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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 e -mail: nebel@db0tuill.bitnet published in Artificial Intelligence 34: 371-383, 1988
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Computational Complexity of Terminological Reasoning in BACK Authors: Bernhard Nebel, TechnischeUniversitätBerlin, CIS/KIT Sekr. FR 5-8, Franklinstraße28/29 D-1000 Berlin 10, West-Germany e-mail: nebel@db0tuill.bitnet published in Artificial Intelligence 34: 371-383, 1988 Presented by Jordan Bradshaw and Corey White
Introduction • Complexity of Subsumption • A Formal Treatment of Subsumption • Definition of FLN • Proof of NP-hardness • Formulation of the Problem • Consequences of Results Overview
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
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…
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…
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
BNF notation of introduction example: A Formal Treatment of Subsumption
Partially defined concepts can be modeled by assuming additional anonymous atomic concepts: A Formal Treatment of Subsumption
Here is the extension, for the objects described by their particular concept: 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, FLby 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
It can be seen that FLN allows the introduction of incoherent concepts: • More can be inferred from this example: 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... 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 Definition of FLN
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. Proof of NP-hardness
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||)} 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 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. Formulation of the Problem
Where CPi,j is defined as: 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. Formulation of the Problem
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 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 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 Consequences