An Improved Critical Diagnosis Reasoning Method Yue Xu and Chengqi Zhang Distributed Arti cial Intelligence Centre Department of Mathematics, Statistic and Computing Science The University of New England Armidale, NSW 2351, Australia fyue,
[email protected]
Abstract
Model-based diagnosis mechanism has the disadvantage of a high computational complexity. One way to overcome this disadvantage is to focus the diagnosis on a reduced diagnostic space. In this paper, we propose an improved critical diagnosis reasoning method based on the method proposed by Raiman, de Kleer and Saraswat. The method proposed here focuses the diagnosis on nding out the kernel diagnoses instead of the whole diagnoses. We give an updated de nition of critical cover which we call \critical partition". The conditions satis ed by critical partition are relaxed compared with the conditions for critical cover. Correspondingly, a non-backtracking algorithm called Searching Critical Partition (SCP) to nd out the critical partition is also proposed here.
1. Introduction The Model-based diagnosis mechanism is one of the dominant paradigms for diagnosis systems. It performs diagnostic tasks with deep knowledge or deep models. By taking into account the di erence between the behavior expected by the models and the real behavior observed, model-based diagnosis systems can isolate the faulty components that can explain the abnormal behavior being observed. The model-based mechanism is more complete because of the use of the deep knowledge, but may su er from computational complexity. Many e orts have been made to overcome the disadvantage[1,2,3,4,5]. In 1993, Raiman, de Kleer and Saraswat proposed an architecture, IMPLODE [5], which uses a sensitivity analysis of assumptions to assign a critical level to assumptions. With the criticality, IMPLODE represents the environments, con icts and diagnoses with critical
environments, critical con icts and critical diagnoses respectively, and performs diagnostic tasks through nding out the critical diagnoses. The diagnosis ef ciency is dramatically improved by focusing diagnosis on the reduced space, because the set of critical diagnoses is a subset of the whole diagnostic space. However, some problems still exist in IMPLODE. First, the conditions satis ed by critical covers that will guarantee the consistency of the critical reasoning is somewhat higher, and the critical covers do not always exist. Second, the IMPLODE algorithm performs the critical reasoning using a backtracking algorithm. When critical covers do not exist, IMPLODE can not nd the critical covers after much backtracking. In this case, the diagnosis results obtained by IMPLODE are not critical ones and also can be found by other non-critical reasoning algorithms, but the eciency of IMPLODE may be lower than that of non-critical algorithms due to the backtracking. In this paper, we give an updated de nition of critical cover, which we call critical partition. The conditions satis ed by critical partition are relaxed compared with the conditions for critical cover. Correspondingly, we propose a non-backtracking algorithm that nds the critical partition with a heuristic method. The eciency of the algorithm is higher than that of IMPLODE, so the algorithm is more general and suitable for practical diagnostic tasks. The paper is organized as follows. In Section 2, a brief review of the critical reasoning method proposed by Raiman, de Kleer and Saraswat is given. In Section 3, we de ne the critical partition, and prove that the critical reasoning based on the critical partition is consistent. In Section 4, we present a non-backtracking algorithm to perform the critical reasoning. Finally, Section 5 concludes this paper.
2. Raiman and de Kleer's Critical Reasoning At rst, let us brie y review the critical reasoning method proposed by Raiman, de Kleer and Saraswat. Suppose ? be a propositional theory over a set of literals , which describes the devices to be diagnosed and ^ stands for the critical envithe observation data; S(n) ronment of n (n designates a literal which is an atom or its negation); "(n) is the environment set of n; and A stands for the set of assumptions. Then the critical abstraction of the theory ? relative to S, where S A, is de ned as: ^ = ? [fS(n) ^ ?! n j n 2 [f?gg S(?) ^ = \fE j E 2 "(n); E S g S(n) where `?!' refers to implication and `?' refers to a contradiction. The critical diagnosis reasoning based on ? is equivalent to the non-critical diagnosis reasoning (de Kleer ^ If S(?) ^ is and Williams's method [1, 7]) based on S(?). inconsistent, the critical reasoning based on ? will result in contradictions. Raiman, de Kleer and Saraswat ^ is consistent if and only if have proved that [5] S(?) ^ ?) 6= fg. Further, they proved that, if there is a S( critical cover fS1; : : :; Sk g relative to a subset S of A, Si S, i=1, : : :, k, then [ ^ fSi (?) j Si 2 fS1 ; : : :; Sk gg is consistent.
De nition 1 (Diagnosis) :SA diagnosis ofS? is a set , A, which makes ? f?a j a 2 g fb j b 2 A ? g consistent. De nition 2 (Critical Cover) : fS1; : : :; Sk g is a critical cover relative to S , S A, Si S , i=1, : : :, k, if the following conditions are satis ed by Si , i=1, : : :, k, (1) 8i, S^i (?) 6= fg; (2) 8C S, 9i (C is a con ict and subsumed by S^i (?)); (3) 8S^i (?); S^i (?) T(Skj=1 Sj ) = fg. j 6=i
De nition 3 (Critical Diagnosis) : A critical diagnosis of ? relative to S is a set , S , such that ^ ; (1) is a diagnosis of S(?)
^ , 0 ). (2) not( 90 of S(?) Reiter has proved that is a diagnosis of ? relative to S if and only if is a minimal hitting set of all con^ ?), since a can
icts in S [6]. For each element a 2 S( hit every con ict in S and fag is a minimal set, then ^ and according to De nition fag is a diagnosis of S(?), 3, fag is also a critical diagnosis of ? relative to S. It is not dicult to prove that, if fS1 ; : : :; Sk g is a critical cover, and i = fai g is a critical diagnosis ofS? relative S to Si , ai 2 S^i (?), i=1, : : :, k, then = 1 : : : k is a diagnosis of ?. Brie y, for ? and A, if a critical cover fS1 ; : : :; Sk g exists, then from the critical con ict set fS^1(?); : : :; S^k (?)g relative to the critical cover, the diagnoses of ? can be obtained from S^1 (?); : : :; S^k (?).
3. Critical Partition and Kernel Diagnosis In this section we will de ne a new critical cover called critical partition, and also de ne the concept of kernel diagnosis. Then we prove that the critical reasoning based on the critical partition through calculating the kernel diagnoses is consistent. De nition 4 (Critical Partition) : fS1 ; : : :; Sk g is a critical partition relative to S , S A, Si S , i=1, : : :, k, if the following conditions are satis ed by Si , i=1, : : :, k, (1) 8i, S^i (?) 6= fg; (2) 8C S, 9i (C is a con ict and subsumed by S^i (?)); (3) 8Si ; S^i (?) 6 Skj=1 Sj . j 6=i
Condition (3) in De nition S 4 tells us that, for each S^i (?), 9a 2 S^i (?), a 62 kj=1 Sj . It can be derived that j 6=i condition (3) in De nition 2 becomes a special case of condition (3) in De nition Sk 4, because it is equiva^ lent to 8a 2 Si (?), a 62 j=1 Sj . Therefore, we can j 6=i see that, condition (3) in De nition 4 is weaker than condition (3) in De nition 2, which makes the chances of nding out the critical partition higher than the chances of nding out the critical cover. However, on the other hand, De nition 4 cannot guarantee that, forTeach S Sj , there is at least one con ict C Sj , C ( kj=1 Si (?)) = fg. That is , there may be an j 6=i Sj in a critical partition, for all con icts C Sj ,
TS C ( ki=1 Si (?)) 6= fg. In such a case, for all coni6=j
icts C j , C must be hit Sby some Sk S^ (?). STherefore, S element in = 1 : : : k cannot be i=1 i i6=j guaranteed to be a minimal hitting set, and the conS S sistency of ? f?a j a 2 g fb j b 2 A ? g cannot be guaranteed. In order to do critical reasoning based on critical partition consistently, the non-minimal hitting sets in S^1 (?); : : :; S^k (?) should be discarded. Then the remaining sets in S^1 (?); : : :; S^k (?) can be guaranteed to be the diagnoses of ?, which guarantees the consistency of critical reasoning based on the critical partition. Below, we de ne the concepts of kernel environments and kernel con icts. Through calculating the kernel environments and kernel con icts, we can obtain a subset of minimal hitting sets from S^1 (?); : : :; S^k (?).
De nition 5 (Kernel Environment and Con ict)
Suppose fS1 ; : : :; Sk g is a critical partition relative to S , for n 2 , fS^n (1); : : :; S^k (n)g is the critical environment set of n, then the kernel environment S^i (n) of n is de ned as: S
S^i (n) = fa j a 2 S^i (n); a 62 kj=1 Sj g j 6=i correspondingly, S^i (?) is the kernel con ict for Si , i=1, : : :, k. Theorem 1 : Suppose fS1 ; : : :; Sk g is a critical partition relative to S , then, 8 2 S^1 (?); : : :; S^k (?), is a minimal hitting set that hits all the con icts subsumed by S . Proof: 1. is a hitting set to S. 8a 2 S^i (?), a hits all con icts in Si . According to condition (2) in De nition 4, for any con ict C subsumed by S, C must be subsumed by some Si . Therefore, 8 2 S^1 (?); : : :; S^k (?), hits all con icts in S. Because S^1 (?); : : :; S^k (?) is a subset of S^1 (?); : : :; S^k (?), 8 2 S^1 (?); : : :; S^k (?), hits all con icts subsumed by S. 2. is minimal. Assume that is not minimal, then there must be at least one ai 2 , ai 2 S^i (?) , let 0 = ? faig, 0 can hit every con ict in S. On the other hand, since 2 S^1 (?); : : :; S^k (?), 8a 2 0 , a 2 S^j (?), for some j, j 6= i. And according to the de nition of kernel con ict, 8a 2 S^j (?), a 2 S^j (?) and a 62 Si , i 6= j. Therefore, 0 cannot hit the con icts subsumed by Si , i.e., 0 cannot hit every con ict in S. From the contradiction, we can see that is minimal. 2
According to Theorem 1 and Reiter's theory [6],
8 2 S^1 (?); : : :; S^k (?), is a diagnosis of ? rela-
tive to S. Therefore, through searching a critical partition of S and calculating the kernel con icts, we can obtain a set of diagnoses of ?. The cost of searching a critical partition is lower than that of searching a critical cover, because the conditions of critical partition are weaker than that of critical cover.
4. Searching Critical Partition Algorithm It should be mentioned that critical partitions do not always exist, and the case is the same for critical covers. When critical partitions do not exist, and a backtracking algorithm is used to search the critical partitions, the backtracking cost can be wasteful. In order to avoid such waste, we use a non-backtracking algorithm called SCP to search critical partitions. Suppose L = fA1 ; : : :; Am g is an environment set, the task of SCP is to segment L into several parts, then, from each part, to get a critical environment by combining the environments in the part, nally, from each critical environment, to get a kernel environment. In SCP, an environment is represented as a vector with its component values being 1 or 0. Assume that the assumption set A = fa1 ; : : :; anA g, E is an environment, E A, E~ = [e1; : : :; enA ]T , then, ei = 10 aai 262 E i E
De nition 6 (Degree of Assumption Element)
Suppose L = fA1 ; : : :; Am g is an environment set, A~i = [ai1; : : :; ainA ]; i = 1; : : :; m, then 8ai 2 A, the degree of ai relative to L is de ned as
L (ai ) = mj=1aji To segment L properly is a crucial task for SCP. Here, the task is performed using the base elements (the de nition is given below) of L , which are selected according to the current degree of assumption elements.
De nition 7 (Base Element ) : Suppose fA1 ; : : :;TAm g is an environment set, Ai A, if 9a 2 A, a 2 m i=1 Ai , then a is a base element of fA1; : : :; Am g. Assume that a 2 A is the current selected base ele-
ment, according to a, L can be divided into two subsets La and L0 , and a is a base element of La . Because of the base element, the intersection set of all environments
in La must not be empty. Therefore, a critical environment can be obtained through intersecting the environments in La . Furthermore, from the critical environment, a kernel environment can be obtained through discarding the elements whose degrees are changed to be lower than their primary degrees (this will be discussed below). Then, SCP is used repeatedly to divide L0 until L0 becomes empty. The selection of base elements is based on the following two heuristic considerations. 1. The element with a higher degree should be preferably selected as a base element. 2. The element that is unlikely to exist in L0 and in La simultaneously should be preferably selected as a base element. Assume that E = fE1; : : :; Em g is an environment set, the corresponding environment matrix is < = [E~1; : : :; E~m]T , then, C =
