22nd IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems
A-Diagnosis: A Complement to Z-Diagnosis Irith Pomeranz1 School of Electrical & Computer Eng. Purdue University W. Lafayette, IN 47907, U.S.A.
[email protected]
and
Sudhakar M. Reddy2 Electrical & Computer Eng. Dept. University of Iowa Iowa City, IA 52242, U.S.A.
[email protected]
Abstract Z -diagnosis was proposed for speeding up diagnostic fault simulation by identifying in an efficient manner fault pairs that are guaranteed to be distinguished by a fault detection test set. Z -diagnosis is based on z -sets, which capture information about the outputs to which fault effects may be propagated. We introduce a dual concept of a -diagnosis that is based on a -sets, which capture fault activation conditions. More generally, a -sets include necessary assignments for the detection of target faults. We use a -sets to speed up diagnostic fault simulation in two ways, as part of a test set independent process and as part of a test set dependent process. The test set dependent process uses only logic simulation of the test set to identify fault pairs that are guaranteed to be distinguished by the test set. We present experimental results to demonstrate the speed up in diagnostic fault simulation obtained by using a -sets in addition to z -sets.
1. Introduction Diagnostic test generation and diagnostic fault simulation, which are important for effective defect diagnosis, require consideration of target fault pairs [1]. Typically, stuck-at faults are used as targets for diagnostic test generation and diagnostic fault simulation. Consideration of stuck-at faults is important even when the defects that need to be diagnosed do not behave as stuck-at faults. This is due to several reasons. (1) Defect diagnosis procedures use stuck-at faults to identify potential defect sites even when the defects are not stuck-at faults [2]-[3]. (2) Diagnostic test sets for stuck-at faults are likely to be more effective than fault detection test sets in diagnosing defects. Efficient procedures for handling large numbers of stuck-at fault pairs during diagnostic test generation and diagnostic fault simulation were described in [4]-[15]. In [14], a process referred to as z diagnosis was introduced. Z -diagnosis allows us to determine efficiently that certain fault pairs will be distinguished by a test set that detects them without requiring diagnostic fault simulation. It can thus speed up diagnostic fault simulation. The simplest form of z -diagnosis is based on the use of z -sets. The z -set of a fault f contains all the outputs to which the effects of f may be propagated. In [14], z -sets were derived based only on structural analysis of the circuit. Using only structural analysis, the z -set of a fault f contains all the outputs to which there is a path from the fault site of f . Let us consider two faults f 1 and f 2. Suppose that the z -set of f i is z (f i ), for i = 1,2. Suppose in addition that z (f 1) ∩ z (f 2) = φ. Let us consider a test t that detects f 1. The test t activates f 1 and propagates its fault effects to one of the outputs in z (f 1) (no other outputs are reachable from the fault site). Since z (f 1) ∩ z (f 2) = φ, even if t detects f 2, it cannot propagate fault effects of f 2 to any of the outputs in z (f 1). Therefore, t distinguishes f 1 and f 2. This conclusion is based only on structural analysis that yields their z -sets, and the existence of a test that detects at least one of them. The concept of z -sets captures the possible outputs to which fault effects may be propagated. It thus focuses on fault propagation conditions. In this work we introduce a dual concept of a -sets that captures fault activation conditions. More generally, the concept of a -sets uses the necessary assignments [16] for target faults to identify faults that are guaranteed to be distinguished when one of them is detected. We 1. Research supported in part by SRC Grant No. 2004-TJ-1244. 2. Research supported in part by SRC Grant No. 2004-TJ-1243.
1550-5774/07 $25.00 © 2007 IEEE DOI 10.1109/DFT.2007.9
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use a -sets to speed up diagnostic fault simulation in two ways, as part of a test set independent process and as part of a test set dependent process. The test set independent process is faster but it identifies smaller numbers of fault pairs that are guaranteed to be distinguished by a test set that detects at least one of the faults in a pair. The test set dependent process uses only logic simulation of the test set to identify large numbers of additional fault pairs that are guaranteed to be distinguished when the test set detects at least one of the faults in a pair. To simplify the discussion, we note that diagnostic test generation and fault simulation are only applied to detectable faults. In addition, diagnostic test generation is typically applied starting from a fault detection test set, and diagnostic fault simulation is typically applied to a test set that contains a fault detection test set. Therefore, throughout the discussion we only consider pairs of detectable faults, and we assume that a fault detection test set is available. Under these conditions, z -sets and a -sets are used to identify detectable fault pairs that are guaranteed to be distinguished by a fault detection test set. The test set may be considered when the analysis is test set dependent, or ignored when the analysis is test set independent. When the test set is ignored, the only information we use is that it is a fault detection test set, i.e., that it detects all the detectable faults. We define a -sets in Section 2. The test set independent process for using a -sets to identify fault pairs that are guaranteed to be distinguished by a fault detection test set is described in Section 3. The test set dependent process for using a -sets to identify fault pairs that are guaranteed to be distinguished by a fault detection test set is described in Section 4. In Section 5 we describe the use of z -diagnosis and a diagnosis as part of a diagnostic fault simulation process and present experimental results.
2. A-sets In its most basic form, the a -set a (f i ) of a fault f i consists of the condition for activating the fault. If f i is the fault line gi stuck-at vi , a (f i ) consists of the assignment gi = v′i , where v′i is the complement of vi . To improve the effectiveness of a -sets for diagnosis, we add to a (f i ) other conditions that are necessary for the detection of f i (necessary assignments [16]). To keep the analysis simple, we only identify necessary assignments, which involve lines that are close to gi . Specifically, we consider the path between gi and the next fanout stem or circuit output. For a gate G along this path, let the inputs of G that are not on the path be h 0,h 1, . . . ,hk −1. Let the non-controlling value of G be v . To propagate a fault effect from gi along the path, it is necessary to set h j = v for 0 ≤ j < k . We include in a (f i ) the assignment h j = v for every 0 ≤ j < k . For illustration, we consider the combinational logic of ISCAS-89 benchmark circuit s 27 shown in Figure 1. In Table 1 we show the sets a (f i ) for the collapsed single stuck-at faults of s 27. The fault gi stuck-at vi is denoted by gi /vi in Table 1. The necessary assignment h j = w j is denoted by [h j ]w j . 2 7
+
9
3 10
+
15
12 22 11 17 1
8
13 6
&
5 +
18
14
+ &
16 4
+
21
20
23
25
+ 26
24
19
Figure 1: ISCAS-89 benchmark circuit s27 For example, to activate the fault f 2, line 4 stuck-at 0, it is necessary to set line 4 to 1, contributing the entry [4]1 to a (f 2). In addition, detection of the fault requires that line 16 would be set to 0, line 18 would be set to 1, and line 5 would be set to 0. These necessary assignments contribute the entries [16]0, [18]1 and [5]0 to a (f 2).
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Table 1: a -sets for s 27 i f i a (f i ) 0 2/0 [2]1 [7]0 1 3/0 [3]1 [10]0 2 4/0 [4]1 [16]0 [18]1 [5]0 3 5/0 [5]1 [20]0 4 6/1 [6]0 [13]1 5 7/0 [7]1 [2]0 6 8/0 [8]1 7 8/1 [8]0 8 9/0 [9]1 9 9/1 [9]0 10 10/0 [10]1 [3]0 11 11/0 [11]1 [17]0 [19]1 [5]0 12 12/0 [12]1 [22]0 13 13/1 [13]0 [6]1 14 14/0 [14]1 15 14/1 [14]0 16 15/0 [15]1 17 15/1 [15]0 18 16/0 [16]1 [4]0 [18]1 [5]0 19 17/0 [17]1 [11]0 [19]1 [5]0 20 18/1 [18]0 [19]1 [5]0 21 19/1 [19]0 [18]1 [5]0 22 20/0 [20]1 [5]0 23 21/0 [21]1 24 21/1 [21]0 25 22/0 [22]1 [12]0 26 24/0 [24]1 27 24/1 [24]0 28 25/0 [25]1 29 25/1 [25]0 30 26/0 [26]1 31 26/1 [26]0 It is possible to extend the set a (f i ) by adding values that are implied by the necessary assignments of f i . For example, for the fault f 0 of s 27, the implications of [2]1 and [7]0 are [9]0, [10]0 and [11]0. We do not use this option since it increases the computational effort, and it is not needed when a -sets are used in the test set dependent process.
3. Test set independent process that uses a-sets Independent of the specific fault detection test set applied to the circuit, a -sets provide information about fault pairs that are guaranteed to be distinguished when one of them is detected. In this section we discuss such fault pairs. Let f i 1 and f i 2 be two faults with a -sets a (f i 1) and a (f i 2), respectively. Suppose that a (f i 1) and a (f i 2) contain conflicting assignments, i.e., there is a line h j such that h j = 0 (1) in a (f i 1) and h j = 1 (0) in a (f i 2). This implies that a single test cannot satisfy the necessary assignments of both faults. Therefore, f i 1 and f i 2 cannot be detected by the same test. As a result, a test that detects f i 1 (f i 2) will not detect f i 2 (f i 1), and will thus distinguish the faults. We conclude that f i 1 and f i 2 are guaranteed to be distinguished by a fault detection test set. In the example of s 27, the a -sets of faults f 0 (2 stuck-at 0) and f 5 (7 stuck-at 0) conflict on lines 2 and 7, and the faults are guaranteed to be distinguished by a fault detection test set. Similarly, the a -sets
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of faults f 2 (4 stuck-at 0) and f 3 (5 stuck-at 0) conflict on line 5, and the faults are guaranteed to be distinguished by a fault detection test set. In general, test set independent a -diagnosis allows us to exclude from the set of target fault pairs P for diagnostic fault simulation every pair of faults f i 1 and f i 2 such that a (f i 1) and a (f i 2) conflict on any line. We use Procedure 1 given below to define a set of target fault pairs P for diagnostic fault simulation. The procedure excludes from P fault pairs that are guaranteed to be distinguished by a fault detection test set based on test set independent z -diagnosis or a -diagnosis. Specifically, the procedure excludes a fault pair (f i 1,f i 2) if z (f i 1) ∩ z (f i 2) = φ, or if a (f i 1) and a (f i 2) conflict on any line. The procedure is independent of the specific test set, and uses only the fact that it is a fault detection test set. We use the following notation [14]. For a z -set z j , F (z j ) is the set that includes every detectable fault f i such that z (f i ) = z j . If z j 1 ∩ z j 2 = φ, z -diagnosis implies that every fault pair (f i 1,f i 2) such that f i 1 ∈ F (z j 1) and f i 2 ∈ F (z j 2) is guaranteed to be distinguished. Procedure 1 considers only z -sets z j 1 and z j 2 such that z j 1 ∩ z j 2 ≠ φ. This includes the case where j 1 = j 2. Procedure 1: Target fault pairs based on z -sets and a -sets (1) (2)
Let F be the set of detectable faults. Let the faults in F have z -sets z 0,z 1, . . . ,zm −1. Set P = φ. For j = 0,1, . . . ,m −1: 1
(a) (b)
For every fault pair f i 1,f i 2 ∈ F (z j 1), if a (f i 1) and a (f i 2) do not conflict on any line, add (f i 1,f i 2) to P . For j 2 = j 1+1,j 1+2, . . . ,m −1, if z j 1 ∩ z j 2 ≠ φ: For every fault pair f i 1,f i 2 such that f i 1 ∈ F (z j 1) and f i 2 ∈ F (z j 2), if a (f i 1) and a (f i 2) do not conflict on any line, add (f i 1,f i 2) to P .
Considering s 27, the circuit has 496 detectable fault pairs. Using only z -sets and ignoring conflicts between a -sets, Procedure 1 will include in P 380 fault pairs. Using both z -sets and a -sets, Procedure 1 defines 354 fault pairs. Some of the fault pairs excluded from P based on z -sets are (f 1 = 3/0,f 2 = 4/0), (f 1 = 3/0,f 3 = 5/0), (f 1 = 3/0,f 4 = 6/1), (f 1 = 3/0,f 6 = 8/0), (f 1 = 3/0,f 7 = 8/1) and (f 1 = 3/0,f 11 = 11/0). Some of the fault pairs excluded from P based on a -sets, which are not excluded based on z -sets, are (f 0,f 5), (f 1,f 10), (f 2,f 3), (f 2,f 18), (f 2,f 20) and (f 3,f 11).
4. Test set dependent process that uses a-sets In this section we consider the use of a -sets to determine fault pairs that are guaranteed to be distinguished by a fault detection test set when the test set is considered. In the case of z -diagnosis, deriving additional information from a test set about fault pairs that are guaranteed to be distinguished requires fault simulation with limited fault dropping of the test set. In the case of a -sets, we show that fault free logic simulation of the test set is sufficient to extract additional information. We denote the given test set by T . Using fault free logic simulation only, we find for every fault f i the subset of tests S (f i ) ⊆ T that satisfy the necessary assignments of f i . We say that tk ∈ T satisfies the necessary assignments of f i if for every assignment h j = w j in a (f i ), tk assigns the value w j to line h j . It should be noted that based on logic simulation we do not know whether the tests in S (f i ) detect f i , only that they satisfy its necessary assignments. For illustration we consider s 27 under the test set shown in Table 2. For every test tk , where 0 ≤ k ≤ 5, we show in Table 2 the set of values V (tk ) assigned by tk to all the lines in the circuit. An assignment where h j = w j is shown using the notation [h j ]w j . Considering the fault f 0 = 2/0 with a (f 0) = {[2]1, [7]0}, we find that t 2 is the only test that assigns a 1 to line 2 and a 0 to line 7. Therefore, S (f 0) = {t 2}. Considering the fault f 12 = 12/0 with a (f 12) = {[12]1, [22]0}, we find that tests t 2 and t 3 both assign a 1 to line 12 and a 0 to line 22. Therefore, S (f 12) = {t 2,t 3}. In a similar way we find the sets S (f i ) shown in Table 3 for all the faults of s 27. The sets S (f i ) can be used to identify fault pairs that are guaranteed to be distinguished by T as follows. Let us consider faults f i 1 and f i 2 that are detected by T . Suppose that S (f i 1) ∩ S (f i 2) = φ. This
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Table 2: Test set for s 27 V (tk ) k 0 [1]0 [2]0 [3]0 [4]0 [5]0 [6]1 [7]1 [8]1 [9]0 [10]0 [11]0 [12]1 [13]1 [14]1 [15]1 [16]1 [17]1 [18]1 [19]1 [20]0 [21]1 [22]1 [23]1 [24]1 [25]0 [26]0 1 [1]1 [2]0 [3]0 [4]1 [5]0 [6]1 [7]0 [8]0 [9]1 [10]1 [11]1 [12]0 [13]0 [14]0 [15]0 [16]0 [17]0 [18]1 [19]1 [20]0 [21]1 [22]1 [23]1 [24]1 [25]0 [26]0 2 [1]0 [2]1 [3]0 [4]0 [5]1 [6]1 [7]0 [8]1 [9]0 [10]0 [11]0 [12]1 [13]1 [14]1 [15]1 [16]1 [17]1 [18]1 [19]1 [20]0 [21]0 [22]0 [23]0 [24]0 [25]0 [26]1 3 [1]0 [2]1 [3]1 [4]1 [5]0 [6]0 [7]1 [8]1 [9]0 [10]0 [11]0 [12]1 [13]1 [14]0 [15]0 [16]0 [17]0 [18]0 [19]1 [20]1 [21]0 [22]0 [23]0 [24]0 [25]0 [26]1 4 [1]1 [2]1 [3]0 [4]1 [5]0 [6]1 [7]1 [8]0 [9]0 [10]0 [11]0 [12]0 [13]0 [14]0 [15]1 [16]0 [17]0 [18]0 [19]1 [20]1 [21]0 [22]0 [23]0 [24]0 [25]1 [26]1 5 [1]1 [2]0 [3]1 [4]0 [5]0 [6]0 [7]0 [8]0 [9]1 [10]1 [11]1 [12]0 [13]0 [14]0 [15]0 [16]0 [17]0 [18]1 [19]0 [20]1 [21]0 [22]0 [23]0 [24]0 [25]1 [26]1
implies that under T , the necessary assignments of f i 1 and f i 2 are never satisfied by the same test. Consequently, f i 1 and f i 2 are never detected by the same test of T . Therefore, f i 2 is not detected by the test that detects f i 1, and vice versa. The tests that detect the faults thus also distinguish them. It is important to note that S (f i 1) ∩ S (f i 2) = φ may happen even if a (f i 1) and a (f i 2) do not conflict. The example of s 27 will demonstrate such cases. In the example of s 27, we have a (f 0) = {[2]1, [7]0}, a (f 1) = {[3]1, [10]0}, S (f 0) = {t 2} and S (f 1) = {t 3}. Although a (f 0) and a (f 1) do not conflict, we have S (f 0) ∩ S (f 1) = φ. This implies that although f 0 and f 1 may have common tests that detect both of them, such tests are not included in T . Therefore, f 0 and f 1 are guaranteed to be distinguished by T . As another example, we have a (f 0) = {[2]1, [7]0}, a (f 7) = {[8]0}, S (f 0) = {t 2} and S (f 7) = {t 1,t 4,t 5}. Although a (f 0) and a (f 7) do not conflict, we have S (f 0) ∩ S (f 7) = φ. This implies that f 0 and f 7 are guaranteed to be distinguished by T . Of the 354 fault pairs included for s 27 in P by Procedure 1, we find that 157 fault pairs are distinguished based on the sets S (f i ) of Table 3. This leaves 197 fault pairs to be considered during diagnostic fault simulation of the test set. The following procedure accepts the set of fault pairs P computed by Procedure 1. It excludes from P fault pairs that are guaranteed to be distinguished by a given test set T based on a -sets. Procedure 2: Excluding fault pairs from P based on a -sets (1)
Let T be the given test set. Let F be the set of target faults detected by T . Perform logic simulation of T and find for every fault f i ∈ F the set S (f i ) ⊆ T of tests that satisfy the necessary assignments of f i .
(2)
For every fault pair (f i 1,f i 2) ∈ P , if S (f i 1) ∩ S (f i 2) = φ, remove (f i 1,f i 2) from P .
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Table 3: Sets S (f i ) for s 27 i f i S (f i ) 0 2/0 t 2 1 3/0 t 3 2 4/0 t 1 3 5/0 t 2 4 6/1 t 3 5 7/0 t 0 6 8/0 t 0,t 2,t 3 7 8/1 t 1,t 4,t 5 8 9/0 t 1,t 5 9 9/1 t 0,t 2,t 3,t 4 10 10/0 t 1 11 11/0 t 1 12 12/0 t 2,t 3 13 13/1 t 1,t 4 14 14/0 t 0,t 2 15 14/1 t 1,t 3,t 4,t 5 16 15/0 t 0,t 2,t 4 17 15/1 t 1,t 3,t 5 18 16/0 t 0 19 17/0 t 0 20 18/1 t 3,t 4 21 19/1 t 5 22 20/0 t 3,t 4,t 5 23 21/0 t 0,t 1 24 21/1 t 2,t 3,t 4,t 5 25 22/0 t 1 26 24/0 t 0,t 1 27 24/1 t 2,t 3,t 4,t 5 28 25/0 t 4,t 5 29 25/1 t 0,t 1,t 2,t 3 30 26/0 t 2,t 3,t 4,t 5 31 26/1 t 0,t 1 The logic simulation process required for computing the sets S (f i ) can be combined with the fault simulation process that always precedes diagnostic fault simulation to identify detected faults.
5. Experimental results We applied Procedures 1 and 2 to the combinational logic of ISCAS-89 and ITC-99 benchmark circuits with at least one million detectable fault pairs. We used fault detection test sets for single stuck-at faults. The results are shown in Table 4 as follows. After the circuit name we show the total number of fault pairs. Under column indep z-diag we show the number of fault pairs included in P by Procedure 1 when only z -sets are used to exclude fault pairs from P . A -sets are ignored in this case (indep stands for test set independent analysis, which is the type of analysis performed by Procedure 1). Under column indep a/z-diag we show the number of fault pairs included in P by Procedure 1 when both z -sets and a -sets are used to exclude fault pairs from P . Under column dep a-diag we show the number of fault pairs included in P after Procedure 2 is used to exclude fault pairs from P (dep stands for test set dependent analysis, which is the type of analysis performed by Procedure 2). Under column dfsim time red(%) we show the reduction in diagnostic fault simulation time due to the use of a -sets in addition to z -sets. Diagnostic fault simulation is applied to a test set T to determine
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Table 4: Experimental results
dfsim fault pairs time indep indep dep circuit total z-diag a/z-diag a-diag red(%) s1423 1125750 369266 367186 255541 14.87 s1488 1103355 182955 174697 101501 5.49 s5378 10408203 1147716 1139294 908442 9.65 s9234 20959575 3000343 2984901 2025824 17.17 s13207 46691616 3446376 3426662 1588337 33.87 s15850 64246780 6437785 6418340 3816892 37.64 s35932 616338495 2996733 2956230 2072749 5.00 s38417 480949605 9709275 9665073 7505759 9.07 b05 1493856 393141 390783 290997 9.14 b12 4570776 534278 527939 374091 16.69 33722578 b14 11490378 11473397 10048691 11.30 194449060 b20 49865503 49826205 45307860 9.56
which fault pairs are distinguished by T , and which fault pairs remain indistinguished. Without using z diagnosis or a -diagnosis, diagnostic fault simulation starts with the set of faults F included in a single equivalence class. By simulating the tests in T one at a time, F is broken into smaller and smaller equivalence classes. Faults in equivalence classes of size one are dropped from simulation. When z diagnosis and a -diagnosis are used to define a set of fault pairs P , which excludes fault pairs that are guaranteed to be distinguished by T , P can be used to break the initial equivalence class F into smaller equivalence classes before applying any tests [15]. P can also be used to break equivalence classes obtained during the diagnostic fault simulation process into smaller classes. This speeds up the diagnostic fault simulation process by reducing the number of simulations needed until faults can be dropped, and by reducing the sizes of the equivalence classes that need to be considered. In the example of s 27, without using z -diagnosis or a -diagnosis, the initial equivalence class is F = {f 0,f 1, . . . ,f 32}. After applying t 0 and t 1, the largest equivalence class is F 0 = {f 0, f 1, f 3, f 4, f 7, f 12, f 13, f 15, f 20, f 21, f 22, f 24, f 27, f 28, f 30}. Using the set of fault pairs P obtained by applying Procedures 1 and 2, the only fault pairs that include f 1 and are not guaranteed to be distinguished by the test set are (f 1,f 9) and (f 1,f 17). Since f 9 and f 17 are not included in F 0, using P , F 0 can be partitioned into F 01 = {f 1} and F 02 = F 0−F 01. After the partitioning, f 1 can be dropped since it is distinguished from all the other faults. The speed up in diagnostic fault simulation time is defined as follows. Let RTz be the diagnostic fault simulation time when only z -sets are used in Procedure 1 to define P . RTz includes the run time for fault simulation with fault dropping to find the set of detectable target faults F , for the computation of z sets, and the run time of Procedure 1, in addition to the time to perform diagnostic fault simulation starting from P . Let RTa /z be the diagnostic fault simulation time when both z -sets and a -sets are used, and both Procedures 1 and 2 are applied to exclude fault pairs from P . RTa /z includes the run time for fault simulation with fault dropping, for the computation of z -sets and a -sets, and the run time for Procedures 1 and 2, in addition to the time to perform diagnostic fault simulation starting from P . The percentage reduction in fault simulation time reported in the last column of Table 4 is computed as (RTz −RTa /z )/RTz .100. From Table 4 it can be seen that test set independent a -diagnosis can reduce somewhat the number of fault pairs included in P by Procedure 1. Test set dependent a -diagnosis reduces significantly the number of fault pairs left in P by Procedure 2. This results in significant reductions in diagnostic fault simulation time, where the run time for z -diagnosis and a -diagnosis is included. The differences between the various circuits are a result of the differences in circuit structure. If fault pairs with overlapping z -sets often have conflicting a -sets, the use of a -sets will be more effective.
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6. Concluding remarks We introduced the concept of a -diagnosis to identify fault pairs that are guaranteed to be distinguished by a fault detection test set. A -diagnosis can be used to speed up diagnostic fault simulation by identifying in an efficient manner fault pairs that can be determined to be distinguished by the test set without performing any fault simulation. A -diagnosis is based on a -sets, which capture fault activation conditions as well as necessary assignments for target faults. The basic idea in using a -sets is that if two faults have conflicting a -sets, the faults cannot be detected by the same test, and they are guaranteed to be distinguished by a test that detects one of them. We used a -sets in two ways, as part of a test set independent process and as part of a test set dependent process. The test set dependent process used only logic simulation of the test set to identify fault pairs that are guaranteed to be distinguished by the test set. We presented experimental results to demonstrate the speed up in diagnostic fault simulation obtained by using a -sets. The speed up was measured relative to an analysis that uses z -sets, which were introduced earlier for the same purpose.
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