1. Reductio ad Absurdum Argumentation in Normal Logic Programs Luís Moniz Pereira and Alexandre Miguel Pinto CENTRIA – Centro de Inteligência Artificial, UNL Lisbon, Portugal ArgNMR’07 Luís Moniz Pereira Alexandre Miguel Pinto (lmp|amp)@di.fct.unl.pt May 14th, 2007 Tempe, Arizona

2. Outline Outline Background and Motivation Revision Complete Scenarios Stable Models and Revision Complete Scenarios Collaborative Argumentation Conclusions and Future Work

3. Background (ground) Normal Logic Program P: set of rules of the form (n,m  0) h  b 1, b 2,..., b n, not c 1, not c 2,..., not c m Motivation In Stable Models (SM) semantics a Normal Logic Program (NLP) not always has a semantics (at least one model) If several NLPs are put together (joining KBs) the resulting NLP may not have any SM. Ex: travel  not mountain mountain  not beach beach  not travel How to ensure that every NLP has at least one 2-valued model?

4. Revision Complete Scenarios Classically, a scenario is a Horn theory P  H, where H is a set of negative (default negated) hypotheses Consider NLPs as argumentation systems Take one set H - of negative hypotheses and draw all possible conclusions from P  H -, i.e., calculate the least model of P  H - – least( P  H - ) If contradictions, ie. pairs {not_L, L}, arise in least( P  H - ) : Revise the initial set H - of negative hypotheses by removing one negative hypothesis L such that {not_L, L} ⊆ least( P  H - ) Repeat until there are no contradictions in least( P  H - ) Add as positive hypotheses to H + the positive literals needed to ensure 2-valued completeness of least( P  H ), where H = H -  H +

5. Revision Complete Scenarios A Revision Complete Scenario is a Horn theory P  H, where H = H +  H - is a set of hypotheses, positive and negative H - is a Weakly Admissible set of negative hypotheses, i.e., every evidence E= {not L 1, not L 2,..., not L n } attacking H - is counter-attacked by P  H -  E H + are the non-redundant and unavoidable positive hypotheses needed to ensure 2-valued completeness and consistency of the model for P  H H + is non-redundant iff there is no h + in H + already derived by the remaining hypotheses, i.e, P  H \ {h + } |-/- h + H + is unavoidable iff for every h + in H +, h + is indispensible to guarantee that P  H is consistent, i.e., least(P  H \ {h + }  {not h + }) is inconsistent – contains a pair {not_L, L}

6.  An example: P = travel  not mountain mountain  not beach beach  not travel H- = {not mountain, not beach, not travel} H + =  least(P  H) = {not mountain, not beach, not travel, mountain, beach, travel} Select one L such that least(P  H)  {L, not L}: L = mountain Remove not mountain from H - H- = {not beach, not travel} least(P  H) = {not beach, not travel, mountain, beach} Select one L such that least(P  H)  {L, not L}: L = beach Remove not beach from H - H- = {not travel} least(P  H) = {not travel, beach}, which is consistent but not 2- valued complete We complete the scenario by adding the positive hyposthesis ‘mountain’ to H+ H=H-  H+={not travel}  {mountain}={not travel, mountain} H + = {mountain} least(P  H) = {not travel, beach, mountain} is consistent and 2-valued complete The other 2 alternative scenarios ({not beach, mountain, travel} and {not mountain, beach, travel}) are simmetrical to this one

7. Stable Models and Revision Complete Scenarios Stable Models and Revision Complete Scenarios Every Stable Model of a NLP P is the Least Model of some Revision Complete Scenario P  H, where H = H +  H -, and H + =  Stable Models do not exist for every NLP, but Revision Complete Scenarios do The least models of Revision Complete Scenarios are the Revised Stable Models of the NLP

8. Collaborative Argumentation Classically, argumentation is viewed as a battle between opponents where each one’s hypotheses attack the others’ Our approach facilitates collaborative argumentation in the sense that it provides a method for finding a consensus solution of two (or more) opposing arguments. This is done by Merging the different arguments H 1, H 2,..., H n into a single H Revising the negative hypotheses needed to eliminate inconsistencies in P  H Adding the unavoidable and non-redundant positive hypotheses needed to ensure 2-valued completeness

9. Conclusions Revision Complete Scenarios extend the Stable Models semantics guaranteeing existence of a 2-valued complete and consistent model Stable Models can be viewed as the result of an iterative process of belief revision (revising hypotheses from negative to positive) Revision Complete Scenarios provide a framework for Collaborative Argumentation Future Work Extend this approach to Generalized Logic Programs Extend this argumentation approach to rWFS Integration with other Belief Revision methods

10.  Further examples a  not aa is unavoidable b  aa  not a b is redundant, a is non-redundant