and every
  product is not orthogonal to K (the same literals
  have the same index sequence).
• A set Ka represents additional possibilities of
  sensitizing the path a together with the paths ?.
  Subset of Ka in which any product takes the 1
  value on vector v denote as Ka(v). Let M be
  derived in the regular way.
• PDF of rising sequence of a appears as functional
  one under the conditions suggested in the
  definition5. Deriving D, __ is similar to above
  mentioned way for functional PDFs.

29
Functional PDFs. Example
Illustrate functional PDF of rising sequence for
the circuit of Fig 5.
Fig5. Manifestation of functional PDF for the
path ____

• Extract from S product _________ and consider
  literal ____. The product is expansible relative
  to d, xi  d. There is no other literal for which
  the product is expansible. Let xj be equal to e.
  The product is non expansible relative to xj. Ka
  is empty, _____________________________,
  _____________, v2  11011. ________, __________
  v1  01000.

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Functional PDFs. Resume

• Take into account that we considered the
  conditions of functional PDF manifestation only
  for expanding literal xi under suggestion that
  the number of additional sensitizing paths is
  minimal.
• It is possible to increase the number of
  additional sensitizing paths using the approach
  based on the definition 5.
• It is also possible to apply this approach for
  non expanding literal xi. It may be useful, for
  example, when PDF corresponding to xi does not
  manifest itself either as robust or non robust
  one.

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Multiple PDFs. Notions

• Consider multiple fault originated by __, a set
  Xj of its literals and the corresponding set d
  of paths. Let product K be derived from __ with
  elimination of repeated literals. Next change in
  K each literal from Xj for inverse literals and
  obtain product __. __ is called an addition of K
  relative to literals of a set Xj.
• Let Ka be subset of S in which every product
  contains all literals of a set Xj correlating
  to a set d of the sequences and each product is
  non orthogonal to K.
• Notice that Xj doesnt contain repeated
  literals. Subset Ka represents additional
  possibilities of sensitizing a set d of paths.
  Subset of Ka in which any product takes the 1
  value on a vector v denote as _____. Let M be
  derived in the regular way.

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Multiple PDFs. Defnitions

• Definition 6. Multiple PDF of falling sequences
  manifests itself under the following conditions.
• 1. Product K(__) takes the 1 value on a vector
  v1.
• 2. Sum M of products takes the 0 value on a
  vector v1.
• 3. Product __ takes the 1 value on a vector v2.
• 4. Sum S of products takes the 0 value on a
  vector v2.
• 5. Product k(u) may be not orthogonal to some
  products from M.
• 6. There exists a product from a set _________ in
  which each literal from Xj appears only once.

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Multiple PDFs. Defnitions

• Definition 7. Multiple PDF of rising sequence
  manifests itself under the following conditions.
• 1. Product __ takes the 1 value on a vector v1.
• 2. Sum S of products takes the 0 value on a
  vector v1.
• 3. Product ______ takes the 1 value on a vector
  v2.
• 4. Sum M of products takes the 0 value on a
  vector v2.
• 5. Product k(u) may be not orthogonal to some
  products from M.
• Notice that multiple PDFs may be originated by
  not only one product but also intersection of
  several products and several paths corresponding
  to the same input variables. It is out of our
  consideration here.

34
Multiple PDFs. Example
Illustrate multiple PDF manifestation. The fault
is caused with one product and its literals.
Consider the circuit of Fig.6.
Fig.6. Sensitizing the multiple falling sequences
originated by paths e59, b59.

• _________, multiple fault is represented with two
  paths Xj e59, b59. ________________________
  _______, _____________. _________. __________.
  Consider falling sequence choosing v1  11011,
  v2  00010. On Fig.6 we see multiple PDF
  manifestation of falling sequences. If changing
  v1 for v2 the conditions for rising sequence for
  the same fault are fulfilled.

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Conclusion

• Path delay fault classification based on using
  ENF and a addition of product is suggested.
• It allows getting the nature of single and
  multiple PDF manifestation and finding vector
  pairs that detect PDFs.
• This approach is useful for investigation of
  testability properties of circuits concerning
  PDFs.
• In partly we found out that all single PDFs in a
  circuit obtained by covering Shared ROBDD with
  CLBs manifest themselves as robust ones.
• Test patterns from pairs are contained among test
  patterns for single stuck-at faults at the CLB
  poles of a circuit.

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Thanks for your attention!