measuring incoherence

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  • 7/27/2019 Measuring Incoherence

    1/27

    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

  • 7/27/2019 Measuring Incoherence

    2/27

    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

  • 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

  • 7/27/2019 Measuring Incoherence

    4/27

    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

  • 7/27/2019 Measuring Incoherence

    5/27

    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

  • 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

  • 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

  • 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

  • 7/27/2019 Measuring Incoherence

    9/27

    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

  • 7/27/2019 Measuring Incoherence

    10/27

    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

  • 7/27/2019 Measuring Incoherence

    11/27

    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

  • 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

  • 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

  • 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

  • 7/27/2019 Measuring Incoherence

    15/27

    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

  • 7/27/2019 Measuring Incoherence

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    M f i h f li l

  • 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

  • 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

  • 7/27/2019 Measuring Incoherence

    19/27

    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

  • 7/27/2019 Measuring Incoherence

    20/27

    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

  • 7/27/2019 Measuring Incoherence

    21/27

    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())

    Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs

  • 7/27/2019 Measuring Incoherence

    22/27

    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)

    Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs

  • 7/27/2019 Measuring Incoherence

    23/27

    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)

    Manuel Ojeda-Aciego Incoherence in Multi-Adjoint Programs

  • 7/27/2019 Measuring Incoherence

    24/27

    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

  • 7/27/2019 Measuring Incoherence

    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

  • 7/27/2019 Measuring Incoherence

    26/27

    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

  • 7/27/2019 Measuring Incoherence

    27/27

    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