Comparing arguments using preference orderings for ... - IEEE Xplore

Comparing Arguments Using Preference Orderings for Argument-Based Reasoning Leila AMGOUD, Claudette CAYROL, Daniel Le BERRE I. R. I. T. Universitk Paul Sabatier, 118 route de Narbonne 3 1062 Toulouse Cedex (FRANCE) Abstract

specificity [12],[18], or explicit priority on the beliefs [3],[10],[8]. However, few works have been devoted to preference-based argumentation. We have proposed a methodological approach to the integration of preference orderings in argumentation frameworks [5]. Two problems have been distinguished: defining preference relations to compare conflicting arguments and specifying principles taking into account these relations in order to select acceptable arguments. The purpose of this paper is to investigate the first topic. After a short presentation of argumentation frameworks, we discuss several proposals for comparing arguments. A comparative study is given which suggests the definition of new preference orderings. The last of the paper is devoted to computational issues. We propose techniques for computing and comparing arguments taking advantage of a recent efficient implementation [6] of an ATMS [14,15].

Argument-based reasoning is a promising approach to handle inconsistent belief bases. The basic idea is to just@ each plausible conclusion by acceptable arguments. The purpose of this paper is to enforce the concept of acceptability by the integration of preference orderings. Pursuing previous work on preference-based argumentation, we focus here on the definition of preference relations for comparing conflicting arguments. We present a comparative study of several proposals. Then, we propose techniques f o r computing and comparing arguments, taking advantage of an Assumption-Based Truth Maintenance System (ATMS).

1. Introduction Two promising approaches to reasoning with inconsistent beliefs have recently emerged: argument-based reasoning [171,[ 18],[101,[ 131,[9],[111 and non-monotonic coherence-based reasoning [16],[2],[5],[7]. Both of them are based on classical reasoning with consistent subsets of the belief base. However, these approaches correspond to two different attitudes in front of inconsistent beliefs: one attitude is to accept inconsistency by providing "good" arguments for each conclusion, the other one is to revise the belief base by selecting preferred consistent subbases. Tight relationships have been established between both approaches [41. The work reported here concerns recent developments of argumentation frameworks. Argumentbased reasoning may be characterized by the following points: the basic principle is "inferring without revising". So, plausible inferences that follow from consistent subsets of an inconsistent belief base must be justified, with so-called arguments. Then, due to the inconsistency of the available knowledge, arguments may be constructed in favor of a statement and other arguments may be constructed in favor of the opposite statement. The basic idea is to view reasoning as a process of first constructing arguments, and then selecting the most acceptable of them. The last point concerns the concept of acceptability. It has been most often defined purely on the basis of other constructible arguments. But other criteria may be taken into account for comparing arguments: for instance,

2. Argumentation framework An argumentation framework is defined as a pair of a set of arguments, and a binary relation representing the "defeat" relationship between arguments. Here, we are interested in arguments built from an inconsistent belief base, using the classical inference. Throughout this paper L is a propositional language. t denotes classical entailment. K and E denote sets of formulas of L. K, which may be empty, represents a core of knowledge and is assumed consistent. Contrastedly, formulas of E represent defeasible pieces of knowledge, or beliefs. So K U E may be inconsistent. K is the context and E will be referred to as the belief base. An argument of E in the context K is a pair (H, h), where h is a formula of L and H is a subbase of E satisfying: (i) K U H is consistent, (ii) K U H t h, (iii) H is minimal (no strict subset of H satisfies i and ii). H is called the support and h the conc2usion of the argument. (See [181 and [ 111for similar definitions). The following definitions for the relation "defeat" are borrowed from [lo]. Let (Hl, hl) and (H2, h2) be two arguments of E. (Hl, hl) rebuts (H2, h2) iff h l = ,h2. (Imeans logical equivalence). Such arguments are said conflicting.

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U E2 U E3 where E l = (a, b, a -+ p}, E2 = {b q}, E3 = (p -+ q}. K = 0. The subbase {b, b -+ 1 q} is BDP-preferred to {a, a -+ p, p -+ q}. More generally, preference relations can be aggregated from an underlying priority relation (a partial ordering) on the belief base. Two aggregation modes, which are adequate for the definition of preferences between arguments, have recently appeared in works about argument-based reasoning. We assume that the belief base E is equipped with a partial pre-ordering 2. Let (H, h) and (HI, h ) be two arguments of E in the context K. H is EFK-preferred to H' iff 3k E H such that V k E H k > k [lo]. Then, (H, h) >>EFK (HI, h ) iff H is EFK-preferred to H . The EFK-preference is used exclusively for comparing conflicting arguments. When the underlying priority ordering is total (or equivalently when the belief base is stratified), H is EFK-preferred to H iff deg(H) < deg(H) wheredeg(H) =max {j/H1 U... U Hj-1 = 0 ) .

(Hl, hl) undercuts (H2,h2) iff for some h E H2, h = Thl. Other definitions are discussed in [41.

2- E = E l

+

3. Introducing preference orderings 3.1 Preference-based argumentation The different theories of acceptability are based on the existence of defeating arguments. Preference orderings have recently emerged from studies in non-monotonic reasoning and belief revision as playing a crucial role. They allow for more sophisticated and more appropriate handling of conflict resolution and default information, for instance. Previous work on prioritized coherence-based entailment [7] and the tight relationships established between argument-based reasoning and coherence-based reasoning [4] indicate that preference orderings should provide promising developments of argument-based reasoning. We consider arguments built in the framework (K, E). Two kinds of preference relations are most commonly encountered: implicit relations which are syntactically extracted from the belief base [12],[18] and explicit relations which are most often induced by a priority ordering on the belief base itself. Here, we focus on explicit preference relations (see [11 for a more complete discussion). We present three relations which are defined in the same way. They are induced by a preference relation defined on the supports of the arguments, which is a partial pre-ordering . The preference relation on the supports is itself induced by a priority relation defined on the belief base. We first present a preference based on certainty levels, which has been introduced in the context of possibilistic logic [3]. The belief base E is stratified in E l u E 2 u ... u E n such that beliefs in Ei have the same level of certainty, and are more reliable than beliefs in Ej where j>i. It is equivalent to consider that E is equipped with a total pre-ordering 2 (a 2 b means that a is at least as certain as b and a > b means that a is strictly more certain than b). The certainty level of a non-empty subbase H is defined as level(H) = min (j / l l j l n and Hj+l U ...U Hn = 0 } ,where Hi denotes H n Ei (with min 0 = n). The certainty level can be used to define a total preordering on the subsets of E. Let H and H' be two consistent subbases of E. H is preferred to H' iff level(H) S level(H). Note that if H c H then H is preferred to H . The associated strict partial ordering can be defined by: H is BDP-preferred to H' iff level(H) c level(H') iff 3k E H' such that tlk E H k > k iff 3 i 21 such that Hi = 0 , H i f 0 andforj > i Hj = H'j = 0. The above equivalences are proved in [13.Let (H, h) and (H',h) be two arguments of E in the context K. We define (H,h) >>BDP (H',h) iff H is BDP-preferred to H .

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?3m.?Qb In Examplel above, B 1 Al. In Example2 above, the subbase (b, b -+ q} is not EFK-preferred to the subbase {a, a -+ p, p + q } . 3- E = E l U E2 U E3 where E l = {a}, E2 = {b}, E3 = { c}. K = 0. (a, b, c} is EFK-preferred to (b, c ) (though {a,b, c} strictly contains (b, c}). The =-preference does not respect minimality for setinclusion. The next preference relation [8] corresponds to the wellknown principle of "elitism": "everything kept must be better than something removed". H is ELI-preferred to H iff Vk E H \ H' 3 k E H \ H such that k zk. If the underlying priority relation 2 on E is finitely chained (no infinite strictly increasing chain), then the ELI-preference is a partial ordering. Note that if H E H' then H is ELI-preferred to H . Thus, the ELI-preference respects minimality for set-inclusion. When the underlying priority ordering is total, H is ELI-preferred to H' iff 321 such that Hi is strictly included in H i and for each j > i, Hj = Hj. (HI, h ) iff H is ELI-preferred to H . Now, (H, h) Example 4- E = E l U E2 U E3 U E4 where E l = (a}, E2 = {b, c}, E3 = (d}, E4 = {e}. K = 0. H = (a, b, e} is ELIpreferred to H = (c, d, e}. H and H' are not comparable by the BDP-preference (level(H) = level(H) = 4).

3.2 Comparative study Duality between EFK-preference and BDPpreference E being equipped with a partial pre-ordering 2, we define the dual relation 2' by: a 2 ' b iff b 2 a. Then, H is BDP-preferred to H' in the structure (E, 2) iff H' is EFK-preferred to H in the structure (E, 2'). Moreover, these two relations may lead to contradictory results as shown by Examplel. Since we emphasize the

ExamDles

1- E = E l U E2 U E3 where E l = (p 4 b}, E2 3 (p 4 7f},E3= { b - + f ) . K = (p}. >>BDP B1: ((b 4 f, p -+ b}, f). Al: ((p -+ l f ) ,

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S denotes a set of propositional symbols, &f?, U 2?3 denotes a partition of S with elements of called assumptions, and B denotes a set of clauses built from S. A set of assumptions is called an environment. A n environment U is incoherent with respect to B iff B U U is logically inconsistent (if no ambiguity on B, U is said incoherent). An environment U is coherent iff it is not incoherent. A nogood is a minimal (for set-inclusion) incoherent environment. Let d be a literal built from S. Let U be a coherent environment. UE label(d) w.r.t. B iff B U U t d and no U' c U satisfies B U U F d. Moreover, the label is complete in the sense that each environment U such that B U U i- d contains at least one element of label(d).

minimality for set-inclusion as a primordial criterium for preference relations between arguments, we drop the EFKpreference in the following. BDP-preference versus ELI-preference These two preference relations have been introduced quite independently, but have analogous behavior in many situations. Indeed, we have proved [l]: T h e o r e m l : Let E = E l U E2 u... U En be a h') be two stratified belief base. Let (H, h) and arguments of E. If (H, h) >>BDP (HI, h') then (H, h) (HI, h'). The converse holds iff level (H n H') < level ( H \ H) (common elements are of high importance). The proof of Theoreml relies upon the following characterlzation of ELI-preference: H is ELI-preferred to H iff level (H \ H') < level (H' \ H). That property shows that the ELI-preference may be viewed as a refinement of the BDP-preference in the sense that the common elements (of H and H') are excluded before taking the level into account. (HI,

entation and AT S The support H of an argument (H, h) in the framework (K, E) is a minimal (for set-inclusion) consistent subset of E such that KuH t h. This definition is very close to the definition of a label. Indeed, an exact correspondence between an argument support and an environment can be obtained by assigning a new assumption to each clause, in a unique way. Then, label computation can be substituted to argument computation. In the following, E is a set of clauses built from 9(so clauses of E are assumption-less). Let E = {Cl, C2, .._Cn}. We define Mod@) = {TAlv C1, 7A2 v C2, ..., TAn v Cn} the modified set of clauses where each clause of E is completed by a unique negative assumption such that if H c E then Mod(H) c Mod(E). Ass(Mod(E)) retums the set of assumptions added to E. In Examplel, E = {p + b, p f +, b + f} and K = {p}. Mod(E) = {A1 p -+ b, A2 p f + , A3 b + f }. Ass(Mod(E)) = {Al,A2,A3} Proposition: (H,d) is an argument of E in the context K iff Ass(Mod(H)) E label(d) w.r.t. B = K U Mod(E). The proof relies upon the following property: Let {Cl, ..., Cp} be a set of clauses built from 3, and Al, ..., Ap be assumptions. { i A l v C1, 7A2 v C2, ... i A p v Cp} U (Al, ..., Ap} F d iff {Cl, ..., Cp} -I d. The equivalence holds because the assumptions Al, . .. Ap do not appear in the clauses C1, .. , Cp. Examdel (continuedl: ({ p -+ b, b + f} , f) is an argument of E in the context K. { {AI, A3}} is the label of f w.r.t. B = K U Mod@). ({p + 7f}, 4)is an argument of E in the context K. { {A2}} is the label of w.r.t. B = K U

The weak-BDP-preference Similarly, we propose another refinement of the BDP-preference which relies upon a kind of lexicographic comparison of the levels. Examvle 5- E = E l U E2 U E3 U E4 where E l = {a], E2 = {b}, E3 = {c, d}, E4 = {e}. K = 0. Let H = {a, d, e} and H' = { b, c, e}. H and H' are neither comparable by the BDPpreference, nor by the ELI-preference. The idea is to extend the concept of level as follows. Let E = E l U E2 u... U En be a stratified belief base and H a consistent subbase of E. For each lain, define levelk(H) = level (Hl U H2 u... U Hk) = min { j/ l l j e and Hj+l U ...U Hk = 0}(with min 0 = k). Then, level(H) = leveln(H). .uEn be a stratified belief base. H Let E = E l u E 2 u.. is weakly-BDP-preferred to H' iff 3 1 a I n s.t. levelk(H) k, level$H)= level$H'). Examples In Example5 above: levelq(H) = levelq(H')= 4; level3(H) = level?(H') = 3; level2(H) = 1, level2(H) = 2, Hence H is weakly-BDP-preferred to H'. 6- E = E l U E2 U E3 U E4 where El = {a}, E2 = (b, c}, E3 = {dl, d2}, E4 = {e}. K = 0.Let H = {a, b, d l , e} and H' = (a, c, d l , d2, e}. H is ELI-preferred to H' but H and H' are not comparable by the weak-BDP-preference. So, ELI-preference and weak-BDP-preference are distinct refinements of BDP-preference.

~ ~ ~ l e ~ e ~ tIna [6], t i we a ~ propose

a

new

implementation of ATMS mechanisms. The major advantage of our methodology is that the computation of the label of a particular literal does not require the computation of other labels. As a consequence, we can use our ATMS on larger bases than De Kleer's original ones. Here, we propose to apply our particular ATMS to argument-based reasoning. Our implementation consists in adding one layer over the ATMS to take into account the stratified base of clauses. The ATMS retums the set of

In this section, we explain why and how an ATMS can be used in order to compute and compare arguments. We briefly recall the basic concepts of ATMS, first introduced by De Kleer [14,151.

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supports of a given conclusion (through the label). The different environments which are thus obtained can be ordered according to a given preference relation. The final step is to decode the information, by replacing each assumption by the corresponding clause. Examde 7- K=(), E = E l U E2 U E3 U E4 where E l = (-+ a, -+ b, a + p), E2 = (b q +), E3 = ( p -+ q, + k) and %-{qk+,+kq). Coding step Mod(E)= (A1 -+a, A 2 + b , A 3 a + p , A 4 b q - + , A5 p -+ q, A6 + k, A 7 q k - + , A S -+ k 4). The stratification of E is taken into account by assigning a rank to each assumption: rank(A) = i iff A is associated to C where C belongs to the stratum Ei. rank(A1) = rank(A2) = rank(A3) = 1; rank(A4) = 2; rank(A5) = rank(A6) = 3; rank(A7) = rank(A8) = 4. Label computation using the ATMS label(ik)= { (Al, A3, A5, A7)) label@)=((A2,A4, AS},(A6) ) Ordering step Let H = {Al, A3, A5, A7}, H' = (A2, A4, A8) and H ' = { A6 1. We obtain: level(H) = 4; level(H) = 4;level(H") = 3. Then H" is BDP-preferred to H' and H is BDP-preferred to H. We obtain the same results with ELI-preference since H' n H" = H n H" = 0. For testing weak-BDP-preference, we group all the assumptions with the same rank: H = (A7) U {A5) U {Al, A3); H = (A8) U (A4) U (A2); H" =(A6). Then, H is weakly-BDP-preferred to H. Conclusion With respect to the weak-BDP-preference, the argument ((-+ b, b q -+ , + k q), k) is better than the argument ((+ a, a + p, p -+q, q k -+I, W. We point out the advantage of an ATMS-based implementation: with an appropriate coding of the initial base of clauses, the ATMS engine produces the desired arguments in a simple form (the support of an argument is a set of assumptions). Due to a unified presentation of the considered preference relations, comparisons of arguments are made easier: we only have to compare levels.

That enables an easy computation of argument comparisons.

References [ l ] L. Amgoud. Etude comparative de relations de preference dans les systkmes d'argumentation.Technica1 report, IRIT, May 1996. [2] S. Benferhat, C. Cayrol, D. Dubois, J. Lang, H. Prade. Inconsistency management and prioritized syntax-based entailment. Proc. IJCAIP3, 640-645. [3] S . Benferhat, D. Dubois, H. Prade. Argumentative Inference in Uncertain and Inconsistent Knowledge Bases. Proc. 9" Conf. on Uncertainty in Artificial Intelligence, 411419, 1993. 141 C. Cayrol. On the relation between Argumentation and Non-monotonic Coherence-based Entailment. Proc. IJCAI'95, 1443-1448. [5] C. Cayrol. From Non-monotonic Syntax-based Entailment to Preference-based Argumentation. Proc. ECSQARU'95 (C. Froidevaux, J. Kohlas Eds.), LNAI 946, Springer Verlag, 99106, 1995. [6] T. Castell, C. Cayrol, M. Cayrol, D. Le Berre. Using the Davis & Putnam procedure for an efficient computation of preferred models. Proc. ECAI'96, to appear. [7] C. Cayrol, M.C. Lagasquie-Schiex. Non-monotonic Syntax-Based Entailment : A Classification of Consequence Relations. Proc. ECSQARU'95 (C. Froidevaux, J. Kohlas Eds.), LNAI 946, Springer Verlag, 107-114, 1995. [8] C. Cayrol, V . Royer, C. Saurel. Management of preferences in Assumption-Based Reasoning. In: Advanced Methods in Artificial Intelligence (B. Bouchon-Meunier, L. Valverde, R.Y. Yager Eds.), LNCS 682, Springer Verlag, 1322, 1993. [9] P.M. Dung. On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77: 321-357. 1995. [lo] M. Elvang-Goransson, J. Fox, P. Krause. Dialectic reasoning with inconsistent information. Proc. 9' Conf. on Uncertainty in Artificial Intelligence, 114-121, 1993. [113 M. Elvang-Goransson, A. Hunter. Argumentative logics : Reasoning with classically inconsistent information. Data & Knowledge Engineering, 16: 125-145, 1995. [12] H. Geffner. Default Reasoning: Causal and Conditional Theories. MIT Press, 1992. [ 131 A. Hunter. Defeasible reasoning with structured information. Proc. KR'94, 281-292. [14] J. de Kleer. An assumption-based TMS. Artificial Intelligence, 28: 127-162, 1986. [15] J. de Kleer. Extending the ATMS. Artificial Intelligence, 28: 163-196, 1986. [16] G . Pinkas, R.P. Loui. Reasoning from inconsistency: a taxonomy of principles for resolving conflicts. Proc. KR'92, 709-71 9. [17] J.L. Pollock. How to reason defeasibly. Artificial Intelligence, 57: 1-42, 1992. [18] G.R. Simari, R.P. Loui. A mathematical treatment of defeasible reasoning and its implementation. Artificial Intelligence, 53: 125-157, 1992.

5. Concluding remarks The work reported here concerns preference-based argumentation from an inconsistent belief base. We have investigated the definition of preference orderings for comparing arguments, and more particularly conflicting arguments. We have discussed several proposals in a unified framework, and the comparative study we have done has suggested us the definition of new preference orderings. All the preference relations we have kept can be aggregated from an underlying priority relation on the belief base. Then, we have proposed computational techniques for implementing three preference relations. With an appropriatecoding of the straufied belief base, the arguments are produced by an ATMS, in a simple form.

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