Using Predicate Logic to Integrate Qualitative Reasoning and Classical Decision Theory

JOHN FOX, DOMINIC A. CLARK, ANDRZEJ J. GLOWINSKI, AND MICHAEL J. O’NEIL

A L t r m t -Classical decision theop can be ten restrictive when formu- lating decision models for practical situations. The principal group of restrictions, inherent in numerical decision theories, is that they make no provision for reasoning about the decision process itself. Classical proce- dures cannot reflect on what the decision is. what the options are, what methods ma! be used in making a decision, what knowledge ma) be relevant, and w forth. An approach that accommodates classical decision theory within a framework of first-order logic with nonmonotonic exten- sions is described. Among the benefits offered by the approach are: the potential to express qualitative urguments about the desirabilih of deckion options in the absence of probability or utilih parameters: to automate techniques for generating decision options; and to initiate, control. and terminate a decision process autonomously. The rationality of a logical decision framework is discussed. A particular benefit of implementing decision procedures in a nonmonotonic logic is that decision systems can adapt automatically as beliefs change. contradictions are encountered. or new knowledge is acquired.

I . INTRODUCTION

INDLEY [13] summarizes the classical theory of deci- L sionmaking under uncertainty as follows. “ . . . there is essentially only one way to reach a decision sensibly. First, the uncertainties present in the situation must be quanti- fied in terms of values called probabilities. Second, the consequences of the courses of actions must be similarly described in terms of utilities. Third, that decision must be taken which is expected on the basis of the calculated probabilities to give the greatest utility. The force of ‘must’ used in three places there is simply that any deviation from the precepts is liable to lead the decision maker into procedures which are demonstrably absurd-or as we shall say, incoherent” [13, p. vii]. Elsewhere, [13, p. 51 he adds that “Our recipes will be limited to selection amongst a given set of decisions and it is important to make sure that the set reasonably exhausts the possibilities” and “we can offer no scientific advice as to how (the skill of selection) is to be developed.”

It seems to us that this view is seriously and unnecessar- ily restrictive. First of all, circumstances do not always permit quantification of uncertainties yet a decision may still be urgently required. Second, the complete set of decision options and relevant indicants is frequently un-

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known when the need for a decision is first recognized, and may only emerge as decisionmaking proceeds. Since circumstances in which such knowledge is lacking are commonplace, the claim that there is only one coherent decision framework is unacceptably exclusive. We cannot simply abandon situations which do not satisfy particular mathematical assumptions but must, rather, develop deci- sion procedures which can permit adaptive selection and modification of practital decision strategies as knowledge accumulates.

Knowledge-based or “expert” systems were recently in- troduced with the aim, in part, to provide decision support systems with greater versatility than traditional numerical systems [l]. Expert systems contrast sharply with classical systems in that they emphasize the representation of knowledge required to make a decision in an explicit qualitative form rather than implicitly in a numeric or algorithmic form. Countless expert systems have been de- veloped which explicate knowledge of problems in medicine, engineering, commerce, etc. It is now generally acknowledged that knowledge-based approaches have much to offer to the design of decision support systems. However, the methods used to manage decision parameters like probability and utility in many systems have been widely criticized as ud hoc.

Much effort has been expended on these problems and significant refinements in expert system design have been suggested [14], [12]. Improved methods for combining qualitative and quantitative reasoning are to be welcomed, but these developments do not address the problems out- lined above and the purpose of this paper is to argue for a more radical development. The heart of decision technol- ogy is the theory of decisionmaking itself and it seems to us that insufficient attention has been paid to correcting the fundamental shortcomings of classical theory. Devel- opments in computational logic and knowledge engineer- ing suggest ways of doing this. In particular, although expert systems have attempted to capture the knowledge of many specialist fields, few attempts have been made to explicate knowledge of decisionmaking itself.

We begin by identifying a number of distinct types of knowledge that are required to make decisions: knowledge of the application or field in which the decision is to be taken; knowledge about what is and is not known; knowl-

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edge of uncertainty and its causes: knowledge about the decision itself and what it is required to achieve, and knowledge of procedures that can be used during decision making and their applicability.

1 ) Application-Specific Knowledge: This is frequently represented in the form of propositional inference rules (e.g., “advanced age and weight loss suggest cancer”). This presupposes the use of specific inference or decision strate- gies, which can introduce exactly the inflexibility that we wish to circumvent. We propose separating the applica- tion-specific propositions embedded in such rules by ab- stracting an application-independent decision procedure expressed in first-order logic.

2) Knowledge About Knowledge: It may be important to be able to determine whether or not available facts are relevant to a decision: whether they are complete, reliable and so on. The importance of using metaknowledge in decisionmaking will be emphasized.

3) Knowledge About Uncertainty: Uncertainties are typi- cally represented in expert systems by modifying categori- cal propositions with quantitative coefficients, e.g., “ high intra-ocular pressure causes retinal damage (0.5);” “ad- vanced age and weight loss imply cancer (0.2).” We shall describe the separation of logical from probabilistic infer- ence and in addition advocate the use of modal sentential operators characterizing logical states of belief.

4 ) Knowledge About Procedures: An advantage of sepa- rating different types of knowledge (like logical and nu- merical knowledge aforementioned in 2)) is that the sepa- rate components can be applied independently. For example we may have knowledge of alternative uncertainty management techniques and their applicability conditions, so they can be enabled or disabled as task constraints become known [2], [4], [6], [lo], [15].

5) Knowledge About Decisions: Practical decision sys- tems may require the capability to operate autonomously, without needing to depend upon a human decision analyst to anticipate or deal with unusual circumstances. An im- portant part of decisionmaking is the knowledge needed to make decisions about decisions, viz when a decision is required, what type of decision it is, when the decision should be terminated, and so forth.

11. A LOGICAL FRAMEWORK FOR DECISION THEORY

Decisionmaking is viewed here as a process that creates and maintains a database of propositions, which, although logical, may express qualitative or quantitative facts. Deci- sion processes maintain this database using generic knowl- edge of decision types (e.g., diagnosis or treatment deci- sions) to construct specific instances (cases) of decisions in the database (e.g., Fred’s diagnosis). Case-specific data (findings) are interpreted using theories (e.g., causal, anatomical, and functional theories) and procedures (e.g.. statistical procedures, simulations) within a framework of predicate logic. The decision procedure is implemented as

(a) propositional case knowledge decisions of fred include diagnosis findings of fred include weight loss

(b) propositional domain knowledge colon cancer is a kind of cancer gastric ulcer is a kind of peptic ulcer gi bleeding is a kind of abnormal condition melaena is a kind of gi bleeding.

elderly IS a posiuve association of m u

delayed epigasmc pam is a negauve assnstion of gasmc cancel

cancer is a caw of weight loss colon cancer is a cause of melaena

conditional probability of weight loss given cancer = 0.4 conditional probability of elderly given cancer = 0.7 probability of gasmc cancer = 0.1 probability of colon cancer = 0.1

(c) propositional meta-knowledge relevant theories in diagnoses include c a w relevant theories in diagnoses include positive aFsociluions

propagation links of diagnoses include superclasses elimination links of diagnoses include subclasses association links of diagnoses include superclasses association links of diagnoses include causes

Fig. 1. Selected fragmcnts of a medical knowledge base and meta- knowledge of diagnostic decisions used the decision procedure for this cwanlplc.

a set of first-order logical schemata that manipulate case data. A concrete example would be reasoning about a patient’s symptoms (case data) in the light of medical knowledge in order to make a diagnostic decision.

In our current implementation propositions are used to represent many relationships commonly seen in AI, such as subclass/superclass relations (e.g., between appli- cation-specific entities like diseases, symptoms, etc), in- stance relationships (e.g., Fred is an instance of the class patient, Fred’s diagnosis is an instance of the class diag- noses), causal links, numerical relationships ( =, > , < ). In addition the decision process generates propositions which indicate explanations of findings, possible decision options, and relationships between them. Some of these are illustrated in Fig. 1; others are introduced as needed.

The database can alternatively be viewed as a graph in which nodes (or connected subgraphs) represent decisions, cases, findings, explanations, justifications for conclusions, and their interrelationships. This view is helpful when considering propagation of decision parameters (e.g., prop- agation of uncertainty through a causal subgraph. [12]) and also for visual illustration. Fig. 2 illustrates in graphi- cal form an example of the decision procedure applied to a particular case (Fred) and a particular decision (diagnosis). The state of the database is illustrated after 1 and 5 findings have been supplied. (NB: A similar example is described in more detail by Fox [lo].)

The decision procedure is modeled as a set of logical processes that modify the database (graph) as specified

349

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I

. candidate: gastric ulcer coverage: 100.0 % probability: 0.12 direct support: 1 arguments confirmed,positive.arguments

statistical associations of elderly

decision

case id: fred decision type: diagnosis findings of fred

elderly

candidate: cancer coverage: 100.0 %

\\ c o n t i i d positive arguments: \ statistical associations of elderly \

case id: ked decision type: diagnosis findings:

disabled subprocedures: b y e s rule night pains, haematemesis, melaena, weight loss, elderly

candidate: cancer, possible coverage: 40.0 % direct suppon: 2 arguments confirmed positive arguments:

causes of weight loss statistical associations of elderly

candidate: gastric cancer, possible coverage: 60.0 %,joint highest

\~ direct support: 1 arguments

cancer

direct support: 3 arguments, highest confirmed,positive.aguments

candidate: duodenal ulcer, possible percent coverage 20% direct support: 1 arguments confirmed negative arguments

confirmed positive arguments statistical associations of elderly

causes of night pains L I I I

Fig. 2. (a) This illustrates the results of initiating a diagnostic d e c i h n for c a ~ called Fred. who is knoun to be elderly. An argument on (qualitative) associations grounds supports possible diagnoses of canccr and gastric-ulccr. Each candidate (triviallv) accounts for or covers 1007; of the known findings at this stage. and sincc conditional probabilities arc available. a numerical probabilitv can be calculated. ( b ) This shows the case after a number of findings have been added. Here additional candidates have been identified. and aasociations between the altcrnativcs arc dctcctcd. !idding a candidate hierarch!. By this time the system no longer has a complete conditional prohahilit! table for all the candidates and findings. s o that calculation of probabilities by Bayes’ rule is no longer valid. so the method is disabled in t h i h context: all pre\iousl\ calculated \slues are deleted and the decision procedure must depend upon other figures of merit. such as coverage or unueightcd support. Italicized terms show some simple logical annotations of the data: others are diwusscd in the tcxt.

by a number of schemata. First we give a brief, formal summary of the central decision processes. In what follows F, a finding, is any item of case data and 0 any decision option which might be considered for the case. A , is an argument based on some theory (rules of reasoning) about the truth of P, a proposition. The braces { } are normal set notation, right arrow - is logical implication and all capitalized terms are universally quantified vari- ables. Finally the predicates provable( Db, A , P, Q ) and mode(C, P, M ) mean “P may be proved from case database Db with qualification Q by argument A” and “proposition P has modality M for case C ” where a modality is a specific class of qualification. We now define several processes which combine to provide a decision

procedure. It may be helpful to return to the definitions after discussion of specific examples in the next section.

A. Constructing Arguments

Argumentation is the process of constructing lines of reasoning for some proposition about a specific case. A confirmed argument, conf-arg(C, A , , S ) . is an assertion that a general line of argument A , applies to the current case, C, where the sign, S, is a qualification of the argued proposition and may be one of: supporting ( + ), weaken- ing ( - ), confirming ( + + ) and excluding ( - - ):

F provable( F , A , , P , S ) + conf-arg( C , A , , P , S ) . (1)

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B. Introducing Decision Options

We will particularly discuss the construction of argu- ments about decision options: where there is a reason for considering the option but no reason to exclude i t then it is added to the option set for the case, {option(C, 0 ) } :

conf-arg( C, A , , O , s ) A s E { +, ++ } A

- Ut,: conf-arg( C, A,,, 0, - - ) + option( C, 0 ) . (2 )

C. Associating Decision Options

Introducing decision options by local arguments cannot be guaranteed to ensure the options are distinct. A deci- sion procedure must identify relationships, R , among op- tions. R may represent any of a number of relationships between options, such as one medical treatment being subsumed by another, one disease causing another, one test option applies to a system which is a part of the system to which another test option applies, etc. We sim- plify here by treating them all as association:

let Pair = { 0, O ’ }

option( C , 0) /\option( C, 0’) A

U,, R : provable(Pair, A, , R , ++) -+associated( C,Pair). (3)

D. Annotation of Options

A base proposition P such as “diagnosis of Fred is cancer,” can have distinct modalities such as possible P; unlikely P ; suspected P . It is useful to compute certain modalities such as “possible diagnosis of Fred is cancer” (or more generally annotation (Decision of Case is Option, Mode)) from the pattern of support for the option 0 recorded in the database:

let Support = {conf-arg(C, A , , O , S ) } then:

option( C , 0 ) A U,,: provable( Support, A, , , 0, M )

-, annotation( C, 0, M ) . (4)

E. Aggregation of Arguments

Procedures are required to aggregate the arguments for and against the current decision options in order to yield a total or partial ordering on the option set, thereby permit- ting a decision. We may view aggregation as a form of argumentation which yields a numerical annotation (such as a probability of a hypothesis being true), and we may derive further qualitative annotations from such annota- tions (e.g.. most probable). Aggregation procedures typi- cally yield a measure (or measures) to give the ordering(s) on the decision options but we do not exclude qualitative methods for establishing the ordering without intermediate quantification.

The remainder of the paper develops these ideas with examples taken from one implementation of the decision procedure. Examples are presented in Props2, a language in which predicate calculus expressions can be expressed in

a readable pseudo-English form. Note that we have in- cluded case-specific parameters within terms in the afore- mentioned formalization but for legibility these are fre- quently excluded from examples.

F. Summary of Notation and Intended Execution Scheme for Examples

Notation: Data and knowledge are represented in an English-like notation in which reserved terms (which deter- mine the formal structure of sentences) are italicized. The emphasis is omitted from examples in the text:

1) symptoms of gastric cancer include weight loss; 2 ) gastric cancer is U kind of cancer; 3) probability of gastric cancer = 0.55; 4) confirmed supporting arguments of cancer include

statistical association of advanced age.

The examples translate into standard propositional nota- tion which instantiate first-order logical schemata, i.e. in- ference rules defined over (open) classes. In the following example:

findings of Case include Finding symptoms of Disease include Finding potential diagnoses of Case include Disease.

The capitalized variables (Case, Finding, Disease) will be instantiated to specific values in specific cases; if the antecedents of the schema, above the bar, are satisfied then the (instantiated conclusion) below the bar is as- serted. If, for example, we knew that

5 )

6) findings of Fred include weight loss

then by (1) and (5) we may conclude

7) Data-Driven Interpretation: Inferences are explicitly as-

serted in the database as soon as the antecedent conditions of a schema are fully instantiated by known facts. Vari- ables are universally quantified, i.e., all valid inferences defined by the schema are asserted. An assertion may cause further schemata to be instantiated, and with one exception described below, the propagation is exhaustive (i.e., the forward closure is normally computed).

Truth Maintenance: If propositions are retracted or con- tradictions generated during forward propagation, depen- dent inferences are retracted and the consequences of the new data state recomputed. The closed world assumption is made during both forward computation and consistency maintenance: propositions which are not in the database are assumed to be false.

Theorv-Bused Derivations: For technical and presenta- tional reasons we have implemented a scheme in which most lines of reasoning are followed exhaustively. As a consequence the database will normally be complete, and the truth or falsity of a proposition is determined simply

potential diagnoses of Fred include gastric cancer.

Environment 4.0

Knowledge acquisilion

Case data

Knowledge base of propositions

Domain specific knowledge

Decision strategy knowledge I t

schemata Control

Fig. 3. Schematic view of some of the knowledge required for decision- making, with decision and control knowledge maintaining a database of ‘ I facts” and data/knowledge acquisition and reports required for communication with environment.

findings include Finding provable Arguments of Option include Theory of Finding

confirmed Arguments of Option include Theory of Finding

4.1

Finding is a abnormal condition causes of Finding include Cause

supporting arguments of Cause include cause of Finding

4.2

statistical associations of Hypothesis include Finding

supporting arguments of Hypothesis include statistical associations of Finding

4.3

known sites of Finding include Site Finding is a abnormal condition location of Structure is Site abnormalities of Structure include Process causes of Finding include Process

supporting arguments of Process include sites of Finding

by determining its explicit presence or absence. An excep- tion is in the use of proofs, which is demand-driven rather than data-driven, as illustrated in 8):

Fig. 4. Schema 4.0 is a schema for generating arguments for and against decision options using theories. here illustrated by simple schemata (4.1-2) and a more elaborate form of argument incorporating causal, functional. and structural antecedants.

findings of Case include Finding provable causes of Finding include Disease; possible diagnoses of Case include Disease.

Here, if a finding is asserted then the set of all causes of the finding that can be proved by the system is derived by proof. This may involve retrieving facts as aforementioned or using a first-order theory (e.g., a causal logic), a proce- dure simulating causal processes, or some other deriva- tional technique.

Fig. 3 is a schematic illustration of this framework as realized in a medical application [Ill.

The central knowledge base consists entirely of proposi- tional knowledge as shown in Fig. 1. Other entities are first-order processes, including mechanisms designed to provide dynamic control of the decision process and user interaction. We shall give relatively little detail of the latter as they are not central to the development of the decision framework.

G. Arguing For and Against Decision Options

8)

In general decision theories say how observations should be interpreted to provide arguments for and against deci- sion options. Calculations based on evidence theories, in- formation, and utility theory which yield degrees of belief, expected information yield of tests, expected value of actions, etc., have been given special attention because they provide weights on certain arguments. As observed earlier, however, it may not always be possible to estimate the required parameters for calculating reliable weights so

we wish to establish other sound strategies of argument. Many underlying theories by which qualitative arguments can be established are intuitively familiar but have re- ceived little formal development. Although qualitative, they deserve attention because they may offer information which cannot easily be provided by the classical quantitative met hods.

Fig. 4 illustrates several ways in which logical argument can add information which is not provided by traditional quantitative calculation. First and foremost there are schemata which incorporate knowledge of material rela- tionships (as distinct from statistical relationships) be- tween objects and events. The relationships of time and causality are well known. A causal schema is illustrated in Fig. 4.1. Less well explored are complex arguments like Fig. 4.3. An indefinite number of other arguments may also be formalized.

A logical approach may also help to overcome a weak- ness common to classical decision support systems and current expert systems, namely that all relevant parameters and facts must be identified in advance. In fact it would frequently be desirable to be able to argue about features of the decision situation even when specific knowledge about such situations is not available. Knowledge in the form of theories can be introduced here to interpret, hypothesize and predict circumstances that have not been encountered before. “Common sense” understanding (of cause and effect; structure and organization; function and purpose, etc.) are included in the term theory. More so-

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Observation is observed at Time manifestations of Abnormality include Observation provable at Time

5.0

possible explanations of Observation at Time

possible damage to Smcture components of System include Smcturc

5.1

arguments for abnormality of System include possible damage to Smcturc.

possible abnormality of System functions of System include Operation

5 2

arguments for malfunction of Operation include possible abnwmality of System.

possible malfunction of Recess causes of State include malfunction of F‘rccess

5.3

arguments for State include possible malfunction of Process.

data: breathlessness is observed at 12.45 possible damage to heart.

plus factual knowledge:

components of cardiovascular system include heart. functions of cardiovascular system include oxygen distribution. causes of hypoxia include malfunction of oxygen distribution. manifestations of hypoxia include breathlessness.

and theories 5.1-5.3 yield:

possible explanations of breathlessness include hypoxia.

Schema 5.0 chains Specialized schemata in order to yield an Fig. 5. explanation of the observation of breathlessness.

phisticated frameworks are also admitted (e.g., theories of biological and physical processes which may be needed in medical diagnosis) and of course, probability theory which provides a well-developed basis for arguing about expecta- tions. The point of all theories is that they capture regulari- ties which can be used for making interpretations or predictions in unfamiliar circumstances. Less naturally, perhaps, we shall also include simulations, and databases (of facts) within the term theory. The former also capture predictive regularities, and the latter may be viewed as merely a variant form (the extension) of a logical theory.

Some commonplace patterns of reasoning about causal- ity, structure, and function can be formalized as first-order theories. Fig. 5 shows an example of simplified theories of structural, functional, and causal reasoning expressed as logical schemata. These can be used to construct argu- ments as required in specific situations. We observe that a patient is breathless, then schema 5.0 finds all the abnor- malities that could be manifested by the observed event. These are then checked to see if they are supported by any other available data and logically follow from knowledge. Schemata 5.1-3 are used by a standard logical proof procedure to show that a state of hypoxia could be pro- duced by suspected heart damage and also explain the breathlessness and should, therefore, be regarded as a possible diagnosis.

Our central point is that the calculation of probabilities and utilities in traditional decision theory is only one form

current decisions include Decision confirmed supponing arguments of Option include Argument Option is not eliminated

6.1

possible Decision is Option

possible Decision is Option propagation links of Decision include Relation Relation of Option is LinkedOption LinkedOption is not eliminated

6.2

possible Decision is LinkedOption

propagation links of diagnoses include superclasses propagation links of diagnoses include causes

Fig. 6. Examples of schemata for generating decision options. 6.1 gen- cratcs options on the basis of direct arguments which support them. 6.2 gcneratcs options indircctly by propagating logical possibility over defined relations such as superclass relationships. causal relationships, etc. (sec text and [lo]. for more details).

of argumentation. It uses a particular theory of regularity, probability theory, to argue for and against possible deci- sion options, but there is nothing especially privileged about that theory and others deserve similar development.

H . Determining Decision Options

Traditionally a human decision analyst is responsible for determining all decision options and relevant criteria for choosing among them, though the execution of the deci- sion process may be carried out mechanically. Lindley asserts that the option set must be established at the outset when designing a decision tree or similar structure. In our view the identification of possible decision options is itself a process that must be formalized and mechanized. Fur- thermore, it must be designed to operate progressively as knowledge of the decision circumstances is accumulated.

Schema 6.1 supplies a direct justification for adding a decision option to the set of possible options. It says that if there is any argument that supports an option, and there is no reason to exclude i t , then the option should be included in the set of possibilities, a fortiori. The term possible is in fact an instance of an annotation or modality of the base template Decisions of Cuse include Option, as in possible diagnoses of Fred include cancer; possible treatments of Fred include analgesics. Here the modality is used to admit decision options, i.e. for further specialized analysis.

Rules for proposing decision options can be based on modalities other than a fortiori possibility. We could adopt the less restrictive criterion of a priori possibility, which is to say “assume everything is possible until proved other- wise.” This is the standard meaning of the possible opera- tor in modal logic, where

possible( P ) i f f not( necessary( not( P ) ) ) . Unlike a fortiori possibility no positive argument for P is required in standard modal logic, only that P is not provably false.

Elsewhere we discuss another annotation that is a condition for introducing new options which we call con-

sidered; we consider an option on indirect grounds (e.g.. i t is a subclass of a possible option) even i f it cannot be argued to be directly supported by case findings. In a medical treatment decision we might propose a drug on the grounds that i t has a similar action to another that is already being considered. There are many criteria that may be of value for introducing new decision options, but here we shall confine discussion to criteria based on explicit supporting arguments.

Finally the scheme for proposing decision options is designed to be nonmonotonic, that is, i f for any reason the decision option is later eliminated then the possibility will be retracted, together with all dependent conclusions. These mechanisms alone satisfy many requirements for the deci- sion procedure to be progressive. They can add and re- move decision options from the option set at any time, as findings are asserted into the database and arguments developed from them.

I . Associations Among Options

Association is concerned with relationships between op- tions. Options are generated by local arguments. For ex- ample it may be proposed that a patient is possibly suffer- ing from arthritis on the basis of one observation, and chronic arthritis on the basis of another. Clearly, these two alternatives are not independent. Arthritis and gout may be diagnostic options with their own supporting argu- ments, but since gout may cause arthritis they should not be treated as independent diagnostic options. Alternatively if we reason that steroids and non-narcotic analgesics are possible treatments, but also develop an argument for aspirin, which is a kind of analgesic, the latter two should be treated as a single possibility, while steroids should be kept separate. The following schema says that if two options are independently supported but they can be shown to be related in ways that are germane to the current decision then their association will be noted in the database:

decision is Decision; confirmed supporting arguments of 01 include

confirmed supporting arguments of 0 2 include

linking relations of Decision include Relations; provable Relations of 01 include 0 2 ; associates of 01 include 02 .

Argument1 ;

Argument2;

In Fig. 2(b), for example, colon cancer and gastric cancer are linked as associates with cancer. This is derived from the fact that the current decision is a diagnosis, and the metaknowledge of diagnoses that linking relations for di- agnoses include superclass relationships (Fig. 1 ).

the debate mentioned earlier and Lindley places clear constraints on acceptable quantitative rules of evidence combination. Since we are placing decisionmaking in a qualitative framework we need to extend the approach, perhaps by using multiple logical and quantitative rules, to be used as appropriate [ 5 ] . The most obvious belief revi- sion rules, for example, are:

a) absorption: e.g., arguments which strengthen or weaken the support for an option do not modify its possibility;

b) precedence: e.g., an argument that an option is logi- cally excluded takes precedence over any statement of possibility, probability, plausibility, etc.;

contradiction: contradictory or anomalous proposi- tions should not be “averaged,” ignored, or otherwise combined. They are important signals that the decision process is unsound so contradictions should be “trapped” and evoke metalevel decisionmaking;

d) numerical aggregation: the classical combination technique. The example (Fig. 2) illustrates the simultane- ous use of two ordering parameters, probability and cover- age. Quantitative revision rules can easily be represented within a qualitative framework. In most respects these rules can be viewed as equivalent to logical schemata though in certain respects they have to be designed with care.’

c)

The following rule implements the multiplicative revision of probability (Bayes’ rule). The rule does not normalize the probabilities (normalization can be carried out by another production, or qualitative schemata may be used to maintain a qualitative ordering on the options):

findings include Finding Bayes’ rule is not disabled possible Options include Hypothesis probability of Hypothesis = P conditional probability of Finding given Hypothesis =

CP

then probability of Hypothesis = P*Cp.

Revision of probability is separated from qualitative rea- soning. This permits the decision procedure to manipulate (control and reason about) its qualitative theories and numerical procedures independently. For instance the sec- ond antecedent of the above rule checks whether Bayesian revision is appropriate for revising a set of probabilities. The rule is invalid for example if the prior probabilities are unknown, or the option proposal generates a decision option for which the {Findings, Hypotheses} conditional probability table is incomplete. Should the validity condi-

‘The problem is that the procedurcs output propositions in the same form as one of the antcccdcnt conditionh. The Prop52 forward propaga- tion mechanism conscqucntly cause!, the rule to cycle. Thia can bc

Mathematical decision theory directly addresses the suppressed b> use of additional control rules but the effects arc that rules that iinplcmcnt numcrical functions may need to be rcgardcd as proccdu- ral rather than strictly logical. so wc indicate this with a minor niodifica-

J . Combining the Pros and Cons for the Decision Options

problem of how to combine information. One kind of combination, revision of belief, has been the main focus of tion to the syntax.

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tion be violated then any probability statements that have been computed for the case should be retracted, along with all their logical dependencies, and this revision procedure is disabled. Under these circumstances the procedure must find a different valid decision rule. One such rule is the “coverage” rule which is more general (it can be applied to treatment decisions as well as diagnostic decisions, for example) but is weaker (since it does not calculate weights for different arguments):

findings include Finding coverage is not disabled possible Options include Option total support of Option is TotalArguments number of (findings include Finding) = Data direct support then percent coverage of Option is (TotalSupport/Data

where total support is a combination of the number of direct arguments for the option and those inherited indi- rectly from associated options (discussed in more detail in

Whatever numerical procedure we use, the final decision rule is straightforward; the option with the highest score is selected. What is much harder is deciding when a decision should be terminated, which we return to later.

x 100)

[W.

K. Annotation

As a decision process develops, patterns of argument may emerge whose support structure carries information which may be of value for directing lines of enquiry, generating new hypotheses, new decisions, etc. As these patterns emerge they can be recognized and added to the database as annotations of decision options. As we saw above, annotation of the definite proposition “diagnoses of Fred include cancer” to “possible diagnoses of Fred in- clude cancer” is based on an explicit supporting argument and the absence of an excluding argument.

A more expressive terminology has many uses in deci- sionmaking. For instance the concept of possibility plays a pivotal role in our examples of decision schemata. Modal terms can also help in communication between decision- making agents. Good communication requires a refined vocabulary to permit, say, a supervisor to ask questions like “what diagnoses are presently suspected?”, “why is such and such implausible?” or to assert “damage to heart is possible” (as in the example in Fig. 5) or “treatment of breathlessness is urgent.” Fig. 7 defines a number of belief terms as logical modalities.

The meaning of the operators is defined in terms of proof rules, which are applied to the propositional database. Linguistic uncertainty terms have often been assumed to have a probabilistic interpretation, e.g. to say:

“cancer is possible” really means that cancer has a

“cancer is probable” really means that the probability is nonzero probability,

greater than 0.5 but less than 1.0, or some such.

If argvmcnfs can be idenufied which suppat an assenIon. P. lhen P is suppurred

wonfirmed supporting argumena lor P include Argumenl+ supported P P is posstblc d here is al least one suppomng argument and P cannot be logically eliminated. RIB This is an a fornori &finition. A pmposition can also be (1 prwri possible if lhe requirement lor a supporting argument IS relaxed.)

supported P A no1 eliminated P 4 possibk P Arguments me ubi- pmfs OYCT a body of knowledge. Eg we may argue that a disease is a possrblc cxplnnnrion of an observation by reasoning from physiology or cl he^ * d e s , as in figure 5. Similarly we argue an aserlion is impossible (and hence chnIinnredl beEause it implies OIal an established theay is vmlated.

confirmed argumena against P include violation of Theory + impossible P

Arguments can he m n l e d lo yield an unweighled measure of rupporf (eg fM a diagnosis, treatment etc.):

possible P A number of conrwmed arguments fa P =PDS

A number oleonrumed arguments against P=Nq + support of P = Pos - N q

which can be vanslated inm qualiwve m s by testing simple anlhmetic relations:

possible P * support 01 P S P A S M A not(suppat of Q S Q A SQ>SP) 4 most wpporled P

possibk P A support of P b Support Support < 0

We can &fie the concep of plausibility as “a pmpoSition IS plausible d It is suPporled and h e are no supgorting f x w orargumene that are themselves dubious’:

+ dubious P

supported P A aot(At supports P A dubious A,) + plausible P

S a e n @ which have a convenuonal p-obablllty assignment can be tested lo yleld qualitauve k n p - tors (smcdy a pMal oniemg)

probabitily of P b ProbP A not( probability of Q is ProbQ A RobQ > RobP )

probabitily of P i8 ProbP A probability of Q is ProbQ A RobQ > RobP

--f probable P

+ improbable P

Intereslingly. pbbilities can also have a second-ader logical inWWelalion, as when we view a high protabilly asa kind of argument fa an uncenain judgement:

probable P + supporting arguments for P include bighest probability of P

So hatprobobrrrryargumcnn can peldjudgements ofploustbrlq, as mtuuon requues

Fig. 7. Logical dcfinitions of il number of uncertainty annotations. Thcsc modify thc truth conditions of a base proposition P .

Such interpretations of uncertainty terms raise theoretical, empirical and technical problems. A logical formulation may capture the natural use more closely [7]; this suspicion has received experimental support recently from a series of psychological experiments by Clark [ 2 ] . He was able to show by scaling analyses of the use of some 50 belief terms that the underlying semantic space is both multidimen- sional, highly structured, and apparently consistent for English speaking subjects.

Logical analysis may also clarify the meaning of utility, the other parameter of classical decision theory. As with uncertainty an exclusively numerical interpretation of util- ity causes practical and theoretical problems for decision theory, compounded by high levels of value subjectivity and limited scope for developing an externalized frame- work for personal values (analogous to the frequentistic interpretation of probability). It would be desirable, and we suspect that it may be possible, to develop logical interpretations of terms expressing urgency, importance, value and so on, which can be viewed as annotations of base propositions and complement utility scales.

111. CONTROL OF DECISIONMAKING

Classical decision theory assumes that the requirement for a decision has already been established. A decision procedure is merely a process which determines the option that will maximize expected utility. We observed earlier that decision strategies must be developed for conditions where the assumptions of the standard theory cannot be

met. We add here that it may be necessary to reason about and revise the decision strategy dynamically. A simple example of this was the disabling of Bayes’ rule illustrated earlier.

The development of more sophisticated decision theories must acknowledge the need to reason about the decision procedure, and in particular to reason about when and how decision procedures should be applied. Our approach is to represent strategies as an explicit part of the knowl- edge base of the decision support system (see Fig. 3) , in such a way that i t is possible to decide when and how the general decision procedure should be applied. Among the attributes that may be explicitly required for controlling all types of decision are the following.

Initiating Conditions: Conditions for initiating a diagno- sis, for example, could be that a patient has a complaint and the cause of the complaint is unknown.

Terminating Conditions: A diagnosis may be terminated because the cause is known with certainty, or there is no further information that could change the currently pre- ferred diagnosis.

Relevant Theories: This specifies the classes of argument which are potentially relevant to the decision, such as causal, statistical, anatomical theories, as illustrated above.

Propagation Relations: What relations should be used for propagating decision options? The use of causal and superclass relations were discussed in the diagnostic exam- ple.

Associations: What relations should be investigated in

Part of control specification for diagnosis decision

evoking conditions of diagnoses include diagnosis is unknown evoking conditions of diagnoses include treatment is required.

termination conditions of diagnoses include diagnosis established.

subprocedures of diagnoses include define problem. subprocedures of diagnoses include acquire evidence. subprocedures of diagnoses include review decision.

constraint of diagnoses is define problem precedes acquire evidence.

~

Initiating, scheduling and terminating a task if tasks include Task

and not(initiating conditions of Task include Condition 8.1 and not Condition)

then plan Task in Task.

if plan Operation in Task and subprocedures of Operation include Step 8 .2

then plan Step in Task.

if plan Operation in Task

then schedule Operation in Task.

if plan Operation in Task

and not subprocedures of Operation include Step 8.3

and constraint of Task is Stepl precedes Step2 8.4 and planned Step2 precedes Stepl

then schedule Stepl precedes Step2.

if termination conditions of Task include Condition

then terminate AllSteps in Task.

Part of a propozitional zpecification for a diagnostic decision (top) and schemata which use such specifications to instantiate and control cxccution o f decision processes (bottom).

8.5 and known Condition

Fig. X.

order to determine whether options are strict alternatives or in some way logically associated?

Input Parameters: What classes of information does the decision class require as input? A diagnostic procedure

with a constraint which indicates that the problem must be defined before evidence is collected.

may accept findings and possible diugnoses as input. Iv. “ R A H O N A L I T Y ” OF DECISION P R O C E D U R E S

Output-Parameters: A diagnostic procedure may deliver possible diagnoses as output.

Subprocedures: If there are any, they include other deci- sions that may be required.

Constraints on Subprocedures: If there are any, they include necessary data conditions for the use of a subpro- cedure and the order in which subprocedures should be carried out.

Fig. 8 illustrates schemata which use parts of such specifications to control execution of the general decision procedure. Schema 8.1 is satisfied if all the evoking condi- tions of a specific decision procedure are satisfied. Schema 8.5 on the other hand will terminate the procedure if any of its termination conditions are satisfied. Schemata 8.2-4 embody a simple recursive scheduler which creates and maintains an ordered sequence of subprocedures where these are specified.

The diagnosis specification requires that the condition for initiating a diagnostic decision is that a treatment is required and the diagnosis is unknown. The termination condition for the procedure is, trivially. that the diagnosis is known. The procedure consists of three steps, “define the problem,” “collect evidence,” and “review decision,”

Lindley [ 131 places two principle requirements on deci- sion procedures; that uncertainty and utility be quantified and that decision alternatives be exhaustively identified. We have described ways of generating arguments for and against decision alternatives which do not depend upon quantitative weights, and ways of extending !he set of decision options as case data are acquired. Now we must ask how such procedures meet another requirement that; “(a decisionmaker) ought to be coherent both in his prefer- ences among consequences and in his opinions about un- certain events” [13, p. 179).

“Coherence” of opinions about uncertain events means that uncertainty calculations must obey three laws: namely the law of convexity (probabilities must lie in the 0-1 interval), the law of additivity (the probability of either of two independent events occurring is the sum of their individual probabilities), and the law of multiplication (the probability of both of two independent events occurring is the product of their individual probabilities). We see no technical or theoretical difficulty with these requirements; i f probabilities can be calculated for all decision options then our decision procedure reduces to the classical proce- dure with, as i t were, some added embellishments for

3 56

introducing new decision options. constructing arguments in support, etc.

Problems appear when the decision procedure is re- quired to consider, or generates for itself, decision options for which the prior or conditional probabilities are incom- plete. Lindley offers no help here. In our example we used a measure of cooerage (the number of findings explained by a hypothesis), elsewhere of plausibility (see above). We do not advocate any particular measure because we think that different measures will be appropriate in different circumstances [6] , [15]. For now we wish to note that it is largely a technical matter to ensure that measures of coverage or plausibility have the desirable arithmetic prop- erties expressed in the three laws. In our view a more difficult problem is to develop mechanical techniques for reasoning about the measures which can be validly used by a decision procedure when, say, the data environment degrades.

The decision procedure must also be coherent in its preferences among consequences-principally preferences must be transitive (if A i s preferred to B and B is preferred to C then A is preferred to C ) . If a complete set of utilities is available then the standard expected utility procedure may be employed, but if preference weightings are not known we must use weaker measures of preference, such as the number of reasons for preferring A to B minus the number of reasons for preferring B to A . Such elemen- tary preferences may not have any identifiable underlying scale. For example a noninvasive medical treatment may be preferred to surgery wholly on the grounds that it is noninvasive. However, we would still expect a rational decision procedure to preserve transitivity of preferences, whatever measure is used. Again in our view the more difficult theoretical problem is in developing good ways of generating preferences rather than on good ways of count- ing them.

The laws of probability and the requirement for transi- tivity of preferences are specific expressions of the more general demand that a decision procedure be “rational.”2 In one important respect we think that a logical approach has something novel to offer to discussion of rational decisionmaking. The latter assumes that the available data or knowledge are consistent, complete and correct: case data are not both true and false; the table of statistical associations is complete; the values in the table are cor- rect; the assumptions about the properties of the data which are made by the decision procedure will remain justified. Minor violations can be managed on an ad hoc basis (e.g., we can recompute a set of probabilities i f findings prove incorrect, estimate the occasional missing value in a conditional probability table by averaging its row and column, etc.).

If, however, sparsity of information is the norm, “facts” are typically unreliable, precise parameters that are rare, and data change rapidly, such repairs seem unsatisfactory.

We are indebted to an anonvmous referee for raising the important subject of rationality.

In some kinds of decisionmaking almost any prior belief. whether case data, domain facts, or assumptions about the decision procedure itself, may come into question. In such circumstances we should introduce a notion of rationality that has not been considered in the classical decision theory literature. viz a rational decision procedure should be rationally aduptiue in that it should be able to cope with incomplete and contradictory propositions and theories, ensuring that its conclusions remain at least logically con- sistent as its beliefs change.

We cannot present a general treatment of consistency management here, but we shall mention the techniques used to automatically maintain a consistent and logically sound database in the above work.

I ) Assumption Maintenance: If case specific data, do- main specific facts, or even theories or procedures are added to the database then the set of all inferences that are valid consequences of the current data state and the up- date are generated automatically. This “completed” database permits decisionmaking in the absence of firm knowledge but may be based on assumptions for which there is no direct justification. If such justifications are subsequently obtained the database will not change but if the assumption is subsequently denied then all dependent conclusions will be retracted and all implications of the new data state generated.

2) Closed World Assumption: Schemata of the form P & not (Q) implies R are held to mean “while P is true and Q is not ... .” The negation of Q is not strict but means that Q cannot be proved to be true on the current database at the time the inference is made. This is viewed as an assumption, and is maintained as above.

3) Logical Exclusivity: Many assertions are constrained to one of a set of values (e.g., P is true/false; present/ absent; a hypothesis can have only one probability value, etc). If a new value is generated this could either be interpreted as a contradictory state and invoke some ap- propriate action or, the choice made here, interpreted as a denial of the old value and maintained as above.

Our current system demonstrates that is possible to revise decision options. beliefs, and assumptions while maintaining the logical consequences of revisions without recourse to a human decision analyst. We have not, of course, developed all the necessary techniques for au- tonomous decisionmaking but we consider it a vital area for future research in rational decision theory. Methods for automatically initiating and controlling decision proce- dures, classifying decisions, formalizing arguments, and the use of qualitative methods for soundly representing and propagating beliefs, are topics of active investigation.

V. CONCLUSION

The principle conclusions from our work are that meta- knowledge and metalevel reasoning must be integral ele- ments of a decision theory because of the versatility and operating flexibility they offer. Here we have illustrated the use of knowledge about information that is known,

both qualitative and quantitative; metaknowledge about [ 151 A SrifflOtti An AI \ I C \ \ of thc trcatnient of unccrtaint\.“ Tlrc AIIOLI I idyc I /rq Rc.1 \ ( > I 1. 110 2 I Y X S beliefs and the arguments that justify them; and knowl-

edge about procedural methods and when they can validly be applied. John Fa\ recciked the fir\t and Ph D degrees i n

p\\cholog\ Ironi Durham and Cambridge Uni-

During 1971-1Y75, he worked in artiflclal in- tclligcncc at Carncgie Mcllon and Cornell Uni- \er\itie\ In IY75. he joined the Medical Rc- \e,ircli Council. whcrc tic worked on dcci\ion

paper. \upport \\\tern\ and cognitice inodeling Since 19x1. he ha\ been rc\pon\iblc for re\carch in Artificial Intelligence and Knowledge Engincer-

ACKNOWLEDGMENT \cr\itic\. re\pccti\el\

The authors would like to thank Saki Hajnal and Paul Krause for their valuable comments on earlier drafts of

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H. G. Iluchanan and R. G. Smith. “Fundamentals of expert sys- tems.” Ariri . Rris. Conipur. .%I.. vol. 3. pp. 23-58. 19XX. D. Clark o r U/.. Discussion of “An AI view of the treatment of uncertainty:’ Tlir Kiiow/ed,ye G I , ~ . Rei . .. vol. 3. no. 1. I Y X X . P. Cohen. Ileirrisrrc. Reo.soiiiri,y A h i t r L ’ i i ~ ~ t ~ r r u ~ ~ i f i ~ : A 1 1 A I Appi.ouc./r. 13oston: Pitman. 19x5. J . Fox. “Formal and knowledge based methods in decision technol- ogy.” in Prrw. Yrli CorI/. o i i Sul>jec~rrrw Pro/whi/it\~. L ‘ r i h r i . urid Diwsrori MoXrirg. <ironingen. The Netherlands. Aug. 19x3: cx- tended version in Ac.ru P.r,~c~/io/i,,yic~u. IYX4. ~. “Knowledge. decision making and uncertainty.” A r r ~ / ~ ( ~ r u / Iiire//r,qiwcr u i i d Srurr.sric.\. W. Gale. Ed. Reading, MA: Addison- Wesley. 19x5. pp. 57-72. -, “Three arguments for extending the framework of probahil- it!.” in L. N. Kana1 and J. F. Lemmcr. Eds.. L’irwrrrririri. 111

Arrij(.ru/ / r i r d / i , y u i w . Amsterdam. The Netherlands: North Hol- land. 19x6. ~. “Making decisions under the influence of knoRledgc.” in P. Morris. Ed.. Modd/;ii,y Co,yriirrori. London. U.K.: Wile!. 1YX7. ~. “Symbolic decision procedures for knowledge based sys- tenis.” in H. Adeli. Ed.. IluriihooX o/ Kriow/<++, Eii,yrriwriri,y. to bc published. -. “Decision mpport systems for safety critical applications.” Phrl. Troii.\. Rei. Soc. .. to he published. J . Fox. A. J . Glowinski. M. J . O’Neil and D. A. Clark. “Decision making as a logical process.” in B. Kelly and A. Rector. Eds.. Rescorc h oiid Dwelopiiieiir 111 I : . y v r Si..\reim V . Cambridge. MA: Cambridge University, I Y X X . A. J . Glowinski. M. O’Neil and J . Fox. “Design of a generic information system and its application to primary care.” in J . Hunter. Ed. Proc. of ?rid Europeoii Corr[c.rrircx~ o i i A I i r i Mtdic.riir

Berlin: Springer-Verlag. 19x9. S. L. Lauritzen and D. J . Spiegelhalter, “Local computations with probahilities on graphical structures and their application to expert systems.” J . Rei,. Storrsr. Soc. 8. vol. 50. no. 2, 1YXX. D. V. Lindlev. MUXII I ,~ Dec~i.sioiis. second cd. London. U.K.: Wile?. I Y X 5 . J . Pearl. I lc io i . \ r ic . \ . l i r rd / i , y~wr Sc.crrc.h Srrore,yrc~ jhr ( ’ o r i i p u r n . P roh- / m i So/ivii,y. Reading. MA: Addison-Wesley. 19x4.

ing. the Imperial Cancer Research Fund.

Ihiiitiic A. Clark received the graduate degree from thc University o f Wales and the Ph.D. dcgrcc i n I Y X X .

Hc is currently a scientific research fellow at thc Imperial Cancer Research Fund. working on know ledge-based systems in rnolecular biology. His interests include reasoning under uncertainty and knowledge cnginccring.

Andriej ,J. Gloainrki received the medical degree from Oxford University in 19x0.

Since 1YX6. he has been a clinical research fellow at the Imperial Cancer Research Fund. whcrc hc is currently working on medical deci- sion support systems. His interests include the applications o f knowledge-hased techniques in priinar! carc.

Michael .I. O’Neil received the medical degree from University College of London.

Sincc I9Xh. he has been a member of the Oxford Systcm of Medicine project team at the Imperial Cancer Research Fund. where he is currcntl! a clinical research fellow. His interests includc the dcsign of very large knowledge-based s!stems and symbolic methods of decision- making.

Dr. O’Ncil is a mciiihcr of the Royal College of Physicians