1. The Logic(s) of Logic Programming João Alcântara Carlos Viegas Damásio Luís Moniz Pereira Centro de Inteligência Artificial (CENTRIA) Depto. Informática, Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2829-516 Caparica, Portugal Altea, 23-25 October, 2003

2. Outline 1.Objectives/Motivation 2.Overview of Substructural Logics 3.Frame Semantics for Logic Programming 4.Equilibrium Logics 5.Embeddings of Logic Programming Semantics 6.Conclusions and Future Work

3. 1. Objectives/Motivation Definition of a logical framework general enough to capture SM, AS, PAS; WFS, WFSX and WFSX P –Present version is limited to programs without disjunction and without embedded implications Challenge of characterising a logic that deals with these semantics uniformly Inspired by Pearce and Cabalar's works, and by Greg Restall's proposals for Substructural Logics

4. Objectives/Motivation (cont) Preliminary difficulties –WFS-based semantics use partial interpretations, whilst SM, AS and PAS use only total ones. –Coherence principle is not satisfied in PAS –WFSX P and PAS are paraconsistent –In WFSX and WFSX P, '  ' frustates some expected properties of the implication

5. 2. Overview of Substructural Logics They allow us to draw many conclusions collapsed in classical logic –More complex semantics We can inflate the number of values for a sentence We can provide more places at which sentences are evaluated (points in frames)

6. Preliminary Definitions Point set P =  P,   is a set P together with a partial order  on P. The set of propositions on P is the set of all subsets X of P which are closed upwards: if x  X and x  x’ then x’  X. The extensional connectives disjunction and conjunction have the usual interpretation Accessibility relations define intensional connectives: necessity and possibility, negation, and conditionals.

7. Accessibility Relations for Intensional Connectives Plump positive two-place (S) – for any x,y,x',y'  P, where x S y, x'  x and y  y' it follows that x' S y' Plump negative two-place (C) – for any x,y,x',y'  P, where x C y, x'  x and y'  y it follows that x' C y' Plump three-place (R) – for any x, y, z, x',y',z'  P, where Rx y z, x'  x,y'  y and z  z' then Rx' y' z'

8. Frames for Substructural Logics A Frame F is a point set P together with any number of accessibility relations on P. Evaluating formulae –Intensional connectives with accessibility relations in the frame –Plump conditions guarantee that  satisfies heredity: If (M,x)  F and x  y then (M,y)  F

9. 3. Point Set for LP (Motivation) thnttnthpttp thn ttn thpttp thnttnthpttp thnttnthpttp A, not  A A,  A, not A not A, not  A AA ww A  w A  w Symbology: a) b) c) d)

10. Frame for Logic Programming thn ttn thpttp P =  [hhp,htp,thp,ttp,hhn,htn,thn,ttn],   hhnhtnhhp htp 81 possible “propositions” Enough to capture SM, AS, PAS; WFS, WFSX and WFSX P

11. Frame for Logic Programming Syntax: given a set of atoms , if ,  are formulae, p  , , not , ^ ,  (  ),   ,   ,   ,   ,  and  are also formulas Interpretation in a point w: I w - set of atoms HT 4 -Interpretation (B h,B t ), in which B h = (I hhp,I hhn,I htp,I htn ) B t = (I thp,I thn,I ttp,I ttn ) and I hxp  I txp, I txn  I hxn, x  {h,t}

12. Evaluation on Frames Given an HT 4 -Interpretation M = (B h,B t ), in P =  {hhp,htp,thp,ttp,hhn,htn,thn,ttn},   with the set of points W = {hhp,htp,thp,ttp,hhn,htn,thn,ttn} we say 1. (M,w)  p iff p  I w, where p   2. (M,w)   for all w  W 3. (M,w)   for no w  W 4. (M,w)     iff (M,w)   and (M,w)   5. (M,w)     iff (M,w)   or (M,w)  

13. Explicit Negation - R  hhn thn htn ttn hhp thp htp ttp R  is plump negative two- place 6. (M,w)   iff for all w' s.t. w R  w' (M,w')  

14. Default Negation - R not htphtn hhphhn ttp ttn thp thn 7. (M,w)  not  iff for all w' s.t. w R not w' (M,w')   R not is plump negative two- place

15. Semi-normality Operator - R ^ htphtnhhphhn ttpttnthpthn 8. (M,w)  ^  iff exists w' s.t. w R ^ w' (M,w')  

16. Possibility Operator - R  htphtnhhphhn ttpttnthpthn 9. (M,w)   (  ) iff exists w' s.t. w R  w' (M,w')   R  is plump positive two- place

17. Conditional - R hhp, hhp, hhp hhn, hhn, hhn htp, htp, htp htn, htn, htn thp, thp, thp thn, thn, thn ttp, ttp, ttp ttn, ttn, ttn hhp,hhp,thp hhn,thn,hhn thn,hhn,hhn thn,thn,hhn htp,htp,ttp htn,ttn,htn ttn,htn,htn ttn,ttn,htn thp,hhp,thp hhp,thp,thp hhp,hhp,thp thn,thn,hhn ttp,htp,ttp htp,ttp,ttp htp,htp,ttp ttn,ttn,htn 10. (M,w)     iff for all w',w'' s.t. R ww'w'' if (M,w')  , then (M,w'')   R is plump positive three-place

18. Model An HT 4 -Interpretation M is a model of a theory T iff for all w  W and all formulae  in T, then (M,w)  

19. 4. Equilibrium Logics General Stable Model General Well-founded Model A belief set B is a general stable model of a theory T iff (B,B) is h-minimal among models of T A belief set B is a general well-founded model of T iff (B,B) is t-minimal among the general stable models of T

20. Minimality Conditions HT 4 -Interpretation (B h,B t ), in which (B h,B t )  h (C h,C t ) iff B t = C t and B h  C h (B h,B t )  t (C h,C t ) iff B t  F C t B h  C h iff I hxp  J hxp and J hxn  I hxn, x  {h,t} B t  F C t iff I thp  J thp, J thn  I thn, J ttp  I ttp and I ttn  J ttn B h = {I hhp,I hhn,I htp,I htn }B t = {I thp,I thn,I ttp,I ttn } Standard ordering: Fitting’s ordering:

21. 5. Embeddings of LP Semantics Difficulties –WFS-based semantics use partial interpretations, whilst SM, AS and PAS use only total ones. –Coherence principle is not satisfied in PAS –WFSX P and PAS are paraconsistent –In WFSX and WFSX P, '  ' is not interpreted in the same way as in AS and PAS.

22. Axioms  A  not A  (A)  A  (  )  A   (A) not  (  ) Default Consistency (DC) Definedness (DE) Coherence Principle (CP) No Negative Information (NNI)

23. Interpreting logic programs rules AS and PAS (Nelson's implication) B  A A  (B \/ not A) WFSX and WFSX p B  A A  (B \/ ^B)

24. Truth-values for SM V 1 = [hhn,htn,thn,ttn] V 2 = [hhn,htn,thn,thp,ttn,ttp] ; V 3 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp] not  A not A, not  A A, not  A Axioms: NNI + DC + CP + DE

25. Truth-values for AS V 1 = [] ; V 2 = [hhn,htn] ; V 3 = [hhn,htn,thn,ttn] ; V 4 = [hhn,htn,thn,thp,ttn,ttp] ; V 5 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp] not A, not  A A, not  A  A, not A not  A not A Axioms CP + DC + DE

26. Truth-Values for PAS V1 = [] ; V2 = [thp,ttp] ; V3 = [hhp,htp,thp,ttp] ; V4 = [hhn,htn] ; V5 = [hhn,htn,thp,ttp] ; V6 = [hhn,htn,thn,ttn] ; V7 = [hhn,htn,thn,thp,ttn,ttp] ; V8 = [hhn,hhp,htn,htp,thp,ttp] ; V9 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp] ; Axioms DC + DE

27. Truth-values for WFS V 1 = [hhn,htn,thn,ttn] ; V 2 = [hhn,htn,thn,ttn,ttp] ; V 3 = [hhn,htn,thn,thp,ttn,ttp] ; V 4 = [hhn,htn,htp,thn,ttn,ttp] ; V 5 = [hhn,htn,htp,thn,thp,ttn,ttp] ; V 6 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp] Axioms NNI + DC + CP

28. Truth-values for WFSX and WFSX p WFSX Axioms: DC + CP 15 truth-values WFSX p Axioms: CP 25 (coherent) truth-values In WFSX p semantics, all operators are closed w.r.t. available truth-values !

29. Results (1) SM: NNI + DC + CP + DE A belief set B is a stable model of a program P iff (B,B) is a general stable model of P AS: CP + DC + DE A belief set B is an answer set of a program P iff (B,B) is a general stable model of P PAS: DC + DE A belief set B is a paraconsistent answer set of a program P iff (B,B) is a general stable model of P

30. Results (2) WFS: NNI + DC + CP A belief set B is a well-founded model of a program P iff (B,B) is a general well-founded model of P WFSX: DC + CP A belief set B is a WFSX of a program P iff (B,B) is a general well-founded model of P WFSX p : CP A belief set B is a WFSX p of a program P iff (B,B) is a general well-founded model of P

31. 6. Conclusions We have defined a logic general enough to capture SM, AS, PAS; WFS, WFSX and WFSX P It allows to characterise logically the interrelationship among the semantics

32. Future Work Study of the disjunction for WFS based semantics Characterisation of the notion of logical consequence and entailment Detection of minimal properties in our frame to capture the cited semantics