### Table 3.3: Complexities in case of Horn bases Proof for m = T We may still use the previously stated algorithms. Using the fact that the complexity of the entailment problem is reduced, we conclude that Uni-T-Horn is in co-NP, Exi-T-Horn is in NP and Arg-T-Horn is in p 2. The completeness proofs are the following: For Uni-T-Horn, we use an idea previously proposed in [14]: Sat can be polynomially transformed to co-Uni-T-Horn (Sat is the satis ability problem for any set of clauses, not only Horn clauses). Let C = fCjg for j 2 f1; : : :; qg a given set of clauses, let V (C) = fx1; : : :; xng the set of propositional variables used in C, take: 14

### Table 3: Complexities in case of Horn clauses Class membership for m = T: We may still use the previously stated algorithms. Using the fact that the complexity of the entailment problem is reduced, we conclude that Uni-T-Horn is in co-NP, Exi-T- Horn is in NP and Arg-T-Horn is in p 2.

1994

Cited by 3

### Table 5. Explanation of non-horn clause introduction

2004

"... In PAGE 8: ... This seems to be due to the increased number of complex clauses generated by the absorption optimisation (as we can see in Table 6, the fraction of non-horn clauses grows 4 times in the absorbed case for the Wine ontology), which makes the proving process harder for Vampire. An example illustrating how absorption can increase the number of non-horn clauses is given in Table5 . Here C, D, E and F are concept names.... ..."

Cited by 14

### Table 4: Converting a Horn Clause to a Closure Operator ^ .

1991

"... In PAGE 41: ... We can convert this math- ematics for ATMS computation into pseudo-code in the way Tables 1 and 2 did for version spaces. Pseudo-code for the operation 7! ^ , which takes a Horn clause to the corresponding closure operator on la- bel functions, is given in Table4 . In the program there, the following functions are taken from the anti-chains interface: upper_union upper_homogeneous_intersection singleton empty The rst two of these we have encountered before.... In PAGE 42: ... The Common Lisp function apply is similar but does not use the `end value apos; x. In Table4 , the end value used is f;g (which is not to be confused with the empty set itself). In particular, note the case in which has the form ) a so that S is the emptyset.... ..."

Cited by 6

### Table 6.5: Local state graphs used to generate the FSM for modular veri cation of properties concerning restriction of machine mobility

### Table 4. Horn axioms for describing equality.

"... In PAGE 9: ... However, it is well known that one can axiomatically describe an equality predicate in Horn logic, as long as only finitely many predicate symbols are considered. The corresponding clauses are depicted in Table4 . In order to ensure that these rules do not impair decidability, one must slightly extend them to become DL- safe, as explained in Section 5.... ..."

### Table 3.6: Equivalent queries to be operationalized into Horn-clause rules

1996

Cited by 5