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Justificationbased TMSs JTMS

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Title: Justificationbased TMSs JTMS

1
Justification-based TMSs (JTMS)

JTMS utilizes 3 types of nodes, where each node
is associated with an
assertion
Premises. Their justifications (provided by the
IE) have no antecedents.
Contradictions. They are no different from other
nodes except that the IE has explicitly
designated them as contradictions. It a
contradiction node becomes believed, JTMS must
signal the IE and the IE must assure that the
contradiction node is no longer believed.
Assumptions. These are also designated by the IE.
An assumption is enabled if the IE has instructed
the JTMS to believe it. If an assumption node has
a valid justification, the it is treated as a
"regular" node.
In the dependency networks premise nodes are
marked as
contradiction nodes are marked as
assumption nodes are marked as

2
JTMS node's labels

Nodes can be either IN or OUR, where
IN means "believed".
OUT means "not believed"
Consider the following 4 cases
N1 is "IN"
N1 is "OUT"
(not N1) is "IN" Contradiction (not
N1) is true
(not N1) is "OUT" N1 is true
Unknown
Note A node being "IN" does not mean a node is
"true".

3
JTMS justifications

Example justification
Here N1 and N2 are the antecedents of
justification, J1, and N3 is the
consequent. The informant is ignored or it may
record information from the
external systems.
A justification is valid if its antecedents are
IN.

N1
N3
J1
N2
4
Propositional specification of a JTMS

JTMS is defined by the following 2 sets
The set of enabled assumptions, A.
The set of justifications, J.
The set of justifications grows monotonically,
because justifications cannot be
removed. Justifications are Horn clauses ,
therefore we can think of them as
implications in PL.
The monotonicity property suggests that if KB1 ?
?, then KB1 ? KB2 ? ?.
However, KB2 may be inconsistent with KB1 and the
set of beliefs, KB1 ? KB2,
will be contradictory. On the other hand, the
monotonicity of J makes justifications
local, I.e. they depend only on their
antecedents, which ensures their processing
in polynomial time.

5
Propositional specification of a JTMS (cont.)

JTMS nodes are propositional symbols. They form
the JTMS belief set, Bel.
Set A may grow and shrink.
Retraction of enabled assumptions
causes the complexity of JTMS
algorithms.
The fundamental task of the JTMS is to answer
queries about whether a given
node holds (is "IN") in a given set of beliefs
and justifications. JTMS classified
node, N, as "IN" iff
The node is an enabled assumption or a premise.
Bel ? J --PL N
Otherwise, node N is "OUT".

Set of Enabled Assumptions, A.
Belief set, Bel.
6
Example of JTMS work how JTMS helps the IE to
improve its efficiency

Consider the example on Figure 7.2 in the
textbook.

IN
OUT
IN
IN
IN
Computation module
OUT
OUT
IN
G
OUT
OUT
IN
IN
Computation module
IN
IN
7
Example (cont.)

JTMS maintains the records of previous IE work,
which is why there is no need
to re-compute the label for G in the third case.
It has already been computed in
case (a). That is, it is already known that if D
and F and IN, G will be IN.
Changing the JTMS state can be caused by
Adding a new assertion (enabled assumption or
premise) or a justification.
Retracting an enabled assumption.
Whenever the IE changes the JTMS state, the later
must
Identify the change exactly.
Carry as much work forward as it is logically
possible.
Next, we consider possible changes of the JTMS
state in more detail.

8
Adding information

There are three ways to add information to the
JTMS state
Add a justification Check whether the consequent
of the justification is IN. If yes, do nothing.
If no, check if the justification is valid (that
is, its antecedents are IN) and if yes, make the
new justification the supporting justification
for the consequent.
Enable an assumption
If the node was not an assumption and was IN (for
which it must have had a valid justification),
remove its valid support and mark it as an
enabled assumption.
Check if this assumption was IN. If yes, do
nothing.
Check any justification where this assumption is
an antecedent to see if it now becomes valid and
if yes change the status of the consequent.
Declare a premise.
In all these cases, the JTMS must
Create a new node or justification, if necessary.
Propagate the consequences (run the
"Propagate-Inness" algorithm).

9
Propagating Inness example

Consider the following dependency network

OUT
IN
J1
J2
J3
10
Propagating Inness example (cont.)

Let A becomes an enabled assumption, The network
changes as follows
Note that we only change nodes from OUT to IN.
Because there is a finite
number of nodes, this process must terminate at
some point. See book, page
183 for another example.

11
Retracting information

The only nodes that can be retracted are
assumption nodes. Premises and
justifications are never retracted (recall that
justifications are records of the IE's
work, and premises by definition must always be
true).
To retract an assumption node
Make the assumption node OUT.
Retract all nodes that have it as an antecedent
(run Propagate-Outness algorithm).
Check all nodes that became OUT as a result of
the previous step to see if they have an
alternative support. For those that do have an
alternative valid justification, make them IN and
run Propagate-Inness algorithm.

12
Propagating Outness example

Consider the example network
Let C be
retracted, i.e. its
status changes
from IN to
OUT.
The result of
this change
E becomes OUT
and
F becomes
OUT.

13
Propagating Outness example (cont.)

Consider following modified version of the
example network
Let C be
retracted, i.e. its
status changes
from IN to
OUT.
The result of
this change
J2 is
invalidated, however
F is still IN
making J4 an
alternative
support
justification
for E now J3
remains valid
because of
E is IN via
J4 gt the result
E is IN,
because F is IN,
and F is IN
because E is IN.
That is, we have an undesirable circularity. To
ensure that this is not going to
happen, we require that each belief marked IN has
a well-founded support.

J2
IN
E
J4
J3
F
IN
14
Well-founded support

Recall that the main task of the JTMS is to
answer queries about whether a
node is IN or OUT. The second responsibility of a
JTMS is to provide a well-
founded explanation as to why a node is IN or
OUT. The notion of a well-
founded explanation relies on the following
definition
A well-founded support for node Ni is a sequence
of justifications J1, , Jk
such that
Jk is the supporting justification for Ni.
All of the antecedents of Jk are justified
earlier in the sequence J1, , Jk-1.
No node has more than one justification in the
sequence.
Note that a node may have more than one valid
justification, therefore more
than one well-founded support. JTMS computes only
one well-founded support
for a node.

15
Well-founded support example

Consider the following network
Here neither A nor B have a well-founded support
because their antecedents
are not justified earlier in the sequence as
required. What we see here is a
case of circularity, which is a highly
undesirable property of a belief set.

IN
IN
J1
J2
16
Detecting contradictions

Consider the following network
When a contradiction node becomes IN, the JTMS
must
Find underlying assumptions.
Resolve the contradiction between the underlying
assumptions by automatically retracting one of
them, or asking an external system (IE or human
user) for help.

IN
This node is explicitly defined as a
contradiction node.
IN
IN
IN
IN
17
How JTMS simulates default reasoning

Consider the following default rules
MA --gt A
MB --gt B
MC --gt C
A B --gt
B C --gt
Assume that A, B and C are all declared as
enabled assumptions. Here is the
corresponding network
A contradiction is produced. To retract it,
the IE must retract one of the two antecedents
of J1.

IN
J1

IN
18
Example (cont.)

Assume the IE decides to retract A. The resulting
network is the following
A new justification, J2, is
recorded and
another contradiction becomes
IN.
To get rid of the new contradiction, assume that
the IE decides to retract B.

OUT
OUT
J2
J1

OUT
OUT
19
Example (cont.)

Note that the resulting network does not satisfy
the requirement that in a default
theory an assumption must be IN unless it causes
a contradiction. To correct
this situation, A must be enabled again. The
resulting network is the following

IN
OUT
J2
J1

OUT
OUT
20
Non-monotonic JTMS

Justifications have 2 types of antecedents
Antecedents that belong to the so-called IN-LIST.
Antecedents that belong to the so-called
OUT-LIST.
IN-LIST
OUT-LIST
For a node to be IN, it must be
Enabled assumption or premise.
It must have a justification with all nodes in
the IN-LIST IN, and all nodes in the OUT-LIST OUT.

21
Problems with non-monotonic JTMS

Beliefs are order-sensitive, which may result is
belief sets are not unique. Consider the
following network
There are two
belief sets
corresponding to
this network
Bel1 A, Bel2
B
Such circularities are called even
non-monotonic loops.
2. There may not be a (stable) belief set
at all as a result of a so-called odd
non-monotonic loop. Here is an example of such
a loop