Fuzzy Answer Sets...

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Fuzzy Answer Sets Approximations

Mario Alviano1 and Rafael Peñaloza2

1Department of Mathematics and Computer ScienceUniversity of Calabria

2Center for Advancing Electronics DresdenDresden University of Technology

ICLP 2013Istanbul, 27 August 2013

Mario Alviano and Rafael Peñaloza Fuzzy Answer Sets Approximations

Outline

1 Introduction

2 Syntax and Semantics

3 Approximation OperatorsImmediate Consequence OperatorMinimal SatisfiabilityWell-founded Operator

4 Implementation and Experiment

5 Conclusion

Introduction

Answer Set Programming (ASP)Overcomes a weakness of classical logic for KR:monotonicityNaturally handles reasoning by defaults, abductivereasoning, belief revisons, ...

ASP makes logic closer to the real world

However...Everything is either true or false in ASPASP is based on precise information

Can we always make these assumptions?

Mario Alviano and Rafael Peñaloza Fuzzy Answer Sets Approximations 1 / 19

Introduction

Answer Set Programming (ASP)Overcomes a weakness of classical logic for KR:monotonicityNaturally handles reasoning by defaults, abductivereasoning, belief revisons, ...

ASP makes logic closer to the real world

However...Everything is either true or false in ASPASP is based on precise information

Can we always make these assumptions?

Is everything true or false? (1)

Barber of Seville paradoxIn the small town of Seville, all and only those men who do notshave themselves are shaved by the barber (who is a man).Who shaves the barber?

shaves(barber ,X )← not shaves(X ,X )shaves(X ,X )← not shaves(barber ,X )

Classical set theory can neither prove nor disprove that thebarber shaves himselfAn odd loop makes the program incoherent for ASPshaves(barber ,barber) is undefined according to thewell-founded semantics

Barber of Seville paradoxIn the small town of Seville, all and only those men who do notshave themselves are shaved by the barber (who is a man).Who shaves the barber?

shaves(barber ,barber)← not shaves(barber ,barber)shaves(barber ,barber)← not shaves(barber ,barber)

Classical set theory can neither prove nor disprove that thebarber shaves himselfAn odd loop makes the program incoherent for ASPshaves(barber ,barber) is undefined according to thewell-founded semantics

A F D G C H E I BHow many truth degrees?

Let’s associate false statements with white, and truestatements with blackUndefined statements can be associated to grayCan we model statements which are more likely to be falseor to be true?... and we can go on!

Only precise information?

When a person stops to be young and becomes old?

Compare it with the classical, crisp approach

How to Handle Vagueness?

Fuzzy logic interprets propositions with a truth degree in [0,1]

Classical (crisp) set vs. fuzzy set

Fuzzification

Fuzzy inference engine

Defuzzification

This work deals withthe engine part

Possible Application

Possible solutiontop(X ,high)← top(X ,medium),

not bottom(X − 1,high).bottom(X ,high)← bottom(X ,medium),

not top(X + 1,high).sign(X , I)← top(X , I).sign(X , I)← bottom(X , I).

Possible Application

Possible solutiontop(X ,high)← top(X ,medium),

not bottom(X − 1,high).bottom(X ,high)← bottom(X ,medium),

not top(X + 1,high).sign(X , I)← top(X , I).sign(X , I)← bottom(X , I).

Contribution

Fuzzy inference engineMinimal satisfiability operatorFuzzy unfounded setsImplementation

Grounder (gringo)Solver: fuzzy answer sets (approximation) andwell-founded model

Experiment

Outline

1 Introduction

5 Conclusion

Syntax and Semantics (1)

Normal FASP program: set of rules of the forma← b1 ⊗ · · · ⊗ bm ⊗ not bm+1 ⊗ · · · ⊗ not bn

Fuzzy atoms are either propositional atoms in a fixed set B ornumeric constants: a, b, speed , 0, 1, 0.5, 1/3, ...

Examplea← not bb ← a⊗ 0.8

⊗ : [0,1]× [0,1]→ [0,1] is a fixed t-norm, e.g.Gödel: x ⊗ y = min{x , y}Product: x ⊗ y = x · yŁukasiewicz: x ⊗ y = max{x + y − 1,0}

Syntax and Semantics (2)

Interpretation: I : B → [0,1]Numeric constant: I(c) = cNegative literal: I(not a) = 1− I(a)Conjunction: I(

⊗ki=1 li) =

⊗ki=1 I(li)

Model: I |= r if I(H(r)) ≥ I(B(r))Program Reduct: in P, replace each not b by I(not b)Fuzzy Answer Set: M |= P and there is no I ⊂ M s.t. I |= PM

Fuzzy Sets: Operations and Relations

I ⊆ J: I(a) ≤ J(a) for each a ∈ BI ⊂ J: I ⊆ J and I 6= JI ∩ J: a 7→ min{I(a), J(a)} for each a ∈ BI ∪ J: a 7→ max{I(a), J(a)} for each a ∈ BI \ J: a 7→ max{I(a)− J(a),0} for each a ∈ B

Outline

1 Introduction

5 Conclusion

Goal: Compute Approximations

Start with the largest approximation (0,1)Lower bound: all propositions associated with 0Upper bound: all propositions associated with 1

Apply operators to restrict the approximationIncrease the lower boundDecrease the upper bound

All the answer sets must be between these bounds

Immediate Consequence Operator

DefinitionFor each a ∈ B, define

T UP (L) : a 7→ max{〈L,U〉 (r) | r ∈ P, H(r) = a}

For r : a← b1 ⊗ · · · ⊗ bm ⊗ not bm+1 ⊗ · · · ⊗ not bn

let 〈L,U〉 (r) := L(b1 ⊗ · · · ⊗ bm)⊗ U(not bm+1 ⊗ · · · ⊗ not bn)

Łukasiewicz t-norm: x ⊗ y = max{x + y − 1,0}a← b ⊗ not c L = {a 7→ 0 ,b 7→ 0.8, c 7→ 0.3}

U = {a 7→ 1 ,b 7→ 1 , c 7→ 0.3}T U

P (L) : a 7→ L(b)⊗U(not c) = max{0.8+(1−0.3)−1,0} = 0.5

Theorem

The fixpoint T UP ⇑0 is reached after a linear number of iterations,

measured on the number of atoms appearing in P.

Immediate Consequence Operator

T UP (L) : a 7→ max{〈L,U〉 (r) | r ∈ P, H(r) = a}

For r : a← b1 ⊗ · · · ⊗ bm ⊗ not bm+1 ⊗ · · · ⊗ not bn

let 〈L,U〉 (r) := L(b1 ⊗ · · · ⊗ bm)⊗ U(not bm+1 ⊗ · · · ⊗ not bn)

Łukasiewicz t-norm: x ⊗ y = max{x + y − 1,0}a← b ⊗ not c L = {a 7→ 0 ,b 7→ 0.8, c 7→ 0.3}

U = {a 7→ 1 ,b 7→ 1 , c 7→ 0.3}T U

P (L) : a 7→ L(b)⊗U(not c) = max{0.8+(1−0.3)−1,0} = 0.5

Theorem

The fixpoint T UP ⇑0 is reached after a linear number of iterations,

measured on the number of atoms appearing in P.

Minimal Satisfiability (1)

SUP (L) : a 7→ inf{I(a) | I |= P ∧ L ⊆ I ⊆ U}

Theorem

If M |= P and L ⊆ M ⊆ U then SUP (L) ⊆ M ⊆ U.

How to compute this operator?For the Łukasiewicz t-norm, rewrite each

a← b1 ⊗ · · · ⊗ bm ⊗ not bm+1 ⊗ · · · ⊗ not bninto

a ≥ b1 + · · ·+ bm − bm+1 − · · · − bn + 1−mand solve a system of linear inequalities.

SUP (L) : a 7→ inf{I(a) | I |= P ∧ L ⊆ I ⊆ U}

Theorem

If M |= P and L ⊆ M ⊆ U then SUP (L) ⊆ M ⊆ U.

How to compute this operator?For the Łukasiewicz t-norm, rewrite each

a← b1 ⊗ · · · ⊗ bm ⊗ not bm+1 ⊗ · · · ⊗ not bninto

a ≥ b1 + · · ·+ bm − bm+1 − · · · − bn + 1−mand solve a system of linear inequalities.

Examplea← not bb ← a⊗0.8

min as.t. a ≥ 1− b

b ≥ a + 0.8− 10 ≤ a,b ≤ 1

S1P(0) : a 7→ 0.6

min bs.t. a ≥ 1− b

b ≥ a + 0.8− 10 ≤ a,b ≤ 1

S1P(0) : b 7→ 0.4

TheoremThe minimal satisfiability operator is computable in polynomialtime for programs over the Łukasiewicz t-norm.

Fuzzy Unfounded Sets (1)

DefinitionA fuzzy unfounded set X for P w.r.t. (L,U) satisfies[U ∩ (1 \ X )](H(r)) >= 〈U ∩ (1 \ X ),L〉 (r)

for each r ∈ P such that X (H(r)) > 0.

Fuzzy unfounded sets evidence lack of (acyclic) support.

Łukasiewicz t-norm: x ⊗ y = max{x + y − 1,0}a← b ⊗ bb ← ab ← 0.9⊗ not cc ← 0.9⊗ not b

Unfounded sets for (0,1):X1 = {a 7→ 0 ,b 7→ 0 , c 7→ 0 }X2 = {a 7→ 0 ,b 7→ 0 , c 7→ 0.1}X3 = {a 7→ 0.2,b 7→ 0.1, c 7→ 0 }X4 = {a 7→ 0.2,b 7→ 0.1, c 7→ 0.1}

LemmaThe union of two (fuzzy) unfounded sets is an unfounded set.

Let GUSL,UP denote the greatest fuzzy unfounded set.

Theorem

M is a fuzzy answer set of a program P iff GUSM,MP = 1 \M.

TheoremFor programs without numeric constants and crisp sets, fuzzyunfounded sets are the unfounded sets by Van Gelder et al.[1991].

LemmaThe union of two (fuzzy) unfounded sets is an unfounded set.

Let GUSL,UP denote the greatest fuzzy unfounded set.

Theorem

M is a fuzzy answer set of a program P iff GUSM,MP = 1 \M.

TheoremFor programs without numeric constants and crisp sets, fuzzyunfounded sets are the unfounded sets by Van Gelder et al.[1991].

Well-founded Operator

WP(L,U) :=(

T UP (L),1 \ RL,U

P ⇓1)

where for every a ∈ B, defineRL,U

P (X ) : a 7→ min{X (a),1−max{〈U ∩ (1 \ X ),L〉 (r) | r ∈ P, H(r) = a}}

TheoremThe fixpoint of WP gives the well-founded semantics byDamásio and Pereira [2001].

Outline

1 Introduction

5 Conclusion

Implementation

FASP program Grounder gringo Numeric format

UpperBoundIncrease UpperBoundDecrease LowerBoundIncrease

MinimalSatisfiability

Bilevel solverYALMIP+Octave

Fuzzy answer setor approximation

https://github.com/alviano/fasp.git

Implementation

FASP program Grounder gringo Numeric format

UpperBoundIncrease UpperBoundDecrease LowerBoundIncrease

MinimalSatisfiability

Bilevel solverYALMIP+Octave

Fuzzy answer setor approximation

https://github.com/alviano/fasp.git

Example

a← not bb ← a⊗0.8

min a− a′ + b − b′

s.t. 0 ≤ a,b ≤ 1min a + bs.t. a′ ≥ 1− b

b′ ≥ a′ + 0.8− 10 ≤ a′,b′ ≤ 1

Experiment

Tested Timeouts Average execution time? Averageinstances Unopt. Opt. Unopt. Optimized perc. gain?

Graph Coloring 60 34 0 247.44 34.45 (2.68) 76.43%Hamiltonian Path 40 33 9 120.51 6.41 (0.02) 81.49%

Stratified 90 10 0 190.07 1.80 (0.02) 96.71%Odd Cycle 90 33 0 186.94 1.95 (0.03) 97.18%

? Computed on the instances solved by both the approaches.

Outline

1 Introduction

5 Conclusion

Conclusion

Approximations provide a sensitive performance gainStill the bilevel approach seems to be too slow

Future WorkMore complex structures in the body (disjunction)

Compile into uninterpreted function symbols in gringoUse a pre-processor in fasp

How to deal with a choice operator?Completion approach by Janssen et al. [2012]Conflict analysis and learning?

Thank you!

Conclusion

Approximations provide a sensitive performance gainStill the bilevel approach seems to be too slow

Future WorkMore complex structures in the body (disjunction)

Compile into uninterpreted function symbols in gringoUse a pre-processor in fasp

How to deal with a choice operator?Completion approach by Janssen et al. [2012]Conflict analysis and learning?

Thank you!

Bibliography

Carlos Viegas Damásio and Luís Moniz Pereira. Antitonic logic programs. InProceedings of the 6th International Conference on Logic Programming andNonmonotonic Reasoning, LPNMR’01, pages 379–392, London, UK, UK, 2001.Springer-Verlag. ISBN 3-540-42593-4.

Jeroen Janssen, Dirk Vermeir, Steven Schockaert, and Martine De Cock. Reducingfuzzy answer set programming to model finding in fuzzy logics. Theory and Practiceof Logic Programming, 12(6):811–842, 2012.

Allen Van Gelder, Kenneth A. Ross, and John S. Schlipf. The Well-Founded Semanticsfor General Logic Programs. Journal of the ACM, 38(3):620–650, 1991.

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