The logic E is determined by the ADS with precisely one unary operator whose class of ADMs consists of all models.  Chellas formulates this completeness result (Theorem 9.8 in Chellas, 1980) for so-called minimal models (alias neighborhood-frames), which are, however, just a notational variant of abstract description models with one unary operator
(Došen, 1988). If the classical modal logic L is determined by an ADS with decidable term satisfiability problem, then L is decidable since t E L iff ¬ t is unsatisfiable.

A classical modal logic L is called normal iff it additionally contains

$f(x_{1},\ldots,x_{j-1},x_{j}\wedge y_{j},x_{j+1},\ldots,x_{n_{f}})$$\leftrightarrow$$f(x_{1},\ldots,x_{j-1},x_{j},x_{j+1},\ldots,x_{n_{f}})\wedge f(x_{1},\ldots,x_{j-1},y_{j},x_{j+1},\ldots,x_{n_{f}})$
and

$f(T,\bot,\ldots,\bot),\ f(\bot,T,\bot,\ldots,\bot),\ \ldots,f(\bot,\ldots,\bot,T),$
for all function symbols f and all j with 1 < j < n f (Jónsson & Tarski, 1951; Jónsson &
Tarski, 1952; Goldblatt, 1989). This definition of normal modal logics assumes that the formulas (terms) are built using only necessity (box) operators. 7 We will work here only with necessity operators; the corresponding possibility-operators are definable by putting

$f^{\circ}(x_{1},\cdot\cdot\cdot\cdot,x_{n_{f}})=\gamma f(-x_{1},\cdot\cdot\cdot\cdot,-x_{n_{f}})$.  
The minimal normal modal logic in the language with one unary operator is known as K
(Chellas, 1980).

We call a function F : Wn → W normal iff for all 1 ≤ j ≤ n and X 1 , . . , X n , Y j ⊆ W

$$F(X_{1},\ldots,X_{j-1},X_{j}\cap Y_{j},X_{j+1},\ldots,X_{n})=F(X_{1},\ldots,X_{j-1},X_{j},X_{j+1},\ldots,X_{n})\cap F(X_{1},\ldots,X_{j-1},Y_{j},X_{j+1},\ldots,X_{n}))$$

and

$$\vartheta,\,.\,.\,.\,,\,\emptyset)\,=\,F(\,\emptyset,\,\Gamma)$$

F(W, 0, . . . , 0) = F(0, W, 0, . . . , 0) = . . = F(0, . . . , 0, W) = W.

Note that a unary function F is normal iff F(W) = W and F(X NY) = F(X) NF(Y), for any X, Y ⊆ W . A function symbol f is called normal in an ADS ( L , M ) iff the functions f™ are normal for all M E M.

For any role R of some DL, the function symbol fvR is normal in the corresponding ADS. To the contrary, it is readily checked that neither f > n R and f < n R nor their duals f ≥ n R and f ≥ n R are normal.

Obviously, an ADS ( L, M ) determines a normal modal logic iff all function symbols of L are normal in ( L, M ). Completeness of K with respect to Kripke semantics (Chellas, 1980) implies that the logic K is determined by the ADS with one unary operator whose class of ADMs consists of all models interpreting this operator by a normal function.