From: AAAI-88 Proceedings. Copyright ©1988, AAAI (www.aaai.org). All rights reserved.

BELIEF MAINTENANCE: AN I TAPNTY MANAGEMENT

Kathryn B. Laskey and Paul E. Lehner

Decision Science Consortium, Inc. George Mason University 1895 Preston White Drive 4400 University Drive Reston, Virginia 22091 Fairfax, Virginia 22030

Abstract In the quantitative approach, uncertainty about a proposition's truth value is expressed Belief maintenance represents a unified by a number or numbers, typically in the range approach to assumption-based and between 0 and 1. Inference rules also have numerical uncertainty management. A numbers associated with them, indicating how formal equivalence is demonstrated strongly belief in the premises warrants belief between Shafer-Dempster belief theory in the conclusions. Various calculi have been and assumption-based truth maintenance proposed for propagating beliefs in chains of extended to incorporate a probability inference. calculus on assumptions. Belief propagation through truth maintenance We argue that a significant advantage may automatically and correctly accounts be had by combining symbolic and numeric for non-independencies among uncertainty management. Specific advantages propositions due to shared antecedents. include the following. Belief maintenance also incorporates an ability to represent and reason with Computational aspects of numeric reasoning. The defaults. The result is a framework numeric approach has been criticized on for non-monotonic reasoning about the efficiency grounds. The number of possible application of a quantitative combinations of truth values grows exponentially uncertainty calculus. with the number of propositions in a system. A numeric calculus maintains propositions, no matter how improbable, unless they are 1.0 INTRODUCTION explicitly proven false. Of course, sophisticated computational architectures and Automated reasoning systems must operate with control strategies can reduce inefficiency incomplete knowledge of the state of the world. (e.g. 9 by not exploring consequents of Much of the work of problem solving or inference improbable propositions). But additional lies in structuring exploration of the system's control strategies open up with the possibility world to reduce this uncertainty. Two general of applying non-monotonic qualitative reasoning approaches to uncertainty management have become to the application of an uncertainty calculus. popular. These approaches--symbolic truth For example, the control strategy might make an maintenance and numeric belief propagation--have assumption, which could be later retracted, been portrayed as rivals in a sometimes assigning a truth value of false to an acrimonious debate. Yet each has an important improbable proposition. Or the structure of an role to play in a comprehensive inference inference network could be subjected to strategy. qualitative reasoning by making retractable conditional independence assumptions. Symbolic truth maintenance [Doyle, 1979; deKleer, 19861 is based on the idea of extending Control of reasoning. An important role of ordinary truth-functional logic to allow the truth maintenance is to control reasoning--to incorporation of defaults or assumptions. A decide what to do next. Additional system based on such a logic can set default possibilities for control are opened up by values for uncertain propositions and reason as including a numeric uncertainty calculus. An if these values were known, but revise its example cited above is to avoid using inference defaults when they give rise to a contradiction. rules with improbable antecedents. Another This capability mimics the non-monotonic strategy involves searching for information that character of intelligent human reasoning. Truth will likely distinguish between two uncertain maintenance increases search efficiency by but competing hypotheses. permitting a control strategy that minimizes regeneration of previously considered search Conflict resolution. The two uncertainty paths. Uncertainty is represented entirely management traditions have very different qualitatively: what is known about propositions approaches to dealing with conflict [Cohen, is whether they are proven true, assumed true, Laskey and Ulvila, 19871. In the symbolic unknown, assumed false, or proven false in a tradition, conflicting conclusions indicate that given context. one of the lines of reasoning is faulty.

2 10 Automated Reasoning Defaults must be changed to restore consistency. reasoning system might incorporate a default By contrast, probabilistic and other numerical assumption that the rule is operating correctly systems regard conflict as an inevitable (i.e., the source is reliable). Receipt of consequence of an imperfect correlation or evidence against q would necessitate dropping causal link between evidence and conclusion. either this default or one of the assumptions Probabilistic reasoning regards inference as underlying the conclusion of wq. The belief weighing a balance of positive and negative calculus, on the other hand, simultaneously evidence; symbolic reasoning adjusts the set of maintains belief that the rule is and is not accepted defaults so that no simultaneous valid, using a number between zero and 1 to arguments exist for both a proposition and its encode the strength of belief in rule validity. complement. Intelligent reasoning, we feel, combines aspects of both these viewpoints. Simultaneously maintaining belief in must be able to entertain conflicting lines of inconsistent propositions is impossible in most argument, adjudicating between them based on the default reasoning systems. When beliefs become strength of each. On the other hand, when the inconsistent, these systems must explicitly conflict becomes too great, we begin to suspect change some of their defaults to regain a problem with one or the other argument, and consistency. Combining beliefs and defaults is examine our implicit beliefs for possible therefore difficult with traditional truth revision. maintenance. But the ATMS [deKleer, 19861 is explicitly designed to reason with multiple This paper represents a preliminary effort contexts. Each proposition is tagged with a toward unifying symbolic and numeric reasoning. label that represents the contexts in which it It has been implemented, but only on small-scale can be proven. Contexts are represented by problems. Much work remains in developing special symbols which deKleer calls assumptions. control strategies and algorithms to circumvent We depart from deKleer's terminology and use the the computational complexity associated with term tokens to refer to these special symbols, large belief networks. But the structure because we prefer the word assumption to retain described here represents an important step the connotation of a proposition that, although toward understanding the relationship between unproven, has been declared to be in the set of probabilistic and belief function models, truth believed propositions. The job of the ATMS is maintenance, and non-monotonic logics. to maintain a parsimonious representation of each proposition's dependence, either directly or through chains of inference, on tokens. Like 2.0 THEATMS AND SHAFER-D STER BELIEFS deKleer, we define &q environment to be a set of tokens and a conteXt to be the entire set of This section introduces an approach to propositions derivable in a particular uncertainty management that unifies the symbolic environment. A proposition's label contains a and numeric approaches. In particular, it is list of environments in which it can be derived. demonstrated that extending assumption-based There may be proofs for a proposition -under truth maintenance [deKleer, 19861 to allow a several mutually inconsistent environments; probability calculus on assumptions leads to a similarly, proofs for a proposition and its belief calculus that is formally equivalent to negation in different environments may be Shafer-Dempster theory. By appropriate entertained simultaneously. construction of inference rules, probabilistic models emerge as a special case. An advantage Consider again the inference from report r of the framework presented here is that to the truth of the reported proposition q. In nonindependencies are automatically and the ATMS, the "noisy" inference rule may be correctly accounted for in the belief encoded as follows. The token V is introduced computations. to represent rule validity, i.e., the proposition that the source is reporting Eet us introduce the theory with an example reliably. (Following deKleer, we use uppercase used by Shafer [1987] to illustrate belief letters to represent tokens and lowercase function theory. Suppose we receive a report r letters to represent other propositions). The that a proposition q is the case. There is some original noisy rule is replaced by the rule probability, say .6, that the source is rAV -, q. When this rule fires, the token V is reporting reliably; otherwise, with probability added to the label of the ATMS node associated .4, the report bears no relation to the truth of with q, indicating that q is valid in any 4. In the first case, the report implies q; in context in which V is true. the second case, both q and its negation are consistent with the report. For Shafer, belief If this token is treated as a default in a proposition is defined as the probability assumption, then the proposition q is‘believed that the evidence implies its truth. Thus, the until this default becomes inconsistent with the above evidence justifies a .6 degree of belief set of known propositions. Alternatively, V may in the proposition q. be assigned a probability, interpreted as the probability that q can be proven--that is, its Another way of looking at this example is Shafer-Dempster belief. in terms of an inference rule (r + q) that may or may not be valid. A symbolic default

Laskeyand Lehner 211 Thus ) belief maintenance is based on a hyperresolution need not be applied to belief simple principle: if probabilities are assigned tokens, because the belief computations are to tokens, these imply probabilities on the performed only on contexts that are maximal with labels for propositions. The probability of a respect to the token sets impacting a label can be interpreted as the probability that proposition's truth value. But de#leer's [1986] the associated proposition can be proven, and is disjunction encoding is needed for other equivalent to its Shafer-Dempster belief. exhaustive sets of propositions (e.g., q and its negation -9).

3.0 A BELIEF MAINTENANCE SYSTEM The belief maintenance system can represent two additional specialized types of token: Our belief maintenance system combines an ATMS default and hidden tokens. Default tokens with a module for computing the probabilities of 'represent propositions that the system chooses ATMS labels, or, equivalently, the to treat as if they were known to be true (i.e., Shafer-Dempster beliefs of the associated probability 1). Hidden tokens correspond to propositions. Belief maintenance is capable of deKleer's ignored assumptions. They are representing the full generality of the manipulated by the ATMS, but environments Shafer-Dempster calculus. The ATMS containing them are ignored in the belief automatically keeps account, in symbolic form, computations and are invisible to the problem of the propagation of beliefs through chains of solver using the belief maintenance system. inference, nonindependencies created through shared premises, and inconsistent combinations Beliefs are computed conditional on the of tokens. The belief computation module current set of defaults (see below). A belief incorporates all this information to compute token may be defaulted (as when a strongly correct Shafer-Dempster beliefs when requested. supported hypothesis is provisionally accepted). Adding to this framework the capability to This causes the token to be treated as if it had represent and compute beliefs with defaults probability 1. The other belief tokens in its results in a fully integrated symbolic and token set become hidden. The default may numeric uncertainty management framework. subsequently be removed, in which case probabilities on tokens in the set revert to Readers familiar with the basics of belief their former values. function theory and assumption-based truth maintenance will more easily follow the Let us return to the inference rule following presentation. Laskey and Lehner rAV + q. If r is observed (i.e., declared as a [1987] include a concise introduction to both premise), this rule fires and adds the theories [see also deKleer, 1986; Shafer, 19761. environment (V) to the label of q. Absent other rules affecting q, the belief in q is equal to 3.1 Combining Beliefs and Chaining the probability of V, or .6. If V is declared Inference Rules as a default, this belief becomes 1.

We have shown how a single uncertain inference Now suppose a report from another source rule could give rise to a Shafer-Dempster indicates the negation of q, and that this belief. But interesting inference problems source is judged to have reliability .8. As involve many propositions, linked together by before, this can be encoded as an inference rule complex chains of inference, involving SAW -,-q, where s stands for the source's report converging and conflicting arguments. In this and W is a belief token with probability .8. section, we show how the ATMS can be used to The label of q remains unchanged, but the manage these inferences, keeping track of the environment (W) is added to the label of -q. tokens on which propositions ultimately depend. The tokens V and W imply inconsistent propositions and cannot both be true. The ATMS The basic unit of belief in a belief represents this inconsistency by a nogood maintenance system is the belief token, a environment (V,W). special token which carries an attached probability. Belief tokens come in sets. Every Belief in q and in --qremain unchanged at 1 extension (maximally specific environment) must and 0, respectively. This is so because the contain exactly one token from each set of label of q, the environment (V), is a default belief tokens. In the above example, the token and so has probability 1. The label of -q V would be paired with another token contains the environment (WI. Belief in this representing the negation of V. This second label is conditioned on the current defaults, token (call it il)would be assigned belief .4, and because W is inconsistent with V its so that its belief and that of V sum to 1. conditional probability given V is zero. Belief tokens are processed by the ATMS exactly as are other tokens. What happened to our belief of .8 in W? Given the default token V, we would have We note here that encoding negations in the assigned the environment (V,W) probability .8 if ATMS involves defining a choose structure and it weren ’t inconsistent . But because beliefs extending the label updating algorithm to are conditioned on the consistent environments, include hyperresolution rules. Actually, this environment, despite its high prior

212 Automated Reasoning probability, is discarded as impossible. The The belief in a node is defined as the probabilities of the consistent environments are probability of its label. But this probability divided by .2 so they will sum to 1 after must be conditioned on the current defaults, and discarding the inconsistent environment. on the consistency of the label. For example, if x had label (A,B) where Pr(A)-.8 and We see that the belief assigned to Pr(B)-.7, then (absent other information) x has inconsistent environments under the current belief .56, the product of these probabilities. defaults can be thought of as measuring conflict But suppose nogood(B,C). This means that the associated with the defaults. In this case, we conjunction of B and C is Jmpossible, so no might decide that .8 is too high a degree of belief can be assigned to it. Belief in x must conflict, and respond by dropping the default V. thus be revised to reflect this. In other Returning to our former belief assignment, the words, belief in x is the conditional prior probability of the nogood environment probability of AAB, given -(BG): (V,W) is reduced to .48. Below we list the propositions of interest, the contexts in which Bel(x) - Pr[AABI-(BAC)] - lPr((A*B,F)) * they can be proven to hold, and their beliefs. - Pr((B,C))

Proposition Context Belief If Pr(C) - .8, this expression evaluates to .25. If C is a default, this evaluates to 0 Q pm .12/.52 - .23 (because the numerator is the probability of an -Q (V,W .32/.52 - .62 envtronment that cannot hold under the default).

In general, the belief in a proposition is These beliefs are the same as obtained by given by using each inference rule to define a belief Be1 (node) - function and combining them by Dempster's Rule. Pr(labelldefaults,-nogood) I Pr(labeln-nogoodldefaults) . Rules may have as antecedents the (*I consequents of other rules. For example, Pr(-nogoodldefaults) consider a third rule qU + t. Firing this rule adds the environment (V,X) to the label of t, A simple algorithm for calculating the belief in a proposition (i.e., the probability that it can indicating that t is true in any context containing both V and X. The probability of be proven) follows. this environment, conditional on the defaults 1. Select all environments in the label that (if any) and the consistent environments, defines the degree of belief in t. The ATHS are consistent with current defaults_ and automatically keeps track of nonindependencies. that contain no hidden tokens. For example, there might be another path of reasoning from q to t. When these rules fire, a 2. Remove default tokens from the environments second environment containing V will be added to containing them (this amounts to treating the label of t. The probability calculus them as if they had probability 1). The described below automatically avoids "double set of tokens thus defined will be referred counting" the impact of V. to as the selected tokens.

3.2 Computing Beliefs 3. Select all nogood environments that are consistent with current defaults, that The probability handler computes beliefs on contain a selected token, and that contain nodes from the probabilities assigned to belief no hidden tokens. Remove all default tokens, given the current set of default tokens. tokens and add the remaining tokens to the The following conditions are assumed: selected tokens.

1. Each belief token is part of a mutually 4. Repeat Step 3 until no more tokens are exclusive and exhaustive set of belief added to the selected tokens. tokens whose probabilities sum to 1. A set of such tokens corresponds to Shafer's 5. The selected token sets are those to which background frame, and carries the basic the selected tokens belong (e.g., if V is a probability for a belief function. selected token, the corresponding token set is (V,V',). Form a list of maximally 2. Each belief token is probabilistically specific environments from the selected independent of all other belief tokens token sets. That is, form all possible except other tokens in the same exhaustive combinations of tokens, taking one from set. This condition may seem restrictive, each selected token set. but in fact it is not. It merely requires that all nonindependencies be represented 6. Remove all nogood maximally specific explicitly as shared information (i.e., the environments (i.e., those containing a labels of two dependent propositions nogood environment, or an environment that contain belief tokens in the same is nogood when coupled with one or more hypothesis set). default tokens). Of the remaining

Laskey and Lehner 213 environments in the maximally specific REFERENCES list, the ones containing a label environment (or which do when combined with one or more default tokens) are those [Cohen, Laskey and Ulvila, 19871 Cohen, M.S., contributing to belief in the node. Add up Laskey, K.B., and Ulvila, J.W. The the probabilities of these (where the management of uncertainty in intelligence probability of the environment is the data: A self-reconciling evidential product of the probabilities of its database (Technical Report 87-B). Falls constituent assumptions) to get the Church, VA: Decision Science Consortium, numerator of (*). Add up the probabilities Inc., June 1987. of the whole list (except the removed nogoods) to get the denominator of (*). [D'Ambrosio, to appear] D'Ambrosio B. A hybrid (An alternate way to compute the approach to uncertainty ,' International denominator is 1 minus the probability of Journal of Approximate Reasoning, to the removed nogoods.) appear.

This algorithm may be modified to simplify [deKleer, 19861 deKleer, J. An assumption-based processing of labels in which environments have truth maintenance system; Extending the little overlap [Laskey and Lehner, 19871. Other ATM; Problem Solving w%th the ATM, efficiency modifications are possible, such as Artificial Intelligence, 1986, 28, 127-162; caching intermediate products so that belief 163-196; 197-223. computation after label changes can be done incrementally. Although the basic algorithm [Doyle, 19791 Doyle, J. A truth maintenance above works regardless of the pattern of system, Artificial Intelligence 12(3), inferential links, 'the improvement gained by 1979, 231-272. these efficiency modifications will be greatest when nonindependencies are few. [Laskey and Lehner, 19871 Laskey, K.B. and Lehner, P.E. Assumptions, beliefs and D'Ambrosio [to appear] has suggested an probabilities, submitted to Artificial approach very similar to the one described here. Intelligence, 1987. His algorithm differs from ours in that his encoding of belief functions is less general, [Shafer, 19761 Shafer, G. A Mathematical his treatment of nogoods that indirectly impact Theory of Evidence, Princeton, NJ: a proposition differs from the network Princeton University Press, 1976. Shafer-Dempster model, and there is no provision for defaults. In the most general networks, [Shafer, 19871 Shafer, G. Belief functions and both algorithms are exponential in the number of possibility measures, in Bezdek, J.E. hypothesis sets in the system. D'Ambrosio puts (Ed.), The analysis of Fuzzy Informafion, forward some suggestions (not yet implemented) Boca Raton, FL: CRC Press, 1987. for decreasing complexity; we are currently exploring others.

4.0 DISCUSSION

Belief maintenance represents a way of implementing a unified approach to uncertainty management. Any Shafer-Dempster or probabilistic inference network can be represented using this formalism. Indeed, Shafer-Dempster belief theory and belief maintenance without defaults are formally equivalent [Laskey and Lehner, 19871. The addition of defaults extends Shafer-Dempster theory to include symbolic non-monotonic reasoning. Defaults may be used to represent working assumptions about how to apply the calculus (such as assuming the truth of propositions with a high degree of belief). A by-product of the belief computation is the prior degree of belief assigned to inconsistent hypotheses, which measures the degree of conflict associated with the current defaults. A high degree of conflict can indicate the need to examine the defaults for possible revision.

214 Automated Reasoning