Now construct an enumerator ℰ that enumerates  
for all  ∈  and all  ∈ ℕ. For each ℭ ∈ ℂ that is limited in time, an arbitrary TM in ℰ*∞+ is similar to ℭ , whereby the theorem statement follows from Item 1 of Lemma 3.1 8. □

## 4.3. Testing For Turing Machines Limited In Memory

Consider a class of communicable TMs limited in memory, namely, TMs with a uniform upper bound for the number of states and a uniform upper bound for the length of the work tape segment that TM may scan in the processing of empty questions.

THEOREM 4.3. *Some tester of the type* (≻, ≻) *with dumb interrogator is successful for* all communicable limited in memory TMs.

PROOF. We need the following definitions:
 Denote by  the set of all TMs that satisfy the specified limitation for the number of states and for the length of the initial content of the work tape.

 Denote by  the set of all TMs from  ∩ ℂ that satisfy the specified limitation for the length of the work tape segment that TM makes use of in the processing of empty questions.

Now calculate such  that  ⊆ *+ and put  ≝ * ∈ :(λ
) ≠ ∞+ (note that  ⊆ ). Observing the work of  ∈  that calculates  
 , it is possible to reject those for which:
 The length of the work tape segment that  makes use of indicates that  ∉ .

 At the processing of current question , the configuration of  (i.e. the combination (state of , position and content of the work tape)) was repeated, whence it follows that  ∉ .

Thus we can reject all TMs from \ and some TMs from \. Then it is possible to construct an enumerator ℛ, based on , on the definition of , and on the described above rejection, such that  ⊆ ℛ*+ ⊆ . Finally, it is possible to construct the following enumerator ℰ
that enumerates generators: 
If  ≤ , then

$$\begin{array}{l}{{\mathcal{E}(n)[k]\ {\stackrel{\mathrm{def}}{=}}\ \left\{\begin{array}{l l}{{\mathcal{R}(n)\left(\lambda^{k}\right)\mathrm{if}\,k\leq N,}}\\ {{\lambda\mathrm{if}\,k>N;}}\end{array}\right.}}\end{array}$$

and if  > , then ∀∈ℕ(ℰ(),- ≝ ).

For each ℭ ∈  there is an arbitrary TM in ℰ*+ that is -similar to ℭ , whereby the theorem statement follows from Item 2 of Lemma 3.1 9. □

## 5. The Strict Turing Test

5.1. Definition of the strict Turing Test TM that is the subject of the test has some advantages: If this TM does not answer the test question simultaneously with SP, it passes the test (see Item 2.6.2). The choice of this condition is rather arbitrary and that allows us to consider the "strict" Turing test:
 TM fails the left (the right) strict test if either TM does not answer some test question 
(regardless whether SP has answered this question) or this test is finished with the correct result.

 TM fails the strict test if it fails both the left and the right strict test.

We introduce for the strict tests notations ,,  , ,,  , ,,∗ , , ,  ⊵ ,
and  ⋭T by analogy with the notations ,,, -, ,,, -, ,,,∗-, ,,-,  ⊳ , and  ⋫T for the ordinary tests.

It is clear that if  ⋫ , then  ⋭ . Therefore, the statements of Lemma 3.1, Theorems 3.2, 4.2, 4.3, and Items 2 and 3 of Theorem 4.1 are extended to the case of the strict test. A different situation arises with the statement of Theorem 3.1 (and its corollary, Item 1 of Theorem 4.1): If SP 
of the tester  reduces to TM , then  may fail the strict test ,  (take for example TM  that 