1. Computational Complexity of Terminological Reasoning in BACK
Authors: Bernhard Nebel, Technische Universität Berlin, CIS/KIT
Sekr. FR 5-8, Franklinstraße 28/29 D-1000 Berlin 10, West-Germany
published in Artificial Intelligence 34: , 1988
Presented by Jordan Bradshaw and Corey White

2. Overview Introduction Complexity of Subsumption
A Formal Treatment of Subsumption
Definition of FLN
Proof of NP-hardness
Formulation of the Problem
Consequences of Results
Overview

3. The BACK system is part of the KL-ONE hybrid knowledge representation system.
Which is a FDL (frame-based description language)
It's used to represent terminological knowledge
Introduction

4. Concept Relationship Example

5. Introduction cont… Important characteristics of FDL
take definition notions seriously
Allows concepts/roles to specify relationships to other concepts
Grandparent is a specialization of parent, although its not explicitly mentioned.
Since there is more represented than what's explicitly written, a reasoner is needed to uncover the hidden relationships.
Introduction cont…

6. Introduction cont…

7. Introduction cont… Some queries can be reduced to other query types
All queries in this case can be reduced to subsumptions provided the concepts/roles.
Classification -> Subsumption (provided O(n2))
Disjointness ->Incoherency
Incoherency -> Subsumption
Property possession -> Classification
Introduction cont…

8. Complexity of Subsumption
Subsumption is a crucial part of terminological reasoning.
Subsumption basic idea:
All detected relationships in KL-ONE are sound, but the detection procedure is incomplete.
FDL used in KL-ONE the subsumption problem can be intractable.
Complexity of Subsumption

9. A Formal Treatment of Subsumption
BNF notation of introduction example:
A Formal Treatment of Subsumption

10. A Formal Treatment of Subsumption
Partially defined concepts can be modeled by assuming additional anonymous atomic concepts:
A Formal Treatment of Subsumption

11. A Formal Treatment of Subsumption
Here is the extension, for the objects described by their particular concept:
A Formal Treatment of Subsumption

12. A Formal Treatment of Subsumption
C1 subsumes C2 if and only if set D and any extension function ε over D, the following will hold:
The language above, FL by Brachman & Levesque is intractable with respect to subsumption.
FL- , without the restr operator was shown to be more acceptable from a computational complexity perspective.
A Formal Treatment of Subsumption

13. Definition of FLN

14. Definition of FLN
It can be seen that FLN allows the introduction of incoherent concepts:
More can be inferred from this example:

15. Definition of FLN
It is therefore necessary to consider the disjointness of role filler concepts.
This can still be handled in polynomial time, as there are n * (n -1) / 2 pairs of sub roles to consider.
There are other more complex situations to consider though...

16. Definition of FLN
These sub roles aren't pairwise disjoint but lead to incoherency when considered together.
This is likely an intractable problem
Even if it wasn't intractable, there are still no sound, complete, polynomial algorithms for subsumption

17. Proof of NP-hardness
Subsumption in FLN can be compared to the complement of the SET-SPLITTING problem
SET-SPLITTING is proven NP-complete
SET-SPLITTING is defined as:
Given a collection C of subsets of a finite set S, is there a partition of S into two subsets S1 and S2 such that no subset in C is entirely contained in either S1 or S2.

18. Formulation of the Problem
Given a special case subsumption problem:
SUBSUMES ((atleast 3 R, X)
X is a description containing atleasts and alls on sub roles of R
Consider a SET-SPLITTING problem with:
S = {s1, s2, … sn}
C = {c1, c2, … cm}
Each ci has the form:
Ci = {sf(i, 1), sf(i, 2), … sf(i, ||Ci||)}

19. Formulation of the Problem
This gives rise to an X of the form:
(and (atleast 1 (androle R Rprim1)) (all (androle RRprim1) π(s1)) (atleast 1 (androle R Rprim2)) (all (androle RRprim2) π(s2)) .... (atleast 1 (androle R Rprimn)) (all (androle RRprimn) π(sn))
Where π is a transformation function such that for each set Ci the conjuction of π(sf(i,k)) for 1 < k < ||Ci|| forms an incoherent concept

20. Formulation of the Problem
Under this formulation, this means that the corresponding sub roles can't be filled with the same instance.
However, if the subset of S doesn't contain a set Ci, then the sub roles can be filled with the same instance.
We then assume m roles in Ri corresponding to sets of Ci.

21. Formulation of the Problem
Where CPi,j is defined as:

22. Formulation of the Problem
This means that a conjunction of π(Sj) is incoherent iff for some role Ri we have more than ||Ci|| -1 different atleast restrictions
This results in the following analogy to the SET-SPLITTING problem:
If a role R of concept X can be filled with 2 or less role fillers, then there is a set splitting.
Else, a there is no set splitting.
Since this solves an instance of the SET-SPLITTING problem, subsumption in FLN is co-NP-hard.

23. Consequences This has some unfortunate consequences:
There are no complete, sound algorithms for subsumption on FDLs with this much expressive power that run in polynomial time.
We can improve this by reducing the expressiveness of the FDL: removing atleast, atmost and androle can help
We can settle on algorithms that are not complete instead, but tractable.
This is a common approach for AI algorithms

24. Consequences
To provide completeness, the expressiveness of the FDL must be limited:
Remove all operators relating roles
Alternatively, restrict these operators to some, none and unique
Weakening the semantics has the effect of reducing what inferences can be made
Even somewhat obvious relationships might be missed

25. Consequences This gives us three ultimate choices:
A complete, sound algorithm: extremely slow
Weak semantics: might miss a lot of inferences
Strong semantics and incomplete algorithm: might miss some less obvious inferences
The approach depends on what's needed, but most practical systems would opt for the last option