ALCN

Tableaux Calculus Rules

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

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

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

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

Numeric  (  n R)(x) R(x,y 1 ) … R(x,y n ) y 1  y 2 … y n-1  y n Unless z 1, … z n already exist such that R(x,z i ) (1  I  n) and z i  z j (1  I  j  n). The y i ’s must be new distinct variables....

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

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

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 Note: observe that the (  2 R) rule is applicable, but fails....

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

ALCN Tableaux Calculus Sound Terminates Complete Satisfiability is Decidable Satisfiability is PS PACE -complete. See The Description Logic Handbook for details.