Justification-based 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. B F D B D A A D F A F B Computation module OUT IN G IN IN G 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 Computation module IN OUT OUT IN G OUT OUT IN Computation module IN 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. A B C IN OUT IN OUT E F D Propagating Inness: example Consider the following dependency network OUT IN J1 J2 J3

10. IN A B C IN J1 J2 IN J3 IN IN IN D E F 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. IN A B C IN J1 J2 IN J3 IN IN IN D E F 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. IN IN C A B J1 IN IN D 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 => 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. A B 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. C B A IN | E D 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. A B C IN IN 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: A contradiction is produced. To retract it, the IE must retract one of the two antecedents of J1. IN J1 | IN

18. OUT A B C C A B J2 J1 | | OUT IN IN IN IN 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. A B C IN 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. B A 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: B A C

22. B A Problems with non-monotonic JTMS (cont.) • No belief set exists. • Two belief sets are possible BS1 = {A} BS2 = {B} Note that only this set satisfies the requirement for each node to have a well- founded support. A