An Anytime Algorithm for Computing Inconsistency Measurement Yue Ma - - PowerPoint PPT Presentation

An Anytime Algorithm for Computing Inconsistency Measurement

Yue Ma1 Guilin Qi2 Guohui Xiao3,5 Pascal Hitzler4 Zuoquan Lin3

1Laboratoire d’Informatique de Paris-Nord, Universit´

e Paris-Nord CNRS, France

2School of Computer Science and Engineering, Southeast University, Nanjing, China 3Department of Information Science, Peking University, China 4Kno.e.sis Center, Wright State University, Dayton, OH, USA 5Institut f¨

ur Informationssysteme, Technische Universit¨ at Wien, Austria

Yue Ma (LIPN-CNRS), et al. @ KSEM’09 Anytime Algorithm for Inconsistency Degree 1 / 15

Motivation

Consistent KBs serve as useful knowledge resources v.s. inconsistent KBs imply any conclusion (meaningless!) For handling inconsistent KBs:

paraconsistent reasoning (1960s) knowledge diagnose and repair (1980s) Which approach should we take? inconsistency measurement: a guidance to choice different approaches (2000s)

How about the computational aspects of inconsistent measurement?

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Introductive Example

K = {p, ¬q, r} consistent K′ = {p, ¬q, r, ¬p ∨ q} inconsistent K′′ = {p, ¬p, q, ¬q} inconsistent The inconsistency degrees (ID): ID(K) = 0, ID(K′) = 1 3, ID(K′′) = 1

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Related Work and Our Contribution

Related work: Defining (various) inconsistency degrees: (1) syntax-based; (2) semantics-based Algorithms

for restricted KBs: [GrantHunter08] only deals with KBs in the form Q1x1, ..., Qnxn.

i(Pi(t1, ..., tmi) ∧ ¬Pi(t1, ..., tmi)), ;

with high complexity: [MaQiHLin2007] with exponential times of invoking a SAT solver

Our work: To show that computing IDs is intractable generally but can be approximated polynomially

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Inconsistency Degree by 4-valued Semantics

The set of truth values

{t, f, BOTH, NONE}

A 4-model I:

Var(K) → {t, f, BOTH, NONE}

✲ t ✻ k

• ❅

❅ ❅

• ❅

❅ ❅

NONE

f t

BOTH

Figure: FOUR

Conflict(I, K) = {p | p ∈ Var(K), pI = BOTH}, PreferModel(K) = {I | ∀I′ ∈ M4(K), |Conflict(I, K)| ≤ |Conflict(I′, K)|}. ID(K) = |Conflict(I,K)|

|Var(K)|

, where I is a preferred model. K′ = {p, ¬q, r, ¬p ∨ q} : ID(K′) = 1

3

I1 : pI1 = BOTH, qI1 = f, rI1 = t, I2 : pI2 = f, qI2 = BOTH, rI2 = t

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Computational Complexities

Given a propositional knowledge base K and a number d ∈ [0, 1]: ID≤d (resp. ID<d): is ID(K) ≤ d (resp. ID(K) < d)? ID≥d (resp. ID>d): is ID(K) ≥ d (resp. ID(K) > d)? EXACT-ID: is ID(K) = d? ID: what is the value of ID(K)?

Theorem

ID≤d and ID<d are NP-complete; ID≥d and ID>d are coNP-complete; EXACT-ID is DP-complete; ID is ΘP

2 -complete.

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Formal Definitions of Approximating IDs

Definition (Bounding Values)

Lower bounding value x: x ≤ ID(K); Upper bounding value y: ID(K) ≤ y.

Definition (Bounding Models)

Given a preferred model I: Lower bounding model I′ of K: |Conflict(I′, K)| ≤ |Conflict(I, K)| Upper bounding model I′′ of K: |Conflict(I′′, K)| ≥ |Conflict(I, K)| and I′′ ∈ M4(K)

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Formal Definitions of Approximating IDs

Definition (Bounding Values)

Lower bounding value x: x ≤ ID(K); Upper bounding value y: ID(K) ≤ y.

Definition (Bounding Models)

Given a preferred model I: Lower bounding model I′ of K: |Conflict(I′, K)| ≤ |Conflict(I, K)| Upper bounding model I′′ of K: |Conflict(I′′, K)| ≥ |Conflict(I, K)| and I′′ ∈ M4(K)

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Requirements on Algorithms for Approximating IDs

An anytime approximating algorithm for computing inconsistency degrees should be able to produce two sequences r1, ..., rm and r1, ..., rk: r1 ≤ ... ≤ rm ≤ ID(K) ≤ rk ≤ ... ≤ r1, (1) such that these two sequences have the following properties: Tractability: ∃.f(|K|), g(|K|) s.t. computing ri and rj both stay tractable if i ≤ f(|K|) and j ≤ g(|K|); Convergence: |ID(K) − ri+1| < |ID(K) − ri|, |ID(K) − ri| < |ID(K) − ri+1|; Meaning: each ri (rj) corresponds to a lower (an upper) bounding model, which indicates the sense of the two sequences.

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Requirements on Algorithms for Approximating IDs

An anytime approximating algorithm for computing inconsistency degrees should be able to produce two sequences r1, ..., rm and r1, ..., rk: r1 ≤ ... ≤ rm ≤ ID(K) ≤ rk ≤ ... ≤ r1, (1) such that these two sequences have the following properties: Tractability: ∃.f(|K|), g(|K|) s.t. computing ri and rj both stay tractable if i ≤ f(|K|) and j ≤ g(|K|); Convergence: |ID(K) − ri+1| < |ID(K) − ri|, |ID(K) − ri| < |ID(K) − ri+1|; Meaning: each ri (rj) corresponds to a lower (an upper) bounding model, which indicates the sense of the two sequences.

Yue Ma (LIPN-CNRS), et al. @ KSEM’09 Anytime Algorithm for Inconsistency Degree 8 / 15

Requirements on Algorithms for Approximating IDs

An anytime approximating algorithm for computing inconsistency degrees should be able to produce two sequences r1, ..., rm and r1, ..., rk: r1 ≤ ... ≤ rm ≤ ID(K) ≤ rk ≤ ... ≤ r1, (1) such that these two sequences have the following properties: Tractability: ∃.f(|K|), g(|K|) s.t. computing ri and rj both stay tractable if i ≤ f(|K|) and j ≤ g(|K|); Convergence: |ID(K) − ri+1| < |ID(K) − ri|, |ID(K) − ri| < |ID(K) − ri+1|; Meaning: each ri (rj) corresponds to a lower (an upper) bounding model, which indicates the sense of the two sequences.

Yue Ma (LIPN-CNRS), et al. @ KSEM’09 Anytime Algorithm for Inconsistency Degree 8 / 15

Approximations from Above and Below

For a given w (1 ≤ w ≤ |Var(K)|):

Theorem (Approximation from Above)

If K is w-4 satisfiable, then ID(K) ≤ 1 − w/|Var(K)|.

Theorem (Approximation from Below)

If K is w-4 unsatisfiable, then ID(K) ≥ 1 − (w − 1)/|Var(K)|.

• Definition. K is w-4 satisfiable iff. there is a subset S ⊆ Var(K) such that K is

S-4 satisfiable, i.e., K has a 4-model in the form of pI ∈

• {B}

if p ∈ Var(K) \ S, {N, t, f} if p ∈ S.

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Approximations from Above and Below

For a given w (1 ≤ w ≤ |Var(K)|):

Theorem (Approximation from Above)

If K is w-4 satisfiable, then ID(K) ≤ 1 − w/|Var(K)|.

Theorem (Approximation from Below)

If K is w-4 unsatisfiable, then ID(K) ≥ 1 − (w − 1)/|Var(K)|.

• Definition. K is w-4 satisfiable iff. there is a subset S ⊆ Var(K) such that K is

S-4 satisfiable, i.e., K has a 4-model in the form of pI ∈

• {B}

if p ∈ Var(K) \ S, {N, t, f} if p ∈ S.

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Tractability of the Approximations

Theorem (Complexity)

There exists an algorithm for deciding if K is S-4 unsatisfiable in O(|K||S| · 2|S|) time for any given S ⊆ Var(K). S-4 unsatisfiability can be computed in P-time, if |S| = O(log |K|).

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Tractable Anytime Algorithm

Suppose ri, rj are defined as follows (1 ≤ w ≤ |Var(K)|): rj = 1 − w/|Var(K)|, where K is w-4 satisfiable; ri = 1 − w − 1 |Var(K)|, where K is w-4 unsatisfiable. If w = O(log |K|), computing upper bounds can be done in P-time w.r.t |K|. If w is limited by a constant, computing lower bounds can be done in P-time w.r.t. |K|. ri(rj) corresponds to inconsistency degrees of K w.r.t. its upper (lower) bounding models. Meets all the requirements given previously for tractable anytime algorithms.

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Tractable Anytime Algorithm

Tow main sources of complexity to compute approximating inconsistency degrees:

1

the complexity of w-4 satisfiability solved by previous results

2

the complexity of search space a truncation strategy to limit the search space by the monotonicity of S-4 unsatisfiability: For all S, if K is S-4 unsatisfiable, K is S′-4 unsatisfiable for all S′ ⊃ S.

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Primary Experiment

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 0.9 0.8 0.7 0.5 0.4 0.2 0.1 precision threshold time (millisecond) N=9 N=8 N=7 N=5

0.2 0.4 0.6 0.8 1 1.2 1 13 48 152 264 2830 3972 41749 22815 238837 time (millisecond) approximations

• f ID(K)

upper bound N=5 lower bound N=5 upper bound N=7 lower bound N=7 upper bound N=10 lower bound N=10

N=7 N=10 N=5

Figure: Evaluation results over KBs with |K| = N 2 + 2N and |Var(K)| = 2N for N = 5, 7, 8, 9, 10.

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Conclusion and Outlook

Conclusion

1

Studied the problem complexity of ID (intractable, Θp

2-complete)

2

Defined approximating inconsistency degrees

3

Proposed a tractable anytime algorithm for computing approximating IDs

Outlook

1

To test the algorithm on more benchmark datasets

2

To explore more optimization for the algorithm

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Thanks for Your Attention! Questions?

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