ALCN

1. ALCN

2. Tableaux Calculus Rules

3. Intersection (C D)(x) C(x) D(x) Unless already present.

4. Union (C D)(x) C(x) D(x) Unless already present.

5. Existential Instantiation (R.C)(x) C(y) R(x,y) The y must be a new variable. Unless a z already exists such that C(z) and R(x,z).

6. Universal Instantiation (R.C)(x) R(x,y) C(y) Unless already present.

7. Numeric  (n R)(x) R(x,y1) … R(x,yn) y1 y2 … yn-1  yn The yi’s must be new distinct variables. . . Unless z1, … zn already exist such that R(x,zi) (1  I  n) and zi  zj (1  I  j  n). .

8. Numeric  (n R)(x) R(x,y1) … R(x,yn+1) [yi/yj] The yi’s must be distinct variables. i.e. wherever possible substitute yj for yi where i > j and yi yj is not present. (If not possible to substitute at least one, CLASH.) .

9. Example ((2 R) (2 R))(x) (2 R)(x) (2 R)(x) R(x, y) R(x, z) y  z <COMPLETED> . Note: observe that the (2 R) rule is not applicable.

10. Example ((3 R) (2 R))(x) (3 R)(x) (2 R)(x) R(x, y) R(x, z) R(x, w) y  z y  w z  w <CLASH> . . . Note: observe that the (2 R) rule is applicable, but fails.

11. Example Show: (2 CHILD) |= (CHILD) Reduce to satisfiability: Negate conclusion, Add to the KB, Put in negation normal form. (2 CHILD CHILD. )(x) (2 CHILD)(x) (CHILD. )(x) CHILD(x, y) CHILD(x, z) y  z (y) <CLASH> • (2 CHILD) |= (CHILD) • (2 CHILD) (CHILD) • (2 CHILD) (CHILD. ) • (2 CHILD) (CHILD. ) • (2 CHILD) (CHILD. ) . Note: <CLASH> because we guarantee at least one for . Also note: <CLASH> is “success”.

12. ALCN Tableaux Calculus • Sound • Terminates • Complete • Satisfiability is Decidable • Satisfiability is PSPACE-complete. See The Description Logic Handbook for details.