Belief Revision in Dynamic Abducers Through Meta-Abduction THESIS

Belief Revision in Dynamic Abducers through Meta-Abduction

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Vivek Bharathan

Graduate Program in Computer Science and Engineering

The Ohio State University

2010

Master's Examination Committee:

Dr. John Josephson, Advisor

Professor B. Chandrasekaran, Advisor

Vivek Bharathan

2010

Abstract

Abduction machines (or abducers) infer to the best explanation for the data presented to them, and may accumulate beliefs (the conclusions of the inferences) about the world. An abducer‟s beliefs are justified as being the best explanation in contrast with alternative hypotheses. However, the best explanation available to (achievable by) an abducer need not be the true explanation for a variety of reasons including: incomplete search for alternative explanations, insufficient data, and inadequate background knowledge for evaluating alternatives. In light of this fallibility, algorithms were investigated for detecting and correcting errors, for dynamic abducers, by comparing a range of alternative algorithms with regard to effectiveness and computational costs.

Dynamic abducers interpret an incoming information stream by producing their best explanations at any point in time. They accumulate new beliefs, and update older beliefs, in the light of new information.

In the present work, algorithms were developed for detecting and correcting errors in the accumulated beliefs of such abducers. These algorithms treat the problem of identifying errors as meta-abduction, where certain anomalies that occur during processing are explained as resulting from specific mistakes in previous abductive processing. Errors are then corrected, and beliefs revised, by adopting alternative explanations. A brute-force algorithm for this meta-abduction is computationally intractable, so heuristics were developed with prospects of improving performance. Since a priori mathematical analysis of the algorithms using these heuristics did not reveal ii useful bounds, simulation experiments were conducted, using a specimen domain. The domain was that of multi-object tracking, where an abduction machine, over time, attempts to maintain the track history of mobile entities, based on sensor reports. The experimental results suggest that, using only the heuristics that were investigated, belief revision by meta-abduction enables only small improvements in correctness, and is computationally expensive.

iii

Vita

1997...... Padma Seshadri Bala Bhavan Senior

Secondary School

2001...... B.E. Computer Science and Engineering, Sri

Venkateswara College of Engineering

2002-2003 ...... Graduate Teaching Associate, Department

of Computer Science and Engineering, The

Ohio State University

2003-2008 ...... Graduate Research Associate, Department

of Computer Science and Engineering, The

Ohio State University

Publications

Vivek Bharathan and John. R. Josephson, "Belief Revision controlled by Meta-

Abduction", Special Issue of Logic Journal of IGPL (Oxford University Press),

Abduction, Practical Reasoning and Creative Inferences in Science, edited by L.

Magnani, 2006 (Volume 14)

Vivek Bharathan and John Josephson, “Detecting and Correcting Mistakes in Information

Fusion”, in proceedings of The Army Science Conference, 2006

Vivek Bharathan and John Josephson, "An Abductive Framework for Level One

Information Fusion", in proceedings of The 9th International Conference on Information

Fusion, 2006

John Josephson and Vivek Bharathan, "Abductive Inference Engine for Entity Re- identification and Tracking", in proceedings of The Army Science Conference, 2004

Fields of Study

Major Field: Computer Science and Engineering

Specialization: Artificial Intelligence

Table of Contents

Abstract ...... ii

Vita ...... iv

Table of Contents ...... vi

List of Tables ...... ix

List of Figures ...... x

Chapter 1: Introduction ...... 1

1.1 Dynamic Abduction ...... 4

Chapter 2: Belief Revision in Dynamic Abducers ...... 9

2.1 Detecting Errors in Dynamic Abducers ...... 9

2.1.2 Trigger 2: Predictions Contradicted by Observations: ...... 10

2.1.3 Trigger 3: Clashing Explanations ...... 11

2.2 Correcting Errors in Dynamic Abducers ...... 11

2.3 Problem Space Complexity for Belief Revision in Abductive Agents ...... 14

Chapter 3: The Abductive Approach to Multi-Object Tracking ...... 19

3.1 Description of the ASAS Domain ...... 19

3.2 The Entity Re-identification Algorithm in Smart-ASAS ...... 21

3.3 An Illustrative Example of Abductive Tracking and Error Correction in Smart-

ASAS ...... 24

3.3.1 Initial State ...... 25

3.3.2 Processing Sensor Reports - Generation, Evaluation and Acceptance of

Hypotheses...... 26

3.3.3 Revision of previous conclusions in Smart-ASAS ...... 28

Chapter 4: The Smart-ASAS Testbed ...... 33

4.1 The Class of Meta-Abductive Error-Correction Algorithms ...... 36

4.1.1 The “In Denial” Algorithm ...... 37

4.1.2 The “Acknowledge Discrepancy” Algorithm ...... 38

4.1.3 The “Revision with Recency Heuristic” Algorithm ...... 39

4.1.4 The “Revision with Entrenchment Heuristic” Algorithm ...... 40

4.2 Performance Measures for the Error-Correction Algorithms ...... 42

Chapter 5: Experimental Results from the Simulated ASAS Domain ...... 44

5.1 Performance of the Error-Correction Algorithms ...... 44

Chapter 6: Summary and Discussion ...... 69

References ...... 71

vii

Appendix A: An Illustrative Example of Abductive Tracking and Error Correction in

Smart-ASAS with Alternative Revisions...... 75

Appendix B: The PEIRCE Hypothesis Assembly Algorithm ...... 82

viii

List of Tables

Table 1. Meta-Abductive Belief Revision versus Abductive Processing ...... 13

Table 2. Parameters in Complexity Analysis of Abductive Belief Revision ...... 16

List of Figures

Figure 1. Intial State of the World ...... 25

Figure 2. Decision D and Second Report...... 28

Figure 3. Decision D' and Third Report ...... 30

Figure 4. Final State Estimate ...... 32

Figure 5. Performance Measure - Average Track Length ...... 46

Figure 6. Performance Measure - Number of Miscorrelations ...... 50

Figure 7. Performance Measure – Error Rate ...... 52

Figure 8. Performance Measure – Efficiency ...... 57

Figure 9. Actual Errors versus Observed Errors in different Revision Algorithms ...... 60

Figure 10. Performance of the BR with Recency Algorithm ...... 63

Figure 11. Performance of the BR with Entrenchment Algorithm ...... 65

Figure 12. Initial State of the World ...... 75

Figure 13. Decision D1 ...... 76

Figure 14. Decision D2 ...... 77

Figure 15. Decision D3 ...... 78

Figure 16. First Revision Option ...... 79

Figure 17. Second Revision Option ...... 80

Chapter 1: Introduction

I describe an abductive reasoner that is able to change its mind and appropriately revise previous conclusions when it encounters a reasoning anomaly. Abductive inference (by which I mean “Inference to the Best Explanation”) (Harman, 1965) is an ampliative inferential process; that is, the conclusion goes beyond merely extracting information already present in the premises. The best explanation constructed by an abducer could be incorrect due to paucity of various forms of knowledge. Thus, abductive inference is inherently fallible, and so it is desirable for a reasoning agent relying on abduction to be able to correct mistaken previous conclusions. A number of frameworks have been proposed for how to model and implement the logic of how an agent might change its mind: the non-monotonic reasoning techniques of default reasoning (Brewka et al, 1997), Justification-based (and Assumption-based) truth maintenance systems (Forbus and de Kleer, 1993) and AGM theory (Gärdenfors, 1988), are prominent examples. These approaches, whatever their differences, share the property that they all seek universal solutions to the problem of belief revision. In contrast, I propose to take advantage of the specific structure of abductive reasoning to identify revision candidates among earlier beliefs, to propose specific revisions, to select among possible revisions, and to make the requisite changes to the system of beliefs. This

1 processing is accomplished by meta-abductive processing over the recorded steps in an abductive agent‟s reasoning trace.

A sophisticated rational agent is able to continue to function even when confronted with information that challenges its current beliefs. It is obvious that human beliefs are not static, and that with the passage of time, and the processing of information, beliefs are revised and not simply monotonically extended. Two major communities addressing the problem of belief revision are comprised of mathematical logicians and of artificial intelligence theorists. Among the logicians, a belief system is commonly treated as a logically closed set of sentences whose behavior under revision is maintained by certain postulates that place restrictions and guide the revision, i.e., these are attempts to axiomatize belief revision (at least partially). One of the most popular approaches is the

AGM-theory by Alchourron, Gärdenfors and Makison (Gärdenfors, 1988, and

Alchourrón et al, 1988). In general, the choice of change in these approaches stems from notions of epistemic entrenchment and minimal mutilation, which use rough measures of how the specific changes would minimally alter (or contaminate) the belief system as a whole. In the community of AI-theorists, typical approaches build on the works of

(Doyle, 1979) and (de Kleer, 1986). They introduce and develop the concept of a reasoning subsystem for updating beliefs, which works by associating with each belief, a set of justifications (in essence the reasons or derivations for that belief) and analyzing the size and strength of the associated set in comparatively determining its truth. For a comprehensive overview, see (Forbus and de Kleer, 1993) or (Shapiro, 1998).

While there have been many contributions that interpret abductive inference as a process of epistemic change or belief revision (Aliseda 2006, Boutilier & Becher 1995),

1 we are aware of no reported work on dynamic abducers, i.e., abductive agents that handle an incoming information stream by producing their best explanations at any point in time, and updating their beliefs in the light of new information. This updating of beliefs could occur for several reasons which are enumerated in the next section. Such a dynamic abducer may be said to perform the task of “maintaining situation awareness” (analogous to perception and situation understanding) by coherently assimilating new information and forming best explanations. A simpler sort of dynamic abducer continually updates its best estimate of the situation, as the situation changes, but it does not explicitly correct previous estimates of what has become the past situation. Let us say that this sort of abducer performs “shallow” belief revision. The past situation influences and constrains the present however, and an agent that has a correct estimate of the past has an advantage in reasoning about the present. Thus, a dynamic abducer that is able to correct previous estimates of the past has a reasoning advantage in estimating the current situation. Let us say that this sort of dynamic abductive agent performs “deep” belief revision. In this thesis, I describe how this sort of abductive agent can work. That is, I describe a reasoning strategy whereby a dynamic abducer is able to appropriately revise previous estimates under some circumstances, and use the revised estimates to improve its estimates of the current situation. I also describe an implementation of this strategy in

Smart-ASAS, a system for entity tracking and re-identification (Bharathan and Josephson

2005, Bharathan and Josephson 2006). The abducer I describe proceeds monotonically, that is, it accumulates beliefs until anomalous data raises significant doubt regarding the correctness of either its beliefs, or the incoming data, and causes it to revise its beliefs in

2 an attempt to improve its estimate of the situation. This belief-revision process makes use of abductive reasoning, where the anomalous data are the to-be-explained, and where mistakes in previous reasoning steps are considered among the alternative hypotheses for doing the explaining.

I shall also show that for such dynamic abducers, due to the large combinatorial growth in the size of the solution space, it is impractical to attempt a comprehensive search of all possible solutions in order to determine the best. Consequently, I develop certain heuristics that may need to be employed to optimize computational effort and arrive at satisficing solutions. These heuristics are in part influenced by psychological, and epistemological, theories of human cognition. Finally, I shall show via empirical simulation in the domain of multi-object tracking, the efficacy of, and various tradeoffs among, the suite of meta-abductive revision algorithms.

In the context of this document, I shall treat the accepted hypotheses of the abducers as equivalent to the notion of “beliefs” as is commonly used in the literature in both AI, and the Philosophy of Science. These accepted hypotheses also have the justification of being the inferences drawn, albeit potentially fallibly, from the observations. For the rest of this document I shall use the terms “belief,” “accepted hypothesis,” and “result of previous inference,” interchangeably. I make a distinction between these accumulated beliefs of the abducer, and information that the abducer possesses regarding the world

(e.g. knowledge about causal relations, or other facts), and call the latter “domain

3 knowledge”. Consequently, I use the term “belief system” or “theory” to refer to the union of the set of all accepted inferences accumulated by the agent thus far, and the domain knowledge, i.e., the belief system is the collection of all information that the abducer has accepted (even those tentatively accepted) regarding the world.

1.1 Dynamic Abduction

In this section I provide a detailed and formal description of the class of abducers that I work with. There is the more commonly recognized task where it is required to construct/identify a set of causes (or a composite hypothesis) that led to the occurrence of a fixed set of observations (effects). Let us call this “static abduction”. In the second category, which we shall call “dynamic abduction”, we deal with a stream of observations, and the task is to construct, at each instant, a hypothesis of causes that would best explain the stream thus far. As new observations arrive, the hypothesis might change. This task is dynamic in two senses. First, even if the external world has not changed, new observations might give information that could lead to changes in the hypothesis about causes. Thus, the explanation is dynamic. Second, the world itself might be changing; thus, the new observations may call for changes to the hypothesis, not because earlier hypotheses were incorrect about the world corresponding to earlier observations, but the hypothesis needs to be changed to reflect changes in the world.

In dynamic abduction problems, some observations are available at the start, and new observations become available at various later times. In simpler versions of the problem, the set of potential causes remains the same; new observations provide additional evidence for the same underlying state of affairs (situation). In more complex versions of the problem, the underlying world state may itself change, and the causes that explained earlier observations may no longer explain some of the new observations. In both versions, the pragmatics of the abduction problem may require an “any time” solution, i.e., that the abducer make available a “best” solution for the observations available at any time. Dynamic abduction problems require efficient strategies for adapting the previous solution St at stage t as additional observations (Ot+1) are taken into account at stage (t + 1). Thus sequential updating is part of the task definition for dynamic abduction problem. We do not have the luxury of choosing the order in which we consider subsets of the observations.

A significant attraction of adapting previous solutions is the potential savings on computational costs that come from adapting the solution St at stage t to also explain the set of observations considered at stage (t + 1). At each stage, as new observations are considered, one of three things happens:

1. The solution St is consistent with, and explains, the additional observations

considered at (t + 1). In this case no change is made to the previous solution (the

working set of putative causes).

2. The solution is consistent with the newly considered observations, but does not

explain some or all of them. In this case the previous solution is augmented with

additional causes, which do not contradict existing beliefs, to explain the new

observations.

3. The previous solution is inconsistent with new observations1. In this case, the

previous solution is in error, and is rejected. A safe, but computationally

expensive strategy would be to ignore the previous solution altogether and

construct a solution for the entire set of observations. There might be

computationally less expensive methods that make some use of St, e.g., methods

that efficiently unwind the previous solution until the inconsistency is removed,

and use this unwound solution as a basis to construct a consistent solution. The

steps involved in this undertaking are:

1. Detecting any inconsistency between St and Ot+1.Strategies may differ in

their efficiencies and completeness in performing this task. Note that not

detecting an inconsistency between St and Ot+1 does not imply that St is

consistent or correct - future observations might contradict St. We will

need to characterize the inconsistency detection strategy and its properties

for any proposed algorithm, or solution strategy, for these problems.

1 This includes the case where the only explanation for the observation, or the best explanation by far, is inconsistent with the previous solution 6

2. When an inconsistency is found, the algorithm or strategy needs a way to

"unwind" St, so that the inconsistency is eliminated, and to construct an

St+1 that is consistent with, and explains Ot+1.

Formally, the dynamic abducer is an entity that receives, either due to the dynamic nature of the domain, and/or due to the nature of the input information stream, a sequence of observations from which the abducer attempts to estimate the state of the underlying situation. The dynamic abducer processes a stream of inputs, O1, O2,…,On, each reporting an observation or a set of observations. It generates and maintains a changing estimate,

S1, S2,…,Sn, of the state of the situation (that is, of the reality underlying the observations, at a certain level or levels of description). A dynamic abducer may also maintain beliefs about the past, especially the recent past, to constrain possibilities and thus help to interpret new observations. The ideal solution would be a function that uses the observations to produce a set of beliefs that is, as accurately and completely as possible, a representation of the current state. This solution is characterized by a function

(fdynamic) whose arguments may be a combination of other states and/or observations, such that

Sn = fdynamic (Sp, Op+1…, On) , for any p < n

Ideally, we would like the state estimate Sn to be independent of the order in which the observations (Op+1…, On) are processed. However, one primary reason for the fallibility of a dynamic abducer is that the (tentative) acceptance of explainers that seem promising with respect to the current observations sometimes lead the formation of subsequent explanations away from the globally optimum (desirable) solution, as in the case of garden path sentences in NLP. Thus the drawback to constructing a theory by augmenting a previous state estimate to account for subsequent observations is that the process is sensitive to the order in which the inputs are considered. Nevertheless, this approach of sequentially considering observations and constructing (partial) explanations is still useful. This is because it is presumably computationally much cheaper to integrate a new observation into an existing belief system and produce a consistent, albeit less than best possible, belief system, than it is to recreate from scratch one that accounts for all the observations thus far.

This class of abduction machines is the subject of the present investigation.

Chapter 2: Belief Revision in Dynamic Abducers

2.1 Detecting Errors in Dynamic Abducers

Obviously, the first step in rectifying possible errors in the belief system of a dynamic abducer is detecting when errors have been committed, i.e. detecting that the current set of beliefs and the new observations contradict in some manner. In abducers, the detection of contradictions between the belief system and observations is not as straightforward as in traditional formulations of truth maintenance systems. This is because the beliefs do not logically entail the observations, but instead describe a world state in which the observations are plausible (or likely). In consequence, new observations that do not logically contradict the belief system could still affect the overall plausibility of the belief system (by casting doubt on its correctness), and these non-contradicting observations could still induce a need for revision2. Thus, in addition to the obvious case of the new observations actually logically contradicting the belief system, there are multiple situations in which the belief system and the new observations can still be in conflict. The following is an open-ended list of triggers that may be employed to initiate error correction in an abductive agent. This alteration to the traditional logical formulation is in order to allow for belief revision to be triggered by less drastic states of reasoning

2 This is distinct from logic-based belief systems in which logical contradictions are the only trigger for revision 9 difficulty, since these states are more representative of errors in ampliative inferential strategies.

2.1.1 Trigger 1: No Explanation Anomalies:

When new observations arrive, the sequential updating strategy might reach an impasse: no explanation can be constructed for the observations. This state of affairs, i.e. the inability of the belief system to account for the observation, coupled with the inability of the problem-solving strategies of the abducer to produce a satisfactory explanation, suggests that the belief system of the agent could be incorrect. I use this easily detectable anomaly as the most common trigger for error correction.

2.1.2 Trigger 2: Predictions Contradicted by Observations:

While it is impractical to maintain all possible predictions of the belief system

(analogous to a closed set of beliefs in traditional logic-based belief systems) due to the computational expense, it is still possible for the abducer to be cognizant of some of the more obvious predictions in the course of problem solving. The contradictions between these predictions and subsequent observations may be used as yet another trigger for error correction. In abductive agents, these predictions do not have to be logical implications, therefore the import of these contradictions vary.

2.1.3 Trigger 3: Clashing Explanations

In rare situations, apparently contradictory, but independently confident alternative explanations compete to account for the same set of observations. This is almost certainly indicative of a flaw in the domain knowledge and not in the accumulated beliefs per se3, however this could still be used as a trigger for correction.

2.2 Correcting Errors in Dynamic Abducers

As already described, a dynamic abducer assimilates information from senses or sensors, and tries to maintain an accurate and consistent representation of the world. The desired state of the agent is a description as close to reality as is possible, or at least those aspects of reality that are important. Due to the fallibility of the inferential process, and the inherent uncertainty in the problem, there will be situations in which the abducer makes an informed, but incorrect, guess based on the available evidence, makes additional inferences dependent on this guess, and continues until it encounters a discrepancy between new information and its estimate of the situation. Cognitively, the anomalies in the observables (depending on how severe they are) may accumulate to cross some threshold before the agent is aware of the existence of a problem, and is motivated to correct it. The challenge then, in the case of an abductive reasoner, is to confidently identify the incorrect, but abductively supported, conclusions, to retract them,

3 A similar deadlock could arise when the abducer is unable to decide between, less confident, alternate explanations. However this is usually due to a dearth of information at that time and is typically resolved by postponing that decision until the requisite information becomes available. 11 and those conclusions that follow from them, and provide convincing alternative explanations for the previous data and the new data that brought about this process of error correction, in such a way as to maintain consistency, coherence, plausibility and explanatory coverage in the agent‟s view of the universe, as much as possible.

A plausible architecture for a sophisticated reasoning agent ought to provide some mechanism for belief revision. For the agent to perform “deep” belief revision, this would seem to require a representation for recording justifications for beliefs (or at least some information from which such justifications may be generated) along with a method or methods for effecting revision of beliefs called for by the acquisition of new information.

A reasonable approach would be to: detect an anomaly, analyze the recorded justifications for tenuous conclusions, reassess them with the advantage of hindsight, and choose a best way out of the difficulty. In a dynamic abducer, these basic components of a belief revision strategy correspond to five distinct phases in generic abductive processing, as summarized in the following table.

Components of a meta- Phases of generic abductive belief revision abductive processing strategy for a dynamic abducer

I. Detecting a reasoning anomaly Determining what is to be (possibly anomalous new data) explained

II. Identifying relevant, Generating Hypotheses lower-confidence past decisions

III. Determining whether the Evaluating Hypotheses current anomaly would be resolved by suspension of belief, combined with assertion of an alternative explanation, at the point of one or more of these past decisions

IV. Choosing the best method Choosing the best of resolving the anomaly, if explanation more than one is available

V. Performing the indicated Accepting the best repairs on the system of explanation beliefs

Table 1. Meta-Abductive Belief Revision versus Abductive Processing

Thus, meta-abductive processing is a plausible strategy for belief revision in abductive processing.

2.3 Problem Space Complexity for Belief Revision in Abductive Agents

While the meta-abductive revision strategy appears plausible as described in the previous section, I shall show that the computational effort involved in identifying and rectifying errors in the belief system of an abductive agent is far from trivial. In traditional formulations, belief systems are taken to be sets of sentences that are logically closed under a consequence relation, and revisions are prescribed by well-defined operators4 such that these revisions uniquely produce an alteration to the belief system which resolves logical contradictions between the beliefs and new information. These procedures typically exploit the logical dependencies between beliefs (propositions) in the theory. In the abductively-developed belief systems, there exist minimal, if any, of these dependencies that may be used in backtracking, for the purpose of locating and correcting errors. Instead, the revision process is presented with an unstructured set of beliefs, and is tasked with altering it in order to resolve conflicts between new information and the existing set of beliefs. It seems reasonable to assume that such a dependency-free belief system is characteristic of most ampliative reasoning systems, except for some instances (subsets) of reasoning chains involving logical implications or possibly, statistical or probabilistic inferences. This makes the problem much more challenging, and in this section I describe the size of the problem space that the meta- abduction revision strategy needs to handle.

4 These operators may be accompanied by some other information, such as preference relations, to help disambiguate between different possible revisions 14

Consider the generic case of a dynamic abducer that has received, and processed n observations. Let the current belief system of the abducer be the theory T. T consists of the domain knowledge (D) of the abducer, and the beliefs or inferences (B) that it has accumulated thus far (T = B ∪ D). Let us assume that there is a conflict between (n+1)th observation (on+1) and the current theory T that needs to be resolved.

I.e., {T + (relevant abductive processing)} is unable to account for on+1.

Without loss of generality, for simplicity, I posit that the set of potentially impugnable subset of the belief system, in the event of conflict or revision, is the set of accumulated beliefs B. This is in effect an implicit, minimal, assumption of preference, or entrenchment, where the abducer assumes that the domain knowledge is less likely to contain errors in comparison to the beliefs accrued through the inferential process.

Now, there is a subset of B that contains the beliefs that are suspected of being in error, and it is these beliefs that the belief revision process is interested in scrutinizing, and possibly rectifying. Let the cardinality of this subset be n'.

Let us assume that on an average each abductive belief was chosen out of k alternative hypotheses, and that on average k' of these alternate hypotheses are plausible enough to be a potential candidates for revision.

The following table summarizes these useful parameters that will aid the analysis.

Symbol Semantics

n Number of observations processed

n' Number of revision candidates, the alteration of

one of which could resolve the conflict

k Number of atomic hypotheses considered as

alternatives in one abductive decision

k' Number of alternative next best hypotheses that

may be substituted for the revision candidate

c Cost of evaluating and assessing confidence of

one atomic hypothesis

Table 2. Parameters in Complexity Analysis of Abductive Belief Revision

The computational effort involved in making one abductive decision includes determining the confidence with which each potential explainer explains the data (k*c), in addition to the effort involved in the hypothesis assembly process.

Therefore the total effort in making one abductive decision is O(kc + PEIRCE),

16 where O(PEIRCE) is the complexity of the hypothesis assembly task (See the PEIRCE algorithm in Josephson & Josephson, Chapter 9, also included in the appendix)

Now consider the evaluation of a revision candidate, i.e. the examination of a suspect belief bi in B. This process consists of the forward trial of the k' plausible alternates for bi, which for a dynamic abducer includes the re-processing of all the observations received subsequent to the formation of bi. For each step of this reprocessing of the information in the light of the altered theory, the computational effort involved is O(kc + PEIRCE). Each of these trials could lead to conflicts that could in turn lead to triggering revision, bounded in the past by bi. One straightforward approach is to recursively evaluate these conflicts, abandoning them when anomaly resolution via this path is determined untenable, or when attempted resolution leads to implausible theories, and thus determine whether the substitution of any of the k' alternates for bi lead to the resolution of the latest conflict between T and on+1. Thus the effort involved in evaluating

(i-1) bi is BR(i) = (k'-1). k' . (kc + PEIRCE).

Therefore the computational cost of comprehensive belief revision, i.e., the evaluation of all possible revision candidates for the current conflict between T and on+1, and choosing the best revision, is given by

BR = ∑BR(i) = (k'-1). (kc + PEIRCE). (k'i – 1)/(k-1)

and this reduces to O(k' (n'+1)), ignoring the lower order terms5.

In the worst case, all the beliefs in the set B are suspect, and all alternate hypotheses in the comparative evaluation of each bi are plausible revision candidates; i.e., n= n' and k= k'. Therefore in the worst case the computational effort required is O(k(n+1)).

As just described, the exhaustive search strategy of brute-force enumeration of all possible revision candidates and their comparative, abductive evaluation in determining the best possible revision is clearly exponential and impractical. One way of addressing this issue would be to devise satisficing approaches that enable an abductive agent to handle the complex process of revising its beliefs, and I present, and evaluate, a repertoire of such schemes.

5 The complete expression for this complexity is (k'c + PEIRCE) (n'+1)

Chapter 3: The Abductive Approach to Multi-Object Tracking

3.1 Description of the ASAS Domain

The purpose of Smart-ASAS is to perform entity tracking and re-identification in an area under surveillance. The system works by analyzing the reports from an Area Of

Interest (AOI) populated with sensors, scouts etc., and by fusing the reports of sightings of different types of entities at different locations and times to form a consistent picture of the AOI. Legacy ASAS (Harmon and Webb, 1987) works by considering one report at a time and trying to associate it with previously sighted entities in that area based on time, location, and type reported in the new sighting. Thus, the fusion problem is already framed as one of abductive inference – the top-level question being „How best can this sighting be explained?‟ in terms of known entities or hypothesized new entities. In ASAS and Smart-ASAS, automated fusion takes place under human supervisory control.

The application may be enhanced by spatial reasoning about entities near a particular location, and by route or path planning. In my system, support for this spatial reasoning comes from the Diagrammatic Reasoning System (DRS). This is a domain-independent module that provides functionality to the abductive reasoner in support of generating, evaluating, and refining hypotheses. The DRS consists of “Perceptual Routines” that act

19 on diagrams (maps of the AOI in our system) and derive “symbolic” information, together with a “Problem Solver” that drives the routines and interfaces with the abduction engine (Chandrasekaran, et. al., 2005).

The abduction engine is a goal-directed problem solver that uses domain knowledge, according to a problem solving strategy, to determine a best explanation for the incoming information. This is done by setting up subgoals in the form of rival candidate hypotheses that are explored and evaluated and then selectively combined or independently chosen as the best explanation. Some of the novel attributes of the abduction engine in fusing information occur in the phases of critiquing hypotheses and in deciding about their acceptance. The evaluation of these hypotheses includes a limited form of prediction or expectation-based reasoning, which derives implications based on the assumption that the individual hypotheses are true, and scores the hypotheses depending on the failure or confirmation of the expectations. That is, the presence (or absence) of expected consequences of a hypothesis strengthens (or weakens) confidence in that hypothesis. In the case of highly ambiguous situations, the system delays making a decision and waits for sufficient information to confidently resolve the ambiguity. When incoming information is sufficiently anomalous, abductive reasoning identifies those parts of the belief system that plausibly cause the difficulty, and the abduction engine attempts to construct alternative plausible explanations to resolve the difficulty.

3.2 The Entity Re-identification Algorithm in Smart-ASAS

The main algorithm used in Smart-ASAS is best described by dividing it according to the main phases of abductive processing.

A. Determining what is to be explained. A report arrives asserting the presence of a

certain type of entity at a certain place and time. This entity, or more precisely,

this report, constitutes data to be explained.

B. Hypotheses generation. The abduction engine generates candidate hypotheses by

polling its database for entities of the same type known to be in the AOI, and by

hypothesizing that any such entities could have moved to the location of the new

report since their last sighting. It also considers the possibility that the newly

reported entity is a previously unseen one, and the possibility of the report being

false (due to a mistake or due to deception).

C. Hypotheses evaluation. The generated hypotheses are subjected to a series of

tests, each of which affects its confidence score.

1. First, impossible hypotheses are filtered out using a simple speed-distance-

time screen based on the last known locations of the entities and the times at

which they were last observed. This depends on knowledge of the maximum

speed of an entity of a given type.

2. Next, the abduction engine uses the DRS to determine, for each hypothesized

entity, whether or not there is a viable route from the location of its last

sighting to the location of the new sighting. Hypotheses without a viable route

are given very low confidence scores.

3. Finally, an observational consequences test is performed, where each of these

routes is checked to see if it crosses some sensor field in the area. If it does,

the DRS attempts to modify the route to determine if another route exists that

avoid the sensors but satisfies the length (i.e., time) constraint. If such a

modification is not possible, any crossed sensor field is queried to determine

whether an object of the type under consideration passed through it. The

confidence of the hypothesis is adjusted depending on the answer. If a path

cannot be modified to avoid a sensor field, and the sensor field did not report

an entity of that type at approximately the time in question, the confidence

score of the hypothesis is significantly degraded. However, it is known that

sensor fields are fallible, and hence a refinement of the hypothesis is possible

wherein the entity in question followed the route crossing the sensor field, and

the sensor field failed to report it. Since this is effectively the conjunction of

two hypotheses, one of which (sensor failure) is presumably a low probability

event, the overall confidence score of this hypothesis will be low.

During this process, a hierarchy of hypotheses may be dynamically created.

For example, a hypothesis of a specific known entity can take on the added

refinements (possibly more than one) of the route that it might have taken, and

one or more sensor fields crossed by that route that might have failed to report.

These three tests of hypothesis evaluation are instances of a more general

strategy according to which expectations of a hypothesis are analyzed (possibly

leading to other expectations) until they are confirmed or refuted by domain

constraints or by observables. The observational consequences test allows for

conclusions to be drawn from negative reports, i.e., evidence of absence may

sometimes be inferred from absence of evidence.

D. Hypothesis acceptance. Once the hypotheses have been evaluated, the confidence

scores are compared, and, if possible, a best explanation is selected. This is

subject to two conditions formulated as thresholds: the PLAUSIBILITY-

THRESHOLD, which is a user-defined value that must be exceeded by the

confidence score of a hypothesis for it to be considered to be plausible; and the

CLEAR-BEST-THRESHOLD, which is a user-defined value that must be

exceeded by difference of two confidence scores for one hypothesis to be 23

considered to be distinctly better than another. If there is a unique best

explanation, it is accepted, and becomes a belief6.

Every such acceptance decision is given a confidence score (not to be

confused with the confidence score of the hypothesis), which is a function of: how

plausible the hypothesis is, how decisively the hypothesis surpasses its nearest

rival, how many rival explanations remain plausible, and how exhaustive the

search was for alternative explanatory hypotheses. Acceptance decisions are

stored with their confidence scores in a reasoning trace to make them available for

subsequent reconsideration.

3.3 An Illustrative Example of Abductive Tracking and Error Correction in Smart-ASAS

I describe here a high-level walkthrough of the processing in Smart-ASAS that illustrates the functioning of the underlying tracking algorithm, and the error detection and correction in a simple situation. In appendix A I provide a more complicated example that showcases more intelligent behavior in the error correction strategy.

6 In the formulation of the problem just given, there is just one item of data to be explained, and possibly several contending hypotheses. However, a more general treatment of the problem would treat it as hypotheses assembly, where the conjunction of a subset of the generated hypotheses will be chosen to explain a finite set of findings. See the PEIRCE algorithm in Abductive Inference – Chapter 9. 24

3.3.1 Initial State

Figure 1. Intial State of the World

Problem solving in Smart-ASAS is begun with the situation as is shown in Figure

1. The portrayed map is the Area Of Interest (AOI), and the system has knowledge of two tanks, T1 at location l1, at time t1, and T2 at location l2, at time t2, on either side of a river. Assume that the river is un-crossable, except via the bridge. The system is also aware of a sensor field (S1) covering the entrance to the single bridge across the river, as shown in the diagram. To the best knowledge of Smart-ASAS, the sensor field S1 is functioning properly, but it is also known in general that sensor fields are fallible, albeit with a low probability of false negatives. A new sighting of a tank T3 (tentatively so

25 labeled) is reported at location l3 at time t3. Smart-ASAS is given the goal of explaining the new report.

3.3.2 Processing Sensor Reports - Generation, Evaluation and Acceptance of Hypotheses

The hypothesis-generation phase, described in section 3.2, constructs for the system, four rival hypotheses: T1, T2, Previously Unseen Entity, and Noise - any of which could explain the new report. Initial evaluation of these hypotheses by Smart-

ASAS establishes that both the T1 and T2 hypotheses pass the first test, the speed- distance-time screen, and are recognized for the time being as equally highly plausible explanations for the new report, while the other two are less so7. In the process of checking to see whether there are viable routes from the two entity locations to the location of the new report, the DRS determines that T1 has a straight-line path to the destination, while T2 would have had to follow a path across the river, and hence across the bridge, that entails crossing sensor field S1. Since sensor field S1 does not have any memory of a tank having crossed it in the time frame in question, the only way that T2 could explain the new report is by hypothesizing that S1 has malfunctioned. Thus, the hypothesis T2 is refined by the addition of the route it must have followed across the bridge, and the additional assumption that S1 must have malfunctioned. This is type of

7 For the sake of the example, these hypotheses are assumed to be less plausible than those of known entities having traveled to the location of the new report; this can be presumed to be based on sensor coverage in that region being good enough to prevent entities from moving in without being observed, etc. In particular, we assume that these hypotheses are below the PLAUSIBILITY-THRESHOLD described previously. 26 expectation-based processing we refer to as “The Dog Did Not Bark” reasoning8, wherein the failure of the predictions of a hypothesis (absence of report of T2 crossing S1) enables abductive inference to turn evidence against one hypothesis into a boost in confidence for its rivals. Since the malfunctioning of S1 is presumed to have low probability, the refined hypothesis is given a low plausibility score, thus concluding the evaluation phase for this observational report.

The hypotheses-acceptance phase now considers the candidate hypotheses, whose confidence scores still pass the PLAUSIBILITY-THRESHOLD, and chooses „T1 following a straight path to l3‟ as the best explanation for T3. That is, Smart-ASAS associates the new report with T1, and decides that the tentatively labeled T3 is the entity previously identified as T1, and that it has moved to location l3. This decision is based on the viability and confidence scores of the hypotheses. This abductive decision is given a confidence score that is modest, but not strong, since there are alternative plausible explanations for the data, though not very good ones. The estimated state of the world is then updated to show this acceptance, as depicted in Figure 2 (ignore Tank T4 for the moment).

8 The subtle but critical evidential importance of anomalously missing information is illustrated by the Sherlock Holmes story of Silver Blaze (Arthur Conan Doyle, 1892), where the key to the solution was that the dog did not bark, indicating that the thief must have been known to the watch dog.

“Is there any point to which you would wish to draw my attention?"

“To the curious incident of the dog in the night-time."

“The dog did nothing in the night-time."

“That was the curious incident," remarked Sherlock Holmes. 27

Figure 2. Decision D and Second Report

3.3.3 Revision of previous conclusions in Smart-ASAS

Assume that there now comes another report, that of a tank T4 at location l4 at time t4, as shown in Figure 2. However, this time, while the hypotheses-generation produces four rival explanatory hypotheses - T1, T2, Previously Unseen Entity and Noise

- the speed-distance-time screen filters out T1 and T2, since they could not have traveled the required distance in the available time, and just as in the previous case, Unseen Entity and Noise are not acceptable since their confidence scores are below the threshold of plausibility. This leaves the agent in an “abductive anomaly” - there is no plausible explanation for the incoming data. If there were no alternative, the agent would have to

28 suspend processing and request for more information, or ignore the report and proceed, or choose arbitrarily among the implausible hypotheses. However, it is plausible under the circumstances that a previous abductive decision was incorrect; so the agent decides to question its previous decisions. This constitutes, especially in systems deployed for a while, an extensive search for hypotheses among its previous processing, to explain the observed data, but it is undertaken only as required, since it is computationally expensive, and to begin with there seemed to be better explanations available. At a “meta” level, it is quite reasonable to extend the same approach of abductive inference to attempt to identify, and rectify, the cause for the anomaly. Meta-abductive reasoning considers the occurrence of the abductive anomaly as what needs to be explained, and hypotheses about mistakes in previous reasoning steps are among the potential explanatory hypotheses.

The abducer, on deciding to question its previous decisions, searches its reasoning trace to locate those decisions with low confidence scores. In this case there is only one: the decision that associated the report of T3 with tank T1. Let us label this decision D. It is left to determine whether any alteration of D would permit a better explanation for T4, and how plausible that alteration would be, in comparison with the current list of implausible hypotheses associated with this report (i.e. Unseen Entity, Noise, etc.).

As an attempt at repair, Smart-ASAS proceeds to re-evaluate confidences of the alternative hypotheses of decision D in the light of rejecting or suspending belief in D.

Recall that there is only the one: `T2 along a route over the bridge, and S1 has malfunctioned', since Previously Unseen Entity and Noise were below the plausibility threshold. The choice of this second-best hypothesis as a replacement to D (let us call this new decision D') involves associating tank T2 with the then new report, T3. D' proposes a specific alteration to a previous decision, substituting a second-best explanation for what was the best explanation at the first pass, amounting to a hypothesis that is worth pursuing further as a possible explanation of the abductive anomaly.

Decision D' entails the estimated world state shown in Figure 3. The correlation of tank T1 with the report T3 has been retracted, and T2 associated with it instead. This is thus the situation at which the report of tank T4 is received.

Figure 3. Decision D' and Third Report

Smart-ASAS is now left with the task of determining whether D' resolves the abductive anomaly. Therefore decision D' is tentatively accepted and the entity re- identification algorithm is rerun on all inputs received after decision D was made; in this case just the one report of tank T4. This time around, the system determines that T1, i.e.

„T1 along a straight line‟, is a plausible, even fairly confident, explanation for the report

T4, that is, that T1 has moved to l4 at t4; resulting from this hypothesis passing the tests of the re-identification algorithm and being chosen as significantly better than all alternatives at the acceptance phase (suppose). The system thus concludes that the anomaly introduced by the report T4 is resolved by accepting D' instead of D, and since

D' (along with the consequence of correlating T4 with T1) is sufficiently better than either of the implausible hypotheses available for explaining T4 (suppose), the decision is made to accept D', and to accept the identification of T4 with T1 that follows. The final view of the Area Of Interest is then the state depicted in Figure 4. Note that the system has inferred that the sensor has failed, as a part of its best explanation for the observational reports.

Figure 4. Final State Estimate

Chapter 4: The Smart-ASAS Testbed

In order to empirically explore the computational gain and potential pitfalls in error correction in dynamic abducers, I simulate the ASAS domain and allow the abducer to perform continual information fusion, with occasional belief revision, over reasonably protracted periods. The performance of various versions of the meta-abduction technique of error correction is then assessed along multiple dimensions of goodness. Biases toward any artificial features of the simulation are then minimized by varying certain control parameters, and repeating the simulations while introducing some degree of randomness.

The testbed aims to model the essential features of legacy ASAS by modeling it in the form of entities, their transitions, and some quasi-reliable estimate of the entities‟ current positions given their recent track histories. The simulation is modeled as the motion of a number of entities (say, ground-based) over a pre-defined Area Of Interest

(AOI), a specific fraction of whose surface (the AOI‟s) is populated by sensors that detect and report these entities passing through them. This motion of the entities is implemented as a random walk along cells in a grid (the AOI), and a predetermined percentage of the cells are randomly selected as sensors. The random walk is fashioned as each entity, in each time click, randomly selecting one of five choices of motion: to remain in the same

33 cell, move one cell over to either the one above, below, left, or right, ultimately constrained by the boundaries of the grid defining the AOI. The initial plausibility score

(see algorithm in section 3.2) given to a potential explainer of the sensor report, is estimated as the probability of any entity present in the knowledge-base of the agent having moved from its previous location to the location of the report in the time available to it since its last sighting. This is calculated as the ratio of the total number of paths that the entity could have followed to reach its destination to the total number of paths the entity could have taken in the time available. The initial plausibility estimate of an entity last seen at point P1(x1, y1) at time t1, and hypothesized as being at point P2(x2, y2) at time t2 is thus calculated as

The dynamic abducer then proceeds to follow the correlation algorithm outlined previously and chooses from among the candidate explainers that which it deems best.

The abductive inference that is drawn is then assigned a confidence score based on the existence of alternate explainers, and how decisively this explanation triumphs over others.

The testbed just described captures the critical aspects of legacy ASAS regarding a) motions of entities in the world, and b) some predictive model that may be used to estimate likelihoods of locations given some track history, without loss of generality. It is easy to see that given a more realistic model for entity movements (compared to a random walk), and a corresponding predictive model (possibly employing entity intentions, or based on features of the terrain), these can be incorporated into the testbed.

The performance of the various correlation-with-revision algorithms can be compared by measuring their performance over multiple runs, while varying the control parameters such as number of entities, density of sensors, etc.

4.1 The Class of Meta-Abductive Error-Correction Algorithms

The tradeoff involved in revising abductive belief systems is the optimization of two factors: (a) the adequacy of the best revision candidate (i.e. how well the new explanation resolves the anomaly, and the other observations), and (b) the computational cost. In the meta-abductive strategy, most of the computational expense is demanded by

36 the process of identifying plausible, suitable hypotheses whose revision, and subsequent reprocessing of future reports, resolves the anomaly. Therefore I rank the revision candidates based on the confidence scores of their corresponding abductive decisions, and then evaluate them. This ranking, coupled with some hypothetical “inexpensive” test for determining relevancy to the current anomaly would cut through a significant amount of the processing involved in exhaustive belief revision. The ordering of revision candidates by confidence also enables pruning away from the search list those beliefs that are either irrelevant or immutable (i.e. below a certain confidence threshold) with respect to the current anomaly. The aim here is to reduce the burden in the computational complexity of abductive belief revision (O(k' (n'+1))).

In the following subsections, I describe in detail the various methods of detecting and revising mistakes in the information fusion process, and submit multiple measures of performance across which I compare these techniques. I start with the extremely simple

“In Denial”, and the “Acknowledge Discrepancy” algorithms that set a baseline for performance, but perform entity tracking without revisions of belief states in case of doubt. Then follow the revision algorithms that attempt to achieve a better estimation of the reality underlying the observations.

4.1.1 The “In Denial” Algorithm

One simple way of ensuring progress in an abductive task is to just explain those observations that it can. It is always possible for a dynamic abducer to persistently ignore 37 information that it cannot explain, and proceed to attempt to account for subsequent input. In spite of its obvious drawbacks, i.e. the inconsistency of the representation with the underlying system, produced by effectively ignoring certain inputs, such a crude approach provides us with an initial benchmark from which to gauge more sophisticated reasoning behaviors. Also, this method has the virtue of being computationally inexpensive. In the case of a cognitive agent, this is tantamount to rejecting sensory information that conflicts with current beliefs.

The In Denial algorithm, tailored to the ASAS domain, is simply the Entity

Correlation Algorithm (section 3.2) that passes over sensor reports it cannot explain with its current belief system. This situation is recognized by the abducer as an abductive anomaly, by the inability of the Hypotheses Generation phase to formulate even one plausible explanation for a sensor report.

4.1.2 The “Acknowledge Discrepancy” Algorithm

In the Acknowledge Discrepancy algorithm, the dynamic abducer recognizes its inability to explain certain data, and acknowledges the inexplicable by assimilating it into its knowledge state, but does so without attempting to determine the cause for its inability. Operationally this is just a technique of infinitely deferring dealing with the intricacies of suitably altering the belief system. However the cost of this move is the risk incurred in expanding the belief system to include potentially conflicting pieces of information. 38

Tailored to the ASAS domain, the agent simply spawns a new track (with a new label), and associates this with the new report when it is unable to explain it (the new report) any other way. The abducer is not committed to interpreting it as an actual new entity distinct from the others in the database. A prudent operational measure is to periodically cull the belief system of stale reports in order not to clutter up the processing9. At first glance this appears to be more intelligent behavior compared to the previous algorithm; however, as I shall display and elaborate with the simulation results, this algorithm does not perform as well as the previous one according to the measures that were used.

4.1.3 The “Revision with Recency Heuristic” Algorithm

From (section 2.3) we see that the search space is of a high order exponent, and one avenue of investigation is to determine how effective some common-sense heuristics could be in aiding the computationally intensive correction strategy. The first approach I attempt is based on the conjecture that anomalies arise in a dynamic abducer due to errors in processing that occurred in the recent past in the time-ordered belief system of the abducer. The procedure is to identify as revision candidates, low confidence abductive decisions that occurred close in time to the anomaly-causing-report and to determine whether positing plausible alternative explainers, coupled with subsequent processing,

9 This may run the risk of eliminating actual entities and the precise threshold is determined based on trial and error to minimize this effect and to maintain a reasonable description of the AOI.

39 enables the construction of a belief system that that can account for what was previously inexplicable. I leave undefined for now what I mean by “low confidence” decisions, and at what point to curtail the search as unproductive, since these are specific to the domain.

4.1.4 The “Revision with Entrenchment Heuristic” Algorithm

The concept of “entrenchment”, which refers to some ordering, in terms of attachment and confidence, of the concepts or propositions in the human mind, is a notion that I borrow from human Epistemology/Psychology (Gärdenfors and Makison, 1988). I fashion a measure of entrenchment as a combination of the time elapsed since the abductive decision was made, and how consistent and confident the subsequent decisions that relate to it turned out to be. This produces an ordering (at least a partial one) of the revision candidates, which leads to the less entrenched decisions being considered before others as revision candidates. Also, in time-sensitive cases, it is reasonable to start processing candidates that do not cross a confidence threshold, and return a possible

“best-so-far answer” thus produced.

Exploiting dependencies between past beliefs in realizing degrees of entrenchment might seem contradictory, at first glance, since I have already made the case that no such dependencies exist in a purely abductive reasoner. The proposal actually involves re-estimation of the intrinsic confidence of the past beliefs in the light of subsequent evidence. Typically the abducer stores an abductive decision and the plausible alternative hypotheses with which it was comparatively evaluated in the 40 abduction process and the confidence of the abduction decision may be considered as its initial measure of entrenchment. It is quite possible that sensor information subsequently received actually alters the confidence of the decision, by virtue of eliminating one or more of its alternative explainers10. The elimination of alternate explainers is made possible by beliefs adopted subsequently by the abducer. Hypothetically, the confidence scores of earlier beliefs could be bolstered by the decision-making process by utilizing them in the adoption of later beliefs (i.e., the later beliefs are contingent on some of the earlier ones being true) or by making implausible (possibly by means of a mutual exclusion relation, or a softer version of it), some of the contending alternative explainers to earlier beliefs. These iterative re-estimations happen as the first stage of the belief revision process when the abducer searches for plausible revision candidates, and provide the required measure of entrenchment in the form of altered confidence scores for abductive decisions.

In a simpler version of this algorithm, we could use purely the confidence scores of the previous decisions, presumably in a pre-defined interval starting from the current anomaly and going back a certain number of decisions/time, without re-estimating these confidences, to order them in order of increasing entrenchment. Processing could then commence from the low-confidence revision alternates. A more sophisticated version is one in which the revision candidates are ordered after the confidence scores of the previous decisions are re-estimated in the light of subsequent information. For e.g.,

10 Of course the elimination itself has, associated with it, a measure of confidence 41 consider a previous decision with a confidence score „c‟, and three alternate explanations in addition to the chosen one. As previously described, the confidence score c is calculated as the initial probability of the leading explanation, normalized over the existing alternates. Now assume that subsequent decisions indicate that two of the alternate explanations could not have been plausible contenders. In this situation the confidence score c is re-estimated by normalizing the score over just the two remaining explanations. Except for this alteration in choosing the revision candidates, the remainder of the revision algorithm is the same as the previous one.

4.2 Performance Measures for the Error-Correction Algorithms

In order to quantitatively compare the performance of these error-correction algorithms, I define two measures that are indicative of the correctness of the algorithms for the ASAS domain – that is, of their ability to track intermittently sensed entities.

These two measures are - number of miscorrelations and the average length of a track.

(i) The number of miscorrelations: is the number of times an actual object

undergoes a change of label over an interval of time

(ii) The length of a track: is the stretch of time an actual object is associated with

the same label

It is possible, as the experimenter, to take these measures from simulation runs, because they provide access to the “ground truth”, as well as to the state being estimated by algorithms being tested, which has access only to sensor reports.

As mentioned earlier, from an operational perspective, there exists a tradeoff between improving the accuracy of the inferential steps, and the expenditure of resources to achieve this improvement. To gauge this I also measure the computational expense of these various strategies. I do this by treating the basic cycle of abductive tracking – determining what is to be explained, hypotheses generation, hypotheses evaluation, and hypotheses acceptance, as atomic11 and count the repetitions of this abductive cycle. This count signifies the measure of computational expense.

11 I describe in (section 2.3) that the time complexity of the atomic abductive inference step is linear in the number of alternate explainers.

Chapter 5: Experimental Results from the Simulated ASAS Domain

In this section I present the statistics obtained from attempting to continually fuse into a coherent whole, the sensor reports triggered by the simulated walks of entities in the ASAS domain. The control parameters in this experiment are the number of entities in a random walk in the area under surveillance, the fraction of the surface area covered by sensors, and the time period of the surveillance. The simulation is conducted on a

100×100 grid that represents the AOI. The performance measures presented here are from analyzing entity tracking and re-identification for a period of 20000 time-clicks. For each of the algorithms described in the previous chapter, I simulate random walks for 5,

10, and 15 entities for this time period. For each of these walks I also vary the density of sensors in the AOI from 10% coverage all the way to 100% in steps of 10%. I then summarize the performance of each algorithm by calculating the mean of the different performance measures across the various walks with different numbers of entities.

5.1 Performance of the Error-Correction Algorithms

The following charts show the performance of ASAS-Smart, the basic correlation algorithm, coupled with the various error-correction strategies described previously. I describe each performance measure in the following outline. For each algorithm (In

Denial, Acknowledge Discrepancy, Belief Revision with Recency, & Belief Revision with Entrenchment), I plot the performance measure obtained from each run of 5, 10, and

15 entities. These four charts are followed by a plot of the average measure obtained from each of these algorithms, in comparison to each other. Finally, I show a representative performance comparison of these algorithms for the run involving 10 entities.

60 In Denial Algorithm 50

30 5 Entities

Track LengthTrack 10 Entities 20 15 Entities

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

60 Acknowledge Discrepancy Algorithm 50

30 5 Entities

Track LengthTrack 10 Entities 20 15 Entities

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 5. Performance Measure - Average Track Length

Figure 5 continued

60 Revision With Recency 50

30 5 Entities

TrackLength 10 Entities 20 15 Entities

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

60 Revision With Entrenchment 50

30 5 Entities

Track LengthTrack 10 Entities 20 15 Entities

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 5 continued

45 Performance Measure - Average Track Length 40

30 In Denial 25 Acknowledge Discrepancy

20 TrackLength 15 Revision with Recency

10 Revision with 5 Entrenchment

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

18 Performance Measure - Average Track Length 16 for 10 Entities 14

12 In Denial 10 Acknowledge Discrepancy

8 Track LengthTrack 6 Revision with Recency

4 Revision with 2 Entrenchment

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

As stated, I define the average track length of a run as the average time a particular label is correctly associated with an entity. The phrase “correctly associated” indicates not just the amount of time the initial label is attached to the entity, it also includes the amount of time any subsequent label is rightly associated with an entity.

Figure 5 depicts this measure of correctness of the various algorithms calculated as the mean of various walks of different numbers of entities in the Area Of Interest. The results are easy to read and show a steady improvement in the correctness of the algorithms as the revision strategy gets more intelligent in the progression: Acknowledge Discrepancy,

Revision with Recency, and Revision with Entrenchment12.

12 The In Denial algorithm at first glance appears to perform better than Acknowledge Discrepancy algorithm. This is because abductive anomalies do not trigger label changes, which results in longer label associations than the other algorithms, and so this performance measure has a bias in favor of this algorithm. 49

Performance Measure - Number of Miscorrelations 40000

35000

30000

25000 In Denial 20000 Acknowledge 15000 Discrepancy Revision with

10000Miscorrelations # Recency Revision with 5000 Entrenchment

0 1 2 3 4 5 6 7 8 9 10 Sensor Density

Figure 6. Performance Measure - Number of Miscorrelations

I use a raw count of the number of miscorrelations, or the number of label changes incurred in tracking an entity as a measure of correctness of the tracking algorithms. As can be observed in Figure 6, similar to the algorithms performances‟ with the other measure of correctness, the behavior improves across the same three algorithms;

Acknowledge Discrepancy, Revision with Recency, and Revision with Entrenchment13.

Since this measure of performance seems counterintuitive at first glance, i.e., the number

13 In one instance (sensor density of 100%) the accuracy of Revision with Recency appears better than that of Revision with Entrenchment, causing the lines to cross over at the end of the plot. My best explanation for this behavior is that it is a statistical fluke caused by the random nature of the entities‟ movements. 50 of errors committed seems to increase when there is increased sensory coverage, I provide a normalized version of this performance measure in Figure 7. This normalized measure is the ratio of number of miscorrelations to the number of reports that the agent attempted to explain. Not surprisingly this ratio seems to improve (decrease) in the following order: In Denial, Belief Revision with Recency, and Belief Revision with

Entrenchment. The In Denial algorithm appears to perform well only until we recognize that in refusing to acknowledge the reports that it is unable to explain, it winds up making markedly lower number of correlation decisions and consequently fewer errors.

30% In Denial Algorithm 25%

20%

15% 5 Entities 10 Entities 10% 15 Entities

# Miscorrelations # / Decisions # 5%

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

60% Acknowledge Discrepancy Algorithm 50%

40%

30% 5 Entities 10 Entities 20% 15 Entities

# Miscorrelations # / Decisions # 10%

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 7. Performance Measure – Error Rate

Figure 7 continued

30% Revision with Recency Algorithm 25%

20%

15% 5 Entities 10 Entities 10% 15 Entities

# Miscorrelations # / Decisions # 5%

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

30% Revision with Entrenchment Algorithm 25%

20%

15% 5 Entities 10 Entities 10% 15 Entities

# Miscorrelations # / Decisions # 5%

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 7 continued

30% Average Error Rate 25%

20% In Denial

15% Acknowledge Discrepancy

10% Revision with Recency

# Miscorrelations # / Decisions # 5% Revision with Entrenchment

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

25% 10 Entities - Error Rate

20%

In Denial 15% Acknowledge Discrepancy 10% Revision with Recency

5% # Miscorrelations # / # Decisions Revision with Entrenchment

0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

Figure 8 depicts the efficiency of the different algorithms measured as the amount of processing expended on the individual runs. (Josephson 1995, and Josephson and

Josephson, (1994, 1996), Chapter 9) show that by redefining the problem of abductive reasoning to explain as confidently as possible the available data, the complexity of the simple abductive step of composing composite explanatory hypotheses is a low-order polynomial in time. Hence, if we consider the abductive correlative step as an atomic unit, the number of correlations attempted, as depicted on the Y-axis provides for a measure of cost. The cost of the two algorithms that do not attempt revision, the In Denial and the Acknowledge Discrepancy, are the same. The most expensive algorithm is the

Revision with Recency, while the cost of the revision algorithm that uses entrenchment as a heuristic, while still expensive, appears to be between these two extremes14.

350000 In Denial Algorithm 300000

250000

200000 5 Entities 150000 10 Entities

100000 15 Entities # Attempted Correlations# 50000

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

14 In the final chart in Figure 8, titled “10 Entities Efficiency”, the curves representing the computational expenditure of the “In Denial” and the “Acknowledge Discrepancy” algorithms overlap, and similarly the curves for “Revision with Recency” and “Revision with Entrenchment” overlap. 56

350000 Acknowledge Discrepancy Algorithm 300000

250000

200000 5 Entities 150000 10 Entities

100000 15 Entities # Attempted # Correlations

50000

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 8. Performance Measure – Efficiency

Figure 8 continued

1800000 Revision with Recency Algorithm 1600000

1400000

1200000

1000000 5 Entities 800000 10 Entities 600000 15 Entities

# Attempted # Correlations 400000

200000

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

1800000 Revision with Entrenchment Algorithm 1600000

1400000

1200000

1000000 5 Entities 800000 10 Entities 600000 15 Entities

# Attempted # Correlations 400000

200000

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 8 continued

800000.00 Average Efficiency 700000.00

600000.00 In Denial 500000.00

400000.00 Acknowledge Discrepancy

300000.00 Revision with Recency

200000.00 # Attempted Correlations# Revision with 100000.00 Entrenchment

0.00 10 20 30 40 50 60 70 80 90 100 Sensor Density

400000 10 Entities Efficiency 350000

300000 In Denial 250000

200000 Acknowledge Discrepancy

150000 Revision with Recency

100000 # Attempted Correlations# Revision with 50000 Entrenchment

0 10 20 30 40 50 60 70 80 90 100 Sensor Density

Actual Errors versus Observed - In Denial Algorithm 100% 90% 80% 70% 60% 50% Number of Abductive 40% Anomalies 30% Number of 20% Miscorrelations 10% 0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

Actual Errors versus Observed - Acknowledge Discrepancy Algorithm 100%

Number of Abductive Anomalies Number of Miscorrelations 100% 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 9. Actual Errors versus Observed Errors in different Revision Algorithms

Figure 9 continued

Actual Errors versus Observed - Revision with Recency Algorithm 100% 90% 80% 70% 60% 50% Number of Abductive 40% Anomalies 30% Number of 20% Miscorrelations 10% 0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

Actual Errors versus Observed - Revision with Entrenchment Algorithm 100%

80%

60% Number of Abductive 40% Anomalies Number of 20% Miscorrelations 0% 10 20 30 40 50 60 70 80 90 100 Sensor Density

Figure 9 represents the relative numbers of mistakes made in processing, versus the number of abductive anomalies encountered by the agent among the different algorithms. The significance of this comparison is explored in the next section.

Figures 10 and 11 represent the performance of the algorithms that attempt revision as a corrective strategy to rectify discrepancies in the belief system. The parameters that are used for comparison are the number of miscorrelations committed, the number of abductive anomalies encountered, the number of errors correctly identified, and the number of errors rectified. Again, I explore the implications of these statistics in the next section15.

15 Consistent with the information represented in Figure 9, the number of abductive anomalies is much lesser than the other measures and does not show significantly in Figures 10 and 11. 62

0.16 Belief Revision with Recency - 5 Entities 0.14

0.12 Number of Miscorrelations

0.1 Number of Abductive 0.08 Anomalies Number of Insoluble 0.06 Anomalies

0.04 Errors Correctly Identified Normalized#decsionswrt

0.02 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

0.25 Belief Revision with Recency - 10 Entities

0.2 Number of Miscorrelations

0.15 Number of Abductive Anomalies

0.1 Number of Insoluble Anomalies Errors Correctly Identified Normalized#decisionswrt 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 10. Performance of the BR with Recency Algorithm

Figure 10 continued

0.3 Belief Revision with Recency - 15 Entities 0.25 Number of Miscorrelations 0.2 Number of Abductive 0.15 Anomalies Number of Insoluble 0.1 Anomalies

Errors Correctly Identified Normalizedwrt #decisions 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

0.25 Belief Revision with Recency - Average

0.2 Number of Miscorrelations

0.15 Number of Abductive Anomalies

0.1 Number of Insoluble Anomalies Errors Correctly Identified Normalized#decisionswrt 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

0.16 Belief Revision with Entrenchment - 5 0.14 Entities 0.12 Number of Miscorrelations

0.1 Number of Abductive 0.08 Anomalies Number of Insoluble 0.06 Anomalies

0.04 Errors Correctly Identified Normalizedwrt #decsions

0.02 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

0.25 Belief Revision with Entrenchment - 10

0.2 Entities Number of Miscorrelations

0.15 Number of Abductive Anomalies

0.1 Number of Insoluble Anomalies Errors Correctly Identified Normalized#decisionswrt 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

continued

Figure 11. Performance of the BR with Entrenchment Algorithm

Figure 11 continued

0.3 Belief Revision with Entrenchment - 15 0.25 Entities Number of Miscorrelations 0.2 Number of Abductive 0.15 Anomalies Number of Insoluble 0.1 Anomalies

Errors Correctly Identified Normalizedwrt #decisions 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

0.25Belief Revision with Entrenchment - Average

0.2 Number of Miscorrelations

0.15 Number of Abductive Anomalies

0.1 Number of Insoluble Anomalies Errors Correctly Identified Normalized#decisionswrt 0.05 Errors Rectified 0 10 20 30 40 50 60 70 80 90 100 Sensor Density

5.2 Significant Lessons from the Simulation Results

The most prominent observation, as seen in Fig. 9, is that the number abductive anomalies encountered is significantly lower than the number of errors committed by the correlation algorithm. This is true across all versions of the error-correction algorithms.

This strongly suggests that abductive anomalies are not sensitive enough to detect a large fraction of the mistakes committed by the dynamic abducer. In section 2.1, I have outlined some ideas for a variety of more sensitive error detectors that ought to detect more mistakes by maintaining a vigil for abnormal patterns of inferential reasoning16.

Now ideally, we would prefer to detect every miscorrelation and attempt to repair it, however it is possible that this is too ambitious a goal. Consequently, it might be constructive to envisage more easily attainable operational standards of reasoning; such as a dynamic abducer that detects and strives to correct obvious errors in reasoning, but is robust enough to recover and continue processing in the presence of undetectable or uncorrectable errors.

The second striking observation, from Figures 5, 6, and 7, is the performance of the two revision algorithms compared with the other two baseline algorithms. There appears to be limited improvement in performance, despite the expenditure of significant computational resources, and this is apparent across both measures – track length and

16 Nevertheless, this leads to the intriguing speculation that in a large number of real-world situations, agents might perform much as the dynamic abducer, and sail along blissfully unaware of past mistakes. 67 number of miscorrelations. On the assumption that these are reasonable metrics for correctness, we are led to the conclusion that, while considerable effort may be required to revise the belief system in response to abductive anomalies, these revisions quite often do succeed. The revisions are quite often (undetectably) incorrect. Figures 10 and 11 clearly show the limited success of the revision algorithms to converge on the underlying reality. Evidently, there often exist a number of plausible alternative explanations that will do the job of removing (albeit incorrectly) the anomalies, with no effective way to distinguish between them. It is an empirical question, given the goals of the user, whether such costly and limitedly effective revision is a desirable in any domain17.

17 Of course, if we were to adopt our new definition for success – the reasoner that attempts to correct only gross errors in reasoning, the performance does not remain that bleak. 68

Chapter 6: Summary and Discussion

I have described the workings of a dynamic abducer, an abductive reasoning agent that continually fuses information from multiple sources to “maintain situation awareness,” that is, to maintain a best estimate of the situation based on incoming data.

In particular, I have described a special class of problems of entity tracking and re- identification, and described in some detail how a dynamic abducer for this problem is able, under some circumstances, to appropriately “change its mind,” and explicitly revise the conclusion of an earlier reasoning step. This ability is achieved by performing “meta- abductive” reasoning in situations where no plausible hypotheses are available for explaining a new observation report. I have also described Smart-ASAS, an implementation of such a dynamic abducer, and illustrated its capabilities by tracing its reasoning on example cases. Although the present system is in many ways simple, idealized, and domain-specific, it illustrates principles of reasoning that may have broad applicability.

Certain strategies for static abduction, where the abducer is required to construct/identify a set of causes (or a composite hypothesis) that led to (account for) the occurrence of a fixed set of observations (effects), consider observations sequentially and incrementally construct a composite explanation for the observations. A solution is

69 produced for a subset of observations; and then additional observations are considered. A solution that explains the previous and current observations in constructed, and the process is repeated until all observations are explained, or until some other termination condition is reached, such as being unable to proceed without “guessing”. While the discussion in this document is limited to dynamic abducers, it seems reasonable to posit that these sequential strategies could be subject to the same process of meta-abductive revision in the event of errors, and would benefit from similar corrective procedures.

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Appendix A: An Illustrative Example of Abductive Tracking and Error Correction in Smart-ASAS with Alternative Revisions

Problem Setup

Figure 12. Initial State of the World

In the current scenario, Figure 12, assume that the database contains tanks T1, T2, and T3, at location l1, l2, and l3, at times t1, t2, and t3 respectively. There exist two bridges across the river, each of them monitored by one of the sensor fields S1 and S2.

Among these two sensor fields, let us posit that one of them, say S1, is more reliable than

75 the other. A new sighting of a tank T4 (tentatively so labeled) is reported at location l3 at time t3. Smart-ASAS is tasked with trying to explain this and subsequent reports.

Processing a Series of Sensor Reports

As in the first example, Smart-ASAS has among its potential explainers for report

T4, tanks T1 and T2 that survive the speed-distance-time filter. Among these, T2 is adjudged to be the best explanation since the hypotheses T1 would require further refinement of the sort „Sensor S1 malfunctioned‟ that would prove detrimental to its overall plausibility. Therefore report T4 is correlated with tank T2, resulting in the following estimated world state, Figure 13. This is decision D1. Similar to the first example, D1 is given a modest confidence score since there is at least one plausible alternative hypothesis that would if true, explain the data.

Figure 13. Decision D1

The next report received by Smart-ASAS is that of a tank T5 (tentatively so labeled). As with the previous report, T1 and T3 are among the possible explainers that survive the speed-distance-time screen, and by the same reasoning, Smart-ASAS concludes that report T5 is actually tank T3 having moved to l5 (Figure 14). This is decision D2 and its confidence score is the same as D1.

Figure 14. Decision D2

The third report is that of tank T6, at location l6, at time t6. On this occasion let us suppose that tank T3 (at location l6, at time t6) is the only candidate that survives the speed-distance-time screen. Tank T3 is thus correlated with the report T6

77 and thus its location and time are updated to l6 and t6 respectively (decision D3). This belief is recognized as one with high confidence since it does not have alternative plausible hypotheses. This leads to the world state in figure 15.

Figure 15. Decision D3

The agent runs into a snag with the next report, tank T7, as there are no hypotheses that pass the speed-distance-time filter and let us also assume that the other hypotheses such as Previously Unseen Entity, Noise, etc. are also below the threshold of plausibility. Again, we face an abductive anomaly - there is no plausible explanation for the incoming data.

On scanning its reasoning trace for low confidence decisions to consider as possible revision candidates, the agent finds two of them – D1 and D2, and since they

78 have the same (low) degree of confidence, proceeds to determine whether altering either of them would resolve the anomaly. Smart-ASAS discovers that the retraction of either candidate followed by acceptance of their next most confident hypothesis would help resolve the anomaly. The revision of D1 (Tank T2 is correlated with report T4), by positing its alternative (Tank T1 is correlated with report T4 and Sensor S1 is malfunctioning) leads to the world state in Figure 16. Let us call this the first revision.

Figure 16. First Revision Option

The revision of the other candidate, decision D2 (Tank T3 is correlated with report T4), leading to Tank T1 being correlated with report T5 and the additional assertion that Sensor S2 is malfunctioning, results in the world state shown in Figure 17.

This is second plausible revision.

Figure 17. Second Revision Option

On the face of it these are two equally desirable revisions (mutually incompatible, of course) and either of these series of decisions is preferable to the original, initially more plausible, track history. Smart-ASAS now faces the question of which of these to choose. Recall that during set up of the problem I claimed that the reliability of Sensor S1 was higher than that of S2. Now that information may be used to determine that the second revision has a higher overall plausibility, since among the extra commitments these two revisions have to make, i.e. malfunctioning sensors, the first revision has a more far-fetched addendum in its history. Provided the difference is adequate, Smart-

ASAS would then choose the second revision as better and conclude that its best estimate of the world is as shown in Figure 17.

Appendix B: The PEIRCE Hypothesis Assembly Algorithm

The Peirce algorithm presupposes that there is a means of generating or obtaining the findings for the case, and a means of generating hypotheses to explain the findings.

This also means that the generated hypotheses must have some associated scoring of their plausibility based on the current situation, and a list of findings that each can explain.

Further, Peirce allows for already explained components to be inserted into the composite automatically. Therefore, some work in solving the problem needs to have already taken place before Peirce begins.

The steps of the algorithm are:

generate hypotheses (with their score) and generate or obtain findings

start the composite with any hypotheses which have been predetermined to

be in the composite (this can be set up at run time when it is discovered

that certain hypotheses should be included in this case, or set up by the

tool user when it is determined that certain hypotheses should always be

included or by the system user interactively while he/she is exploring

alternative hypotheses).

82 expand expectations from a higher level in the abductive process (this means that some higher level knowledge has implied either positively or negatively for some hypotheses). The expectations will cause the hypotheses in question to be rescored by taking their current score and adapting it to reject the positive or negative interactions.

propagate the effects of hypotheses being accepted into the composite.

This may rule out other hypotheses incompatible to those in the composite or it may alter scores of other hypotheses which may be implied or causally connected to hypotheses in the composite.

Loop on the following until either all findings are accounted for or until no more progress is made in the explanatory coverage

o find all confirmed hypotheses and include them in the composite.

A confirmed hypothesis is one which receives the highest

confidence score possible and is assumed to be present in the case

(this is an optional feature which can be turned off if confirmed

hypotheses should not be automatically included in the composite).

83 o if confirmed hypotheses are found then propagate the effects of the

latest additions and go back to loop beginning, else continue.

o find all essential hypotheses and add them to the composite. An

essential hypothesis is one that is the only possible hypothesis to

explain a finding (i.e. no other plausible hypotheses can account

for a finding).

o if essential hypotheses are found, then propagate the effects of

their inclusion into the composite and go back to loop beginning,

else continue.

o find all clear best hypotheses. In this case, a clear best hypothesis

is one which explains some finding better than any other

hypothesis. To be a clear best, the hypothesis must have a score

higher than a given threshold and must surpass all other hypotheses

by a given distance (thresholds are given by the tool user at the

time the system is build, however, they can be easily modified

during or between cases. There are defaults if no thresholds are

specified).

o if clear best hypotheses are found, then propagate the effects of

their inclusion into the composite and go back to loop beginning,

else continue.

o find all of the weak bests hypotheses. Here, we may relax the

conditions on the clear bests criteria mentioned above. This step is

optional.

o if weak best hypotheses are found, then propagate the effects of

their inclusion into the composite and go back to loop beginning,

else continue.

end loop

if there are still some unaccounted findings, attempt to guess out of the remaining hypotheses which have not been ruled out. Guessing is accomplished by letting the unexplained findings vote on hypotheses which they are most likely to be explained by. This allows hypotheses which explain more to have better chances of being chosen than hypotheses which explain less.

if any guessed hypotheses are found, then propagate the effects of their

inclusion into the composite and go back to the loop beginning, else end.

At this point, either all findings should be accounted for, or there are no more hypotheses available to explain findings, or the only remaining hypotheses are too close in plausibility and explanatory coverage to allow for a better explanation. In the last case, the unexplained findings have no way to discriminate with significant confidence between alternative explainers and so no decision is made. In such a case where unexplained findings cannot discriminate between hypotheses, guessing can take place.

This is a last resort attempt to explain some unexplained findings but a complete coverage can be achieved as long as there exist hypotheses to explain the remaining findings.

It should be noted that each pass through the loop part of the algorithm is considered relative to the previous passes. This means, for example, that a hypothesis which is considered essential because a competing hypothesis is ruled out due to it being incompatible to a clear best is only an essential hypothesis relative to clear bests. Thus, an essential from the first pass through the loop is more confidently an essential than an essential which is relative to a clear best. Similarly, any newly included hypothesis which is relative to some guessing (that is, a hypothesis is included due to the effects of the inclusion of a guessed hypothesis) must be regarded as less confident than any

86 hypotheses included before guessing took place. So, each pass through the loop in the algorithm further limits the system's confidence in any new hypothesis included into the composite as it is included relative to whatever previous propagations have occurred due to previously included hypotheses. Hypotheses may be confirmed, essential, clear bests, weak bests, disbelieved (due to incompatibility), guessed, or ruled-out because of a low confidence rating, and these statuses may be relative to confirmeds, essentials, clear bests, weak bests, disbelieved or guessed hypotheses.