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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
On the Measure of Incoherent Information inExtended Multi-Adjoint Logic Programs
Nicolas Madrid Manuel Ojeda-Aciego
Ostrava Univ (Czech Republic) Univ de Malaga (Spain)
April 12, 2013
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
Aims of this research
We are studying the introduction of two kind of negations intofuzzy frameworks:
Default negation: This negation enables non-monotonicreasoning and is introduced by generalizing the semanticsof Answer sets and Equilibrium logic.
Strong negation: Including this kind of negation intopositive fuzzy theories does not affect to the
monotonicity of them. In this sense, it is necessary toprovide a suitable generalization of the notion ofconsistency.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
3/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
Overview
In this work we consider several strong negations. So:
We start by giving the notion of extended multi-adjointlogic programs.
To deal with strong negations we recall (and extend) thenotion of coherence given on residuated logic programs.
Subsequently, we measure the incoherence by:
focusing firstly on measuring the incoherence on
interpretationsand later by extending the measures defined oninterpretations to extended multi-adjoint logic programs.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
Preliminaries
Definition
A multi-adjoint latticeL is a tuple (L, , i, i) such that:
1 (L, ) is a complete bounded lattice, with top and bottomelements 1 and 0.
2 for all i, for all x the equations 1 i x = x i 1 = x hold.3 for all i, and for all x, y, z L the tuple (i, i) forms an
adjoint pair, i.e. z (x i y) iff y i z x.
A negation operator on L is any decreasing mapping n : L Lsatisfying n(0) = 1 and n(1) = 0.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
SyntaxLiterals
The language considered in this approach is propositional.Thus, given L a multi-adjoint lattice with a finite set of
negation operators {j}jJ:We denote by the set of propositional symbols
We define the set of literals by
Lit = {jp | p and j J}
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
6/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extended
programs
Conclusions
SyntaxExtended multi-adjoint logic program
Definition
Given a multi-adjoint lattice with negations (L, , i, i, j),an extended multi-adjoint logic program P is a finite set of
weighted rules of the form F; satisfying the followingconditions:
F is a formula of the form i B where is a literal(called the head of F) and B (called the body of F) isbuilt from literals
1, . . . , n and operators i.
the weight is an element of the underlying multi-adjointlattice L.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
7/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Semantics
Definition
Let L = (L, ) be a bounded lattice, an L-interpretation is amapping I: Lit L.
The domain of the interpretation is the set of literals, and itcan be lifted to any rule by homomorphic extension:
Definition (Model)
We say that I satisfies a rule i B; if and only ifI(B) i I() or, equivalently, I( i B).
Finally, I is a model ofP if it satisfies all rules in P.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
8/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The immediate consequences operator
The immediate consequences operator defined on positivemulti-adjoint logic programs can be applied straightforwardly toextended programs. We recall its definition below:
Definition
Let P be an extended multi-adjoint logic program and let I bean L-interpretation. The immediate consequence operator of Iwrt P is the L-interpretation defined by
TP(I)() = sup{I(B) i : i B; P}
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The least model
The immediate consequences operator TP is monotonic.
By the Knaster-Tarski fix-point theorem, TP has a leastfix-point; lfp(TP).
lfp(TP) coincides with the least model ofP
.The least model semantics of an extended multi-adjointlogic program P is given by the lfp(TP).
However,
one has to take into account the possible interactionbetween opposite literals.
For this purpose we define the notion of coherence.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
CoherenceThe definition
Definition
Let L be a multi-adjoint lattice. An L-interpretation I iscoherent if the inequality I(i ) i I() holds for every
literal and all strong negation i L.
The notion of coherence coincides with consistency in theclassical framework.
It only depends on negation operators.
It allows to handle missing information (i.e I such thatI() = 0 for all Lit is always coherent).
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
CoherenceExtension to multi-adjoint programs
Definition
Let P be an extended multi-adjoint logic program, we say that
P is coherent if its least model is coherent.
Theorem
An extended multi-adjoint logic program is coherent if and only
if it has at least one coherent model.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
12/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Incoherent pair of literals
DefinitionLet L be a multi-adjoint lattice and let I be an interpretation.We say that (, i ) is coherent w.r.t. I if and only if theinequality I(i ) iI() holds. Otherwise the pair (, i )is called incoherent.
However, it is convenient to provide degrees of incoherence.Consider the following two interpretations on {p, p}:
I1(p) = 0.5 I2(p) = 1
I1( p) = 0.6 I2( p) = 0.9
and the usual negation (x) = 1 x to determine thecoherence. Certainly, the pair (p, p) seems to be moreincoherent w.r.t. I2 than w.r.t. I1.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
13/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Information measure
We propose to assign a value to each element in the latticecorresponding to the inherent information it contains.
Definition
Let (L, ) be a lattice, an information measure is an operatorm : L R+ such that the following holds for all x, y L:
m is monotonic.
m(x) = 0 if and only if m(x) = 0.
m(sup(x, y)) m(x) + m(y) m(inf(x, y)).
Note that the third item imposes no restriction if the lattice islinearly ordered.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
14/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The set of coherent pairs w.r.t. i
Hereafter we will assume that our multi-adjoint lattices have anassociated information measure m.In order to measure the degree of incoherence of a pair (, i )w.r.t. I we focus on the minimal amount of information we
have to remove from I() and I(i ) in order to recover thecoherence of the pair (, i ).
Definition
Let i be a negation operator. The set of coherent pairs w.r.t.
i is the set:
i = {(x, y) L L : y i(x) }
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Measure of incoherence for pairs of opposite literals
Definition (IL)
We define the measure of incoherence of a pair (, i ) w.r.t.one interpretation I (denoted by IL((, i ); I)) as follows:
inf(x,y)i
(x,y)(I(),I(i ))
m(I()) m(x) + m(I(i)) m(y)
where the ordering within i is considered componentwise.
IL((, i ); I) measures how much information we have toremove from I() and I(i ) in order to obtain a coherent pairof literals (, i ) w.r.t. I.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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M f i h f li l
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7/27/2019 Measuring Incoherence
17/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Measure of incoherence for literalsDefinition of coherent literals
Here, we define a general degree of incoherence by consideringall possible negations of a literal.
Definition
Let L be a multi-adjoint lattice and let I be an interpretation.
We say that a literal is coherent w.r.t. I if and only if theinequality I(i ) i I() holds for all strong negationi L. Otherwise the literal is called incoherent.
Definition
Given a multi-adjoint lattice with negations {1, 2, . . . , n},we define the set of coherent tuples w.r.t. L as
L = {(x, y1, . . . , yn) Ln+1 : yi i(x) }
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
18/27
Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Measure of incoherence IG(; I)
Now, we define a measure of incoherence for literals, similar toIL, but by considering the set L instead of i. Formally,
Definition
We define the measure of incoherence IG(; I) by:
inf(x,yi)L
xI()yi I(i )
m(I()) m(x) +
ni
m(I(i)) m(yi)
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Relationship between the measures IL and IG
PropositionLet L be a multi-adjoint lattice and let I be an interpretation.Then for all Lit and i L we have that:
jL
IL((, j); I) IG(; I) IL((, i); I)
We need to remove less information from I if we deal directlywith the incoherence of all negated literals {i }i of than ifwe deal independently with each incoherent pair (, i ).
Corollary
Let L be a multi-adjoint lattice, let I be an interpretation andlet be a literal. Then, IG(; I) = 0 if and only ifIL((, i); I) = 0 for all pair (, i)
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
P i f IL d IG
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Properties of IL and IGNull measure
Proposition
If the literal (resp. the pair (, i )) is coherent w.r.t. I thenIG(; I) = 0 (resp. IL((, i ); I) = 0).
We can ensure the equivalence between coherence and nullmeasure of incoherence in the following frameworks:
Whether the multi-adjoint lattice is finite and theinformation measure used is injective.
Whether the multi-adjoint lattice is the unit interval [0, 1]and the information measure is continuous and injective.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
P ti f IL d IG
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7/27/2019 Measuring Incoherence
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Properties of IL and IGMonotonicity and bounds
Proposition
Let I J be two L-interpretations. Then
IL((, i ); I) IL((, i ); J) for all pair ofliterals (, i ).
IG(; I) IG(; J) for all literal .
The following proposition shows that IL((, i ); I) isbounded by the inherent information in I() and I(i ):
Proposition
Let I be an L-interpretation, then
IL((, i); I) min
m(I(i)), m(I())
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Incoherence in
Multi-AdjointPrograms
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The average number of incoherences
Definition
Let P be an extended program. We denote the number ofincoherent literals w.r.t. MP (least model ofP) as N I(P) andthe number of incoherent pairs of opposite literals w.r.t. MP as
N IP(P). So we can consider the measures of incoherenceIL1 (P) and I
G1 (P) as:
IL1 (P) =N IP(P)
|LitP| |P|(1)
IG1 (P) =N I(P)
|LitP|(2)
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Incoherence inMulti-Adjoint
Programs
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The maximal size of incoherence
The measures IL2 and IG2 focus on estimating the maximal size
of incoherence in P.
Definition
Given an extended program P, we consider:
IL2 (P) = maxiLitP
{IL((, i); MP)} (3)
and
IG2 (P) = maxLitP
{IG(; MP)} (4)
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Incoherence inMulti-Adjoint
Programs
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
The average size of incoherence
The measures IL3 and IG3 can be defined by focusing on
estimating the average size of incoherence in P.
Definition
Given an extended programP
, we can consider:
IL3 (P) =
i LitP
IL((, i); MP)i LitP
IL((, i); I)(5)
and
IG3 (P) =
LitPIG(; MP)LitP
IG(; I)(6)
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
W i h d f i h
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25/27
Incoherence inMulti-Adjoint
Programs
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Weighted measures of incoherence
Definition
Given an extended program P and a set of weights (with theform {(,i )}, resp. {}, with LitP) we consider:
IL4 (P; {(,i )}) =
i LitP
(,i ) IL((, i); MP) (7)
and
IG4 (P; {}) =
LitP
IG
(; MP) (8)
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
C l i
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Incoherence inMulti-Adjoint
Programs
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
Conclusions
We have extended the semantics of multi-adjoint logicprograms to allow the use of several strong negations.
We have recalled the notion of coherent L-interpretations.
We have measure the incoherence on interpretations by:
determining the incoherence in each pair (, i )measuring the incoherence related to each literal
Finally we have extended the measures of incoherencedefined on interpretations to extended programs by:
determining the average number of incoherences.measuring the greatest degree of incoherence.quantifying the average degree of incoherences.assigning weights to literals.
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs
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Incoherence inMulti-Adjoint
Programs
ManuelOjeda-Aciego
Introduction
Preliminaries
Least modeland coherence
Incoherence onInterpretations
Incoherenceon extendedprograms
Conclusions
On the Measure of Incoherent Information inExtended Multi-Adjoint Logic Programs
Nicolas Madrid Manuel Ojeda-Aciego
Ostrava Univ (Czech Republic) Univ de Malaga (Spain)
April 12, 2013
Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs