Computability and Logic

19.2. SKOLEM NORMAL FORM 247
both satisfiable or both unsatisfiable, though generally when we prove the existence
of such equivalents our proof will actually provide some additional information,
indicating a stronger relationship between the two sets. Two different results on the
existence of equivalents for satisfiability will be established in sections 19.2 and
19.4. In each case, Ɣ will be shown to have an equivalent for satisfiability Ɣ* whose
sentences will be of a simpler type syntactically, but which will involve new nonlogical
symbols.
In this connection some terminology will be useful. Let L be any language, and L +
any language containing it. Let M be an interpretation of L, and M + an interpretation
of L + . If the interpretations have the same domain and assign the same denotations to
nonlogical symbols in L (so that the only difference is that the one assigns denotations
to symbols of L + not in L, while the other does not), then M + is said to be an
expansion of M to L + , and M to be the reduct of M + to L. Note that the notions
of expansion and reduct pertain to changing the language while keeping the domain
fixed.
19.2 Skolem Normal Form
A formula in prenex form with all quantifies universal (respectively, existential) may
be called a universal or ∀-formula (respectively, an existential or ∃-formula). Consider
a language L and a sentence of that language in prenex form, say
(1)
∀x 1 ∃y 1 ∀x 2 ∃y 2 R(x 1 , y 1 , x 2 , y 2 ).
Now for each existential quantifier, let us introduce a new function symbol with
as many places as there are universal quantifiers to its left, to obtain an expanded
language L + . Thus in our example there would be two new function symbols, say
f 1 and f 2 , corresponding to ∃y 1 and ∃y 2 , the former having one place corresponding
to ∀x 1 , and the latter two places corresponding to ∀x 1 and ∀x 2 . Let us replace each
existentially quantified variable by the term that results on applying the corresponding
function symbol to the universally quantified variable(s) to its left. The resulting
∀-formula, which in our example would be
(2)
∀x 1 ∀x 2 R(x 1 , f 1 (x 1 ), x 2 , f 2 (x 1 , x 2 ))
is called the Skolem normal form of the original sentence, and the new function
symbols occurring in it the Skolem function symbols.
A little thought shows that (2) logically implies (1). In any interpretation of the
expanded language L + with the new function symbols, it is the case that for every
element a 1 of the domain there is an element b 1 , such that for every element a 2 there
is an element b 2 , such that a 1 , b 1 , a 2 , b 2 satify R(x 1 , y 1 , x 2 , y 2 ): namely, take for
b 1 the result of applying to a 1 the function denoted by f 1 , and for b 2 the result of
applying to a 1 and a 2 the function denoted by f 2 .
We cannot, of course, say that conversely (1) implies (2). What is true is that (2)
is implied by (1) together with the following:
(3.1)
(3.2)
∀x 1 (∃y 1 ∀x 2 ∃y 2 R(x 1 , y 1 , x 2 , y 2 ) →∀x 2 ∃y 2 R(x 1 , f 1 (x 1 ), x 2 , y 2 ))
∀x 1 ∀x 2 (∃y 2 R(x 1 , f 1 (x 1 ), x 2 , y 2 ) → R(x 1 , f 1 (x 1 ), x 2 , f 2 (x 1 , x 2 ))).