Approved Models for Normal Logic Programs Luís Moniz Pereira and Alexandre Miguel Pinto

1. L. M. Pereira and A. M. Pinto Approved Models forNormal Logic Programs Luís Moniz Pereira and Alexandre Miguel Pinto Centre for Artificial Intelligence Universidade Nova de Lisboa

2. L. M. Pereira and A. M. Pinto Approved Models forNormal Logic Programs • Motivation • Notation • The Argumentation Perspective • Our Argumentation • Program Layering • Collaborative Argumentation • Properties • Conclusions and Future Work

3. L. M. Pereira and A. M. Pinto Motivation • Generalize the Argumentation Perspective to all Normal Logic Programs (NLP) by permitting inconsistency removal • Allow revising arguments by Reductio ad Absurdum (RAA) • Identify the 2-valued complete, consistent, and most skeptical models of any NLP • Identify those models respecting the layered stratification of a program

4. L. M. Pereira and A. M. Pinto Notation and Example • An NLP is a set of rules of the form h ← b1, ..., bn, not c1, ..., not cm 'not' denotes default negation • Ex: intend_to_invade ← iran_will_have_WMD iran_will_have_WMD ← not intend_to_invade • An argument A is a set of negative hypotheses (default literals). Above, the argument {not intend_to_invade} attacks itself, i.e., leads to the conclusion intend_to_invade, and so cannot be accepted • This program has no Stable Models

5. L. M. Pereira and A. M. Pinto The Argumentation Perspective • Though {not intend_to_invade} cannot be accepted, by applying RAA in a 2-valued setting, its contrary intend_to_invademust be true • For 2-valued completeness and consistency iran_will_have_WMD is false • In general, using an RAA-inclusive Argumentation Perspective, how to specify and find 2-valued complete, consistent and most skeptical models?

6. L. M. Pereira and A. M. Pinto Received Wisdom • Classically, an Admissible Argument is such that: • it does not attack itself • it counter-attacks all arguments attacking it • Dung's Preferred Extensions are set-inclusion Maximal Admissible Arguments • In general, Preferred Extensions are 3-valued • in the example above the only Preferred Extension is the empty argument {}, yielding a 3-valued model whose literals are all undefined • There are no 2-valued Classical Arguments for all NLPs !

7. L. M. Pereira and A. M. Pinto Our Argumentation • We wish to provide an Argumentation Perspective where all NLPs have a 2-valued semantics based on a 2-valued Argument • Dung's 2-valued Arguments for NLPs correspond exactly to their Stable Models (SMs) • By completing Dung's Arguments via RAA, we obtain conservative 2-valued extensions for the SMs of any NLP

8. L. M. Pereira and A. M. Pinto Approved Models (AMs) • Our approach allows adding positive literals as argument hypotheses, but only insofar as to settle RAA application • Positive hypotheses resolve the Odd Loops Over Negation (OLONs) addressed by RAA. Similarly, they resolve the Infinite Chains Over Negation (ICONs) too • Intuitively, AMs are 2-valued, maximize default literals and minimally add positive literals so as to be complete • AMs without positive literals are the SMs

9. Top-down querying • When top-down querying we can detect OLONs ”on-the-fly” and resolve them with RAA • SM cannot employ top-down query procedures because the semantics is not Relevant, but our extension to SM permits them because it is so • A query literal is supported by the arguments found in its top-down derivation • Relevancy of AM guarantees that any supporting arguments are extendable to a complete model

10. L. M. Pereira and A. M. Pinto ICONs An ICON: p(X) ← p(s(X)) p(X) ← not p(s(X)) Ground version: p(0) ← p(s(0)) p(0) ← not p(s(0)) p(s(0)) ← p(s(s(0))) p(s(0)) ← not p(s(s(0))) ... ... • Approved Models (AMs): • {p(X)} • Ground Approved Models: • {p(0), p(s(0)), p(s(s(0))),...} • This program has no Stable Models!

11. L. M. Pereira and A. M. Pinto Program Layering • Example: • d ← not c • c ← not b • b ← not a • a ← not a • Approved Models • (the first is an RSM): • {a,c} {a,b,d} – Given a, then b is false in the WFM • There are no Stable Models • The Approved Models do not necessarily respect the Layering (≠ from stratification) • Respect of Layering is an optional further requirement • The complying Approved Models are the Revised Stable Models

12. L. M. Pereira and A. M. Pinto Program Layering WFM = < WFM+, WFMu, WFM- > • Program division P // I by interpretation I: • remove from P rules with not a in body, where aI • remove from bodies of remaining rules positive literals aI • M respects the Layering of P iff given some aM let L={bM: b is in the call-graph of a but not vice-versa}; then a is True or Undefined in the WFM of P // L

13. L. M. Pereira and A. M. Pinto Collaborative Argumentation • Collaborative Argumentation caters for consensus arguments wrt an NLP • Our approach enables it, e.g.: • merge arguments into one – possibly self-attacking • build AMs from it by non-deterministically revising (to positive) negative hypotheses leading to self-attacks • An AM is found when a negative maximal and positive minimal argument is reached

14. L. M. Pereira and A. M. Pinto Properties • AMs are consistent 2-valued completions of Preferred Extensions • AM existence is guaranteed for NLPs • AM is Relevant (bonus: and Cumulative) • Layer respecting AMs are the Revised SMs • AMs, RSMs and SMs coincide on programs with neither OLONs nor ICONs

15. L. M. Pereira and A. M. Pinto Conclusions and Future Work • Argumentation approach provides general flexible framework • The framework can adumbrate seamlessly other cases of inconsistency, namely arising from Integrity Constraints and Explicit Negation, and thus encompass (collaborative) Belief Revision • Results could be generalized to Argumentation settings not specific to Logic Programs, keeping to the Occam precept, i.e., skepticism maximizing negative assumptions with the help of minimal positive ones