Semantics of FO Formulas

1. Semantics of FO Formulas

2. Semantics • Given a vocabulary R1, …, Rk • A model is D = (D, R1D, …, RkD) • D = a set, called domain, or universe • RiD D  D  ...  D, (ar(Ri) times) i = 1,...,k

3. Semantics • Given: • A model D = (D, R1D, ..., RkD) • A formula  • A substitution s : {x1, x2, ...}  D • We define next the relation:meaning “D satisfies with s” D |= [s]

4. Semantics If (s(t1), ..., s(tn))  RD D |= (R(t1, ..., tn)) [s] D |= (t= t’) [s] If s(t) = s(t’)

5. Semantics D |= (  ’) [s] If D |= ()[s] and D |= (’) [s] D |= (  ’) [s] If D |= ()[s] or D |= (’) [s] D |= (’) [s] If not D |= ()[s]

6. First Order Logic: Semantics If for all s’ s.t. s(y) = s’(y) for all variables y other than x, D |= ()[s’] D |= (x.) [s] D |= (x.) [s] If for some s’ s.t. s(y) = s’(y) for all variables y other than x, D |= ()[s’]

7. FO and Databases • FO:a sentence  is true in D if D |=  • Databases:a formula  with free variables x1, ..., xn defines the query:(D) = {(s(x1), ..., s(xn)) | D |= [s]}