João Alcântara, Carlos Damásio and Luís Pereira Centro de Inteligência Artificial (CENTRIA) Depto. Informática, Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa Caparica, Portugal Paraconsistent Logic Programs WoPaLo 2002 Trento, August 2002

2Outline Motivation Bilattices Paraconsistent Logic Programs Example Embedding Conclusions and Further Work

3Motivation 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

4Motivation 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)

5Motivation 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:

6Motivation 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

7Motivation 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

8Motivation 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

9Bilattice Given two complete lattices and the structure B (C,D) = is a complete bilattice, where the partial orderings are defined as follows:  k if c 1  1 c 2 and d 1  2 d 2  t if c 1  1 c 2 and d 2  2 d 1 To each ordering are associated join (\oplus) and meet (\otimes) operations according to the following equations: Knowledge ordering (  k ) \otimes k = \oplus k = Truth ordering (  t ) \otimes t = \oplus t =

10 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.

11 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

12 Paraconsistent Logic Programs A Paraconsistent Logic Program P is a set of the form A  [A 1,..., A m |B 1,..., B n ]  is isotonic w.r.t. A 1,..., A m  is antitonic w.r.t. B 1,..., B n  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 \oplus t -  C monotonicantitonic

13 Paraconsistent Logic Programs Interpretation: I :   C  C Lattice of intepretations Partial intepretations Î : Form(  )  C  C is a complete lattice where I 1  I 2 iff  p  I 1 (p)  k I 2 (p) Valoração truetrue or undefined

14 Paraconsistent Logic Programs Standard Ordering Fitting Ordering I 1 s I 2 iff I 1 t I 2 t and I 1 tu I 2 tu I 1 f I 2 iff I 1 t I 2 t and I 1 tu I 2 tu Given I 1 = and I 2 = 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)

15 Paraconsistent Logic Programs Extending the Classical Immediate Consequences Operator Let P be a monotonic logic program T P (I)(A) = lub k {Î(  ) such that A    P} In Paraconsistent Logic Programs, we have to eliminate the antitonic part Program Division P/I = {A  [A 1,..., A m |I(B 1 ),..., I(B n )} s.t. A  [A 1,..., A m |B 1,..., B n ]  P

16 Paraconsistent Logic Programs Gamma Operator – Let P a paraconsistent program and J an interpretation  P (J) = lfp T P/J = T P/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  \oplus k -A We have to guarantee the coherence principle: A  k not  A

17 Paraconsistent Logic Programs (Semantics) We say M = 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 = 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)

18 Paraconsistent Logic Programs (Semantics) A Coherent Answer Set is a partial paraconsistent model of the form 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

19Example Using a paraconsistent logic program to enconde a rather complex decision table based on rough relations fevercoughheadachemuscle-painflu no no (in 99% of the cases) yesno no (in 80% of the cases) yes no no (in 30% of the cases) yes no yes (in 60% of the cases) yes yes (in 75% of the cases) We resort to the bilattice B ([0,1],[0,1]) to encode this decision table, where  ( ) =, -( ) =, and \otimes k (, ) =

20Example fevercoughheadachemuscle-painflu no no (99%) The first case can be represented by  flu  ( \otimes k  fever \otimes k  cough \otimes k  headache \otimes k  muscle-pain ) flu   ( \otimes k  fever \otimes k  cough \otimes k  headache \otimes k  muscle-pain ) flu   ( \otimes k fever \otimes k cough \otimes k headache \otimes k muscle-pain )

21Example fevercoughheadachemuscle-painflu yesno no (80%) Similarly, the second case flu  ( \otimes k  fever \otimes k cough \otimes k headache \otimes k muscle-pain ) fevercoughheadachemuscle-painflu no no (99%) The last case flu  ( \otimes k fever \otimes k cough \otimes k headache \otimes k muscle-pain )

22Example flu  ( \otimes k fever \otimes k cough \otimes k headache \otimes k muscle-pain ) flu   ( \otimes k fever \otimes k cough \otimes k headache \otimes k muscle-pain ) flu  ( \otimes k fever \otimes k cough \otimes k headache \otimes k 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.

23Example fevercoughheadachemuscle-painflu yes no no (30%) yes 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  ( \otimes k  fever \otimes k  cough \otimes k headache \otimes k muscle-pain ) flu  ( \otimes k fever \otimes k cough \otimes k  headache \otimes k  muscle-pain )

24Example Assume antibiotics are prescribed when flu is not concluded. We will compare two possible translations of this statement: antibiotics   fluantibiotics  -  flu The rules for diagnosing flu are flu  ( \otimes k  fever \otimes k  cough \otimes k headache \otimes k muscle-pain ) flu  ( \otimes k fever \otimes k cough \otimes k  headache \otimes k  muscle-pain ) flu  ( \otimes k  fever \otimes k cough \otimes k headache \otimes k muscle-pain ) flu  ( \otimes k fever \otimes k cough \otimes k headache \otimes k muscle-pain )

25Example We illustrate the behaviour of WFMp in several situations fevercoughheadachemuscle- pain flu  flu-  flu fever  cough  headache  muscle-pain  By the coherence principle, Î(  flu)  k Î(-  flu)  k Antibiotics should be prescribed according to antibiotics  -  flu

26Example fevercoughheadachemuscle- pain flu  flu-  flu <0.7,0.3><0.1,0.9> <0.3, 0.4> 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

27Example fevercoughheadachemuscle- pain flu  flu-  flu <0.7,0.3> <0.3, 0.4> It illustrates how paraconsistency is handled by our semantics flu T TU flu is inconsistent 

28Embedding Extended Logic Programs are defined as a set of rules of the form A  B 1,..., B m, not C 1,..., not C n where A, B i, and C j (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  ( ) = and -( ) =. Define P w as a paraconsistent logic program such that For each rule A  B 1,... B m, not C 1,..., not C n (m,n  0) belonging to P, we have A  [1,0] \otimes k B 1 \otimes k... \otimes k B m \otimes k -  C 1 \otimes k... \otimes k -  C n in P w ; For each rule  A  B 1,... B m, not C 1,..., not C n (m,n  0) belonging to P, we have A  [0,1] \otimes k  B 1 \otimes k... \otimes k  B m \otimes k - C 1 \otimes k... \otimes k - C n in P w.

29Embedding Moreover, let I and I w be respectively interpretations in WFSX p and WFM p senses. We say I is a translation of I w (and vice-versa), denoted by I  I w, iff for each atom A, I w (A) = iff {A,  A}  I; I w (A) = iff A  I and  A  I; I w (A) = iff A  I and  A  I; I w (A) = iff A  I and  A  I. Theorem: Let P be an extended logic program with well- founded model T  not F, and P w the corresponding paraconsistent logic program with the model WFM p (P w ) =. Then we have T  M t and  s T  M tu.

30Results 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.

31Results 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

32 Further 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.

33Questions???