1. Testing of Synchronous Sequential CircuitsBy
Dr. Amin Danial Asham

2. An Introduction to Logic Circuit Testing
References
An Introduction to Logic Circuit Testing

3. A Model of Sequential Circuits
Test generation for sequential circuits is extremely difficult because the behavior of a sequential circuit depends both on the present and on the past input values.
The mathematical model of a synchronous sequential circuit is usually referred to as a sequential machine or a finite state machine (FSM).
The following figure shows the general model of a synchronous sequential circuit. As can be seen from the diagram, sequential circuits are basically combinational circuits with memory to remember past inputs.
The combinational part of the circuit receives two sets of input signals:
primary (coming from the external environment)
and secondary (coming from the memory elements)
The secondary input variables are also known as state variables.
If there are m secondary input variables in a sequential circuit, then the circuit can be in any one of 2m different present states

4. Testing of Synchronous Sequential Circuits
Sequential circuits can be tested by checking that such a circuit functions as specified by its state table. This is an exhaustive approach and is practical only for small sequential circuits.
Given the state table of a sequential circuit, find an input/output sequence pair (X,Z) such that the response of the circuit to X will be Z if and only if the circuit is operating correctly.
The application of this input sequence X and the observation of the response, to see if it is Z, is called a checking experiment; the sequence pair (X, Z) is referred to as a checking sequence.
The derivation of checking sequence for a sequential circuit is based on the following assumptions:
The circuit is fully specified and deterministic. In a deterministic circuit the next state is determined uniquely by the present state and the present input.
The circuit is strongly connected; that is, for every pair of states qi and qj of the circuit, there exists an input sequence that takes the circuit from qi to qj.
The circuit in the presence of faults has no more states than those listed in its specification. In other words, the presence of a fault will not increase the number of states.

5. Definitions:
Homing Sequence (HS): An input sequence is said to be a homing sequence for a sequential circuit if the circuit’s response to the sequence is always sufficient to determine uniquely its final state.
Example:
New state
Output
Input Sequence: 1 1 D C C B D A C B B A C C

6. Definitions (cont.):
Distinguishing sequence (DS): A distinguishing sequence is an input sequence that, when applied to a sequential circuit, will produce a different output sequence for each choice of initial state.
Every distinguishing sequence is also a homing sequence because the knowledge of the initial state and the input sequence is always sufficient to determine uniquely the final state as well.
On the other hand, not every homing sequence is a distinguishing sequence.
The output sequence that the machine produces in response to 101 uniquely specifies its initial state

7. Obtaining Homing and Distinguishing sequences
If a Finite state machine with the following state table.
We shall refer a collection of uncertainties, such as (B)(ACD) above, as an uncertainty vector (U).
(B) and (ACD) in this example, are called the components of the uncertainty vector.
For a component, the order of states is not important. Repetition of states are allowed and meaningful.
Initially be in any of its four states; hence, the initial uncertainty is (ABCD)
Consider the case that we just applied a 1 input, then the successor uncertainties will be (B) or (ACD), depending whether the output is 0 or 1, respectively. In this case:
( ) ( )0( )1 1 A B C D B A C D

8. DEFINITONS (for components only)
Singleton: A component having a single state without duplication. Ex: (A), or (B). Note: (AA) is not a singleton.
Homogeneous: A component having only one state, with or without duplications. Ex: (A), or (AA), or (BBB). Note: (AAB) is not homogeneous.
Nonhomogeneous: A component containing at least two non-identical states. Ex: (AB) or (AAB) or (AAAC) or (AABC). Note: A component is either homogeneous or nonhomogeneous.
Multiple: A component containing duplicate states. Ex: (AA) or (AAB) or (AAA) or (AABB) or (AABC).

9. DEFINITIONS (for vectors only)
Trivial vector: All components are singletons. Ex: (A)(B)(C) or (A)(A)(B) or (C)(C)(C).
Homogeneous vector: All components are homogeneous. Ex: (AA)(BBB)(C) or (AA)(B)(C) or (A)(B)(C) or (AA)(A)(B).

10. SUCCESSOR TREE
Successor Tree: assuming we have a sequence machine with the following state table.
(ABCD)
Loop 1
(AB)0 (DD)1
(ABCD)0
Loop 1
(AB)0(DD)1
(BD)0(CC)0
Same 1
(A)0 (D)1 (DD)1
(BC)0 (AA)0 1
Homing
(AB)0 (DD)0
(A)0 (D)1 (BB)0

11. Distinguishing Sequence
Example
Successor Tree
Init. State In/Out Final state
Homing Sequence
Homing
Init. State 00/01 Final state A 1 1
Init. State 01/00 Final state B
A or B as initial state with input sequence 00 produces 01 as output sequence. 1 1 1 1 1 1
Distinguishing
A 100/100 C
B 100/101 A
State Table 1 1 1 1 1 1 1 1
Distinguishing Sequence

12. Properties of the successor tree
P1-Multiple component > multiple component. Ex. (BCC)(A) → (C)(AA)(C); (BCC)(A) → (BB)(A)(D)
P2-Identical nonhomogeneous components > similar sub- trees [repeat]. Ex. (ABB), (AB), (AABB) are considered identical
P3-Trivial vector: We have found a DS. Each state in the initial uncertainty vector responds to the input sequence leading to the trivial vector with a distinct output sequence.
P4- Homogeneous vector: We have found a HS. The output sequence of the input sequence leading to the homogeneous vector allows us to uniquely determine the final state.

13. Homing Experiment
Rule H1 [Repeat]: A node becomes terminal if the nonhomogeneous components are associated with some node in a preceding level (P2).
Ex. (AB)(C)(D) is “identical” with (AAB)(B)(E)]
Rule H2 [HS found]: All kth level nodes become terminal if any kth level node is associated with a homogeneous vector (P4) [only if we are seeking the shortest, or one of the shortest, HS].

14. Distinguishing Experiment
Rule D1 [Repeat]: Identical to Rule H1.
Rule D2 [Dead end]: A node becomes terminal if it is associated with a vector having a multiple component (P1).
Rule D3 [DS found]: All kth level nodes become terminal if any kth level node is associated with a trivial vector (P3) [only if we are seeking the shortest, or one of the shortest, DS].

15. Thanks