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This document provides a comprehensive overview of advanced data modeling concepts with a focus on first-order logic. It covers essential topics including the syntax and semantics of first-order logic, Herbrand interpretations, terms, formulas, substitutions, literals, and clauses. Readers will find examples illustrating the abstract and concrete notations, foundational principles of Datalog, and the use of substitutions in various contexts. This guide serves as a valuable resource for mastering the complexities of logical structures in data modeling.
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Advanced Data Modeling Steffen Staab with Simon Schenk TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA
Overview • First-order logic. Syntax and semantics. • Herbrand interpretations; • Clauses and goals; • Datalog.
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 §.
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
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).
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
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 {}
Application of substitution • Application of a substitution µto a term t: • xµ = µ(x) • cµ = c • f(t1, … , tn)µ = f(t1µ, … , tnµ)
Herbrand interpretation • A Herbrand interpretation of a signature § is any set of ground atoms of this signature.
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 §
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
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
Goal • Goal (also normal goal) is any conjunction of literals L1Æ … ÆLn • Definite goal: all Li are positive • Empty goal □: when n = 0
