A probabilistic ATMS

Knowledge Systems Laboratory Report No. KSL 94-13

A probabilistic ATMS

by Sampath Srinivas [email protected]

KNOWLEDGE SYSTEMS LABORATORY Department of Computer Science Stanford University Stanford, CA 94305

February 1994

Abstract

Truth maintenance systems (TMS) provide a method of improving the eciency of search during problem solving. The problem solver uses the TMS to record the reasons that facts are derivable so that facts need not be rederived during the course of the search. De Kleer's Assumption Based Truth Maintenance system (ATMS) [deKleer 86] overcomes the limitations of many earlier systems, such as not being able to switch states swiftly and not being able to consider multiple solutions to a problem at once. We describe a probabilistic extension to the ATMS { An ATMS structure is augmented with a probability distribution over the set of assumptions. A probabilistic model is then constructed in the form of a Bayesian network from the ATMS structure. The probabilistic ATMS provides signi cant new functionality such as the derivation of the probability of a fact being derivable, the posterior probability over the assumptions given that a fact is derivable and the most probable context in which a fact is derivable. Our technique does not require the probability distribution of an assumption to be independent of the distributions of other assumptions. As an example of the use of the probabilistic ATMS, we show that it can be applied to construct probabilistic models to do multiple fault diagnosis. This generalizes some aspects of de Kleer and Williams' work on model based diagnosis [deKleer et. al. 87, deKleer et. al. 89]. The probabilistic ATMS has been implemented in IDEAL [Srinivas et. al. 90], a Bayesian network solver.

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1 Introduction

solver are derivable, then the statement associated with the node nc is also derivable. We allow for two special kinds of justi cations that the problem solver might want to make. The problem solver may want to say that a sentence (associated with ATMS node ns ) is always true irrespective of any assumptions or any other justi cations of the node associated with the sentence. Such a sentence is called a premise. In other words the problem solver wants to inform the ATMS of the implication t ) ns1 . In this case the ATMS creates a justi cation from a special node n (call it the \Top node") to ns , i.e., the ATMS creates the justi cation n ! ns2 . The node n is associated with the statement t in the problem solver. Also, the problem solver may want to state that the conjunction of a particular set of sentences (with associated nodes n1 ; n2 : : :; nm ) is impossible. In other words the, problem solver wants to inform the ATMS of the implication n1 ^ n2 ^ : : :; nm ) f . In this case the ATMS creates a justi cation to a special node n (call it the \Bottom node"). The justi cation created is n1 ; n2; : : :nm ! n . The node n is associated with the statement f in the problem solver. A node n is derivable from a set of assumptions A if n if A; J j= n in the propositional calculus. Here J is the set of justi cation statements. An environment is a set of assumptions. Logically, an environment is the conjunction of assumptions in the set. A label for a node n is the set of environments which satis es the following:  It is sound, i.e., node n is derivable from every environment in the label.  It is minimal, i.e., no environment in the label is a subset of another.  It is complete, i.e., every environment that derives the node n using the justi cations given to the ATMS is either in the label or a superset of some environment in the label. If we interpret an environment in a label as a concise representation for itself and all the sets of assump-

An ATMS is meant to be used with a problem solver to record what inferences are derivable given varying sets of assumptions. Speci cally, the problem solver has a set of sentences in some language which are interrelated in the sense that the derivability of each sentence follows from the derivability of some other sentences. Some sentences are primitive; i.e., their truth does not follow from any other sentence. The problem solver has the freedom to choose whether it believes a primitive sentence is true or not. Such sentences are called assumptions. For ecient context switching, the problem solver would like to know for each sentence S in the database the sets of assumptions which, when true, make S derivable. The ATMS provides exactly this service to the problem solver. This paper generalizes the ATMS to the situation where the assumptions have an associated probability distribution. With this information, instead of the question \Under what assumption sets is this fact derivable?", the generalized ATMS answers, among other things, the question \What is the posterior probability distribution over the assumption sets under which this fact is derivable?".

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2 de Kleer's ATMS We sketch here the essentials of an ATMS for review purposes and to establish terminology. In doing so, we closely follow de Kleer [deKleer 86]. The interaction between the problem solver and the ATMS is as follows: The problem solver incrementally gives the ATMS a set of nodes. Each sentence that the problem solver is interested in is associated with a unique node. Each node is treated by the ATMS as a propositional variable. If a node is associated with a primitive sentence we call it an assumption { note that the problem solver decides what sentences it wants to consider primitive. The problem solver also incrementally informs the ATMS of the interrelations of the sentences it is manipulating. It does so by incrementally giving the ATMS a set of justi cations. A justi cation is an implication of the form n1 ^ n2 ^ : : : ^ nm ) nc where each ni is a node. The intent of this statement is to inform the ATMS that when the statements associated with the nodes n1 ; n2 : : :; nm in the problem

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1 We use t to designate a valid problem solver statement (i.e., a statement that is always true) and f to designate a statement that is always false. 2 de Kleer does not explicitly have the concept of the Top node. The Top node is useful in the Bayesian net construction described later.

2

sometimes use B ! b. B can be interpreted as the conjunction of b1, b2 , : : : , bn or the set of them depending on context. We will use B = t to mean \bi = t for all bi 2 B ". Say that we consider each node in the ATMS structure as a binary random variable and are able to construct a joint probability distribution of these random variables which satis es the following criteria:  A node b is true i a justi cation for it is satis ed: { If a justi cation B ! b exists in the ATMS structure then P (b = t j B = t) = 1. { Let B1 ! b, B2 ! b : : : , Bn ! b be the set of justi cations of node b. Then P (bjBi 6= t for all Bi ! b) = 0.  P (n = t) = 1. Premises are always true.  P (n = f ) = 1. Contradictions are not possible.  For an assumption a, P (a = t) = pa . We see that this joint distribution can be used to answer the query we are interested in. The answer to the query is a marginal probability that can be calculated from this joint distribution. We now show a construction procedure which constructs a joint distribution of the form described above in the form of a Bayesian network. The query we are interested in can then be answered by a manipulation of the Bayesian network. We also show that various other interesting quantities can be computed from the Bayesian network.

tions that are its supersets, we see that the label represents all possible environments in which the node is known to be derivable by the ATMS and no other environments. If we interpret every environment as a conjunction of assumptions and every label as a disjunction of conjunctions, then the node is derivable if and only if the label statement is true. An environment is consistent if it does not derive the Bottom node and it is inconsistent otherwise. Note that by our de nition of labels above a node label may include inconsistent environments 3 . A context in an ATMS is the set of assumptions of a consistent environment and the set of all nodes derivable from the environment. The consistent environment that de nes a context is called the characterizing environment of the context. The primary purpose of the ATMS is to provide an ecient query operation derivable(assumptionset, node). This operation returns t if node is indeed derivable from the nodes in assumption-set and the assumption-set does not result in a contradiction (i.e., the assumption set does not derive the Bottom node). The operation returns f otherwise. The ATMS algorithm ensures that correct labels are maintained for each node after a new justi cation or node is added to the ATMS. The derivable operation then just reduces to checking that assumptionset is indeed subsumed by some environment in the label of node and is not subsumed by some environment in the label of n . The ATMS operations are fully de ned in Appendix A.

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3 Probabilistic extension

3.1 Mapping an ATMS structure to a Bayesian Network

We proceed to generalize the ATMS in the following way { For every assumption a, we assign a probability P (a = t). Call this quantity pa . This assignment would come from the problem solver that uses the ATMS. Our initial goal is to answer the query \For a node n in the ATMS structure, what is the probability that n is derivable ?". This is a generalization of the derivable operation of the ATMS. We introduce some syntax here. We will write a justi cation whose antecedents are b1 , b2, : : : , bn and whose consequent is b as b1 ; b2; bn ! b. We will also

We rst introduce an important restriction in the structure of the ATMS { we specify that circularities in justi cations are not permitted. Speci cally, let children(node) be the set of consequent nodes of all justi cations that have node as an antecedent. The non-circularity restriction states that for every node n in the ATMS, n is not a member of the descendants of the node obtained by the closure of the children function. This is a reasonable assumption when the problem solver is being used to reason about a causal model [Pearl 88], as, for example, in model 3 De Kleer [deKleer 86] omits inconsistent environments from the label. We do not omit then from the label since this based diagnosis. We also introduce another minor restriction, viz, makes description of his approach to model based diagnosis easier in Sec 4. that there be no justi cations with an assumption as 3

3.2 Using the Bayesian network

a consequent. This requirement is not a large price and in fact, is the preferred method of using an ATMS [deKleer 86]. If we now consider each node of the ATMS as a node in a graph, each justi cation as an AND link linking the set of antecedent nodes of the justi cation to the consequent node, and the set of justi cations coming into a node as being ORed, we obtain a acyclic AND-OR graph. Now we convert the AND-OR graph into a Bayesian network as follows:

3.2.1 Probability of a node being derivable

Apply any standard Bayesian evidence propagation algorithm to the network (for example [Jensen et. al. 89, Lauritzen et. al. 88]). This yields as a result, for each node n, the posterior probability P (n = t), i.e., the conditional probability that node n is derivable given that a contradiction does not occur and that the premises are always true. This generalizes the ATMS operation derivable. Given an assumption set, derivable determines 1. Consider each node in the graph as a discrete whether n is derivable from the assumption set. Now, binary random variable. given a probability distribution over assumptions, we get the probability of n being derivable. 2. For each justi cation, create an arc from each of the antecedents to the consequent.

3.2.2 Posterior probabilities of assumption sets

3. For each node n: Let the set of parents of node n be P = fp1; p2 : : :; png. Let a truth assignment  be an assignation of true or false to each of the parents. There are 2 P such truth assignments. For each possible truth assignment  of the parents, check whether the justi cations assign node n a truth value of t. If so, in the conditional probability table of node n, set P (n = t j  ) = 1 and P (n = f j  ) = 0. If on the other hand, the justi cations assign node n a truth value of f for the truth assignment  , set P (n = f j  ) = 1 and P (n = t j  ) = 0. j

Given a particular node n, say we want to determine the probability that some set B of assumption nodes is in the characterizing environment of the context of the node n. In other words, we want to sum the probabilities of every context the node n is in such that B is a subset of the characterizing environment of the context. This is done as follows. Create a new node in the ATMS called b. Add a justi cation B ! b, i.e., a justi cation in which the antecedent contains all the nodes in B and the consequent is b. Now declare evidence n = t. Propagate the evidence through the network. Look up the posterior probability P (b = t), which is a direct result of the propagation. Note that b = t exactly represents the conjunction of events we are looking for, i.e., that every assumption in B is true. Determining the posterior probability of assumption sets is a generalization of the label concept in the ATMS. Given that a node n is derivable, the label (with all inconsistent environments deleted) tells us what consistent assumption sets are possible. Given a prior probability distribution over assumptions, we now get a posterior probability distribution over the possible assumption sets given that node n is derivable and no contradiction is entailed.

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4. Enter evidence n = f . This enforces the condition that contradictions should not occur. Enter evidence n = t. This enforces the condition that premises are always true. These two pieces of evidence are never retracted and are always present in the network. ?

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Note that the assumption nodes (and the Top node) are necessarily root nodes in the Bayesian network DAG. Furthermore, no other nodes are root nodes. The Bayesian network is now fully speci ed except for the marginal distribution of each of the root nodes. Assign any arbitrary strictly positive marginal distribution to the Top node (since the Top node has evidence assigned to it, the marginal is irrelevant as long as it does not contradict the evidence). For each assumption a, assign the distribution P (a = t) = pa . The Bayesian network is now completely speci ed.

3.2.3 Most probable context Given a set of marginals assigned to the assumption nodes of the network, say we want to discover the most probable context that any particular node n 4

belongs to and the characterizing environment of the context. In other words, of all contexts in which the node n is true, we want to nd the context that is most probable and read o the environment of this context. This is easily done as follows. Declare evidence for node n as n = t. Then run a Bayesian network algorithm for most probable extension [Pearl 87]. An extension is a truth assignment for all nodes in the network. Each extension has an associated probability { viz., the probability of the joint event that each node in the network has the truth value associated with it in the extension. The algorithm for most probable extension nds the most probable of all such extensions consistent with the declared evidence. Since the evidence n = t has been declared, the Bayesian extension algorithm nds the most probable extension in which n = t. The set of nodes assigned truth value t in this extension is precisely the most probable context. The assumption nodes in this context form the characterizing environment of the context.

I1

I4

A I2

X I6

O I3

I5

Figure 1: A digital circuit.

4 An example: Multiple fault diagnosis We consider multiple fault diagnosis [deKleer et. al. 87, deKleer et. al. 89, Gener et. al. 87] as an example of the use of the probabilistic ATMS. Consider a problem solver which is trying to perform diagnosis on the digital circuit shown in Fig 1. The problem is that each of the gates may fail; i.e., their input-output behaviour may deviate from the expected behaviour. Let us assume we have no fault model for each gate; i.e., we have no information about the input-output behaviour if the gate fails. The problem we want to solve is to determine which gates have failed given some anomalous input-output readings of the circuit. The problem solver represents the fact \Gate G is assumed to be working normally" with a propositional variable G. There are three such variables in this example, A, O and X (see Fig 1). The problem solver represents each of the state variables in the circuit with a propositional variable (see Fig 1 for the names of variables). Now the problem solver database may initially have the following sentences, which represent a complete model for the circuit when it is working correctly4 :

3.2.4 Non-independent assumptions

Note that in all the discussion so far, marginals are assigned to each assumption independently; i.e., we assume that the problem solver is comfortable with making the transition from assigning truth values to assumptions independently to assigning probabilities to assumptions independently. If the assumptions are indeed correlated, then one method of handling this when the correlation is deterministic is to add justi cations to the ATMS. For example, \Either assumption A or assumption B can be true, not both" is handled by creating a justi cation A; B ! ?. However, if we want non-deterministic constraints, for example, \If A is true, then the chances of B being true are less", then we can add additional structure A ^ (I1 ^ I2 ) ) I4 (S 1) to the Bayesian network created by the ATMS to encode such constraints. The assumption interactions O ^ (I2 _ I3 ) ) I5 (S 2) can be modeled as a Bayesian network BA which conX ^ ((:I4 ^ I5 ) _ (:I5 ^ I4 )) ) I6 (S 3) tains a node corresponding to each assumption. This network BA can then be integrated with the main ATMS network by just copying the links in BA to links between the corresponding assumption nodes in 4 We use Ii to represent both the state variable and the corthe ATMS network and then copying the probability responding propositional variable in the problem solver. The intent will be clear from context. tables over. 5

A f t f t

A X

I1 = 1 I4 = 1 Top

I6 = 0

I2 = 1

I3 = 0

P (AjL)

0.8 0.2 0.0 1.0 L P (L) pc 0.1 wc 0.9

X f t f t O t f

L pc pc wc wc

P (X jL)

P (O)

0.8 0.2 0.0 1.0

0.8 0.2

Table 1: Failure priors.

I5 = 1

O

L pc pc wc wc

node represents exactly all those behaviours which are inconsistent with the observations and device models. Therefore the diagnosis is exactly the negation of the label of the Bottom node (interpreting the label as a propositional sentence). If the negation is cast as a disjunct of conjuncts, each conjunct represents one set of device failures which could cause the observation. In the above example, the only situation which causes a contradiction is if both A and X are working, since then the output would have to be I6 = 1, contradicting the observation. The label of the bottom node in this case is precisely ffA; X gg. In propositional sentence form this is A ^ X . The negation gives us the answer :A _ :X , which translated into sets is the answer we want: ff:Ag; f:X gg. Note that each set of negated assumptions also represents all supersets of negated assumptions. DeKleer and Williams generalize their diagnosis method to the situation where all failures are assumed to occur independently and compute a posterior probability on each possible set of device failures that is consistent with the observations. We generalize this further and compute the posterior probability in the case that the device failures are not independent. Say we have the following information about the failures of the gates (see Table 1). Gates A and X are on the same power line L. L has a probability of 0:1 of being poorly conditioned. If L is poorly conditioned. each of X and A fail independently with probability 0:8. If L is well conditioned. then A and X do not fail. The gate O fails independently of the other gates and its failure probability is 0:2. The question we now ask is \Given the observation, what is the probability distribution over the possible failures?". The Bayesian network corresponding to the ATMS (and including the failure interactions and probabilities) is shown in Fig 3. The node F is created to

Bot I6 = 1

Figure 2: ATMS structure for the diagnosis example. Now, say an observation is made. An observation, in this example, involves going out to the circuit, giving it values for input variables and observing some output and intermediate variable values. Technically, in this context it is a partial value assignment to the state variables of the circuit. Assume the following observation is made: I1 = 1, I2 = 1, I3 = 0, I5 = 1, I6 = 1. Note that there is something wrong with the circuit, because, with this set of inputs, the output I6 should actually be 0. The problem solver creates the ATMS structure of the network (see Fig 2) as follows. Each possible failure is made an assumption. In this case the assumptions are A, X and O. Each observation is recorded as a premise since an observation is true regardless of any assumptions of the state of the devices. Then the problem solver uses its theory to derive all the facts it can from all the current observations. Whenever a contradiction occurs between a derived fact (i.e., predicted behaviour) and an observation (i.e., actual behaviour), a contradiction is recorded. In this example, the problem solver discovers a contradiction between the derived fact I6 = 0 and the observed fact I6 = 1. The question addressed by diagnosis is: \What are the sets of device failures which are consistent with this observation?". Each set of device failures is represented by a set of negated assumptions. We rst describe De Kleer and Williams' approach to the problem. Note that the label of the bottom 6

A X O f f f f

f f t t

Prob

f t f t

A X O

0.2 0.05 0.05 0

t t t t

f f t t

f t f t

Prob

and set of observations. Note that with incomplete fault models or non-deterministic fault models, many contexts are possible for a particular fault combination and set of observations. We need to sum the posterior probabilities of all these contexts to nd the posterior probability of that particular combination of faults [Srinivas 94].

0.4667 0.1167 0.1167 0

Table 2: Posterior P (F = axo) of failures. P

5 Conclusions and Related Work

F

A X

Given a probability distribution over the assumptions we have shown how to extend the ATMS to nd the probability of a node being derived, the posterior probability of an assumption set given a node was derived and the most probable context that the node is a member of. We are able to handle non-independent assumption distributions. The probabilistic ATMS described above is implemented in IDEAL [Srinivas et. al. 90], a Bayesian network solver. The implementation has been tested on a variety of examples, including the one in this paper. Inference in Bayesian networks is NP-hard in the worst case. However, when the network is sparse (i.e., has few arcs), inference becomes tractable. This methodology is practical with large ATMS structures (i.e., hundreds of justi cations) when the resulting Bayesian Network topology is sparse { roughly speaking, there should not be too many instances of a node n being dependent (through a justi cation or chain of justi cations) on another node p in multiple ways. Pearl [Pearl 88] discusses a problem similar to the one addressed in this paper and shows how the probability of calculating the probability that a node in the ATMS is derivable given a set of independent distributions over assumptions. He shows that the problem of calculating the probability reduces to that of computing the probability of an arbitrary Boolean expression, for which procedures are available. Laskey et al. [Laskey et. al. 89] show that a formal equivalence exists between the posterior probability of a node being derivable in an ATMS extended with a probability distribution on assumptions and the node's belief in a corresponding Dempster-Shafer inference network. They give an algorithm for computing this probability. This paper provides a direct interpretation for these results in terms of Bayesian networks and generalizes them, in that many more kinds of inferences are available other than the calculation of the probability of a node being derivable. Further, indepen-

I1 = 1 I4 = 1 Top

I6 = 0

I2 = 1

I3 = 0

I5 = 1 Bot O

I6 = 1

Node with declared evidence

Assumption node

Failure interaction node

Aggregation node

Figure 3: Bayesian network created from ATMS structure and the failure interaction information. model the joint state of A, X and O and thus capture the posterior distribution that we seek. It has 8 states, one corresponding to each possible truth assignment to the assumptions. We denote by axo the state of F corresponding to the truth assignment A = a, X = x and O = o. We set the distribution of F such that P (F = axojA = a; X = x; O = 0) = 1 for each a, x and o and all other probabilities in the distribution are zero. We now perform propagation in the Bayesian network producing the updated distribution for F in Table 2. Determining the posterior distribution over all possible combinations of faults, as in the preceding example, is inherently combinatorial. We would generally like to nd the most probable fault combination eciently. We note that in the case that deterministic fault models are speci ed for each component, instead of needing to determine the distribution over all possible combinations of faults, we can determine the most probable context of the Bayesian network instead. Reading o the state of each assumption node in this context gives us the most probable fault combination. This is because in this case there is precisely one context for a particular fault combination 7

dence of assumptions is not necessary. Darwiche [Darwiche 93] de nes an abstract state of belief (of which probability is one and propositional calculus is another) and shows a general framework for belief revision. The Bayesian network framework then, generalizes to an abstract belief network framework and exact analogs exist for Bayesian network inference algorithms. He applies this result to a propositional state of beliefs and shows how one could apply the result to computing ATMS labels more eciently. Darwiche's results are orthogonal and complementary to this work in that it allows the integration of both the ATMS label computations and the probability computations into the same framework. We could thus have an algorithm adapted from abstract belief networks which eciently performs both computations in parallel. D'Ambrosio [D'Ambrosio 90] describes how to do probabilistic inference using an ATMS construction. This is, in a sense, the dual of the construction described in our paper. He constructs an ATMS from a Bayesian network with the purpose of doing probabilistic inference eciently in the Bayesian network. Shachter et al. provide a complete algorithmic description of such inference [Shachter et. al. 90]. The results of this paper provide a generalization to some aspects of de Kleer and William's [deKleer 86, deKleer et. al. 87] work on integrating uncertainty into ATMS calculations in that many additional kinds of inferences are allowed. In addition, non-independence of assumptions can be handled. We describe construction of probabilistic models in the form of Bayesian networks for model based diagnosis in detail in [Srinivas 94, Srinivas 93].

computationally desirable. The ATMS achieves this speed by performing a (potentially exponential) pre-computation operation each time a justi cation is added { this is the label update operation. We are currently exploring whether a similar pre-computation operation is feasible in the case of the probabilistic ATMS. A very interesting possible extension of this scheme is to have non-deterministic justi cations. Consider a problem solver that is using a probabilistic proof method. Given a set of facts it may be able to state that some conclusion is derivable with some speci ed high probability. To handle such a situation the concept of a justi cation has to be generalized. A possibility is to de ne a non-deterministic justi cation to be of the form J ! n, where  is the probability that n is derivable given that the nodes in the set J are derivable. Ideally we would like the non-deterministic extension to work as follows: Given a node n, say K is the set of nodes in the ATMS such that each node in K is an antecedent of some justi cation of n. The Bayesian network constructed from the ATMS therefore would have an analogous node n, nodes K and a probability distribution P (njK ). Say we now add the justi cation J ! n. We would like to now create a modi ed distribution P (njJ [ K ). We currently have one possible de nition for nondeterministic justi cations which is analogous to the Noisy-Or model used in Bayesian network modeling [Pearl 88, Srinivas 93]. Each justi cation J ! j is seen as an independent \cause" for the consequent. Each \cause" can be inhibited with probability 1 ? . When the \cause" is inhibited the consequent is false even though the antecedent is satis ed. In other words, P (j = f j J = t) = 1 ? . If we assume that the inhibition of each justi cation is probabilistically independent of the inhibition of any other justi cation we can de ne a mechanism for nondeterministic justi cations which meets the desiderata sketched above. This independence assumption, however, is very strong. Its applicability depends on the mechanics of the proof method that the problem solver uses. More generally, non-deterministic justi cations are a method of introducing defeasible implications in the propositional calculus. De ning a well founded semantics for non-deterministic justi cations is thus closely related to Epsilon semantics [Adams 75] and work on non-monotonic reasoning. 0

6 Future work In the context of diagnosis the most useful operation of the probabilistic ATMS is the following: Given an assumption set and a node, return the posterior probability that the assumption set is true given that the node was derivable. In the method described above, we need a belief propagation through the network to answer this question. This is potentially expensive. However, the analogous question can be answered using the derivable operation in the basic ATMS. This operation is very quick (requiring only a check of whether the assumption set is subsumed by the label). This speed is a basic feature of the ATMS which makes it 8

References [Adams 75] [Darwiche 93] [D'Ambrosio 90]

[deKleer 86] [deKleer et. al. 87] [deKleer et. al. 89]

[Jensen et. al. 89]

[Gener et. al. 87]

[Laskey et. al. 89]

Laskey, K. B. and Lehner, P. E. (1989/90) Assumptions, Beliefs and Probabilities. Arti cial Intelligence, Volume 41, 65{77. [Lauritzen et. al. 88] Lauritzen, S. L. and Spiegelhalter, D. J. (1988) Local computations with probabilities on graphical structures and their applications to expert systems J. R. Statist. Soc. B, 50, No. 2, 157{224. [Pearl 87] Pearl, J. (1987) Distributed Revision of Composite Beliefs. In Arti cial Intelligence, Volume 33 (2), 173{216. [Pearl 88] Pearl, J. (1988) Probabilistic

Adams, E. (1975) The logic of conditionals. Dordrecht, Netherlands: D. Reidel. Darwiche, A. (1993) Objection Calculus and Clause Management Systems. Submitted to IJCAI-93. D'Ambrosio, B. (1990) Incremental Construction and Evaluation of Defeasible Probabilistic Models. In International Journal of Approximate Reasoning 4, 233{260.

de Kleer, J. (1986) An Assumption-based TMS. Arti cial Intelligence, Volume 28, Number 2, 127{162. de Kleer, J. and Williams, B. C. (1987) Diagnosing multiple faults. Arti cial Intelligence, Volume 32, Number 1, 97{130. de Kleer, J. and Williams, B. C. (1989) Diagnosis with behavioral modes. Proc. of Eleventh International Joint Conference on AI, Detroit, MI. 1324{1330. Jensen, F. V., Lauritzen S. L. and Olesen K. G. (1989) Bayesian updating in recursive graphical models by local computations. Report R 8915, Institute for Electronic Systems, Department of Mathematics and Computer Science, University of Aalborg, Denmark. Gener, H. and Pearl, J. (1987) Distributed Diagnosis of Systems with Multiple Faults. In Proceedings of the

Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann

Publishers, Inc., San Mateo, Calif. [Shachter et. al. 90] Shachter, R. D., D'Ambrosio, B. and DelFavero, B. (1990) Symbolic Probabilistic Inference in Belief Networks In Proceedings of the Eighth National Conference on Arti cial Intelligence, 126{31.

[Srinivas et. al. 90] Srinivas, S. and Breese, J. (1990) IDEAL: A software package for analysis of in uence diagrams. Proc. of 6th Conf. on Uncertainty in AI, Cambridge, MA. [Srinivas 93] Srinivas, S. (1993) A Generalization of the Noisy-Or Model In Ninth Annual Conference on Uncertainty in Arti cial Intelligence.

[Srinivas 94]

3rd IEEE Conference on AI Applications, Kissimmee, FL, February 1987. Also in Readings in Model based Diagnosis,

Morgan Kauman.

9

Srinivas, S. (1994) Building diagnostic models from functional schematics. Technical Report No. 94-15, Knowledge Systems Laboratory, Computer Science

Department, Stanford University, Stanford, CA 94305.

(b) Add 3. Return Envs.

S ea to the set Envs. a

justifications-into(node)

A The ATMS operations

The problem solver uses the following interface to interact with the ATMS: add-node(node-name, assumption-p)

1. Create a node with name node-name. 2. If assumption-p is true, then create a label for the node which contains just one environment. This environment contains only one node, which is the node itself. If assumption-p is false, then create an empty label for the node, i.e., a label containing no environments. add-justification(antecedent-set, consequent-node) 1. Create the data structure for the new justi cation. 2. restore-labels(consequent-node) derivable(assumption-set, node) If(subsumed-by-node-label(assumption-set, node) ^ : subsumed-by-node-label(assumption-set, n? )) return t, otherwise return f .

Return the set of justi cations which have node as the consequent. consequent(j) Return the consequent node of the justi cation j. subsumed-by-node-label(assumption-set, node) If assumption-set is a member of the label of node or a superset of some member of the label return t, else return f .

The Top node and the Bottom node are created during the initialization of the ATMS. The Top node is given a label which consists of only one environment, the empty set. Since the empty set is a subset of all environments, this enforces the condition that the Top node is always true. The Bottom node is created with an empty label. If we assume that all node labels are correct (i.e., sound, minimal and complete) before each add-justification operation, correct updated labels for each node are restored after the add-justification the operation. The ATMS algorithm terminates even in the presence of circularities in the justi cations [deKleer 86]. In the interests of keeping it simple, the algorithm These operations in turn use the following func- described in this section is correct but not as ecient tions: as de Kleer's original algorithm. Speci cally, it proprestore-labels(n) agates inconsistent environments in restore-labels (which is unnecessary) and further, it does not take 1. J = justifications-into(n) care to propagate only new environments not present 2. C S = envs-supporting-justif(j) in the old label when the old label is to be replaced j2J by the new label. 3. If C 6= Label(n) (i.e., the existing label)  Label(n) = C  For each justi cation j in the ATMS in which n is an antecedent, do restore-labels(consequent(j)) envs-supporting-justif(j) 1. Initialize the set Envs = fg. 2. In all possible ways: Choose an environment ea from the label of each antecedent a of justi cation j . If thereSis no E 2 Envs such that E  a ea : (a) Delete all environments S E in Envs such that a ea  E .

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