Full Fault Dictionary Storage Based on Labeled Tree Encoding

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1 Full Fault Dictionary Storage Based on Labeled Tree Encoding Vamsi Boppana, Ismed Hartanto and W. Kent Fuchs Coordinated Science Laboratory University of Illinois at UrbanaChampaign Urbana, IL 6181 Abstract The process of fault dictionary compaction can lead to a loss of information that is potentially useful in locating unmodeled failures. The focus of this paper is on developing alternative storage structures that can efficiently represent full fault dictionaries without discarding any information. We present the problem of storing the full fault dictionary storage as a labeled tree encoding problem. Two labeled trees are introduced to represent the diagnostic experiment. For the first tree, the unlabeled tree is stored using a binary string code, while the second tree is constructed so that the unlabeled tree is regular in structure, thus allowing implicit storage. Eight alternative representations based on the three label components are presented, and two existing full fault dictionary representations (the matrix and the list [12] dictionaries) are shown to be special cases in our general framework. Experimental results on the ISCAS 8 and ISCAS 89 circuits are used to study and characterize the performance of the proposed storage structures. 1 Introduction Fault dictionaries have proven to be useful in the diagnosis process, particularly when repeated diagnosis of different copies of the same circuit is performed [2 4]. One of the problems associated with the dictionary approach to diagnosis is that the size of the dictionary may be large and impractical for large circuits. Full fault dictionaries need to store output information corresponding to each vector and fault pair. Conventionally, they have been stored using a matrix representation. For a circuit with v vectors, o outputs and f faults, the size of the matrix dictionary is vof bits for combinational circuits and 2vof bits for sequential circuits. A listbased representation can also be used as an alternative to the matrix representation [12]. Compaction techniques address the size issue [8 1], but a key problem is that the information they identify as diagnostically useful is for modeled faults. Hence, the diagnostic accuracy of such dictionaries in the presence of unmodeled faults may degrade. The focus of our paper is on This research was supported in parts by the Semiconductor Research Corporation(SRC) under grant 94DP19, by the Joint Services Electronics Program (JSEP) under grant N149J127 and by an equipment grant from HewlettPackard. developing storage structures that enable efficient representation of the information in the full fault dictionary. This approach is orthogonal to the previous techniques, which have achieved dictionary compaction by removing output information. The diagnostic experiments in our approach are represented by labeled trees. An efficient representation of a labeled tree can be obtained by disjointly representing the label information and the underlying unlabeled tree. We provide two different labeled trees that represent the same diagnostic experiment. In the first approach, the unlabeled tree is encoded compactly using a binary string code, while the second labeled tree is constructed such that the resulting unlabeled tree possesses a regular structure. Specifically, the associated unlabeled tree is binary for combinational circuits and ternary for sequential circuits. The amount of storage required for storing the label information is different for the two schemes. In the second scheme, the label information storage is provided in eight different formats. This is possible because there exist three orthogonal (not influencing each other) components determining the label information, namely the fault, vector and output information. 2 The Storage Structure Diagnostic experiment trees serve as the basis for the storage structures described in this paper. The two labeled trees that we use to represent the diagnostic experiment are introduced as special instances of the general diagnostic experiment tree. 1 (General Diagnostic Experiment Tree T (V; E)) A general diagnostic experiment tree consists of a set of vertices V (T ) and a set of directed edges E(T ), i.e., each edge e is of the form (u; v); u; v 2 V (T ) with the direction of the edge being from u to v. Each vertex v 2 V (T ) of the tree is associated with a set of faults F (v) that is a subset of the list of all modeled faults F and each edge e 2 E(T ) is associated with a list of outputs O(e) that is a subset of all the primary outputs of the circuit. 2 (Root of the Diagnostic Experiment Tree r(t )) The unique node r(t ) 2 V (T ) such that there is no edge of the form (u; r(t )) 8u 2 V (T ) is the root of the tree T. 1

2 The fault list associated with such a node is the entire list of modeled faults of the circuit, i.e., the set F. 3 (Level of a node L(v)) For each node v 2 V (T ) of the tree, the level L(v) is defined as the length of the path between r(t ) and v. 4 (Parent of a node v P (v)) For each node in a tree T except the root r(t ), there is a unique node P (V ) such that the edge (P (v); v) 2 E(T ). (Vectorbased Diagnostic Experiment Tree T V (V; E)) A diagnostic experiment tree in which each level represents the application of a test vector, and each edge e 2 E(T V ) is associated with a list of outputs O(e) that is the set of all the primary outputs of the circuit, is called a vectorbased diagnostic experiment tree. 6 (Outputbased Diagnostic Experiment Tree T O (V; E)) A diagnostic experiment tree in which each level represents a (test vector,output) pair rather than a test vector, and each edge e 2 E(T O ) is associated with a single primary output of the circuit, is called an outputbased diagnostic experiment tree. 2.1 Representation of the Diagnostic Experiment Vectorbased Diagnostic Experiment Tree Let T V (V; E) be a vectorbased diagnostic experiment tree. The start of the diagnostic experiment can be represented by the root r(t V ) with the entire fault list F being associated with it. The application of a test vector is represented by a unique level of the tree. The fault list associated with a node v at level i is given by the set of faults associated with P (v) that produce the same primary output response upon the application of vector i. The output response associated with the edge (P (v); v) is the response produced on the complete set of primary outputs by each of the faults in F (v) upon the application of vector number i. Example : In Figure 1, a vectorbased diagnostic experiment tree is shown representing information from the full fault dictionary shown in Table 1, represented in the conventional matrix format Outputbased Diagnostic Experiment Tree Let T O (V; E) be the tree representing the diagnostic experiment. The start of the diagnostic experiment can be represented by the root r(t O ) with the entire fault list F being associated with it. The fault list associated with a node v at level i of the circuit is given by the set of faults associated with P (v) that have the same output response on output pin number (i? bi=oc + 1) upon the application of test vector Table 1: Matrix representation of a full fault dictionary fault output values number (bi=oc + 1). The same output pin response is associated with the edge (P (v); v). Example : In Figure 2, an outputbased diagnostic experiment tree is shown representing information from the full fault dictionary shown in Table 1, represented in the conventional matrix format. The don t care value is treated as a special symbol of the alphabet. Since no information is discarded in this storage structure, don t care values are treated just as the other two symbols and 1. Also clear from the above discussion is the fact that the structure of the outputbased diagnostic experiment tree is regular. 3 Dictionary Representations Hence, by finding an efficient representation for the diagnostic experiment trees discussed above, compact fault dictionary storage may be derived. A labeled tree T (V; E) can be represented using standard graphtheoretic techniques for storing labeled trees like Prüfer sequences [1], but the storage required is not necessarily optimal, due to the inherent redundancy in the natural labeling of the tree. Hence, our storage structures store the label information and the underlying unlabeled tree information disjointly. We also noticed that the unlabeled tree associated with the outputbased diagnostic experiment tree is regular; hence there is no need to develop a separate storage structure. We shall first present the storage structure for representing the vectorbased diagnostic experiment tree. 3.1 VectorBased Diagnostic Experiment Tree Let the tree G(V; E) represent the tree corresponding to a vectorbased diagnostic tree T (V; E), but with its nodes and edges not associated with any information. (In T (V; E), each node has a set of faults that is associated with it and each edge has the set of all primary outputs associated with it.) Let a canonical numbering N umber(v) be associated with each node v 2 G(V; E). The same numbering is used both in T (V; E) and G(V; E). This numbering is obtained from the scheme used to encode the tree G(V; E). The information in the tree T (V; E) is represented by encoding the tree G(V; E) using a binary code [1]. The distinct error responses produced by the circuit are stored in DERRTABLE. The output information 2

3 Outputs Faults Figure 1: Vectorbased diagnostic experiment tree for the dictionary in Table 1 3 Output Faults Figure 2: Outputbased diagnostic experiment tree for the dictionary in Table 1 Code Faults Node number Figure 3: Encoding information for the vectorbased diagnostic experiment tree of Figure 1 is obtained from the table INDEXTABLE, with each entry in it indexing into DERRTABLE. The index into IN DEXTABLE is available from the canonical labeling to each node in the tree. The last piece of information needed is, for each fault, the canonical labeling number of the node at the highest level in the diagnostic tree containing that fault. This array is called the FARRAY. The actual output values do not need to be stored, but rather the errors occurring in the diagnostic process can be identified and indices to errors stored in a table called the DERRTABLE. It has been noted [11] that storing errors is typically more economical than storing outputs. The proof that this method indeed stores the information in the tree T (V; E) is presented next. The following lemma is required to prove the required result. Lemma 1 There exists a unique sequence of node numbers from any node at the highest level to the root r(t ). Proof: This result follows from the fact that there is a unique path from r(t ) to any node in the tree. Property 1 The information in the tree T (V; E) can be represented by the given storage scheme. Proof: It suffices to prove that all the output responses associated with each fault in the tree T (V; E) can be decoded from the present form of storage. From Lemma 1, the path from r(t ) to any node at the highest level is decipherable; and for each fault, there is a scheme for storing the number of the highestlevel node containing it. These numbers can then be used to obtain the actual output sequences, because they are also stored in the canonical numbering order. 3.2 Binary Code for G(V; E) Encoding Algorithm The encoding is defined inductively over the number of levels in the tree. The encoding of a tree with one vertex and no edges is given by the binary code 1. The encoding of every tree G(V; E) with root r(g) is defined as follows: Let C 1 ; C 2 ; : : : ; C k represent the codes of the subtrees rooted at all the nodes in the tree G(V; E) such that r(g) is their parent. Let these codes be written in some preferred order, to be defined, given by a permutation (i 1 ; i 2 ; : : : ; i k ) of the integers 1; 2; : : : ; k. Then the code of the tree is defined to be ; C i 1; C i 2; : : : ; C ik ; 1. A straightforward choice for the preferred order in this application, unlike the one in [1], is the order obtained from diagnostic fault simulation. Example: The binary encoding of each node in the tree in Figure 1 is shown alongside each node in Figure 3. IN DEXTABLE, DERRTABLE and the array FARRAY are shown in Tables 2, 3 and 4 respectively. To illustrate, consider the retrieval of primary outputs produced by fault 3 at test vector 2. FARRAY is used to indicate the canonical 3

4 Table 2: INDEXTABLE Table 3: DERRTABLE node number at which the fault resides after the last test vector as. This is used to index into INDEXTABLE, which in turn implies that the distinct error produced is contained in location 1 of the DERRTABLE. The error is identified as 1 and knowing the good circuit response to be 11, the outputs produced are identified to be Decoding Algorithm 1. Associate a label with each occurring in the code by numbering them in order left to right. 2. Scan the code from left to right until the configuration 1 is found. Note the pair of labels associated with the two s in this configuration, and then delete the second and the 1. The two labels will define an edge of the tree. 3. If the resulting string has more than 2 symbols, repeat from Step 2; otherwise the string is just 1, the label associated with this is that of the root r(t ), and the algorithm terminates. The amount of storage required for this form of storage is 2n bits, where n is the number of nodes in the tree. 3.3 Outputbased Diagnostic Experiment Tree The unlabeled tree associated with the vectorbased diagnostic experiment tree is irregular in nature and hence required the development of the stringbased encoding. In contrast, the outputbased tree has a regular structure, thus obviating the above necessity. This enables us to present a spectrum of dictionaries that store the label information in a unique manner. It is interesting to note that our theoretical framework provides eight dictionaries, six of which were not previously known. It also provides for understanding the two known formats (matrix and list) in a unified framework. The label information has three components in the faults, the vectors and the output bits; i.e., each detection can be represented uniquely by the 3tuple (v; o; f ). Let us define the number of detections and the number of potential detections as below. Table 4: FARRAY fault node (Number of Detections, d) The number of detections in a diagnostic experiment is the total number of 3tuples (v; o; f ) on which there is a detection. 8 (Number of Potential Detections, p) The number of potential detections in a diagnostic experiment is the total number of 3tuples (v; o; f ) on which there is a potential detection. 9 (Total Number of Detections, t (= d + p)) The total number of detections in a diagnostic experiment is the number of 3tuples (v; o; f ) on which there is either a potential or a definite detection. Our experiments with benchmark circuits have revealed a very interesting phenomenon that we present next. Observation : The total number of detections is far less than the total number of detections that are possible, i.e., v o f. This immediately suggests an alternative form of dictionary storage based on recording the detections alone. We have already seen that a detection can be completely recorded by the 3tuple (v; o; f ). The eight dictionary formats that we now present for the storage of detections are based on the 3 possible components. It is obvious that we can exhaustively enumerate a fixed number of components (including the null set) and can list the detections and potential detections in terms of the rest of the 3 components. This explains the eight alternative format choices. The dictionaries presented have names of the form (s1)?(s2) representing, the fact that the components in (s1) are exhaustively enumerated and the components in (s2) are listed across the enumerated components. In the following discussion, it is to be noted that the potential detections list is necessary only for sequential circuits The ()? (vof ) Dictionary In this dictionary, there is no list of exhaustive enumeration for any components. There are lists storing the (v; o; f ) combinations producing detections and potential detections. The size of this dictionary can be enumerated in terms of the number of detections and potential detections as given below. Size = t (dlog(vof )e + 1) 4

5 3.3.2 The (v)? (of ) Dictionary The enumerated component in this dictionary is the vector component. There are lists storing (o; f ) tuples that produce detections and potential detections for each vector. The size of this dictionary for combinational and sequential circuits is given below. Size(combinational) = t (dlog(of )e + 1) + v Size(sequential) = t (dlog(of )e + 1) + 2 v The (o)? (vf ) Dictionary Outputs are enumerated in this dictionary. There are lists storing (v; f ) tuples that produce detections and potential detections for each output pin. The size of this dictionary for combinational and sequential circuits is given below. Size(combinational) = t (dlog(vf )e + 1) + o Size(sequential) = t (dlog(vf )e + 1) + 2 o The (f )? (vo) Dictionary The enumerated component in this dictionary is the fault component. There are lists storing (v; o) tuples that produce detections and potential detections for each fault. Size(combinational) = t (dlog(vo)e + 1) + f Size(sequential) = t (dlog(vo)e + 1) + 2 f 3.3. The (vo)? (f ) (List) Dictionary The tuple enumerated in this dictionary is the (v; o) tuple. There are lists storing the (f ) tuples that produce detections and potential detections for each unique (v; o) tuple. The dictionary is recognizable as the list dictionary [12]. Size(combinational) = t (dlog(f )e + 1) + vo Size(sequential) = t (dlog(f )e + 1) + 2 vo The (vf )? (o) Dictionary The tuple enumerated in this dictionary is the (v; f ) tuple. There are lists storing the (o) tuples that produce detections and potential detections for each unique (v; f ) tuple. Size(combinational) = t (dlog(o)e + 1) + vf Size(sequential) = t (dlog(o)e + 1) + 2 vf The (of )? (v) Dictionary The tuple enumerated in this dictionary is the (o; f ) tuple. There are lists storing the (v) tuples that produce detections and potential detections for each unique (o; f ) tuple. Size(combinational) = t (dlog(v)e + 1) + of Size(sequential) = t (dlog(v)e + 1) + 2 of The (vof )? () (Matrix) Dictionary All the components in this dictionary are enumerated. Hence, there is no listed component. Consequently, for each (vof ) tuple, there are two bits, one indicating a definite detection and the other indicating a potential detection. This is the standard matrix dictionary. The storage requirements are presented below. Size(combinational) = vof Size(sequential) = 2 vof Table : Performance of vector treebased representation Cir. no. nodes # derrs matrix tree tree/matrix c c c c c c c c c c s s s s s s s s s s s s s Experimental Results Experiments were performed on the ISCAS 8 and IS CAS 89 benchmark circuits to study the space requirements of the proposed dictionary storage schemes. The test vectors used were generated by the HITEC test generation package. In Table, sizes of the representations of the full fault dictionary using the vector tree based and the matrix based representations are presented for ISCAS 8 and ISCAS 89 circuits respectively. The columns in these tables indicate respectively the circuit name, total number of nodes in the diagnostic tree, number of distinct errors, size of the matrixbased representation, size of the treebased representation and the ratio of the tree versus the matrix representations. It is useful to note the substantial savings obtained over the matrixbased representation. It is also interesting to note that the vectorbased tree representation performs substantially better than the matrixbased representation when the number of vectors is small. Data supporting this claim is present in Table 6, wherein the sizes of the dictionaries are presented with the number of vectors reduced to 1. This data shows that the number of vectors may be crucial in determining the fault dictionary representation format. Table 7 presents the performance of the outputbased tree representations. The experiments show that better performance than a currently known representation is achieved in all the benchmark circuits by an alternative format. The savings obtained become significant for large circuits. As an example, for s3932 the best known result can be improved from 32M b to 22M b. Summary Our experimental results indicate that full fault dictionaries can be efficiently stored without discarding any in

6 Table 7: Performance of the output treebased representations Cir. ()? (vof ) (v)? (fo) (o)? (vf ) (f )? (vo) (vo)? (f ) (vf )? (o) (fo)? (v) (vof )? () (list) (matrix) c c c c c c c c c c s s s s s s s s s s s s s s s s Table 6: Performance on 1 vectors Cir. no. nodes # derrs mat tree tree/mat c c c c s s s s formation. We have been able to provide fault dictionary representation schemes that provide better results than the best known current formats for all the benchmark circuits. Two labeled trees were associated with the diagnostic experiment. Compact dictionary representations have been arrived at by disjointly storing the label information and the underlying unlabeled tree. A choice between the eight formats that were presented for representing the outputbased tree can be made from the dictionary size formulae that are presented before the creation of the dictionary by an estimation of the total detections parameter and the circuit characteristics. The choice between the vector treebased representation and the output treebased representation may be made by considering the number of vectors that are used for the experiment. This is because the vector treebased representation performs better when the size of the fault list associated with each node of the tree is large. However, it is recognized that for some applications it will be impossible to use even the compact representations described in this paper due to the size of the original full dictionary. For such conditions, a combination of compaction, multiplestage diagnosis, and efficient storage may be required. References [1] R. C. Read, The Coding of Various Kinds of Unlabeled Trees, Graph Theory and Computing, NY: Academic Press, 1972, pp [2] M. Abramovici, M. A. Breuer and A. D. Friedman, Digital System Testing and Testable Design, NY: Computer Science Press, 199. [3] H. Y. Chang, E. Manning and G. Metze, Fault Diagnosis of Digital Systems, NY: Wiley Interscience, 197. [4] J. Richman and K. R. Bowden, The Modern Fault Dictionary, Proc. IEEE Intl. Test Conf., Sept. 198, pp [] P. G. Ryan, S. Rawat and W. K. Fuchs, TwoStage Fault Location, Proc. IEEE Intl. Test Conf., Oct. 1991, pp [6] V. Ratford and P. Keating, Integrated Guided Probe and Fault Dictionary: An Enhanced Diagnostic Approach, Proc. IEEE Intl. Test Conf., 1986, pp [7] R. E. Tulloss, Fault Dictionary Compression: Recognizing when a Fault may be Unambiguously Represented by a Single Failure Detection, Proc. IEEE Intl. Test Conf., Nov. 198, pp [8] I. Pomeranz and S. M. Reddy, On the Generation of Small Dictionaries for Fault Location, Proc. IEEE Intl. Conf. on Computer Aided Design, Nov. 1992, pp [9] P. G. Ryan, W. K. Fuchs and I. Pomeranz, Fault Dictionary Compression and Equivalence Class Computation for Sequential Circuits, Proc. IEEE Intl. Conf. on Computer Aided Design, Nov. 1993, pp [1] V. Boppana and W. K. Fuchs, Fault Dictionary Compaction by the Elimination of Output Sequences, Proc. IEEE Intl. Conf. on Computer Aided Design, Nov. 1994, pp [11] B. Chess and T. Larrabee, Test Pattern Generation for Small Full Dictionaries, Univ. of Santa Cruz, Manuscript, April [12] P. G. Ryan, W. K. Fuchs, and I. Pomeranz, Fault Dictionary Compression, Manuscript,