Computational Complexity of Terminological Reasoning in BACK

Authors: Bernhard Nebel, Technische Universität Berlin, CIS/KITSekr. FR 5-8, Franklinstraße 28/29 D-1000 Berlin 10, West-Germanye-mail: nebel@db0tuill.bitnet published in Artificial Intelligence 34:

371-383, 1988Presented by Jordan Bradshaw and Corey White

Overview

• Introduction

• Complexity of Subsumption

• A Formal Treatment of Subsumption

• Definition of FLN

• Proof of NP-hardness

• Formulation of the Problem

• Consequences of Results

Introduction

• 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

Concept Relationship Example

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…

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

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.

A Formal Treatment 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, 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.

Definition of FLN

Definition of FLN

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

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 S

1 and S

2 such that no subset in C

is entirely contained in either S1 or S

2.

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 = {s

1, s

2, … s

n}

C = {c1, c

2, … c

m}

Each ci has the form:

Ci = {s

f(i, 1), s

f(i, 2), … s

f(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 C

i the conjuction of π(s

f(i,k)) for 1 < k < ||C

i|| 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 R

i corresponding to sets of C

i.

Formulation of the Problem

Where CPi,j is defined as:

Formulation of the Problem This means that a conjunction of π(S

j) is incoherent iff for

some role Ri we have more than ||C

i|| -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.

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

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