Abstract

A key question for research in model-based, qual-itative reasoning is how to predict or analyze thebehavior of complex systems without resorting tocompletely quantitative models . One difficultythat arises is the ambiguity of results due to con-flicting indications. Argumentation has long beenrecognized as a means for resolving issues of beliefin situations characterized by incomplete, uncer-tain, inconsistent, and imprecise knowledge . Weexplore the application of a model of dialecticalargumentation to the domain of qualitative rea-soning . Models take the form of qualitative net-works, with variables connected by strict or de-fault, positive or negative arcs . A notion of de-feat between qualitative arguments representedas paths in a network is defined. Burden of proofis specified as a flexible means of allocating risk,determining relevant argument moves, and decid-ing eventual outcomes . Examples are presentedthat illustrate the semantics of our approach .

IntroductionA basic question addressed by research in model-basedqualitative reasoning is how we can predict or analyzethe behavior of complex systems without resorting tocompletely quantitative models . One difficulty thatarises from the associated loss of precision in qual-itative reasoning is the ambiguity of results due toconflicting indications . In a more general setting, ar-gumentation has long been recognized as an appro-priate means for resolving issues of belief in situationscharacterized by incomplete, uncertain, inconsistent,and imprecise knowledge (Polya, 1968). In this pa-per, we explore the application of a model of dialec-tical argumentation to issues of ambiguity resolutionin qualitative reasoning about complex systems.

Qualitative ModelsModels of qualitative systems and associated ar-gument structures will be specified in a form de-rived from that used for inheritance reasoning (Horty,

Qualitative ArgumentationArthur M. Farley

Computer and Information ScienceUniversity of Oregon

Eugene, Oregon 97403 USAart@cs.uoregon.edu

1994). By this approach, qualitative models of com-plex systems are represented as qualitative networks,being directed graphs of nodes interconnected bytyped, directed links. Nodes of a qualitative network,denoted by (names in) small letters, represent thevariables and parameters of the system ; we will usea small letter from the end of the alphabet (e.g ., x, y,z) to represent an arbitrary node of a network .The nodes of a qualitative network are intercon-

nected by directed links, each link connecting a pairof nodes. There are four link types, corresponding topossible combinations of strength, either strict or de-fault, and sign, either positive or negative . The linktypes =+> and =_> denote strict positive and strictnegative links, respectively . The meaning of a strictlink x =+> y (or x =-> y) is that there is a reliable(i .e ., always) influence of a change in x upon variabley. A positive link means the change in y is in the samedirection as the change in x, while a negative sign indi-cates that the direction is the opposite . A strict link isused to represent definitional relationships in systemmodels . Link types -+ > and -_ > denote defaultpositive and default negative links, respectively . Themeaning of a default link x-+ > y (or x-_ > y)is that there is an expected, but somewhat unreliable,(i .e ., usually) influence of a change in x upon the valueof variable y. The meanings of the signs are the sameas for strict links. Default links are used to representobserved, but unexplained, regularities or tendenciesin system behavior .

Default links have also been called "defensible"(Pollock, 1987), as they can be preempted by stronger,more conclusive indications during reasoning. Weuse the term "default" for these links to capture therather strong connection that is intended . We reservethe term "defensible" for a more general, somewhatweaker influence that exists between nodes intercon-nected by paths containing several default links, i.e .,the default relation is not transitive . We will discussthe construction of allowable qualitative arguments inthe next section .A qualitative network can serve as basis for answer-

ing questions regarding directions of change in modelvariables given external, input perturbations of pa-rameters or for suggesting changes to parameters that

could give rise to a desired (or observed) direction ofchange in a model variable . Parameters are distin-guished from variables in that they have no incominglinks within the given qualitative network. Impacts onparameters will be indicated by connecting a specialnode labeled INPUT to parameters of the network bya set of strict, positive or negative, links . A strict linkfrom INPUT to a parameter indicates a perturbationin the value of the affected parameter in the indicateddirection (i .e ., positive or negative) . We provide anexample of a qualitative network with an associatedINPUT in Figure 1 . In all figures, strict links will beshown as heavy lines .

INPUTt

Figure 1 .

Qualitative ArgumentsGiven a qualitative network modeling a complex sys-tem, we are interested in defining the notion of ar-guments for and against claims regarding changes invalues of variables relative to a given input perturba-tion . An qualitative argument is a directed path in aqualitative network . We denote an argument P con-sisting of the path a,b,c,d,e as P(a,b,c,d,e) ; if nodeu immediately precedes node v in an argument path,then u is directly connected to v by a link of the qual-itative network. An argument from a start node xto a finish node y through an intermediate, possiblyempty, sequence of nodes 7r is denoted as P(x,a,y) .Which directed paths form arguments and with whatstrengths and signs will be defined by argument con-struction rules given below .We characterize an argument in terms of strength

and sign . We introduce defeasible strength for argu-ments between nodes connected by paths involvingmore than one default link ; defeasible arguments cap-ture a qualitative relationship that is weaker than thatestablished by either strict (always) or default (usu-ally) links . If P(x, 7r, y) is a defensible positive argu-ment, it represents the notion that "it is reasonable to

believe that a change in x will result in a change in y inthe indicated direction" . If this path begins at a nodeINPUT, we are saying that "the given input perturba-tion set could reasonably be expected to result in thecorresponding change in y" . Distinguishing betweenstrict, default, and defensible argument strengths willallow us to capture, in a straightforward manner, cer-tain reasoning results that seem intuitively correct .This will be demonstrated by examples presented laterin the paper .

Allowable arguments in an inheritance network,with associated strengths and signs, are defined recur-sively, in the "backward direction" from a given goalnode . Links in the network form direct arguments, asdefinedby the following argument construction rule :

Rule Rl: (direct arguments)A. Given x =+> y, P(x, e, y)

is a strict positive argument .B . Given x =_> y, P(x, e, y)

is a strict negative argument .C. Given x-+ > y, P(x, c, y)

is a default positive argument .D. Given x-_ > y, P(x, e, y)

is a default negative argument.

In the above rule, x refers to an arbitrary variableor parameter node x or INPUT; the symbol e rep-resents the empty sequence of nodes. We extend anargument by adding a new element as start node, cre-ating a compound argument . We consider only acycliccompound arguments, defined by the following rule :

Rule R2: (compound arguments)A. If P(x, ar, y) is a strict positive argument

not containing z, then(i) given z =+> x, P'(z, x, a, y)

is a strict positive argument;(ii) given z-+ > x, P'(z, x, 7r, y)

is a default positive argument ;(iii) given z =_ > x, P'(z, x, a, y)

is a strict negative argument ;(iv) given z- - > x, P'(z, x, a, y)

is a default negative argument .B. If P(x, a, y) is a strict negative argument

not containing z, then(i) given z =+> x, P'(z, x, 7r, y)

is a strict negative argument ;(ii) given z-+ > x, P'(z, x, r, y)

is a default negative argument;(iii) given z =+> x, P'(z, x, a, y)

is a strict positive argument;(iv) given z-+ > x, P'(z, x, r, y)

is a default positive argument .C. If P(x, a, y) is a default positive argument

not containing z, then(i) given z =+> x, P'(z, x, 7r, y)

is a default positive argument ;

(ii) given z-+ > x, P'(z, x, 7r, y)is a defensible positive argument ;

(iii) given z =_ > x, P'(z, x, a, y)is a default negative argument ;

(iv) given z-- > x, P'(z, x, a, y)is a defeasible negative argument .

D. If P(x, r, y) is a default negative argumentnot containing z, then(i) given i =+> x, P'(z, x, r, y)

is a default negative argument ;(ii) given z-+ > x, P'(z, x, 7r, y)

is a defeasible negative argument ;(iii) given z =_ > x, P'(z, x, ar, y)

is a default positive argument;(iv) given z-_ > x, P'(z, x, a, y)

is a defensible positive argument .E . If P(x, 7r, y) is a defeasible positive argument

not containing z, then(i) given z =+> x, P'(z, x, 7r, y)

is a defeasible positive argument ;(ii) given z-+ > x, P'(z, x, 7r, y)

is a defensible positive argument ;(iii) given z =_ > x, P'(z, x, a, y)

is a defeasible negative argument ;(iv) given z-_ > x, P'(z, x, 7r, y)

is a defeasible negative argument .F. If P(x, a, y) is a defeasible negative argument

not containing z, then(i) given z =+> x, P'(z, x, 7r, y)

is a defeasible negative argument ;(ii) given z-+ > x, P'(z, x, a, y)

is a defeasible negative argument,(iii) given z =_ >

is a defeasible positive argument ;(iv) given z-- >

is a defeasible

x, P' (z, x, a, y)

x, P'(z, x, 7r, y)positive argument .

In rule R2, z represents either an arbitrary node ofthe network or INPUT. Unless otherwise specified, avariable subargument, such as 7r, can be empty, i .e .,equal to c . The sign of an argument corresponds tothe multiplicative influence of the signed links alongits path in the qualitative network. Of importanceis how the strength of an argument is impacted byadding new links . Put simply, an argument path withone default link is of default strength ; a path withmore than one default link is of defeasible strength .A strict argument contains only strict links. An argu-ment starting with INPUT is said to be grounded.

For example, given the network in Figure 1, wefind three grounded arguments with respect to vari-able f, as follows : a negative, defeasible argumentPl(INPUT,t,p,f) and two positive, defeasible argu-ments P2(INPUT,t,p,b,f) and P3 (INPUT,t,b,f) .We see that there are both positive and negative

arguments regarding the conclusion that variable f in-

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creases as parameter t is perturbed upward . This isan example of the ambiguity that often arises in qual-itative analyses of complex systems. From an argu-mentation perspective, we can restate this situationas arguments standing in a conflicting relationship toone another . Various definitions of conflict relation-ships between arguments have been given previously(Pollock, 1987 ; Loui, 1987 ; Sartor, 1993 ; Horty, 1994 ;Farley, 1996). Here, two arguments with the samestart and finish nodes, P(x, 7rl, y) and P'(x, a2, y), di-rectly conflict if they differ in sign . More generally,two arguments conflict if one directly conflicts with asubargument of the other, i.e ., one argument is of theform (al, x, 7r2, y, A3), the other is the form (x, a, y),where (x, 7r, y) and (x, 7r2, y) are of opposite sign . Inour example, we have the following pairs of conflicting,grounded arguments: (Pi , P2) and (PI, P3) .

Are there methods for resolving qualitative ambi-guities, now that they are viewed as conflicts betweenqualitative arguments? One helpful notion is that ofdefeat between arguments . Certain pairs of argumentsmay stand in a stronger form of conflict relationship .In such cases, one argument will be said to defeat theother. Defeat between conflicting arguments is deter-mined by comparing their respective strengths . Tomake this possible, we define a strict argument to bestronger than a default argument, which in turn isstronger than a defeasible argument .One argument A defeats another argument B iff

argument B is of the form (x1, x, 72, y, 73), A is ofthe form (x, a, y), argument A and the subargument(x, a2, y) are of opposite sign, and A is of greaterstrength than the subargument (x, 7r2, y) of B. In otherwords, an argument that conflicts with another argu-ment and is of greater strength than the subargumentwith which it directly conflicts defeats the other ar-gument . In our example from Figure 1, the argumentP4(p,c,f) defeats argument P2 .

Defeating arguments are of course vulnerable to be-ing defeated themselves . In addition, there is a vulner-ability associated with intermediate nodes of defeat-ing arguments . If argument A(xi, x, V2, v, x3), wherexi is a sequence of one or more nodes, is defeatedby an argument B(x, 7r, y), where ar is a sequence ofone or more nodes, then the set of arguments start-ing at nodes of al and ending at nodes of a that onlypass through nodes of A and B constitutes the set ofvulnerable arguments for the defeat relation betweenA and B. The vulnerable arguments associated witha defeat are those that start prior to the beginningof the defeating argument and end at intermediatenodes of the defeating argument. Finally, we say thatan argument A ultimately defeats an argument B ina qualitative network Q only if A defeats B, A is notdefeated, and none of the associated vulnerable argu-ments are defeated .

Burden of ProofA qualitative claim is a statement regarding the in-fluence of one element (i .e ., variable, parameter, orINPUT perturbation) of a qualitative network on an-other variable of the network . A positive claim will bedenoted as x+ > y, while x_ > y will denote the nega-tive claim . Each grounded argument P(INPUT, r, y)supports a positive or negative claim regarding the di-rection of influence of a specified input perturbationset on variable y of the model .To determine the ultimate acceptability of a quali-

tative claim, we must decide which type of error, ei-ther of commission (i .e ., false positive acceptance of aclaim) or of omission (i.e ., false negative rejection of aclaim), we are more willing to accept . We would liketo adjust this allocation of risk depending upon esti-mated consequences of wrongly accepting or rejectinga qualitative claim between a particular model ele-ment and a given variable .We provide the ability to allocate risk in a flexible

manner by specifying a burden of proof . A burdenof proof is a parameter to the argument process, nota property of the reasoning system . One domain inwhich the notion of burden of proof has long been ap-plied is that of legal reasoning . A different burden ofproof may be mandated at each stage of a legal processor for a different type of legal action . For example, ar-guments sufficient to indict someone need not be asconvincing as those needed to convict someone . Whenconsidering conviction in criminal cases, we are moreconcerned about errors of commission (i .e ., a findingof guilt when not guilty) and, thus, place a high bur-den of proof on the side arguing for guilt .There are two aspects to the specification of a bur-

den of proof: (i) which side of a claim (positive ornegative) bears the burden and (ii) what level of proofis required for acceptance of the claim . The first as-pect addresses whether one is more concerned aboutaccepting false positives, where the burden is placedon the claim of interest, or false negatives, where theburden is placed on the claim of opposite sign . Some-times, we want a good argument supporting a claimbefore accepting it, where the risk associated withwrongly accepting the claim is perceived to be high .On other occasions, where accepting a claim has po-tentially high value and little perceived risk, we de-mand a good argument against the claim (i .e ., of op-posite sign) before denying its acceptance .The second aspect of burden of proof, that of proof

level, addresses the issue of what is considered to bea good, or sufficient, argument . Proof level will bebased upon the following notions of defendable andjustifiable arguments . A defendable argument is anargument that is not ultimately defeated by any otherargument of a given qualitative network and INPUTperturbation . A justifiable argument is a defendable

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argument with the added condition that every argu-ment that conflicts with it is ultimately defeated .We now define the following three proof levels :

scintilla of evidence (se) : there exists a defend-able argument supporting the claim ;

preponderance of evidence (pe) : there exist moredefendable arguments supporting the claim than itsnegation ;

dialectical validity (dv) : there exists a justifiableargument supporting the claim .Winning a scintilla ofevidence argument for a claim

merely requires that some argument for the claim canbe defended against all attacks . In a dialectical con-text, this means that only defeating arguments can beconsidered by the side opposing the claim ; a conflict-ing argument of lesser or equal strength is irrelevant .Preponderance of evidence requires a means for assess-ing relative strengths of sets of conflicting, defendablearguments . We state that having a greater number ofdefendable arguments for a claim represents a strongercase . Under preponderance of evidence, if the oppos-ing side finds an equally strong argument, this can notbe ignored ; the argument must be counteracted, eitherby defeating it or by finding a new argument of equalstrength . Finally, dialectical validity requires not onlythat some argument be defendable but also that anyconflicting argument be defeated, even though thatargument is not any greater in strength .We can now define the semantics associated with

a given qualitative network Q and INPUT perturba-tion in terms of the sets of claims acceptable underdiffering burdens of proof . Given network Q and aperturbation set as links from INPUT, we denote byD(Q, INPUT) the set of defendable, grounded argu-ments . Similarly, we denote by J(Q, INPUT) the setofjustifiable, grounded arguments . By our definitions,J(Q, INPUT) is a subset of D(Q, INPUT) .We denote by C(Q, INPUT, L) the set of claims

that are acceptable with proof level L, given Q andINPUT . The set C(Q, INPUT, L) is derived from setsD(Q, INPUT) and J(Q, INPUT), as per our defini-tions above . A claim c is an element of C(Q, INPUT,se) iff there exists an argument for c as claim in D(Q,INPUT) . A claim c is an element of C(Q, INPUT, pe)iff there exists more arguments for claim c in D(Q, IN-PUT) than there are arguments for the complementof c in D(Q, INPUT) . A claim c is an element of C(Q,INPUT, dv) iff there exists an argument for claim cin J(Q, INPUT) .The three proof levels defined above create a hier-

archy of acceptable claims based on set inclusion . Forany Q and INPUT, C(Q, INPUT, se) contains C(Q,INPUT, pe), which contains C(Q, INPUT, dv) . OnlyC(Q, INPUT, se) can contain contradictory claims,i .e ., claims between the same model elements with op-posite signs .

ExamplesWe now turn our attention to a number of examplesthat demonstrate general principles of our approachand illustrate the impact that burden of proof hasupon argumentation semantics . Looking back to theexample of Figure 1, both positive and negative claimscan win only scintilla of evidence arguments; there isexactly one defendable, defeasible argument support-ing each claim .

In Figure 2, we find the argumentPI (INPUT,e,d,b,a) of defeasible strength for a pos-itive input perturbation to e resulting in a positivechange in a . However, this argument is defeated byargument P2(d,c,a), a negative argument of defaultstrength defeating subargument P3(d,b,a) of Pl . Asa result, the only grounded, defendable argument isthe negative argument P4(INPUT,e,d,c,a) . While itis only of defeasible strength, it is not attacked byany other defendable argument . Thus, the negativeclaim INPUT _ > a prevails for all three burdens ofproof.

b

a

d

INPUT-~ e

Figure 2 .

We can make the situation slightly more complex byadding parameter f connected to e and letting e havea direct, default negative impact on c, as shown belowin Figure 3 . As before, the defeasible positive argu-ment P,(INPUT,f,e,d,b,a) is defeated by the defaultnegative argument P2(d,c,a) . This time, however, thedefeasible negative argument P3(INPUT,f,e,d,c,a) isdefeated by the default negative argument P4(e,c,c),which defeats subargument P5(e,d,c) of P3 . By de-feating P5, argument P4 reinstates Pl ; it defeats avulnerable argument associated with the defeat of Plby P2 . The only other defendable, grounded argument

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in this example is P6(f,e,c,a), which is a defeasible pos-itive argument .

Thus, we see that for the network of Figure 3, theclaim INPUT + > a is acceptable up through a bur-den of proof of dialectical validity. In the previoustwo examples, if we did not consider the structure ofthe arguments involved and their conflict interactions,the qualitative indications would be ambiguous . Byconsidering defeat relations among arguments, we seethere is a clear prediction for direction of influenceaccording to our argumentation semantics .

b

a

Figure 3 .

We can create more ambiguous situations, such asthe one presented in Figure 4 . Here no arguments aredefeated . As a result, there are three grounded, de-fendable arguments: P,(INPUT,e,b,a), P2(INPUT,ec,a) and P3(INPUT,e,d,a) . Argument Pl indicates anegative influence, while P2 and P3 indicate positive .The claim INPUT _ > a is acceptable with a burdenof proof of scintilla of evidence . However, by beingable to outweigh the set of arguments available forthe negative claim, the claim INPUT + > a can winarguments up through a burden of proof of prepon-derance of evidence .

Recall that scintilla of evidence is the only burden ofproof for which both positive and negative claims canbe considered acceptable . Use of scintilla of evidenceis appropriate for situations where there is little per-ceived risk in making errors of commission . Scintilla ofevidence allows acceptance of a claim even when the

a

Figure 4 .

other side has more arguments in its favor, as longas there exists a defendable argument for the claim .If any arc in Figure 4 were made strict, the argumentcontaining it would become dominant . The associatedclaim would win arguments up through dialectical va-lidity .Economic reasoning is frequently based upon in-

complete, imprecise models of complex interactionsamong markets (Farley and Lin, 1990) . Thus, thisdomain is particularly well-suited to application ofqualitative argumentation . We present a qualitativenetwork designed to allow macroeconomic reasoningabout the interactions between product and moneymarkets in Figure 5, where the nodes are defined asfollows :PD : Product DemandPS : Product SupplyEPD : Excess PDP: PriceMD: Money DemandMS: Money SupplyEMD: Excess MDIR: Interest RateWe have only two parameters in our model, MS

and PS ; the two market cycles are negative in sign,indicating their qualitative stability . The links fromPD and PS (MD and MS) to EPD (EMD) are strictas EPD (EMD) is the defined to be the differencebetween the other two variables . Suppose we con-sider the impact that an increase to these parameterswould have upon variable MD . For a perturbation ofPS there is only one allowable, defeasible argumentPI (INPUT,PS,EPD,P,MD), which is of negative sign,so the negative claim would win arguments under allburdens of proof .

For a perturbation of MS there are twodefeasible arguments P2 (INPUT, MS,EMD,IR,MD)and P3(INPUT,MS,EMD,IR,PD,EPD,P,MD), both ofwhich are positive . Suppose we perturb both PS andMS upward at the same time . At first blush, it ap-pears that the two positive arguments P2, P3 wouldoutweigh the single negative argument Pl , allowing

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MS>w EMDE MD

Figure 5 .

the positive claim to be accepted up through prepon-derance of evidence, while the negative claim couldonly win scintilla of evidence arguments . However,the argument P4 (INPUT,PS,PD) would be of defaultstrength and, therefore, defeat argument P3 . As such,both positive and negative claims can only musterone defendable, defensible argument and are only ac-cepted under the minimal, scintilla of evidence burdenof proof.

Argument ProcessNow that we have defined a structure for arguments asacyclic paths in a qualitative network and character-ized several important properties of and relationshipsbetween arguments, we describe a process whereby wecan decide whether to accept a claim or not . The de-cision will be made through a process of dialecticalargumentation under a given burden of proof . Theprocess model presented here is based upon a modelof dialectical argumentation described earlier (Free-man, 1993; Farley and Freeman, 1995) . modified andadapted to the circumstances of qualitative argumen-tation .A dialectical argument has two sides, where Side-1

argues in favor of an input claim and Side-2 againstthe claim, i .e ., in support of its negation . The argu-ment process begins with Side-1 attempting to finda grounded argument for the input claim . Searchfor a supporting argument proceeds from the goalnode toward the INPUT node in a backward-directedmanner, according to the argument construction rulesgiven above . If no support can be found, the argumentends with a loss for Side-1 ; all burdens of proof requireSide-1 to find at least one grounded, defendable argu-ment .

Subsequently, the two sides alternate as active sideof the argument . A side remains active until it suc-

ceeds in creating a check condition or runs out ofmoves and must concede the argument . A check con-dition for a side S is a situation such that, if the otherside can not successfully respond, side S wins the ar-gument . Except for the initial situation, when Side-1must generate a grounded argument for the claim, theactive side is faced with a set of check arguments forthe other side, i .e ., arguments responsible for the otherside holding the check condition .When active, a side selects an argument move to

apply from a set of possible moves . For this, a sidecan apply one of two primitive functions that searchthe qualitative network for relevant arguments . Thefirst is find-arguments (x, y, s, Q), which searchesfor argument paths from node x to node y of signs in qualitative network Q. The function returns alist of argument paths in order of decreasing strength(i .e ., the empty list if no such paths exist) . The otherfunction, find-conflicting-arguments(A, Q), finds ar-guments that conflict with argument A in network Q,being equivalent to the union of directly conflictingarguments for each pair of nodes in A. This functionis implemented by calls to the function find-argumentswith parameters being pairs of nodes from argumentA and the complement of the sign of the correspond-ing subargument in A. Defeating and rebutting (i .e .,conflicting, but not defeating) arguments can be rec-ognized in the process .

Whether an argument is adequate to generate acheck condition for the active side depends upon theburden of proof specified . For example, under a bur-den of proof of dialectical validity, Side-2 can considerboth defeating and rebutting arguments in response toSide-1's arguments . If Side-2 finds a conflicting argu-ment, Side-1 must defeat Side-2's response or proposea completely new argument for its claim ; otherwise, itmust concede the argument . Side-2 can continue pos-ing counterarguments, all of which Side-1 must defeatif it is to prevail under a burden of dialectical validity.

If the burden of proof is preponderance of evi-dence, then Side-2 must generate a counterargumentof strength equal to that proposed by Side-1 . If it cando this, Side-1's finding another argument in favor ofthe claim is sufficient to regain its check condition .As long as Side-1 can come up with more argumentsof strength greater than or equal to those that Side-2 produces, it will prevail . If the burden of proofis merely scintilla of evidence, Side-2 can only con-sider defeating arguments in response to Side-1's ar-guments. Side-1 need not defeat rebuttals to win theargument under this burden of proof; it must merelydefend some argument against defeat ; if Side-2 candefeat an argument, Side-1 can abandon it in favor ofanother supporting the input claim .We can characterize burden of proof in terms of

where it places the "burden of defeat", i .e ., whichside must defeat the other's arguments. In the case of

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scintilla ofevidence, the burden of defeat is on Side-2 ;while under a burden of proof of dialectical validity,the burden of defeat is on Side-1 . Under preponder-ance of evidence, neither side takes on the burden ofdefeat . This is a free for all, where piling up more ar-guments of equal strength in favor of the given claimis sufficient .We see that burden of proof plays several roles in

the process of dialectical argumentation : (i) as basisfor deciding the relevance of arguments found by theactive side ; (ii) as basis for deciding the sufficiencyof the outcome of an active side's move ; (iii) as ba-sis for determining that an argument is over ; and (iv)as basis for determining whether a claim is acceptedor not . With scintilla of evidence or dialectical va-lidity as burden of proof, the argument process maybe shortened significantly due to the burden of defeatborne by one side of the argument .

DiscussionWe have previously applied our framework for ar-gumentation to the domains of legal reasoning andnonmonotonic inheritance (Farley and Freeman, 1995 ;Freeman and Farley, 1996; Farley, 1996) . The currenteffort is most closely related to the work on inheritancereasoning, where inheritance arguments were definedto be paths in inheritance networks . The semanticsof links in an inheritance network and the set of al-lowable arguments differ somewhat from those definedhere for qualitative argumentation ; the definitions ofargument defeat and burden of proof are adopted di-rectly .More generally, an argument framework consists of

three main elements :(i) a logic for generating allowable, consistent argu-

ments from background knowledge ;(ii) a definition of defeat between pairs (or sets) of

arguments ;(iii) a decision process for determining which side

prevails in a given argument (what claims are accept-able from given background information) .

It is the second two aspects that distinguish argu-mentation from theorem proving and extend the ap-plicability of the underlying logic from generating al-lowable, consistent arguments to making decisions inreal-world contexts .

Under the approach for arguing about the implica-tions of qualitative models presented here, we acceptany acyclic path in a given qualitative nework as anallowable argument . In terms of defeat, we only con-sidered comparisons between the predictability or re-liability of opposing impacts upon a chosen variable,as determined by subpaths of differing arguments . Wedid not consider the possible magnitudes of impactsor the summation of impacts over several paths whendefining defeat between arguments . We could extendour model to include these considerations . Each edge

could be labeled by a coefficient capturing the rela-tive impact that a unit change in the variable at thetail of a directed arc would have in terms of units ofthe variable at the head of the arc. Combining thisimpact label with the types we are currently usingwould allow us to capture the notion of expected im-pact on the variable at the head . Impact coefficientlabels could be specified in qualitative or quantitativeform . The presence of these labels would not alter thegeneral framework we have defined, but would requirea new specification of argument strength propagationand comparison .

The issue of representing additive effects in quali-tative networks is more significant . In such networks,there can be two reasons that differing paths reacha given variable in the network. One is that eachpath represents an alternative explanation of a possi-ble direction of change . Our argumentation semanticsdefined above reflects this view . The other reason isthat each path represents a separate term of an addi-tive or multiplicative impact on the variable . In thiscase, we would need to consider these paths as a sin-gle argument and develop appropriate definitions forstrength computation and defeat determination. Ourargumentation framework provides a structure withinwhich to address this added complexity.

Conclusion

This paper reports results of an initial study applyingnotions developed previously in the field of argumen-tation to issues arising in qualitative reasoning aboutcomplex systems . The domain of qualitative reason-ing is well-suited for the application of argumentation,as qualitative reasoning contexts are often character-ized as being dependent upon incomplete, imprecise,uncertain, often conflicting indications . Dialectical ar-gumentation provides a method for considering bothsides of a qualitative claimandmaking adecision as toits acceptability based upon an appropriate allocationof risk by assigning burden of proof.We have modified our previous Scheme implemen-

tation for the inheritance reasoning domain to per-form the qualitative argumentation outlined in thispaper. This implementation determines sets of claimsacceptable for each of the three burdens of proof bycomputing the sets of justifiable and defendable argu-ments. The next steps for our research will be toreimplement the argumentation system in terms ofthe dialectical argument process described above andto use this to explore problem solving and diagnosticreasoning over complex qualitative networks modelinginteresting, real-world systems. One example domainwould be economics, as we have illustrated, where im-pacts among variables are often not well understood,giving rise to non-strict links in the resultant models.

ReferencesDung, P.M . 1995."On the acceptability of argumentsand its fundamental role in nonmonotonic reasoning,logic programming and n-person games, Artificial In-telligence, 77(2), 321-358.

Farley, A.M . 1996 . "Dialectical Nonmonotonic In-heritance", UO-CIS-TR-96-15, submitted to Compu-tational Intelligence .

Farley, A.M . and Freeman, K . 1995 . "Burdenof proof in legal argumentation", inProceedings ofFifth International Conference on Artificial Intelli-gence and Law, 156-163.

Farley, A.M . and Lin, K.P. 1990, "Qualitative rea-soning in economics", Journal of Economic Dynamicsand Control, 14, 465-490.Freeman, K. 1993 . A Computational Model of Di-

alectical Argumentation, Ph .D . Thesis, Computer andInformation Science Department, University of Ore-gon.

Freeman, K. and Farley, A.M . 1996. "A compu-tational model of argumentation and its applicationto legal reasoning", in Artificial Intelligence and Law,(4), 163-197 .

Horty, J .F . 1994 .

"Some direct theories of non-monotonic inheritance", in Gabbay, D.M., Hogger,C.J, and Robinson, J .A . (eds.),Handbook of Logic inArtificial Intelligence and Logic Programming, OxfordPress : New York, 111-188.

Loui, R. 1987 . "Defeat amongarguments: asystemofdefeasible inference", Computational Intelligence, 3,100-106.

Pollock, J. 1987 . "Defensible reasoning", CognitiveScience, 11, 481-518.

Polya, G . 1968 . Mathematics and Plausible Reason-ing (2nd ed .), Princeton University Press: Princeton,NJ .

Prakken, H . 1993. "An argumentation frameworkin default logic", Annals ofMathematics and ArtificialIntelligence, (9), 93-132 .

Sartor, G. 1993 . "A simple model for nonmono-tonic and adversarial legal reasoning", in Proceedingsof Fourth International Conference on AriificialIntel-ligence and Law, 192-201.