We present a decision procedure for the
∀
∃
\forall \exists
-theory of
D
[
0
,
0
′
]
\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]
, the Turing degrees below
0
′
{\mathbf {0}}’
. The two main ingredients are a new extension of embeddings result and a strengthening of the initial segments results below
0
′
{\mathbf {0}}’
of [Le1]. First, given any finite subuppersemilattice
U
U
of
D
[
0
,
0
′
]
\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]
with top element
0
′
{\mathbf {0}}’
and an isomorphism type
V
V
of a poset extending
U
U
consistently with its structure as an usl such that
V
V
and
U
U
have the same top element and
V
V
is an end extension of
U
−
{
0
′
}
U - \{ {\mathbf {0}}’ \}
, we construct an extension of
U
U
inside
D
[
0
,
0
′
]
\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]
isomorphic to
V
V
. Second, we obtain an initial segment
W
W
of
D
[
0
,
0
′
]
\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]
which is isomorphic to
U
−
{
0
′
}
U - \{ {\mathbf {0}}’ \}
such that
W
∪
{
0
′
}
W \cup \{ {\mathbf {0}}’ \}
is a subusl of
D
\mathcal {D}
. The decision procedure follows easily from these results. As a corollary to the
∀
∃
\forall \exists
-decision procedure for
D
\mathcal {D}
, we show that no degree
a
>
0
{\mathbf {a}} > {\mathbf {0}}
is definable by any
∃
∀
\exists \forall
-formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any
∀
\forall
or
∃
\exists
-formula. The analysis again uses the decision procedure for the
∀
∃
\forall \exists
-theory of
D
\mathcal {D}
. A similar analysis is carried out for the high/low hierarchy using the decision procedure for the
∀
∃
\forall \exists
-theory of
D
[
0
,
0
′
]
\mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ]
. (A jump class
C
\mathcal {C}
is
σ
\sigma
-invariant if
σ
(
a
)
\sigma ({\mathbf {a}})
holds for every
a
{\mathbf {a}}
in
C
\mathcal {C}
.)