## Belief revision and answer-set programming

The revision problem in the context of GIS is represented in propositional calculus and amounts to revise a knowledge base K represented by a finite set of clauses by a new item of information A represented by another finite set of clauses. The revision method is the Removed Sets Revision, which removes the minimal subsets of clauses from the initial knowledge base K, called removed sets, in order to restore consistency, while keeping the new information. We first formalize the Removed Sets Revision in terms of answer set programming, translating the revision problem into a satisfiabily problem.

We first apply a transformation on K, denoted by H(K), introducing for each clause of K, a new variable, called hypothesis variable, which acts as a clause selector and we built a logic program P, in the same spirit of Niemela, corresponding to the union of H(K) and A. The revision of H(K) by A amounts to look for the answer sets of P which minimize the number of hypothesis variables assigned false. We formally established the correspondence between removed sets and answer sets, which minimize the number of hypothesis variables assigned false. We then adapted the S-models algorithm proposed by I. Niemela and P. Simons and proposed an algorithm, called Rsets, in order to compute the answer sets corresponding to removed sets. The main adaptation of the original S-models algorithm consists in stopping the recursive call to avoid certain answer sets and throwing away certain answer sets already found. We conducted an experimental study on the flooding application. This revision problem is represented in propositional calculus and we focused on an area consisting in 120 compartments, which involve 33751 propositional clauses with 2343 propositional variables. The test was conducted on a Pentium III at 1GHz with 256Mo of RAM. Until 20 compartments the Rsets algorithm gave similar results than the ones obtained by the previously proposed REM algorithm, however from 25 compartments the Rsets algorithm is significally more efficient than the REM algorithm. Even if Rsets algorithm is significally better than the REM one, it can only deal with 80 compartments. In order to deal with the whole area we introduced the Prioritised Removed Sets Revision.

Prioritised Removed Set Revision (PRSR) generalizes the Removed Set Revision to the case of prioritised belief bases. Let K be a prioritised finite set of clauses, where K is partitioned into n strata, such that clauses in Ki have the same level of priority and are more proprietary than the ones in Kj where i is lower than j. K1 contains the clauses which are the most proprietary beliefs in K, and Kn contains the ones which are the least proprietary in K. When K is prioritised in order to restore consistency the principle of minimal change stems from removing the minimum number of clauses from K1, then the minimum number of clauses in K2, and so on. We introduce the notion of prioritised removed sets, which generalizes the notion of removed set in order to perform Removed Sets Revision with prioritised sets of clauses. This generalization requires the introduction of a preference relation between subsets of K reflecting the principle of minimal change for prioritised sets of clauses. We then formalize the Prioritised Removed Sets Revision in terms of answer set programming. We first construct a logic program, in the same spirit of Niemela but, for each clause of K, we introduce a new atom and a new rule, such that the preferred answer sets of this program correspond to the prioritised removed sets of the union of K and A. We then define the notion of preferred answer set in order to perform PRSR. In order to get a one to one correspondence between preferred answer sets and prioritised removed sets, instead of computing the set of preferred answer sets of $P_{K \cup A}$ we compute the set of subsets of literals which are interpretations of Rk and that lead to preferred answer sets.

The computation of Prioritised Removed Sets Revision is based on the adaptation of the smodels system. This is achieved using two algorithms. The first algorithm, Prio, is an adaptation of smodels system algorithm which computes the set of subsets of literals of Rk which lead to preferred answer sets and which minimize the number of clauses to remove from each stratum. The second algorithm, Rens, computes the prioritised removed sets of the union of K and A, applying the principle of minimal change for PRSR, that is, stratum by stratum. In the flooding application we have to deal with an area consisting of 120 compartments and the stratification is useful to deal with the whole area. A stratification of S1 is induced from the geographic position of compartments. Compartments located in the north part of the valley are preferred to the compartments located in the south of the valley. Using stratification, the Rens algorithm can deal with the whole area with a reasonable running time.

We first apply a transformation on K, denoted by H(K), introducing for each clause of K, a new variable, called hypothesis variable, which acts as a clause selector and we built a logic program P, in the same spirit of Niemela, corresponding to the union of H(K) and A. The revision of H(K) by A amounts to look for the answer sets of P which minimize the number of hypothesis variables assigned false. We formally established the correspondence between removed sets and answer sets, which minimize the number of hypothesis variables assigned false. We then adapted the S-models algorithm proposed by I. Niemela and P. Simons and proposed an algorithm, called Rsets, in order to compute the answer sets corresponding to removed sets. The main adaptation of the original S-models algorithm consists in stopping the recursive call to avoid certain answer sets and throwing away certain answer sets already found. We conducted an experimental study on the flooding application. This revision problem is represented in propositional calculus and we focused on an area consisting in 120 compartments, which involve 33751 propositional clauses with 2343 propositional variables. The test was conducted on a Pentium III at 1GHz with 256Mo of RAM. Until 20 compartments the Rsets algorithm gave similar results than the ones obtained by the previously proposed REM algorithm, however from 25 compartments the Rsets algorithm is significally more efficient than the REM algorithm. Even if Rsets algorithm is significally better than the REM one, it can only deal with 80 compartments. In order to deal with the whole area we introduced the Prioritised Removed Sets Revision.

Prioritised Removed Set Revision (PRSR) generalizes the Removed Set Revision to the case of prioritised belief bases. Let K be a prioritised finite set of clauses, where K is partitioned into n strata, such that clauses in Ki have the same level of priority and are more proprietary than the ones in Kj where i is lower than j. K1 contains the clauses which are the most proprietary beliefs in K, and Kn contains the ones which are the least proprietary in K. When K is prioritised in order to restore consistency the principle of minimal change stems from removing the minimum number of clauses from K1, then the minimum number of clauses in K2, and so on. We introduce the notion of prioritised removed sets, which generalizes the notion of removed set in order to perform Removed Sets Revision with prioritised sets of clauses. This generalization requires the introduction of a preference relation between subsets of K reflecting the principle of minimal change for prioritised sets of clauses. We then formalize the Prioritised Removed Sets Revision in terms of answer set programming. We first construct a logic program, in the same spirit of Niemela but, for each clause of K, we introduce a new atom and a new rule, such that the preferred answer sets of this program correspond to the prioritised removed sets of the union of K and A. We then define the notion of preferred answer set in order to perform PRSR. In order to get a one to one correspondence between preferred answer sets and prioritised removed sets, instead of computing the set of preferred answer sets of $P_{K \cup A}$ we compute the set of subsets of literals which are interpretations of Rk and that lead to preferred answer sets.

The computation of Prioritised Removed Sets Revision is based on the adaptation of the smodels system. This is achieved using two algorithms. The first algorithm, Prio, is an adaptation of smodels system algorithm which computes the set of subsets of literals of Rk which lead to preferred answer sets and which minimize the number of clauses to remove from each stratum. The second algorithm, Rens, computes the prioritised removed sets of the union of K and A, applying the principle of minimal change for PRSR, that is, stratum by stratum. In the flooding application we have to deal with an area consisting of 120 compartments and the stratification is useful to deal with the whole area. A stratification of S1 is induced from the geographic position of compartments. Compartments located in the north part of the valley are preferred to the compartments located in the south of the valley. Using stratification, the Rens algorithm can deal with the whole area with a reasonable running time.