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Advanced ALCN Tableaux Calculus: Intersection, Union, Instantiation, and Numerical Constraints

This document discusses the ALCN tableaux calculus, covering rules for intersection and union, existential and universal instantiation, and numeric constraints in logic. It details how to apply these rules effectively, ensuring new variables are used unless already present, and explores the Clash phenomenon when specific substitutions fail. Examples illustrate the application of these rules, showcasing satisfiability in decoupling complex statements in knowledge bases. The ALCN system is characterized as sound, terminating, complete, and capable of decidable satisfiability, essential for computational logic.

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Advanced ALCN Tableaux Calculus: Intersection, Union, Instantiation, and Numerical Constraints

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  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.

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