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The Logic(s) of Logic Programming
João AlcântaraCarlos 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
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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
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1. Objectives/Motivation
• Definition of a logical framework general enough to capture SM, AS, PAS; WFS, WFSX and WFSXP
– 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
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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
– WFSXP and PAS are paraconsistent
– In WFSX and WFSXP, '' frustates some expected properties of the implication
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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)
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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.
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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'
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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
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3. Point Set for LP (Motivation)
thn ttnthp ttp
thn ttnthp ttp
thn ttnthp ttp
thn ttnthp ttp
A, not A
A, A, not A
not A, not A
A
w wA w A wSymbology:
a)
b)
c)
d)
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Frame for Logic Programming
thn ttnthp ttp
P = [hhp,htp,thp,ttp,hhn,htn,thn,ttn],
hhn htnhhp htp
81 possible “propositions”
Enough to capture SM, AS, PAS; WFS, WFSX and WFSXP
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Frame for Logic Programming
• Syntax: given a set of atoms , if , are
formulae, p , , not , ^, ( ), , , , , and are also formulas
• Interpretation in a point w: Iw - set of atoms
• HT4-Interpretation (Bh,Bt), in which
Bh = (Ihhp,Ihhn,Ihtp,Ihtn) Bt = (Ithp,Ithn,Ittp,Ittn) and Ihxp Itxp, Itxn Ihxn,
x {h,t}
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Evaluation on Frames
• Given an HT4-Interpretation M = (Bh,Bt), 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 Iw , 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)
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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')
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Default Negation - Rnot
htp htnhhp hhn
ttpttn
thpthn
7. (M,w) not iff for all w' s.t. w Rnot w' (M,w')
Rnot is plump negative two-place
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Semi-normality Operator - R^
htp htnhhp hhn
ttp ttnthp thn
8. (M,w) ^ iff exists w' s.t. w R^ w' (M,w')
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Possibility Operator - R
htp htnhhp hhn
ttp ttnthp thn
9. (M,w) () iff exists w' s.t. w R w' (M,w')
R is plump
positive two-place
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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
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Model
• An HT4-Interpretation M is a model of a theory T iff for all w W and all formulae in T, then (M,w)
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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
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Minimality Conditions•HT4-Interpretation (Bh,Bt), in which
(Bh,Bt) h (Ch,Ct) iff Bt = Ct and Bh Ch
(Bh,Bt) t (Ch,Ct) iff Bt F Ct
Bh Ch iff Ihxp Jhxp and Jhxn Ihxn, x {h,t}
Bt F Ct iff Ithp Jthp, Jthn Ithn, Jttp Ittp and Ittn Jttn
Bh = {Ihhp,Ihhn,Ihtp,Ihtn} Bt = {Ithp,Ithn,Ittp,Ittn}
Standard ordering:
Fitting’s ordering:
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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
–WFSXP and PAS are paraconsistent
–In WFSX and WFSXP, '' is not interpreted in the same way as in AS and PAS.
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Axioms
A not A
(A) A
() A (A)
not ()
Default Consistency (DC)
Definedness (DE)
Coherence Principle (CP)
No Negative Information (NNI)
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Interpreting logic programs rules
•AS and PAS (Nelson's implication)
B A A (B \/ not A)
•WFSX and WFSXp
B A A (B \/ ^B)
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Truth-values for SM
V1 = [hhn,htn,thn,ttn]
V2 = [hhn,htn,thn,thp,ttn,ttp] ;
V3 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp]
not A
not A, not A
A, not A
•Axioms: NNI + DC + CP + DE
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Truth-values for AS
V1 = [] ;
V2 = [hhn,htn] ;
V3 = [hhn,htn,thn,ttn] ;
V4 = [hhn,htn,thn,thp,ttn,ttp] ;
V5 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp]
not A, not A
A, not A
A, not A
not A
not AAxioms CP + DC + DE
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Truth-Values for PASV1 = [] ;
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
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Truth-values for WFS
V1 = [hhn,htn,thn,ttn] ;
V2 = [hhn,htn,thn,ttn,ttp] ;
V3 = [hhn,htn,thn,thp,ttn,ttp] ;
V4 = [hhn,htn,htp,thn,ttn,ttp] ;
V5 = [hhn,htn,htp,thn,thp,ttn,ttp] ;
V6 = [hhn,hhp,htn,htp,thn,thp,ttn,ttp]
Axioms
NNI + DC + CP
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Truth-values for WFSX and WFSXp
•WFSX Axioms: DC + CP
15 truth-values
•WFSXp Axioms: CP
25 (coherent) truth-values
In WFSXp semantics, all operators are closed w.r.t. available truth-values !
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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
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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
• WFSXp: CP
A belief set B is a WFSXp of a program P iff (B,B) is a general well-founded model of P
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6. Conclusions
• We have defined a logic general enough to capture SM, AS, PAS; WFS, WFSX and WFSXP
• It allows to characterise logically the interrelationship among the semantics
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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