Paraconsistent Logic Programs

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Paraconsistent Logic Programs
João Alcântara, Carlos Damásio and Luís Moniz
Pereira e-mail jflacdlmp_at_di.fct.unl.pt Centro
de Inteligência Artificial (CENTRIA) Depto.
Informática, Faculdade de Ciências e
Tecnologia Universidade Nova de Lisboa 2825-114
Caparica, Portugal
JELIA'02
Cosenza, September 2002
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Outline

• Motivation
• Bilattices
• Paraconsistent Logic Programs
• Example
• Conclusions

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Motivation

• Uncertain reasoning in Logic Programming
• Probability theory
• Fuzzy set theory
• Many-valued logic
• Possibilistic logic
• Monotonic frameworks without default negation

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Motivation

• General frameworks for uncertain reasoning
• Monotonic Logic Programs rules are constituted
  by arbitrary isotonic body functions and by
  propositional symbols in the head.

A ?? (isotonic function)
Note a function is isotonic (antitonic) iff the
value of the function increases (decreases) when
we increase any argument while the remaining
arguments are kept fixed.
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Motivation

• Because of their arbitrary monotonic and
  antitonic operators over a complete lattice,
  these programs pave the way to combine and
  integrate into a single framework several forms
  of reasoning, such as fuzzy, probabilistic,
  uncertain, and paraconsistent ones

A ??1 (isotonic function)
Antitonic Logic Programs
A ??2 (antitonic function)
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Motivation

• Specific treatment for the explicit negation in
  Antitonic Logic Programs is not provided
• Our approach
• Arbitrary complete bilattice of truth-values,
  where both belief and doubt are explicitly
  represented
• Framework for Paraconsistent Logic Programs
• Based on
• Fitting's bilattice
• Lakshmanan and Sadri's work on probabilistic
  deductive databases

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Motivation

• Our approach (cont)
• Fitting's bilattices
• They support an elegant framework for logic
  programming involving belief and doubt.
• They lead to a precise definition of explicit
  negation operators
• We use these results to characterize default
  negation
• Lakshmanan and Sadri's work convenience of
  explicitly representing both belief and doubt
  when dealing with incomplete knowledge, where
  different evidence may contradict one another

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Motivation

• A semantics for Paraconsistent Logic Programs
• We have to deal with both contradiction and
  uncertain information
• We may have programs with various degrees of
  contradictory information
• Obedience to coherence principle explicit
  negation entails default negation
• We can introduce any negation operator supported
  by Fitting's bilattice.
• Generalization of paraconsistent well-founded
  semantics for extended logic programs (WFSXp)

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Bilattice

• Given two complete lattices lt C, ?1 gt and lt D,
  ?2 gt , the structure B(C,D) lt C ? D, ?k ,
  ?t gt is a complete bilattice, where the partial
  orderings are defined as
• lt c1, d1gt ?k lt c2,d2gt if c1 ?1 c2 and d1 ?2 d2
• lt c1, d1gt ?t lt c2,d2gt if c1 ?1 c2 and d2 ?2 d1
• ?k
  ?t
• We say a bilattice B(C,D) is interlaced if each
  of the operations ?k , ?k , ?t , ?t , is
  monotone w.r.t. both orderings.

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Bilattice (Basic operations)

• Negation B(C,D) has a negation operation if
  there is a mapping
• ? C ? D ? C ? D such that
• a ?k b ? ?a ?k ?b
• a ?t b ? ?b ?t ?a
• ? ? a a
• Conflation B(C,D) enjoys a conflation operation
  if there is a mapping - C ? D ? C ? D
  such that
• a ?k b ? -b ?k -a
• a ?t b ? -a ?t -b
• - - a a

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Bilattice
A complete bilattice B(C,D) is given, and let A ?
C ? D. If
A ?k - A Consistent A
- A Exact A gt k - A
Inconsistent
Given the bilattice B(0,1, 0,1) where - (lt ?,
? gt) lt 1 - ?, 1 - ? gt
A lt0.4, 0.5gt -A lt0.5, 0.6gt A is
consistent A lt0.4, 0.6gt -A lt0.4, 0.6gt
A is exact A lt0.6, 0.7gt -A lt0.3, 0.4gt
A is inconsistent
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Bilattice (Default negation)
Conflation operator results in moving to "default
evidence"
In -L we are to count as "for'' whatever did not
count as "against'' before, and as "against''
what did not count as "for''. Thus, - ? L
resembles not L , or believe not.
Default negation Let B(C,D) be a
bilattice. Consider ? and - , respectively, a
negation and a conflation operator on B(C,D)
. We define not C ? D ? C ? D as the default
negation operator where not L - ? L
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Paraconsistent Logic Programs

• Antecedents
• Extended Logic Programs
• A framework for precisely characterizing explicit
  negation
• A Paraconsistent Logic Program P is a set of
  rules of the form

A ??A1,..., AmB1,..., Bn

• is isotonic w.r.t. A1,..., Am
• is antitonic w.r.t. B1,..., Bn

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Paraconsistent Logic Programs
Given a bilattice B(C,C)

• Interpretation I ? ? C ? C
• Lattice of interpretations
• Partial interpretations

Î Form(?) ? C ? C
Valuation
Let I be a set of interpretations with I1 and
I2 belonging to I. lt I, ? gt is a complete
lattice where I1 ? I2 iff ?p?? I1(p) ?k I2(p)
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Paraconsistent Logic Programs
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Paraconsistent Logic Programs
Extending the Classical Immediate Consequences
Operator
Let P be a monotonic logic program. Then
In Paraconsistent Logic Programs, we must first
eliminate the antitonic part.
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Paraconsistent Logic Programs
? Operator Let P a paraconsistent logic program
and J an interpretation
?P(J) lfp TP/J TP/J?? , for some ordinal ?
We must guarantee the Coherence Principle ?A ?
not A Which is an instance of necessitation
principle what is known is believed
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Paraconsistent Logic Programs (Semantics)
We say M ltM t,M tugt is a partial paraconsistent
model for P iff M t ?P (?Ps(M t)) and M tu
?Ps(M t).
We define the Paraconsistent Well-Founded Model
(WFMp(P)) as the least partial paraconsistent
model under the Fitting ordering.
Proposition All partial paraconsistent models
obey the coherence principle. So too the
paraconsistent well-founded model of any program
P.
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Example
Using a paraconsistent logic program to encode a
rather complex decision table based on rough
relations
We resort to the bilattice B(0,1,0,1) to
encode this decision table, where ? (lt ?, ? gt)
lt ?, ? gt - (lt ?, ? gt) lt 1 - ?, 1 -
? gt and ?k(lt ?, ? gt, lt ?, ? gt) lt
min(?, ? ), min (?, ?) gt
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Example
The first case
can be represented by
?flu ? (lt0.99,0.0gt ?k ?fever ?k
?cough ?k ?headache ?k ?muscle-pain )
flu ? ?(lt0.99,0.0gt ?k ?fever ?k ?cough ?k
?headache ?k ?muscle-pain )
flu ? (lt0.0, 0.99gt ?k fever ?k cough ?k
headache ?k muscle-pain )
?(A ?k B) (?A ?k ?B)
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Example
The last case
flu ? (lt0.75,0.0gt ?k fever ?k cough ?k headache
?k muscle-pain )
flu ? (lt0.0, 0.99gt ?k fever ?k cough ?k
headache ?k muscle-pain )
first last
flu ? (lt0.75, 0.99gt ?k fever ?k cough ?kheadache
?k muscle-pain )
- If a patient has fever, cough, headache, and
muscle-pain, then flu is a correct diagnosis with
0.75 assuredness of belief. - If a patient
doesn't have fever, doesn't have cough, doesn't
have neither headache nor muscle-pain, then he
doesn't have flu with 0.99 of belief assurance.
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Example
The rules for diagnosing flu are
flu ? (lt0.0, 0.80gt ?k ?fever ?k cough
?k headache ?k muscle-pain )
flu ? (lt0.75, 0.99gt ?k fever ?k
cough ?k headache ?k muscle-pain )
flu ? (lt0.0, 0.3gt ?k ?fever ?k ?cough
?k headache ?k muscle-pain )
flu ? (lt0.6, 0.0gt ?k fever ?k cough
?k ?headache ?k ?muscle-pain )
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Example
Paraconsistency in our semantics
fever ? lt0.4, 0.6gt
headache ? lt0.7, 0.9gt
muscle-pain ? lt0.2, 0.7gt
cough ? lt0.7, 0.3gt
flu ? (lt0.0, 0.80gt ?k ?fever ?kcough ?k headache
?k muscle-pain )
P
flu ? (lt0.75, 0.99gt ?k fever ?k cough ?k headache
?k muscle-pain )
flu ? (lt0.0, 0.3gt ?k ?fever ?k ?cough ?k headache
?k muscle-pain )
flu ? (lt0.6, 0.0gt ?k fever ?k cough ?k ?headache
?k ?muscle-pain )
WFMp(P) ?
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Example
T component of WFMp(P) is Mt ?P?Ps(M t)
fever ? lt0.4, 0.6gt
lt0.4, 0.6gt
cough ? lt0.7, 0.3gt
lt0.7, 0.3gt
lt0.7, 0.9gt
headache ? lt0.7, 0.9gt
muscle-pain ? lt0.2, 0.7gt
lt0.2, 0.7gt
flu ? (lt0.0, 0.80gt ?k ?fever ?k cough ?k headache
?k muscle-pain )
lt0.0, 0.3gt
flu ? (lt0.75, 0.99gt ?k fever ?k cough ?k headache
?k muscle-pain )
lt0.2, 0.3gt
flu ? (lt0.0, 0.3gt ?k ?fever ?k ?cough ?k headache
?k muscle-pain )
lt0.0, 0.3gt
flu ? (lt0.6, 0.0gt ?k fever ?k cough ?k ?headache
?k ?muscle-pain )
lt0.4, 0.0gt
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Example
TU component of WFMp(P) is Mtu ?Ps(M t)
fever ? lt0.4, 0.6gt
?k -fever
lt0.4, 0.6gt
lt0.4, 0.6gt
cough ? lt0.7, 0.3gt ?k -cough
lt0.7, 0.3gt
lt0.7, 0.3gt
lt0.1, 0.3gt
lt0.1, 0.3gt
headache ? lt0.7, 0.9gt ?k -headache
lt0.2, 0.7gt
muscle-pain ? lt0.2, 0.7gt ?k -muscle-pain
lt0.3, 0.8gt
flu ? (lt0.0, 0.80gt ?k ?fever ?k cough ?k headache
?k muscle-pain ) ?k -flu
lt0.7, 0.6gt
lt0.0, 0.3gt
flu ? (lt0.75, 0.99gt ?k fever ?k cough ?k headache
?k muscle-pain ) ?k -flu
lt0.7, 0.6gt
lt0.1, 0.3gt
flu ? (lt0.0, 0.3gt ?k ?fever ?k ?cough ?k headache
?k muscle-pain ) ?k -flu
lt0.7, 0.6gt
lt0.0, 0.3gt
lt0.7, 0.6gt
flu ? (lt0.6, 0.0gt ?k fever ?k cough ?k ?headache
?k ?muscle-pain ) ?k -flu
lt0.3, 0.0gt
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Example
WFMp(P) ltM t, M tugt
M t
M tu
M tu (headache) ?k M t (headache)
Inconsistency
M tu (flu) ?k M t (flu)
The truth-value assigned to flu is "contaminated"
by the inconsistent value verified in headache.
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Conclusion - 1

• We have combined and integrated several forms of
  reasoning into a single framework. Namely fuzzy,
  probabilistic, uncertain, and paraconsistent.
• We have introduced, into this rather general
  framework, the concepts to cope with explicit
  negation and default negation. It is certified
  that default negation complies with the coherence
  principle.
• Program rules have bodies corresponding to
  compositions of arbitrary monotonic and antitonic
  operators over a complete bilattice, thus
  providing a precise way to represent belief and
  doubt.

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Conclusion - 2

• A logic programming semantics was defined, with
  corresponding model and fixpoint theory, where a
  paraconsistent well-found model is guaranteed to
  exist for each program.
• Further, we have provided a simple translation of
  Extended Logic Programs under WFSXp into
  Paraconsistent Logic Programs.

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THE END

• THE END