Suppose there is a π-violation of a positive constraint C in G ; this violation ( C i π ) is said to be R-restorable if there exist an R-derivation from G into a SG H and a projection π ′ from H into irr( H ) such that the projection π ′ o π of the trigger of C into irr( H ) can be extended to a projection of C as a whole. The violation of a negative constraint can never be restored. Note that the R-restoration can create new violations, that must themselves be proven R-restorable DEFINITION 13 ( SRC-CONSISTENCY and SRC-DEDUCTION ) A KB K = ( G,R,C ) is consistent if, for any SG H that can be R-derived from G, for every constraint C E C, for ery π-violation of C in H , ( C i π) is R-restorable. A SG Q can be deduced from K if K is consistent and Q can be deduced from ( G,R).

![0_image_0.png](0_image_0.png)

Consider for instance a KB containing the SG G in Fig. 13, expressing the existence of the number 0, a constraint and a rule, both represented by the colored SG K. The constraint asserts that for every integer n, there must be an integer n ′ , successor of n. If the rule is an evolution rule, G is seen as an inconsistent initial world (there is no successor of 0 in G ) and nothing will be deduced from this KB. If the rule is an inference rule, its application immediately repairs the constraint violation, while creating a new integer, that has no successor, thus a new violation. Finally, every constraint violation could eventually be repaired by a rule application, and the KB should be proven consistent.

Let us point out that the SR model is obtained from SRC or SEC when C is empty, and SGC is obtained from SRC (resp. SEC ) when R (resp. E ) is empty.

The SREC model combines both derivation semes of the SRC and SEC models. Now, G describes an initial world, inference rules of R complete the description of any world, constraints of C evaluate the consistency of a world, evolution rules of E try to make a consistent world evolve into a new, consistent one. The deduction problem asks whether G
can evolve into a consistent world satisfying the goal.

DEFINITION 14 ( SREC-DEDUCTION ) A SG G l is an immediate RE-evolution from a SG G
if there exists an R-derivation from G into C 11 and an immediate E-derivation from G 11 into G'. An RE-evolution from a SG G to a SG G' is a sequence of SGs G = G 0 , . . , G k = G'
such that, for 0 ≤ i ≤ k, (G i , R, C) is consistent and, for 1 ≤ i ≤ k, G i is an immediate RE-evolution from G i − 1 . Given a KB K = (G, R, E,C), a SG Q can be deduced from K if there is an RE-evolution G = G0,..., Gk such that Q can be deduced from ( Gk, R ).

When E = 0 (resp. R = 0), one obtains the SRC model (resp. SEC ).