1 / 28

Paraconsistent Logic Programs

Paraconsistent Logic Programs. João Alcântara, Carlos Damásio and Luís Moniz Pereira e-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.

Download Presentation

Paraconsistent Logic Programs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Paraconsistent Logic Programs João Alcântara, Carlos Damásio and Luís Moniz Pereira e-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 JELIA'02 Cosenza, September 2002

  2. Outline • Motivation • Bilattices • Paraconsistent Logic Programs • Example • Conclusions

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

  4. 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.

  5. 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)

  6. 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

  7. 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

  8. 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)

  9. k (meet), k (join) t (meet), t (join) Bilattice 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 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.

  10. Bilattice (Basic operations) • Negation: B(C,D)has a negation operation if there is a mapping : CDCD such that • akb akb; • atb 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kb-bk-a; • atb-at-b; • --a = a.

  11. Bilattice Given a complete bilattice B(C,D) and let A CD. If A k- A Consistent A = - A Exact A k - A Inconsistent Given the bilattice B([0,1],[0,1]) where -(< ,  >) = < 1 -, 1 - > A = <0.4, 0.5> -A = <0.5,0.6> A is consistent A = <0.4, 0.6> -A = <0.4,0.6> A is exact A = <0.6, 0.7> -A = <0.3,0.4> A is inconsistent

  12. Bilattice (Default negation) 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 notL Default negation: LetB(C,D) a bilattice. Consider  and – respectively a negation and a conflation operator on B(C,D) . We define not : C  D  C  Das the default negation operator where not L = - L

  13. Paraconsistent Logic Programs • Antecedents • Extended Logic Programs • A framework for precisely characterize explicit negation A Paraconsistent Logic Program P is a set of rules of the form A[A1,..., Am|B1,..., Bn] • is isotonic w.r.t. A1,..., Am • is antitonic w.r.t. B1,..., Bn

  14. Paraconsistent Logic Programs Given a bilattice B(C,C) • Interpretation: I :  C  C • Lattice of intepretations • Partial intepretations Î :Form()  C  C Valuation Let I be a set of interpretations with I1 and I2 belonging to I. <I,  > is acomplete lattice where I1 I2 iff pI1(p) kI2(p) <It, Itu> true true or undefined

  15. Given I1 = < I1t,I1tu> and I2 = < I2t,I2tu> Standard Ordering I1s I2 iff I1t I2t and I1tu I2tu Fitting Ordering I1f I2 iff I1t I2t and I2tu I1tu A partial interpretation I satisfies a rule A  of P iff Î() k I(A) Satisfaction I is a model of P iff I satisfies all rules of P Models Paraconsistent Logic Programs

  16. TP(I)(A) = lub{Î() such that A   P} Program Division P/I = {A[A1,..., Am|I(B1),..., I(Bn)} s.t. A[A1,..., Am|B1,..., Bn] P Paraconsistent Logic Programs Extending the Classical Immediate Consequences Operator Let P be a monotonic logic program In Paraconsistent Logic Programs, we have to eliminate the antitonic part

  17. Semi-normal program – The semi-normal version of P is the program Ps obtained from P replacing every A  in P by A k - A In a bilattice: Aknot A Ak- A Ak- A 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 have to guarantee the Coherence Principle:A not A Instance of necessitation principle: if something is known then it is believed

  18. Paraconsistent Logic Programs (Semantics) We say M = <M t,M tu> 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; in particular, the paraconsistent well-founded model for a program P.

  19. 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 (< ,  >) = < ,  >, -(< ,  >) = < 1 -, 1 - >, and k(< ,  >, < , >) = < min(,  ), min (, ) >

  20. Example The first case can be represented by flu (<0.99,0.0> kfeverk coughkheadachekmuscle-pain ) flu (<0.99,0.0> kfeverk coughkheadachekmuscle-pain ) flu (<0.0, 0.99> kfeverk coughkheadachekmuscle-pain ) (A k B) = (A kB)

  21. Example The last case flu (<0.75,0.0> k feverkcoughkheadachekmuscle-pain ) flu (<0.0, 0.99> k feverkcoughkheadachekmuscle-pain ) flu (<0.75,0.99> kfeverkcoughkheadachekmuscle-pain ) -If a patient has fever, cough, headache, and muscle-pain, then flu is a correct diagnosis with 0.75 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.

  22. Example The rules for diagnosing flu are flu (<0.0, 0.80> kfeverk coughkheadachekmuscle-pain ) flu (<0.75, 0.99> kfeverk coughkheadachekmuscle-pain ) flu (<0.0, 0.3> kfeverk coughkheadachekmuscle-pain ) flu (<0.6, 0.0> kfeverk coughkheadachekmuscle-pain )

  23. Example Paraconsistency in our semantics fever <0.4, 0.6> headache <0.7, 0.9> muscle-pain <0.2, 0.7> cough <0.7, 0.3> flu (<0.0, 0.80> kfeverkcoughkheadachekmuscle-pain ) P = flu (<0.75, 0.99> kfeverk coughkheadachekmuscle-pain ) flu (<0.0, 0.3> kfeverk coughkheadachekmuscle-pain ) flu (<0.6, 0.0> kfeverk coughkheadachekmuscle-pain ) WFMp(P)?

  24. lubkflu = <0.4, 0.3> Example T component of WFMp(P) – Mt= PPs(M t) fever <0.4, 0.6> <0.4, 0.6> cough <0.7, 0.3> <0.7, 0.3> <0.7, 0.9> headache <0.7, 0.9> muscle-pain <0.2, 0.7> <0.2, 0.7> flu (<0.0, 0.80> kfeverk coughkheadachekmuscle-pain ) <0.0, 0.3> flu (<0.75, 0.99> kfeverk coughkheadachekmuscle-pain ) <0.2, 0.3> flu (<0.0, 0.3> kfeverk coughkheadachekmuscle-pain ) <0.0, 0.3> flu (<0.6, 0.0> kfeverkcoughkheadachekmuscle-pain ) <0.4, 0.0>

  25. lubkflu = <0.3, 0.3> Example TU component of WFMp(P) – Mtu = Ps(M t) fever <0.4,0.6> k -fever <0.4, 0.6> <0.4, 0.6> <0.7, 0.3> <0.7, 0.3> cough <0.7, 0.3> k -cough <0.1, 0.3> <0.1, 0.3> headache <0.7, 0.9> k -headache <0.3, 0.8> <0.2, 0.7> muscle-pain <0.2, 0.7> k -muscle-pain flu (<0.0, 0.80> kfeverk coughkheadachekmuscle-pain ) k -flu <0.7, 0.6> <0.0, 0.3> flu (<0.75, 0.99> kfeverk coughkheadachekmuscle-pain ) k -flu <0.7, 0.6> <0.1, 0.3> flu (<0.0, 0.3> kfeverk coughkheadachekmuscle-pain ) k -flu <0.7, 0.6> <0.0, 0.3> <0.7, 0.6> flu (<0.6, 0.0> kfeverkcoughkheadachekmuscle-pain ) k -flu <0.3, 0.0>

  26. Example WFMp(P) = <M t, M tu> M t = M tu = M tu (headache)kM t (headache) - Inconsistency M tu (flu)kM t (flu) The truth-value assigned to flu is "contaminated" by the inconsistent value verified in headache.

  27. Conclusion • 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 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 a precise way to present belief and doubt.

  28. Conclusion • 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

More Related