on verifying inferences in an influence net - CiteSeerX

ON VERIFYING INFERENCES IN AN INFLUENCE NET

Abbas K. Zaidi Alexander H. Levis Center of Excellence in C3I George Mason University Fairfax, VA 2230

Abstract A methodology for the verification of inferences obtained from an influence net is presented. The methodology is based on transforming an influence net into an equivalent Petri net representation. The application of well-known analytical tools of Petri net theory is shown to reveal chains of inferences in the influence net that are problematic and possibly erroneous. The methodology is an extension of the results presented in Zaidi and Levis (1995) for the validation and verification of decision making rules. The tools and techniques presented in this paper are based on theory and are supported by software tools.

1 INTRODUCTION The behavior of a C2 organization is guided by a collection of policy and procedure rules. An example of such a system could be one that gets inputs from a set of sensors, identifies a task based on these inputs, and on the basis of attributes of the task determines a response. The domain model and the set of rules that govern the system’s behavior can be modeled and analyzed using the influence net formalism (Rosen and Smith, 1994). The influence net formalism—a different approach than that of influence diagrams (Howard and Matheson, 1984; Smith et al., 1993)—has emerged as a preferred tool to represent both a graphic and a mathematical description of the causal relationships between events, decisions, factual information and outcomes in a problem domain. Consistency and completeness of a rule set become desired features — necessary conditions for achieving acceptable performance. A methodology for the verification of inferences obtained from an influence net is presented. The methodology is an extension of the results presented in Zaidi and Levis (1995) for the validation and verification of decision making rules.

*The research was conducted with support provided by Science Applications International Corp. (SAIC) under sub-contract number 26-940079-80 and by the Office of Naval Research under contract No. N-00014-93-1-0912.

An influence net is expected to be an acyclic graph that consists of nodes which represent conditions, beliefs, and actions. The arc between two nodes represents the implication mechanism, i.e., an arc from node x to node y indicates that assertion y can be inferred given the truth of proposition x. An arc between nodes x and y, annotated by a negative sign, indicates that the proposition x affects y in a negative manner. Other features include arc annotations that represent necessity and sufficiency values, certainty factors, probabilities, or strength assigned to both negative and positive inferences. These arc annotations, however, are not considered in this illustration. The empirical development of the causal relationships among the nodes of an influence net can result in inferences that are inconsistent, incomplete and partially erroneous. An influence net may be constructed through an incremental process in which new nodes/arcs are added and/or modified to reflect the requirements of a changing environment, and previous inferences may be deleted if no longer valid for the domain in hand. This incremental process, if carried out without a continuous check on the effects of the changes made to the net on the rest of the inferences, may introduce errors in the influence net. The approach is presented with the help of an illustration, which is taken from a Rand Report containing an analysis of the invasion of Kuwait by Davis and Arquilla (1991). The influence net considered in this illustration was developed at SAIC (see Rosen and Smith, 1994) and ties together the assertions of the Rand report. The assertions used in the influence net consist of information related to the financial, political, cultural, and military situation of Iraq. The nodes representing actions incorporate different players' (Iraq, USA, etc.) potential actions. Some of the nodes represent beliefs of the players regarding the situation. The net is considered for four hypotheses, denoted by Options A-D, that represent potential actions taken by Iraq. The net is evaluated for these possible

hypotheses as part of the model's output assessment. The influence net, INF, is shown in Figure 1. The rectangular nodes in Figure 1 represent the initial value nodes which are referred to as "INIT" nodes in the report. No distinction has been made between regular nodes and these nodes during this presentation. A glossary of the acronyms used in the net is given in Table 1. The notation of the influence net follows the syntax prescribed by the Situation Influence Assessment Module (SIAM), a software for influence net modeling. (Rosen and Smith, 1994)

description of the algorithm is provided in Table 2. The notation p—»x represents the binary relation that a directed path exists from node p to x. The FPSI algorithm, when applied to a node p in a graph G, collects all the nodes to which p has a directed path and returns a subgraph composed of those nodes only. The application of the FPSI algorithm to the virtual input OP, FPSI(OP), results in a subnet of net INF that consists of all those nodes that have a directed path from at least one of the four nodes, OPA, OPB, OPC, and OPD. Any calculation or assessment performed on the original net for these four hypotheses (OPA-D) requires that only those nodes of net INF are considered that are included in the subnet obtained through FPSI(OP). The shaded nodes in Figure 2 and all the connecting arcs between these nodes represent the subnet obtained as a result of the application of the FPSI algorithm to the input place. In case the system is designed only for the four hypotheses, the nodes in the rest of the net characterize either incompleteness (missing inferences) or presence of excessive inferences.

The net INF is considered for four hypotheses, denoted by nodes labeled as OPA, OPB, OPC, and OPD. The four nodes are shown aggregated by a single virtual input place labeled as OP. The set of inputs U, therefore, is defined as: U = {OPA, OPB, OPC, OPD}

(1)

Based on the inputs defined in (1), a problem reduction technique is presented in Section 2 which narrows down the problem of detecting erroneous inferences to those parts of the net that are influenced by the inputs, or in other words influence the outcome of the hypothesis testing. Section 3 presents the specifics of the methodology for detecting problematic and plausible erroneous cases in an influence net. Section 4 concludes the paper with remarks on the applicability of such a methodology and its possible extensions.

Since the analysis performed in this illustration considers only the four hypotheses as the input to the system, the subnet obtained as a result of the FPSI algorithm is extracted from the rest of the influence net. From now on, the term influence net refers to this subnet that corresponds to the set U defined earlier. With no knowledge of the real outputs (main concepts) for the system under consideration, all the sink places in the net are considered as outputs. The set of main concepts, therefore, is given as:

2 PROBLEM REDUCTION Since the influence net is considered for the four inputs in (1), all those nodes in the net that are not directly connected to at least one of these inputs can be ignored for the analysis. An algorithm, called FindPath-to-Sinks (FPSI) (Zaidi, 1994), is used to filter out all such nodes. A formal

Ψ = {CFP, DSS, IIK, IKD, ILN, OMK, OMS, SIC, UDK}

-2-

(2)

NAF MRK ARS

SBL

SFL

IMC

TFS

UKJ

IMV

—

NAF 1

—

DSS

— — SVL ICS—

—

UNR

OPD ILN

—

UIP

SFS

—

— — RMP

—

TFK

IIK

SPR

MRK 1

OP

BRD

—

— OPA

KFD

SFD

KFI

KLP

OPC

OPB

— IKD

ISD

OPR

SLP

IPC

—

INC

—

—— RDW

RSD

OMK 1

—

SAB

UNR 1

SGI

CFP

IEB

SIC

SUP

IHF

SAC

SMD

—

OMS 1

OMS

OMK

UNR 2

—

IIW PSF

UDK

IDW

IIG

Figure 1. The Influence Net INF for SIAM OPR PSF RDW RMP RSD SAB SAC SBL

Oil Prices are Raised Iraq Persuades Saudi Arabia to Forget Debts Iraq Reschedules Debts with West Iraq Reduces Military Purchases US Assists Iraq to Reschedule Debts Saddam is Ambitious Saddam Believes he Can Achieve Ambitions Saddam Believes he Is Destined to be a Great Leader SFD US Asks Saudi to Forget Iraqi Debts SFL Saddam Financially Limited SFS US Senate Sends Significant Forces to Saudi Arabia SGI Saddam Believes he has Good Intentions SIC Saddam Believes he is in Control Of Events SLP Iraq Persuades Saudi Arabia to Lower Oil Prices SMD Saddam Makes the Decisions (he is in control) SPR Senate Passes Resolution to Defend Kuwait SUP Saddam Understands the Use of Power SVL Saddam Viewed as a Great Leader TFK US Sends Tripwire Forces to Kuwait TFS US Sends Tripwire Forces to Saudi Arabia UDK Saddam Believes US will Defend Kuwait UIP Saddam Believes US will Increase Presence in Region UKJ US Kuwait Conduct Joint Exercises UNR1 Saddam Underestimates Negative Risks UNR2 Saddam Believes US has No Resolve in Region

Table 1. Glossary of Terms Acro -nym ARS BRD CFP DSS ICS IDW IEB IHF IIG IIK IIW IKD ILN IMC IMV INC IPC ISD KFD KFI KLP MRK NAF OMK OMS OPA OPB OPC OPD

Corresponding Condition, Belief, or Action Arab Leaders Respect Saddam Bush Resolves to Defend Kuwait Iraq Has Severe Cash Problem Saddam Believes US Will Deploy Forces to Saudi Arabia Iraq Conforms to Arab Behavior Standards Iraq Doubles Wealth Iraq Economy is Bad Iraq has 40% Inflation Iraq Imports Goods and Foods Iraq Ignores Kuwait Debts Iraq Increase Wealth Iraq Ignores Debts to Kuwait Saddam Believes Iraq Destined to be a Great Arab Nation Iraq has Military Conquests Iraq has Military Victories Iraq has No Credit Line Iraq has Poor Credit Iraq Ignores Saudi Debts Iraq Persuades Kuwait to Forget Debt US Asks Kuwait to Forgive Iraqi Debts Iraq Persuades Kuwait to Lower Oil Prices US has Minimal Relations with Kuwait Saudis will Not Allow Entry of US Forces Iraq Owes Money to Kuwait Iraq Owes Money to Saudi Arabia Option A: Iraq Continues Policies Option B: Iraq Coerces Kuwait Option C: Iraq Occupies Section of Kuwait Option D: Iraq Conquers Kuwait

Table 2. FindPath-to-Sinks (FPSI) Algorithm

-3-

-

net. Every directed arc in the influence net is replaced by a directed link (a place with single input and a single output arc); an arc between two nodes x and y in the influence net is replaced by a place labeled as "x>y" with an input arc from the transition labeled as "x" and an output arc to the transition labeled as "y". Finally, an external transition labeled as "EXT" is also created that is connected to all the source and sink transitions of the net with the help of several links each to a source and a sink transition. The algorithm to convert an influence net to the Petri net representation has been implemented on Design CPN™ using ML™. The Petri net, PN, obtained by the application of this procedure to the net in Figure 3, is shown in Figure 4. The net in Figure 4 also contains the external transition, EXT, that is connected to all inputs and outputs of the influence net in Figure 3 with the help of links represented by input and output places labeled as either "IN" or "OUT". Since there is only one input, OPA, there is only one input place labeled as "IN".

For a G = (V, E) p∈V FPSI(p) = (S, G') where S = {x | p—»x} G' = (V', E') V ⊇ V' (= S) G ⊇ G'

In order to apply the analysis presented in this paper, the influence net is converted to an equivalent Petri net. Before carrying out the conversion, however, the FPSI algorithm can again be used to further simplify the problem by extracting smaller subnets, each corresponding to one of the four inputs, OPA, OPB, OPC, and OPD. As part of this problem reduction, the FPSI algorithm is applied to input OPA, FPSI(OPA); this results in a subnet which consists of all those nodes that have directed paths from the input, OPA. The subnet, so obtained, is shown in Figure 3.

Definition Marked Graph A Marked graph is a connected Petri net in which each place has exactly one input and one output transition. Proposition 1 The Petri Net obtained as a result of transforming an influence net to an equivalent Petri net is a Marked graph.

3 PROBLEM SOLUTION The net in Figure 3 is now converted to an equivalent Petri net by converting each node in the influence net into a transition labeled with the name of the node in the influence

NAF

ARS

SBL

SFL

IMC

TFS

UKJ

IMV

—

NAF 1

MRK

—

DSS

— — SVL ICS—

— UNR

OPD

— — —

UIP

ILN

SFS

—

RMP

TFK

IIK

—

SPR

MRK 1

OP

BRD

— OPA

KFD

SFD

KFI

KLP

OPC

OPB

— IKD

ISD

OPR

SLP

IPC

—

INC

—

—— RDW

RSD

OMK 1

OMK

UNR 2

SAB

UNR 1

SGI

IEB

SIC

SUP

IHF

SAC

SMD

—

IIW PSF

UDK

IDW

— —

OMS

OMS 1

CFP

— IIG

Figure 2. Detection of Incompleteness and Extraction of the Net for Analysis

4

SFL

ARS

— SVL

ILN

—

ICS

OPA

KFD

IKD

UNR1

INC

PS F

— RDW

—

IEB

CFP

SIC

—

— OMK

OMK 1

OMS

OMS 1

Figure 3. Influence Net Corresponding to Hypothesis "OPA" ARS

SFL ARS>SVL

SFL>SVL

ICS>ARS SVL>ILN

SVL

OU T

ILN

ICS>ILN

ICS

SFL>IK D

CFP>SFL

OPA>ICS

IN

OPA>KFD

OPA

IEB>ILN

KFD

OPA>PSF

IKD

IKD>INC

UNR1 INC

OPA>RD

INC>CFP

KFD>OMK PSF

IEB>UNR1 UNR1>SIC IKD>OMK1

RDW>CFP

CFP

PSF>OMS

OMK1>CF

CFP>IEB

IEB

SIC

OMS1>CF OUT

RDW

OMK1

OMK1>OM

OMK

OMS

OUT

OUT

OMS1>OM

OMS1

EXT

Figure 4. Equivalent Petri net for the Net in Figure 3 The proof of the proposition directly follows from the If X is an S-invariant, the set of places whose construction of the influence net and the procedure to corresponding components in X are strictly positive is the transform it into an equivalent Petri net representation. Support of the invariant, noted . The Support of an S-invariant is said to be minimal if and only if it does not contain the Support of another SOnce the influence net corresponding to the input OPA invariant but itself and the empty set. is converted to a Petri net, the S-invariant analysis of the Petri net theory is applied to it. The S-invariant analysis looks at all the directed paths in the PN and searches them As part of the analysis, the S-invariant algorithm is for problematic cases. The analysis is shown to reveal certain applied to the net in Figure 4. The calculated minimal Spatterns of PNs that correspond to circular and inconsistent invariants for the Petri net are listed in Table 3. The inferences. technique used to calculate directed paths of the influence net INF with the help of calculated S-invariants of the Petri net representation is described below. Definition S-Invariant Given an incidence matrix C of a PN, an S-invariant is a Consider that the Support of a calculated minimal Sn × 1 non-negative integer vector X of the kernel of C T, invariant is given as: i.e., CT X = 0

= {x>y, u>v, v>w, w>x}

(3)

Definition Support of S-Invariant -5-

The corresponding directed path in the influence net is calculated by reordering the places:

i

u, v, w, x, y

1 CFP, SFL, IKD, INC, CFP 2 CFP, SFL, IKD, –OMK1, –CFP 3 OPA, PSF, OMS1, OMS 4 OPA, PSF, OMS1, CFP, IEB, UNR1, SIC 5 OPA, PSF, OMS1, CFP, IEB, –ILN 6 OPA, PSF, OMS1, CFP, SFL, –SVL, ILN 7 OPA, PSF, OMS1, CFP, SFL, IKD, –OMK1, OMK 8 OPA, KFD, –OMK1, OMK 9 OPA, KFD, CFP, IEB, UNR1, SIC 10 OPA, KFD, –OMK1, –CFP, IEB, –ILN 11 OPA, KFD, –OMK1, –CFP, SFL, –SVL, ILN 12 OPA, RDW, –CFP, IEB, UNR1, SIC 13 OPA, RDW, –CFP, IEB, –ILN 14 OPA, RDW, –CFP, SFL, –SVL, ILN 15 OPA, RDW, –CFP, SFL, IKD, –OMK1, OMK 16 OPA, ICS, ARS, SVL, ILN 17 OPA, ICS, ILN The following Proposition characterizes the presence of cycles inside the net.

If in a directed path node a affects node b negatively, depicted as a negative sign on the arc from a to b, then a negative sign is added to the label of node b. For example, in the directed path u, v, w, x, y, node u affects node v negatively and so does node w to node x. The directed path will be represented as: u, –v, w, –x, y Now the first five rows in Table 3, which correspond to "IN" and "OUT" places, are ignored and the directed paths of INF corresponding to the invariants of the Petri net are calculated with the help of the technique illustrated above. The directed paths of the influence net, so obtained, are listed in Table 4. Table 3. Minimal S-invariants of the Net in Figure 4 Places

Proposition 2 Circular Case A directed path in INF that does not contain the input place (OPA in the example) indicates the presence of a cycle (directed elementary circuit) inside the Influence net.

Minimal S-invariant, X i 1 2 3 4 5 6 7 8 9 1011 1213 141516 17

IN OUT OUT OUT OUT ARS>SVL SFL>SVL SVL>ILN ICS>ARS ICS>INC OPA>ICS OPA>RDW OPA>KFD OPA>PSF RDW>CFP PSF>OMS1 IEB>ILN KFD>OMK1 OMK1>OMK OMK1>CFP IKD>OMK1 CFP>SFL IKD>INC INC>CFP OMS1>OMS OMS1>CFP CFP>IEB IEB>UNR1 UNR1>SIC SFL>IKD

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1

1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0

1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0

1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1

1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0

1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0

1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0

1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0

1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1

1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Directed Paths of the Influence Net

1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

or equivalently, The Support of a minimal invariant with a zero element corresponding to the input place "IN" indicates the presence of a cycle (directed elementary circuit) inside the Petri net representing an influence net. The directed paths 1 and 2 represent the circular cases (cycles) inside the influence net INF. The first two columns in Table 3 present these two circular cases in the Petri net. The two reported circular cases are shown in Figure 5. The cycle shown in Figure 5 shows the circular argument in which one reaches the same assertion again and again after making a finite number of inferences, i.e., starting from SFL (Saddam Financially Limited) one infers IKD (Iraq Ignores Debts to Kuwait), which in turn implies INC (Iraq has No Credit Line). With the truth of INC one can infer CFP (Iraq has Severe Cash Problems), which leads back to the same assertion SFL. The example illustrates the fact that although some of the circular cases may represent the reality of the domain, some of the cycles may have been introduced in the influence net due to the presence of erroneous implications. Therefore, even if the system in hand allows circular inferences, the detection of circular cases might help indicate erroneous implications present in the net. Proposition 3 Subsumed Case

Table 4. Directed Paths of the Influence Net

-6-

Two directed paths in an influence net, where the nodes in one form a subset of nodes in the other, indicate the presence of a Subsumed case.

Based on this definition, the following unary operations are defined:

In Table 4, the set of nodes in directed path 17 is a subset of the set of nodes in directed path 16. Therefore, the directed path 17 subsumes the directed path 16 in the influence net INF. Figure 6 shows these two directed paths. The directed path 17 illustrates the assertion "Saddam believes that Iraq is Destined to be a Great Arab Nation" (ILN) from the proposition "Iraq Conforms to Arab Behavior Standards" (ICS). On the other hand, the directed path 16 represents the chain of inference that first concludes "Arab Leaders Respect Saddam" (ARS) from "Iraq Conforms to Arab Behavior Standards" and from ARS it infers "Saddam Viewed as a Great Leader" (SVL). Only after inferring SVL, the directed path concludes "Saddam believes that Iraq is Destined to be a Great Arab Nation" (ILN). The first path (17) represents a stronger implication (causal relation) from ICS to ILN, where as path 16 shows a weaker association between the two. Depending upon the nature of the application, such an existence of both a weak and a stronger implication between two propositions, or beliefs, the reported subsumed cases may or may not be problematic. However, the detection of such cases might help an analyst in filtering out the redundant implications and in optimizing the net.

• •

u, ¬v, ¬w, x, y The directed paths listed in Table 4 are transformed to their normalized representation. The corresponding normalized directed paths of the influence net are listed in Table 5. Table 5. Normalized Directed Paths

i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

u, –v, w, –x, y Since node v affects node w in a positive manner, the negatively effected assertion v will affect w negatively; the effect of node u on node w will also be negative. Therefore, in the directed path shown, the assertion represented by node u affects all the remaining assertions in a negative manner. The effects of negative signs in a directed path can, therefore, be defined as follows.

ARS

(5)

The application of these operations on the calculated directed paths results in the normalized directed paths. In the illustration, the normalized directed path will be:

Now, consider the directed path:

u, –v, w, –x, y ≡ u, –(v, w, –(x, y))

–(x) ≡ ¬x –(–(x)) ≡ x

(4)

Normalized Directed Paths

CFP, SFL, IKD, INC, CFP CFP, SFL, IKD, ¬OMK1, CFP OPA, PSF, OMS1, OMS OPA, PSF, OMS1, CFP, IEB, UNR1, SIC OPA, PSF, OMS1, CFP, IEB, ¬ILN OPA, PSF, OMS1, CFP, SFL, ¬SVL, ¬ILN OPA, PSF, OMS1, CFP, SFL, IKD, ¬OMK1, ¬OMK OPA, KFD, ¬OMK1, ¬OMK OPA, KFD, CFP, IEB, UNR1, SIC OPA, KFD, ¬OMK1, CFP, IEB, ¬ILN OPA, KFD, ¬OMK1, CFP, SFL, ¬SVL, ¬ILN OPA, RDW, ¬CFP, ¬IEB, ¬UNR1, ¬SIC OPA, RDW, ¬CFP, ¬IEB, ILN OPA, RDW, ¬CFP, ¬SFL, SVL, ILN OPA, RDW, ¬CFP, ¬SFL, ¬IKD, OMK1, OMK OPA, ICS, ARS, SVL, ILN OPA, ICS, ILN

SFL

— SVL

ILN

—

ICS

OPA

KFD

IKD

INC

PSF

— RDW

UNR1

—

CFP

—

— OMK 1

OMK

OMS

-7-

OMS 1

IEB

SIC

Figure 5. Cycles in INF ARS

SFL

— SVL

ILN

—

ICS

OPA

KFD

IKD

UNR1

INC

PSF

— RDW

—

CFP

IEB

SIC

—

— OMK

OMK 1

OMS

OMS 1

Figure 6. Subsumed Case in INF Definition Inconsistency (Galton, 1990) A set of statements (inferences) is said to be inconsistent if they can not all be true at the same time.

• The normalized directed paths 4, 5, 6, and 7 (numbers refer to the listed directed paths in Table 5) all have directed paths of the form OPA—»CFP as part of them; starting from the proposition "Iraq Continues Policies" (OPA) the directed paths infer "Iraq has Severe Cash Problem". The normalized directed paths 12, 13, 14, and 15, on the other hand, have directed paths of the form OPA—»¬CFP, which show chains of inference that infer that Iraq does not have the cash problem from the same input, OPA. Therefore the two sets of directed paths are inconsistent with respect to each other.

Proposition 4 User-defined Inconsistency A normalized directed path of the form x—»y (or y—»x), where µ = {x, y} is a user-defined set of mutually exclusive assertions, indicates the presence of inconsistencies in the influence net. Proposition 5 System-defined Inconsistency Two normalized directed paths of the form x—»¬y and x—»y indicate the presence of inconsistencies in the influence net.

• The set {4, 5, 10} is found inconsistent with {12. 13}, since the members of the first set have OPA—»IEB, and the members of the latter have OPA—»¬IEB; one can infer both "Iraq's Economy is Bad" and "Iraq's Economy is not Bad" from the same input.

With no knowledge of the user-defined mutually exclusive assertions, the calculated normalized directed paths are searched for the inconsistent cases characterized by Proposition 5. The following inconsistent cases are found: ARS

SFL

— SVL

ILN

—

ICS

OPA

KFD

IKD

INC

PSF

— RDW

UNR1

—

CFP

—

— OMK 1

OMK

OMS

-8-

OMS 1

IEB

SIC

Figure 7 Two Inconsistent Paths in INF

inconsistencies among the calculated ones (The analysis is beyond the scope of this paper.).

• The set {4, 9} is found inconsistent with 12, since the members of the first set have OPA—»UNR1 and OPA—»SIC, and the directed path 12 has OPA—»¬UNR1 and OPA—»¬SIC; they represent the inference of "Saddam Believes he is in Control of Events" and "Saddam Underestimates the Negative Risks" and their negations from the input OPA.

In the illustration, a number of errors were intentionally introduced in the influence net to demonstrate all the features of the methodology. In addition to these known errors, a number of other problematic cases were also reported by the approach that were present in the original net. The process proceeds with the application of the methodology to influence nets associated with each of the remaining inputs, OPB, OPC, and OPD.

• The set {5, 6, 10, 11} is found inconsistent with {13. 14, 16, 17}, since the members of the first set have OPA—»¬ILN, and the members of the latter have OPA—»ILN; one can infer both that Saddam believes that Iraq is destined to be a great nation and that he does not believe so from the same input assertion.

An Alternate Approach The theoretical results obtained in the previous section can be used to further simplify the computation of solutions. This section presents an approach that uses the results from Petri net theory but computes the directed paths in the influence net without first converting it to an equivalent Petri net, thus reducing the overhead associated with the transformation of an influence net to an equivalent Petri net. Consider the following definition of the Connectivity matrix associated with an influence diagram.

Figure 7 shows two of the inconsistent cases found in the net. The normalized directed path 4 (OPA, PSF, OMS1, CFP, IEB, UNR1, SIC) represents a chain of inference from proposition OPA (Iraq Continues Policies) to CFP (Iraq has Severe Cash Problem) through PSF (Iraq Persuades Saudi Arab to Forget Debts) and OMS1 (Iraq Owes Money to Saudi Arabia). On the other hand, the normalized directed path 12 (OPA, RDW, ¬CFP, ¬IEB, ¬UNR1, ¬SIC) represents another chain of inference in the net where proposition OPA (Iraq Continues Policies) implies RDW (Iraq Reschedules Debts with West), which in turn results in the assertion ¬CFP (Iraq does not have Cash Problem), which is in contradiction to the inference made through path 12. The analysis of these two cases raise an interesting modeling issue. The implications present in each of the two directed paths seem correct and logical as long as they represent two different (mutually exclusive) instances of the real world, i.e., either Iraq reschedules its debts with the west and as a result solves its cash problems, or it chooses otherwise to continue its policies and end up having severe cash problems. In case the two deductive inferences are considered simultaneously, the reported cases represent the real inconsistency present in an influence net. The detection of such cases in an influence net raises temporal and concurrency related issues that must be resolved before any computation is performed.

Definition Connectivity Matrix An influence net with n directed arcs and m nodes can be represented by a n × m matrix J, the Connectivity matrix. The rows correspond to arcs, the columns correspond to nodes. • Jij = 1 if the directed arc in i-th row originates from the j-th node. • Jij = -1 if the directed arc in i-th row terminates in the j-th node. • Jij = 0 if the directed arc in i-th row is not connected to j-th node. The following proposition interprets the results presented in the previous section in terms of Connectivity matrix of an influence net. Proposition 6 If in an influence net a directed arc from node x to node y is labeled as "x>y", and the labeling scheme is applied to every arc in the net, then the Connectivity matrix J is identical to the Incidence matrix C of the Petri net representation of the influence net.

The other features of the influence nets include arc annotations that represent necessity and sufficiency values, certainty factors, probabilities, or strength assigned to both negative and positive inferences. Therefore, Propositions 4 and 5 only identify the potential inconsistencies present in the influence net. In order to establish a real inconsistent case, one has to consider the arc annotations and the nature of interaction between two nodes connected by the arcs. For example, if it is possible to reach conflicting assertions as a result of executing the influence net, only then the two directed paths are inconsistent. The calculated inconsistent cases, therefore, represent potential inconsistencies, and additional analysis is required to establish the real

The proof of the proposition directly follows from the construction of Connectivity and Incidence matrices and labeling scheme of the respective influence and Petri nets. The directed paths in the influence net can now be directly computed by simply replacing the C matrix by J

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matrix in equation (3) and following the rest of the approach as described for the calculated S-invariants.

4

CONCLUSION

A methodology for the verification of inferences obtained from an influence net is presented. The Methodology is based on transforming an influence diagram to an equivalent Marked Petri net representation. An algorithm has been defined and implemented on Design/CPN™ for an automatic conversion of an influence net to an equivalent Marked Graph Petri net. The verification of the influence net is done by exploiting the structural properties of the Petri net. The results obtained from Petri Net theory are then used directly on the influence net formalism to detect circular, conflicting, and incomplete cases present in the influence net. In addition to detecting the erroneous implications in the influence diagram, the results of this application are shown to raise issues about influence net models that must be resolved for a robust representation of the domain in hand.

ACKNOWLEDGMENTS The authors of the paper are grateful to Dr. Julie A. Rosen of Science Applications International Corp. (SAIC) for providing a challenging real world problem and helpful comments for its solution.

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