Transcript Lecture 13

Justification-based TMSs (JTMS)
JTMS utilizes 3 types of nodes, where each node is associated with an
assertion:
1.
2.
3.
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
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".
JTMS justifications
Example justification:
N1
J1
N3
N2
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.
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.
Propositional specification of a JTMS (cont.)
JTMS nodes are propositional symbols. They form the JTMS belief set, Bel.
Set of
Enabled
Assumptions, A.
Set A may grow and shrink.
Retraction of enabled assumptions
causes the complexity of JTMS
algorithms.
Belief set, Bel.
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:
1.
2.
The node is an enabled assumption or a premise.
Bel  J |--PL N
Otherwise, node N is "OUT".
Example of JTMS work: how JTMS helps the IE
to improve its efficiency
Consider the example on Figure 7.2 in the textbook.
IN
A
OUT
B
IN
D
IN
IN
Computation
module
IN
G
F
OUT
A
IN
B
IN
D
IN
F
OUT
A
IN
B
OUT
D
OUT
F
IN
Computation
module
G
IN
Computation
module
G
IN
OUT
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:
1.
2.
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:
1.
2.
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.
Adding information
There are three ways to add information to the JTMS state:
1.
2.
3.
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:
i.
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.
ii.
Check if this assumption was IN. If yes, do nothing.
iii.
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:
1.
2.
Create a new node or justification, if necessary.
Propagate the consequences (run the "Propagate-Inness" algorithm).
Propagating Inness: example
Consider the following dependency network
OUT
A
IN
IN
B
J1
C
J2
OUT
D
E
J3
F
OUT
IN
Propagating Inness: example (cont.)
Let A becomes an enabled assumption, The network changes as follows:
IN
A
IN
IN
B
J1
C
J2
IN
D
E
IN
J3
F
IN
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.
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:
1.
2.
3.
Make the assumption node OUT.
Retract all nodes that have it as an antecedent (run PropagateOutness 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 PropagateInness algorithm.
Propagating Outness: example
Consider the example network:
IN
A
IN
IN
B
J1
C
Let C be retracted, i.e. its
status changes from IN to
OUT.
J2
IN
D
E
IN
The result of this change:
E becomes OUT
and
J3
F
IN
F becomes OUT.
Propagating Outness: example (cont.)
Consider following modified version of the example network:
Let C be retracted, i.e. its
status changes from IN to
IN
A
IN
IN
B
C
OUT.
J1
J2
The result of this change:
J2 is invalidated, however
F is still IN making J4 an
IN
IN
E
D
alternative support
justification for E; now J3
J4
J3
remains valid because of
E is IN via J4 => the result:
E is IN, because F is IN,
F
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.
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 wellfounded explanation as to why a node is IN or OUT. The notion of a wellfounded explanation relies on the following definition:
A well-founded support for node Ni is a sequence of justifications J1, …, Jk
such that:
1.
Jk is the supporting justification for Ni.
2.
All of the antecedents of Jk are justified earlier in the sequence J1,
…, Jk-1.
3.
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.
Well-founded support: example
Consider the following network:
IN
A
J1
IN
B
J2
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.
Detecting contradictions
Consider the following network:
IN
IN
IN
A
D
IN
B
C
IN
IN
|
This node is explicitly
defined as a contradiction
node.
E
When a contradiction node becomes IN, the JTMS must:
1.
Find underlying assumptions.
2.
Resolve the contradiction between the underlying assumptions by
automatically retracting one of them, or asking an external system
(IE or human user) for help.
How JTMS simulates default reasoning
Consider the following default rules:
MA --> A
MB --> B
MC --> C
A & B -->
B & C -->
Assume that A, B and C are all declared as enabled assumptions. Here is the
corresponding network:
IN
A
IN
J1
IN
|
B
IN
C
A contradiction is produced. To retract it,
the IE must retract one of the two antecedents
of J1.
Example (cont.)
Assume the IE decides to retract A. The resulting network is the following:
OUT
A
IN
IN
J2
J1
OUT
B
|
IN
C
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
A
OUT
IN
J2
J1
OUT
B
|
OUT
|
C
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
A
OUT
IN
J2
J1
OUT
B
|
OUT
|
C
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:
1.
2.
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.
Problems with non-monotonic JTMS
1.
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}
B
A
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:
B
A
C
Problems with non-monotonic JTMS (cont.)
3.
No belief set exists.
A
4.
Two belief sets are possible
A
B
BS1 = {A}
BS2 = {B} Note that only this set
satisfies the requirement for
each node to have a wellfounded support.