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Prenex Normal form, PNF Theorem, Skolem Normal Form
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(a) Prenex Normal Form
A formula is in prenex normal form (PNF) if it has the form
Q 1 xi 1 Q 2 xi 2 · · · Qrxir ψ,
where each Qi is a quantifier (i.e. either ∀ or ∃), and where ψ is a formula containing no quantifiers.
15.1 PNF-Theorem Every φ ∈ Form(L) is logically equivalent to an L-formula in PNF.
Proof: Induction on φ (working in the language with ∀, ∃, ¬, ∧):
φ atomic: OK
φ = ¬ψ, say φ ↔ ¬Q 1 xi 1 Q 2 xi 2 · · · Qrxir χ
Then φ ↔ Q− 1 xi 1 Q− 2 xi 2 · · · Q− r xir ¬χ, where Q−^ = ∃ if Q = ∀, and Q−^ = ∀ if Q = ∃
φ = (χ ∧ ρ) with χ, ρ in PNF Note that ⊢ (∀xjψ[xj/xi] ↔ ∀xiψ), provided xj does not occur in ψ (Ex. 12.5)
So w.l.o.g. the variables quantified over in χ do not occur in ρ and vice versa.
But then, e.g. (∀xα ∧ ∃yβ) ↔ ∀x∃y(α ∧ β) etc. 2
Then ∀x∃yφ(x, y) is satisfiable iff ∀xφ(x, f (x)) is satisfiable. (f is called a Skolem function for φ.)
Proof: ‘⇐’: clear
‘⇒’: Let A be an L-structure with A |= ∀x∃yφ(x, y)
⇒ for every a ∈ A there is some b ∈ A with φ(a, b)
Interpret f by a function assigning to each a ∈ A one such b (this uses the Axiom of Choice!). 2
f (x) =
√ | x | will do.
15.3 Theorem For every L-formula φ there is a formula φ⋆ (with new constant and function symbols) having only universal quantifiers in its PNF such that
φ is satisfiable iff φ⋆^ is.
More precisely, any L-structure A can be made into a structure A⋆ interpreting the new constant and function sym- bols such that
A |= φ iff A⋆^ |= φ⋆.
