Behavioural Dynamics

1
Behavioural Dynamics

• Analysis of the Dynamics of Diagnostic Reasoning
  Tasks
• Part C, Chapter 2

2
Overview

• Analysis of dynamics of reasoning
• Dynamic properties at different levels
• Diagnostic Reasoning (by assumption)
• Simulation
• Human traces
• Checking
• Relationships between dynamic properties

3
Reasoning Trace

• Reasoning state
• characterising state properties
• Reasoning step
• a transition from one reasoning state to the
  other
• Reasoning trace
• a sequence or trajectory of reasoning states over
  time

4
Characterizing Reasoning

• Dynamic properties
• local for elementary reasoning steps
• intermediate for chunks of the process
• global whole reasoning process

5
Reasoning by Assumption Example 1

• Suppose I do not take my umbrella with me.
• Then, if it starts raining at 5 pm, I will get
  wet,
• which I dont want. Therefore I'd better take my
  umbrella with me.

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Reasoning by Assumption Example 1

• Suppose I do not take my umbrella with me.
• Then, if it starts raining at 5 pm, I will get
  wet,
• which I dont want. Therefore I'd better take my
  umbrella with me.

What are the different states in this trace?
7
Example 1 in states

• Suppose I do not take my umbrella with me. Then,
  if it starts raining at 5 pm, I will get wet,
  which I dont want. Therefore I'd better take my
  umbrella with me.

• suppose not umbrella
• (assumed fact)
• if it starts raining
• (assumed fact)
• I will get wet
• (implied fact)
• which I dont want
• (evaluation ? rejection)
• take umbrella
• (assumed fact)

8
Wise Persons Puzzle Example 2

• Two wise persons, A and B, each wear a hat.
• Each hat is either black or white but at least
  one of the hats is white.
• Each wise person can only observe the colour of
  the other wise person's hat.
• Both wise persons are able to reason logically
  and they know this from each other.

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Wise Persons Puzzle Example 2

• Suppose I am wearing a black hat, then he would
  know the solution. But if I ask, he says he
  doesnt know the solution. Therefore I am not
  wearing a black hat.

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Example 2 in states

• Suppose I am wearing a black hat, then he would
  know the solution. But if I ask, he says he
  doesnt know the solution. Therefore I am not
  wearing a black hat.

• suppose I black hat
• (assumed fact)
• he would know solution
• (implied fact)
• he doesnt know solution
• (observed fact)
• but
• (evaluation ? rejection)
• Im not wearing a black hat
• (assumed fact)

11
Reasoning by Assumption Example 3

• Suppose the battery of my car is empty, then the
  lights wont work.
• But if I try, the lights turn out to work.
• Therefore the battery is not empty.

12
Example 3 in states

• Suppose the battery of my car is empty, then the
  lights wont work. But if I try, the lights turn
  out to work.
• Therefore the battery is not empty.

• suppose battery empty
• (assumed fact)
• lights wont work
• (implied fact)
• lights work
• (observed fact)
• but
• (evaluation ? rejection)
• battery is not empty
• (assumed fact)

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Example 3 in steps

• Making assumptions
• Deriving implications
• Making observation
• Evaluating
• Concluding (new assumption)

• suppose battery empty
• (assumed fact)
• lights wont work
• (implied fact)
• lights work
• (observed fact)
• but
• (evaluation ? rejection)
• battery is not empty
• (assumed fact)

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Reasoning by Assumption General Pattern

• Reasoning step Reasoning state
• start initial reasoning state
• an assumption is made assumed fact
• logical implications of this fact are
• derived e.g. by modus ponens implied fact
• these implications are evaluated observed fact
• against information from other
• sources e.g., observation results evaluated
  assumption
• contradicting implied fact ?
• retract assumption and implied fact rejected
  assumption
• make opposite assumption assumed fact

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Example Reasoning Trace State 0
Could be car does not start
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Example Reasoning Trace State 1

• assumed battery empty

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Example Reasoning Trace State 2

• assumed battery empty

derived lights wont work
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Example Reasoning Trace State 3

• assumed battery empty

derived lights wont work
observed lights work
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Example Reasoning Trace State 4

• assumed battery empty

derived lights wont work
observed lights work
rejected assumption battery empty
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Example Reasoning Trace State 5
observed lights work
rejected assumption battery empty
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Example Reasoning Trace State 6
observed lights work
rejected assumption battery empty
assumed battery not empty, sparkling
plugs problem
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State Properties

• assumed(I, S)
• rejected(I, S)
• observation_result(I, S)
• holds_in_world(I, S)
• Examples
• assumed(battery_empty, pos)
• holds_in_world(car_starts, neg)

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Global Properties

• Termination of reasoning
• Correctness of rejection
• Completeness of rejection
• Guaranteed outcome
• Persistence of the world state
• Conditional persistence of assumptions and
  predictions
• Rejection of non-intended world situations

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GP1 Termination of Reasoning

• Informal
• version a) Termination of reasoning
• version b) After some point in time no more
  changes occur in the reasoning state.
• Semi-formal After some point in time t, all
  reasoning states are equal to the reasoning state
  at time t.

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GP1 Termination of Reasoning

• Formal
• ?? TRACE ?tT ?t T
• t gt t ? state(?, t) state(?, t)
• Useful abbreviation
• termination(?, t) ?
• ? t T tgtt ? state(?, t) state(?, t)

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GP2 Correctness of Rejection

• Semi-formal
• In all traces, everything that has been rejected
  does not hold in the world situation.
• Formal
• ?? TRACE ?t T ?A INFO_ELEMENT ?S SIGN
  state(?, t) rejected(A, S) ?
• state(?, t) not holds_in_world(A, S)

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GP3 Completeness of Rejection

• Informal
• After termination, all assumptions that have not
  been rejected hold in the world situation.
• Formal
• ?? TRACE ?t T ?A INFO_ELEMENT ?S SIGN
  termination(?, t)
• state(?, t) assumed(A, S)
• state(?, t) / rejected(A, S) ?
• state(?, t) holds_in_world(A, S)

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GP4 Guaranteed Outcome

• Informal
• In normal cases, after termination there is at
  least one assumption that is not rejected.
• Formal
• ?? TRACE ?t T
• termination(?, t) iws(state(?, t)) ?
• ?A INFO_ELEMENT ?S SIGN
• state(?, t) assumed(A, S)
• state(?, t) / rejected(A, S)

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Local Properties

• LP1 Observation Result Correctness
• LP2 Assumption Effectiveness
• LP3 Prediction Effectiveness
• LP4 Observation Initiation Effectiveness
• LP5 Observation Result Effectiveness
• LP6 Evaluation Effectiveness
• LP7 Rejection Grounding
• LP8 No Assumption Repetition
• LP9 Rejection Effectiveness
• LP10 Rejection Correctness
• LP11 Assumption Uniqueness

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LP1 Observation result correctness

• Informal
• Observations that are obtained from the world,
  indeed hold in the world.
• Formal
• ?? TRACE ?t T ?A INFO_ELEMENT ?SSIGN
• state(?, t) observation_result(A, S) ?
• state(?, t) holds_in_world(A, S)

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LP2 Assumption Effectiveness

• Informal
• In normal cases and if possible, new assumptions
  will be generated for as long as all assumptions
  made in the past have been rejected.

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LP2 (formalisation)

• Semi-formal
• if the world state is an intended world state
• then as long as there are possible assumptions
  that
• have not been rejected
• and as long as all assumptions that have been
• made (in the past) have been
• rejected,
• the agent will keep generating new assumptions

33
LP2 (formalisation cont.)

• as long as there are information elements A and
  signs S such that it is possible to assume that A
  has sign S and the assumption that A has sign S
  has not been rejected in this trace ? at this
  point in time t.
• Formalisation (and abbreviation)
• possible_unrejected(?, t) ?
• ?A INFO_ELEMENT ?S SIGN
• possible_assumption(A, S)
• state(?,t) / rejected(A, S)

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LP2 (formalisation)

• Semi-formal
• if the world state is an intended world state
• then as long as there are possible assumptions
  that
• have not been rejected
• and as long as all assumptions that have been
• made (in the past) have been
• rejected,
• the agent will keep generating new assumptions

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LP2 (formalisation cont.)

• ... all assumptions that have been made
• (in the past) have been rejected
• Formalisation (and abbreviation)
• all_previous_rejected(?, t) ?
• ?A INFO_ELEMENT ?S SIGN ?t T
• t ? t
• state(?,t) assumed(A, S) ?
• state(?,t) rejected(A, S)

36
LP2 (formalisation)

• Semi-formal
• if the world state is an intended world state
• then as long as there are possible assumptions
  that
• have not been rejected
• and as long as all assumptions that have been
• made (in the past) have been
• rejected,
• the agent will keep generating new assumptions

37
LP2 (formalisation cont.)

• ... the agent will keep generating new
  assumptions
• Formalisation (and abbreviation)
• new_assumption(?, t) ?
• ?t T ?A INFO_ELEMENT ?S SIGN
• t gt t
• state(?,t) assumed(A, S)
• state(?,t) / rejected(A, S)

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LP2 (formalisation cont.)

• ?? TRACE ?t T
• iws(state(?, t)) ?
• possible_unrejected(?, t)
• all_previous_rejected(?, t) ?
• new_assumption(?, t)

• if the world state is an intended world state
  then
• as long as there are possible assumptions that
• have not been rejected and
• as long as all assumptions that have been made
  (in the past) have been rejected,
• the agent will keep generating new assumptions

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Dynamic Properties Step Property (LP3)

• For each assumption the agent will derive all
  possible implied predictions about the observable
  part of the world state.

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Dynamic Properties Step Property (LP4)

• All predictions made will be observed.

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Dynamic Properties Step Property (LP5)

• If an observation is made the appropriate,
  correct observation result will be received.

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Dynamic Properties Step Property (LP6)

• Each assumption for which there is a derived
  prediction that does not match the corresponding
  observation result will be rejected.

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Simulation Results
44
Experiments

• Car Diagnosis
• DESIRE
• Leads-To
• Wise Persons Puzzle
• DESIRE
• Leads-To
• Human test

45
Experiment Wise Persons Puzzle

• Two wise persons, A and B, each wear a hat.
• Each hat is either black or white but at least
  one of the hats is white.
• Each wise person can only observe the colour of
  the other wise person's hat.
• Both wise persons are able to reason logically
  and they know this from each other.

46
Human ResultsProtocol for White-White

• a. observation_result(hat_colour(other, white),
  pos) / A is wearing a white hat.
• observation_result(conclusion(dont_know_my_co
  lour, other), pos) / A does not know that he is
  wearing a white hat.
• b. assumed(hat_colour(white, self), neg) / 1.
  A sees either a white hat or a black hat of B.
  2. If he sees a black hat of B
• c. predicted(conclusion(my_hat_is_white, other),
  pos) / 3. then he knows that he wears a white
  one 4. and then he also knows what colour he
  wears 5. that is the white one,

47
Human ResultsProtocol for White-White

• d. rejected(hat_colour(white, self), neg) /
  6. so in that case he doesn't answer "I don't
  know"
• e. assumed(hat_colour(white, self), pos) / 7.
  so A must see a white hat
• f. predicted(conclusion(dont_know_my_hat_colour
  , other), pos) / 8. then he doesn't know
  9. since there can be two white hats involved
  10. it can be also the case that A wears a
  black hat 11. and B a white hat
• 12. so A doesn't know what hat he is wearing

48
Human ResultsProtocol for White-White

• g. assumed(hat_colour(white, self), pos)
• / 13. and that means that A, as I mentioned
  before, must have seen a white hat,
  14. so B can conclude, after A's answer that
  he is wearing a white hat.

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Checking Properties
REASONING TRACES
GLOBAL PROPERTY
TTL CHECKER
50
Relationships Between Dynamic Properties

• GP2 - In all traces, everything that has been
  rejected does not hold in the world situation.

IP1 - If an assumption is rejected, then earlier
on there was a prediction for it that did not
match the corresponding observation result.
IP2 - If a prediction does not match the
corresponding observation result, then the
associated assumption does not hold in the world.
WP1 - If something holds in the world, it will
hold forever.
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Relationships Between Dynamic Properties

• IP1 IP2 WP1 ? GP2
• IP3 IP4 ? IP2
• IP5 WP1 ? IP3
• IP6 WP6 ? IP4
• IP7 ? IP6
• LP3g DK1 ? IP7
• LP6g ? IP1
• LP5g ? IP5

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Relationships Between Dynamic Properties
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Analysis of reasoning processes