Relating Defeasible and Normal Logic Programming through Transformation

Relating Defeasible and Normal Logic Programming through Transformation Properties C. I. Chesñevar – J. Dix – F. Stolzenburg – G. Simari Departamento de Ciencias de la Computación Universidad Nacional del Sur, B. Blanca, ARGENTINA Univeristät Koblenz-Landau, Koblenz, GERMANY CACIC 2000 – Ushuaia, Argentina 1

Outline • Introduction and motivations • Transformation rules in NLP. Notions • De. LP. Transformation rules in De. LP. • Comparison • Conclusions CACIC 2000 – Ushuaia, Argentina 2

Introduction and Motivations • Defeasible Logic Programming (De. LP) is a logicprogramming formalism which relies upon defeasible argumentation for solving queries. • A number of transformation rules have been developed to simplify a normal logic program (NLP) and to compute its well-founded semantics (WFS). • We focus on finding similar transformation rules for De. LP, which allow us to simplify the knowledge encoded in a De. LP program. CACIC 2000 – Ushuaia, Argentina 3

Transformation Rules for NLP Red+ p q, not r q s s Red- Sub Unfold p q, not r p q q s p q, r r q p q, r, s r t, w w p q q s q r s since r does notsince r is a since appear in any fact {q} {q, r} head CACIC 2000 – Ushuaia, Argentina p q, t, w, s r t, w w the first rule was ‘unfolded’ 4

Transformation Rules for NLP Taut p q, not r u u r Obtaining a residual program: p q, not r u u q s s via Taut p q, not r r tautologies can be deleted p q, not r q s s via Red+ p q q s s CACIC 2000 – Ushuaia, Argentina via Unfold p s q s s via Unfold p s q s via Unfold p q s Residual program 5

Defeasible Logic Programming • De. LP allows both strict and defeasible rules. • When defeasible rules are used in a proof, we get an argument (tentative proof) • Arguments may be defeated by counterarguments. • An argument A is deemed better than other argument B when A is more specific than B. • De. LP allows both strict (~) and default negation (not). • We will focus on De. LPneg (only strict negation). CACIC 2000 – Ushuaia, Argentina 6

De. LP: example • Given the following De. LP program. flies(X) —< bird(X) ~flies(X) —< bird(X), wounded(X) bird(X) —< feathered(X), beak(X) feathered(kaiken) beak(kaiken) wounded(kaiken) A 1 Argument A 2 for ~flies(tweety) is more specific (more informed) than A 1 for flies(tweety) flies(k) bird (k) feath’ed(k) beak (k) A 2 bird (k) wounded (k) feath’ed(k) beak (k) CACIC 2000 – Ushuaia, Argentina 7

Our goal dlp P nlp P semanticspreserving Transnlp( P) Transdlp( P) Wfs( P) ? CACIC 2000 – Ushuaia, Argentina Semdlp( P) 8

Transformations in De. LPneg Red+ in NLP innocent not guilty Redneg+ does not hold in De. LP Red+ in NLP: not A holds iff A cannot be derived innocent Redneg+in De. LP guilty ~innocent Redneg+ in De. LP: ~A holds iff ~A can be derived guilty CACIC 2000 – Ushuaia, Argentina 9

Transformations in De. LPneg Tautneg holds for De. LP. Redneg- holds for De. LP. Tautneg Redneg - flies —< bird flies —< bird, ~dead flies —< bird dead CACIC 2000 – Ushuaia, Argentina 10

Transformations in De. LPneg Unfoldneg holds in De. LP only for strict rules. Unfoldneg Subneg holds in De. LP only for strict rules Subneg flies —< bird ~flies —< bird, wounded bird —< feathered, beak feathered beak wounded flies—< feathered, beak ~flies —<bird, wounded bird —<feathered, beak p —< q 1, q 2 p —< q 1 ~p —< q 2 CACIC 2000 – Ushuaia, Argentina ~r q r w 1, w 2 r w 1 ~r q r w 1 11

Behavior of NLP, Delpneg, Delpnot NLP (wrt WFS) RED+ DELPneg DELPnot RED- yes no yes yes SUB yes (for strict rules only) yes yes UNFOLD yes TAUT yes CACIC 2000 – Ushuaia, Argentina 12

Relating NLP and De. LP • Argumentation-based semantics has been given to NLP using an abstract argumentation framework. • Some properties of NLP under wfs are also present in De. LP (Taut and Red-). • Other transformation properties only hold for strict rules (e. g. SUB), sometimes with extra requirements (e. g. UNFOLD). • Some properties (e. g. Red+) do not hold wrt. strict negation, but do hold wrt. default negation. CACIC 2000 – Ushuaia, Argentina 13

Conclusions • We have related classical logical programming semantics to De. LP semantics by comparing analogies wrt. suitable transformation rules. • The differences obtained are to be found in the expressive power of De. LP for knowledge encoding. • These results can be applied to gain a better understanding of links between argumentationbased frameworks and logic programming. CACIC 2000 – Ushuaia, Argentina 14