João Alcântara, Carlos Damásio and Luís Pereirae-mail: jfla|cd|lmp@di.fct.unl.pt

Centro de Inteligência Artificial (CENTRIA)

Depto. Informática, Faculdade de Ciências e Tecnologia

Universidade Nova de Lisboa

2825-114 Caparica, Portugal

Paraconsistent Logic ProgramsParaconsistent Logic Programs

WoPaLo 2002 Trento, August 2002WoPaLo 2002 Trento, August 2002

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OutlineOutline

• Motivation• Bilattices • Paraconsistent Logic Programs • Example• Embedding• Conclusions and Further Work

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MotivationMotivation

• Uncertain reasoning in Logic Programming– Probability theory

– Fuzzy set theory

– Many-valued logic

– Possibilistic logic

• Different ways of dealing with uncertainty• Monotonic frameworks without default negation

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MotivationMotivation

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.

• 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)

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MotivationMotivation

• 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)

A 2 (antitonic function)

Antitonic Logic Programs:

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MotivationMotivation

• Specific treatment for the explicit negation in Antitonic Logic Programs is not provided

• Our approach– Framework for Paraconsistent Logic Programs

– Arbitrary complete bilattice of truth-values, where both belief and doubt are explicitly represented

– Fitting's bilattice

– Lakshmanan and Sadri's work on probabilistic deductive databases

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MotivationMotivation

• 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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MotivationMotivation

• 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)

– A semantics based on Coherent Answer Sets

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BilatticeBilattice

Given two complete lattices < C, 1 > and < D, 2 > the structure B(C,D) = < C D, k, t > is a complete bilattice, where the partial orderings are defined as follows:

< c1, d1> k < c2,d2> if c1 1 c2 and d1 2 d2

< c1, d1> t < c2,d2> if c1 1 c2 and d2 2 d1

To each ordering are associated join (\oplus) and meet (\otimes) operations according to the following equations:

Knowledge ordering (k) < c1, d1 > \otimesk < c2, d2 > = < c1 \sqcap1 c2, d1 \sqcap2 d2 > < c1, d1 > \oplusk < c2, d2 > = < c1 \sqcup1 c2, d1 \sqcup2 d2 >

Truth ordering (t)< c1, d1 > \otimest < c2, d2 > = < c1 \sqcap1 c2, d1 \sqcup2 d2 > < c1, d1 > \oplust < c2, d2 > = < c1 \sqcup1 c2, d1 \sqcap2 d2 >

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Bilattice (Basic operations)Bilattice (Basic operations)

Negation: A bilattice B(C,D) has a negation operation if there is a mapping : C D C D such that

1. a k b a k b;2. a t b b t a;3. a = a.

Conflation: B(C,D) enjoys a conflation operation if there is a mapping - : C D C D such that

1. a k b -b k -a;2. a t b -a t -b;3. --a = a.

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Bilattice (Default negation)Bilattice (Default negation)

Default negation: Let B(C,D) 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

Conflation operator results as moving to "default evidence"

In -L we are to count as "for'' whatever did not count as "against'' before, and "against'' what did not count as "for''. Thus, - L resembles not L

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Paraconsistent Logic ProgramsParaconsistent Logic Programs

A Paraconsistent Logic Program P is a set of the form

A [A1,..., Am|B1,..., Bn]

is isotonic w.r.t. A1,..., Am

is antitonic w.r.t. B1,..., Bn

can be isotonic w.r.t. some occurrence of a propositional symbol A and antitonic w.r.t. other occurrence of the same propositional symbol.

= -C \oplust - C

monotonic antitonic

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Paraconsistent Logic ProgramsParaconsistent Logic Programs

• Interpretation: I : C C

• Lattice of intepretations

• Partial intepretations

Î : Form() C C

<I, > is a complete lattice where I1 I2 iff p I1(p) k I2(p)

Valoração

<It, Itu>true true or undefined

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Paraconsistent Logic ProgramsParaconsistent Logic Programs

Standard Ordering

Fitting Ordering

I1 s I2 iff I1t I2

t and I1tu I2

tu

I1 f I2 iff I1t I2

t and I1tu I2

tu

Given I1 = < I1t, I1

tu > and I2 = < I2t, I2

tu >

Models I is a model of P iff I satisfies all rules of P

Satisfaction A partial interpretation I satisfies a rule A of P iff Î() k I(A)

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Paraconsistent Logic ProgramsParaconsistent Logic Programs

Extending the Classical Immediate Consequences Operator

Let P be a monotonic logic program

TP(I)(A) = lubk {Î() such that A P}

In Paraconsistent Logic Programs, we have to eliminate the antitonic part

Program Division P/I = {A [A1,..., Am|I(B1),..., I(Bn)}

s.t. A [A1,..., Am|B1,..., Bn] P

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Paraconsistent Logic ProgramsParaconsistent Logic Programs

Gamma Operator – Let P a paraconsistent program and J an interpretationP(J) = lfp TP/J = TP/J , for some ordinal

Semi-normal program (PS) – The semi-normal version of P is the program obtained from P replacing every A in P by A \oplusk -A

We have to guarantee the coherence principle: A k not A

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Paraconsistent Logic Programs Paraconsistent Logic Programs (Semantics)(Semantics)

We say M = <M t,M tu> is a partial stable model for P iff M t = P

(Ps(M t)) and M tu = Ps(M tu).

We define the Paraconsistent Well-Founded Model (WFMp(P)) as the least partial paraconsistent model under the Fitting ordering

Given M = < M t,M tu > is a partial stable model, we say an atom A is

true with degree wrt. M if k M t(A) and k M tu (A)

undefined with degree wrt. M if k M t(A) and k M tu (A)

false with degree wrt. M if k M t(A) and k M tu (A)

inconsistent with degree wrt. M if k M t(A) and k M tu (A)

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Paraconsistent Logic Programs Paraconsistent Logic Programs (Semantics)(Semantics)

• A Coherent Answer Set is a partial paraconsistent model of the form <M, M>

• All partial paraconsistent models obey the coherence principle; consequently, all coherent answer set and the paraconsistent well-founded model for a program P observe the coherence principle

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ExampleExample

Using a paraconsistent logic program to enconde a rather complex decision table based on rough relations

fever cough headache muscle-pain flu

no no no no no (in 99% of the cases)

yes no no no no (in 80% of the cases)

yes yes no no no (in 30% of the cases)

yes yes no no yes (in 60% of the cases)

yes yes yes yes yes (in 75% of the cases)

We resort to the bilattice B([0,1],[0,1]) to encode this decision table, where (< , >) = < , >, -(< , >) = < 1 - , 1 - >, and \otimesk(< , >, < , >) = < min(, ), min (, ) >

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ExampleExample

fever cough headache muscle-pain flu

no no no no no (99%)The first case

can be represented by

flu (<0.99,0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.99,0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.0, 0.99> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

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ExampleExample

fever cough headache muscle-pain flu

yes no no no no (80%)

Similarly, the second case

flu (<0.0,0.80> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

fever cough headache muscle-pain flu

no no no no no (99%)The last case

flu (<0.75,0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

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ExampleExample

flu (<0.75,0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.0, 0.99> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.75,0.99> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

-If a patient has fever, cough, headache, and muscle-pain, then flu is a correct diagnosis in 75% of the cases.

-If a patient doesn't have fever, doesn't cough, doesn't have neither headache nor muscle-pain, then he doesn't have flu in 99% of the situations.

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ExampleExample

fever cough headache muscle-pain flu

yes yes no no no (30%)

yes yes no no yes (60%)

For the remaining situation in the decision table

two distinct rules are required for concluding both the patient might have or not have a flu

flu (<0.0, 0.3> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.6, 0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

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ExampleExample

Assume antibiotics are prescribed when flu is not concluded. We will compare two possible translations of this statement:

antibiotics flu antibiotics -flu

The rules for diagnosing flu are

flu (<0.0, 0.3> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )flu (<0.6, 0.0> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

flu (<0.0, 0.80> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )flu (<0.75, 0.99> \otimesk fever \otimesk

cough \otimesk headache \otimesk muscle-pain )

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ExampleExample

We illustrate the behaviour of WFMp in several situations

fever cough headache muscle-pain

flu flu -flu

<1, 0>

<1, 0>

<0, 1>

<0, 1>

<0, 1>

<0, 1>

<0, 1>

<0, 1>

<0, 0.8>

<0, 0.8>

<0.8, 0>

<0.8, 0>

<1, 0.2>

<1, 0.2>

fever <0, 1> cough <0, 1> headache <0, 1> muscle-pain <0, 1>

By the coherence principle, Î(flu) k Î(-flu)

<0.8, 0> k <1, 0.2>

Antibiotics should be

prescribed according to antibiotics -flu

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ExampleExample

fever cough headache muscle-pain

flu flu -flu

<0.4,0.6>

<0.4,0.6>

<0.7,0.3> <0.7,0.3>

<0.1,0.9> <0.1,0.9>

<0.2,0.7>

<0.2,0.7>

<0.4,0.3>

<0.4,0.3>

<0.3, 0.4> <0.3, 0.4>

<0.6, 0.7>

<0.6, 0.7>

The physician is not certain regarding all symptoms

The degree of confidence for flu is obtained by combining the degrees of confidence of several rules

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ExampleExample

fever cough headache muscle-pain

flu flu -flu

<0.4,0.6>

<0.4,0.6>

<0.7,0.3> <0.7,0.3>

<0.7,0.9> <0.1,0.3>

<0.2,0.7>

<0.2,0.7>

<0.4,0.3>

<0.3,0.3>

<0.3, 0.4> <0.3, 0.4>

<0.7, 0.7>

<0.6, 0.7>

It illustrates how paraconsistency is handled by our semantics

flu

<0.4,0.3>

<0.3,0.3>

T

TU

flu is inconsistent

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EmbeddingEmbedding

Extended Logic Programs are defined as a set of rules of the form A B1,..., Bm, not C1,..., not Cn where A, Bi, and Cj (1 i m, 1 j n) are atoms or the explicit negation of atoms.

Let P be an extended logic program. Consider the bilattice B({0,1},{0,1}) with the operators – and , where (< a,b >) = < b,a > and -( < a, b > ) = < 1 - b, 1 - a >. Define Pw as a paraconsistent logic program such that

• For each rule A B1,... Bm, not C1,..., not Cn (m,n 0) belonging to P, we have A [1,0] \otimesk B1 \otimesk ... \otimesk Bm \otimesk - C1 \otimesk ... \otimesk - Cn in Pw;

•For each rule A B1, ... Bm, not C1,..., not Cn (m,n 0) belonging to P, we have A [0,1] \otimesk B1 \otimesk... \otimesk Bm \otimesk - C1 \otimesk ... \otimesk - Cn in Pw.

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EmbeddingEmbedding

• Moreover, let I and Iw be respectively interpretations in WFSXp and WFMp senses. We say I is a translation of Iw (and vice-versa), denoted by I Iw, iff for each atom A, Iw(A) = < 1,1 > iff {A, A} I; Iw(A) = < 1,0 > iff A I and A I; Iw(A) = < 0,1 > iff A I and A I; Iw(A) = < 0,0 > iff A I and A I.

• Theorem: Let P be an extended logic program with well-founded model T not F , and Pw the corresponding paraconsistent logic program with the model WFMp(Pw) = < M t, Mtu >. Then we have T M t and s T Mtu.

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ResultsResults

• We have combined and integrated several forms of reasoning into a single framework, namely fuzzy, probabilistic, uncertain, and paraconsistent.

• Introduction into a rather general framework, of an appropriate kind, of the concepts that 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, and provide an elegant way to present belief and doubt.

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ResultsResults

• A logic programming semantics with corresponding model and fixpoint theory was defined, where a paraconsistent well-found model is guaranteed to exist for each program.

• We further provide a simple translation of Extended Logic Programs under WFSXp into Paraconsistent Logic Programs

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Further WorkFurther Work

• Generalize our structure to consider rules with more complex heads, where we can have, for instance, disjunctions of atoms.

• Study particular instances of our framework to improve the understanding of properties of the concrete instances, and to compare these instances to existing work.

• Study the generalized class of logic programs, extending the Residuated one, where rule bodies can be anti-monotonic functions.

• Study of the various types of negation in our framework, specially if we allow for weak negation operators as well.

• The definition of derivation procedures is also envisaged.

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Questions???Questions???