1. Advanced Data Modeling Steffen Staab with Simon Schenk TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Overview • First-order logic. Syntax and semantics. • Herbrand interpretations; • Clauses and goals; • Datalog.

3. 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 §.

4. 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 t1, … , tn are terms, then f(t1 , …, tn) is a term. • A term is ground if it has no variables • var(t) — the set of variables of t

5. 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).

6. Formulas • Atomic formulas, or atoms p(t1, …, tn). • (A1Æ … Æ An) and (A1Ç … Ç An) • (A ! B) and (A $ B) • :A • 8v A and 9v A

7. 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: { x1 t1, … , xn tn } • empty substitution {}

8. Application of substitution • Application of a substitution µto a term t: • xµ = µ(x) • cµ = c • f(t1, … , tn)µ = f(t1µ, … , tnµ)

9. Herbrand interpretation • A Herbrand interpretation of a signature § is any set of ground atoms of this signature.

10. Truth in Herbrand Interpretations • 1. If A is atomic, then I ² A if A 2 I • 2. I ² B1Æ … Æ Bn if I ² Bi for all i • 3. I ² B1Ç … Ç Bn if I ² Bi for some i • 4. I ² B1! B2 if either I ² B2 or I ² B1 • 5. I ²: B if I ² B • 6. I ²8 xB if I ² B{x  t} for all ground terms t of the signature § • 7. I ²9xB if I ² B{x  t} for some ground term t of the signature §

11. 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

12. Clause • Clause: (or normal clause) formula L1Æ … Æ Ln! A, where • n ¸ 0, each Li is a literal and A is an atom. • Notation: A :- L1Æ … Æ Ln or A :- L1, … , Ln • Head: the atom A. • Body: The conjunction L1Æ … Æ Ln • Definite clause: all Li are positive • Fact: clause with empty body

13. Classification

14. Goal • Goal (also normal goal) is any conjunction of literals L1Æ … ÆLn • Definite goal: all Li are positive • Empty goal □: when n = 0