For each natural number $d$ we consider a finite structure $M_{d}$ whose

universe is the set of all $0,1$-sequence of length $n=2^{d}$, each

representing a natural number in the set $\lbrace 0,1,...,2^{n}-1\rbrace

$ in binary form.

The operations included in the structure are the

constants $0,1,2^{n}-1,n$, multiplication and addition modulo $2^{n}$,

the unary function $ \min\lbrace 2^{x}, 2^{n}-1\rbrace$, the binary

functions $\lfloor x/y\rfloor $, $\max(x,y)$, $\min(x,y)$, and the

boolean vector operations $AND$, $OR$, $NEGATION$ defined on $0,1$

sequences of length $n$ by performing the operations on all components

simultaneously. These are essentially the arithmetic operations that

can be performed on a RAM by a single instruction. We show that there

exists a term (that is, an algebraic expression) $F(x,y)$ built up from

the mentioned operations, with the only free variables $x,y$, such that

for all terms $G(y)$, which is also built up from the mentioned

operations, the following holds. For infinitely many positive integers

$d$, there exists an $a\in M_{d}$ such that the following two statements

are not equivalent: (i) $M_{d} \models \exists x, F(x,a) $, (ii)

$M_{d}\models G(a)=0$. In other words, the question whether an

existential statement, depending on the parameter $a\in M_{d}$ is true

or not, cannot be decided by evaluating an algebraic expression at $a$.

We also show that this theorem remains true if we include the operation

$\min \lbrace x^{y}, 2^{n}-1 \rbrace$ into the structure $M_{d}$. A

general theorem is proved as well which describes sufficient conditions

for a set of operations on a sequence of structures $K_{d}$, $d=1,2,...$

which guarantees that the analogue of the mentioned theorem holds for

the structure $K_{d}$ too.

For each natural number $d$ we consider a finite structure $M_{d}$ whose

universe is the set of all $0,1$-sequence of length $n=2^{d}$, each

representing a natural number in the set $\lbrace 0,1,...,2^{n}-1\rbrace

$ in binary form.

The operations included in the structure are the

constants $0,1,2^{n}-1,n$, multiplication and addition modulo $2^{n}$,

the unary function $ \min\lbrace 2^{x}, 2^{n}-1\rbrace$, the binary

functions $\lfloor x/y\rfloor $, $\max(x,y)$, $\min(x,y)$, and the

boolean vector operations $AND$, $OR$, $NEGATION$ defined on $0,1$

sequences of length $n$ by performing the operations on all components

simultaneously. These are essentially the arithmetic operations that

can be performed on a RAM by a single instruction. We show that there

exists a term (that is, an algebraic expression) $F(x,y)$ built up from

the mentioned operations, with the only free variables $x,y$, such that

for all terms $G(y)$, which is also built up from the mentioned

operations, the following holds. For infinitely many positive integers

$d$, there exists an $a\in M_{d}$ such that the following two statements

are not equivalent: (i) $M_{d} \models \exists x, F(x,a) $, (ii)

$M_{d}\models G(a)=0$. In other words, the question whether an

existential statement, depending on the parameter $a\in M_{d}$ is true

or not, cannot be decided by evaluating an algebraic expression at $a$.

We also show that this theorem remains true if we include the operation

$\min \lbrace x^{y}, 2^{n}-1 \rbrace$ into the structure $M_{d}$. A

general theorem is proved as well which describes sufficient conditions

for a set of operations on a sequence of structures $K_{d}$, $d=1,2,...$

which guarantees that the analogue of the mentioned theorem holds for

the structure $K_{d}$ too.