Information Systems & Semantic Web University of Koblenz ▪ Landau, Germany Advanced Data Modeling Steffen Staab with Simon Schenk TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAA

ISWeb - Information Systems & Semantic Web Steffen Staab Overview  First-order logic. Syntax and semantics.  Herbrand interpretations;  Clauses and goals;  Datalog.

ISWeb - Information Systems & Semantic Web Steffen Staab First-order signature First-order signature § consists of  con — the set of constants of § ;  fun — the set of function symbols of § ;  rel — the set of relation symbols of §.

ISWeb - Information Systems & Semantic Web Steffen Staab Terms Term of § with variables in X: 1. Constant c 2 con; 2. Variable v 2 X; 3. If f 2 fun is a function symbol of arity n and t 1, …, t n are terms, then f(t 1, …, t n ) is a term.  A term is ground if it has no variables  var(t) — the set of variables of t

ISWeb - Information Systems & Semantic Web Steffen Staab Abstract and concrete notation Abstract notation:  a, b, c, d, e for constants;  x, y, z, u, v, w for variables;  f, g, h for function symbols;  p, q for relation symbols, Example: f(x, g(y)). Concrete notation: teletype font for everything. Variable names start with upper-case letters. Example: likes(john, Anybody).

ISWeb - Information Systems & Semantic Web Steffen Staab Formulas  Atomic formulas, or atoms p(t 1, …, t n ).  (A 1 Æ … Æ A n ) and (A 1 Ç … Ç A n )  (A ! B) and (A $ B)  : A  8 v A and 9 v A

ISWeb - Information Systems & Semantic Web Steffen Staab Substitutions  Substitution µ : is any mapping from the set V of variables to the set of terms such that there is only a finite number of variables v 2 V with µ (v)  v.  Domain dom( µ ), range ran( µ ) and variable range vran( µ ): dom( µ ) = {v | v  µ (v)}, ran( µ ) = { t | 9 v 2 dom( µ )( µ (v) = t)}, vran( µ ) = var(ran( µ )).  Notation: { x 1  t 1, …, x n  t n }  empty substitution {}

ISWeb - Information Systems & Semantic Web Steffen Staab Application of substitution Application of a substitution µ to a term t:  x µ = µ (x)  c µ = c  f(t 1, …, t n ) µ = f(t 1 µ, …, t n µ )

ISWeb - Information Systems & Semantic Web Steffen Staab Herbrand interpretation A Herbrand interpretation of a signature § is any set of ground atoms of this signature.

ISWeb - Information Systems & Semantic Web Steffen Staab Truth in Herbrand Interpretations 1. If A is atomic, then I ² A if A 2 I 2. I ² B 1 Æ … Æ B n if I ² B i for all i 3. I ² B 1 Ç … Ç B n if I ² B i for some i 4. I ² B 1 ! B 2 if either I ² B 2 or I ² B 1 5. I ² : B if I ² B 6. I ² 8 xB if I ² B{x  t} for all ground terms t of the signature § 7. I ² 9 xB if I ² B{x  t} for some ground term t of the signature §

ISWeb - Information Systems & Semantic Web Steffen Staab Literals  Literal is either an atom or the negation : A of an atom A.  Positive literal: atom  Negative literal: negation of an atom  Complimentary literals: A and : A  Notation: L

ISWeb - Information Systems & Semantic Web Steffen Staab Clause Clause: (or normal clause) formula L 1 Æ … Æ L n ! A, where  n ¸ 0, each L i is a literal and A is an atom.  Notation: A :- L 1 Æ … Æ L n or A :- L 1, …, L n  Head: the atom A.  Body: The conjunction L 1 Æ … Æ L n  Definite clause: all L i are positive  Fact: clause with empty body

ISWeb - Information Systems & Semantic Web Steffen Staab Classification ClauseClass lives(Person, sweden) :- sells(Person, wine, Shop), not open(Shop,saturday) normal spy(Person) :- russian(Person)definite spy(bond)fact

ISWeb - Information Systems & Semantic Web Steffen Staab Goal  Goal (also normal goal) is any conjunction of literals L 1 Æ … Æ L n  Definite goal: all L i are positive  Empty goal □: when n = 0