does not answer any question). However, the modified Theorem 3.1, given in following subsection, is valid for the strict test.

## 5.2. Successfulness Of The Testers For The Strict Turing Test

THEOREM 5.1.

1. For each tester, which SP is reducible to TM, an arbitrary *TM can pass the strict test* with this tester10.

2. For each tester T of *the type* (≻, ≻) some generator ℌ *can pass the strict* test ,ℌ 11.

3. For each tester T of *the type* (≻, ≻) some communicable TM ℭ *can pass the strict* test , ℭ 12.

4. For a given oracle  and for any TMs ℑ and , *the problem of constructing the TM* 
that can pass the strict test ℑ
,  , *is algorithmically unsolvable*13.

PROOF OF ITEMS 1, 2, AND 3 OF THEOREM 5.1.

For an arbitrary tester , denote by  the TM to which SP of this tester is reduced, and consider the strict test  = , ,∗ . If  answers all test questions, then the proof of Items 1 and 2 of the theorem coincides with the proof of Items 1 and 2 of Theorem 3.1. Assume now that  does not answer the th test question in the test , where  ∈ ℕ. Then  fails the test , but generator ℌ, 
which first - 1 answers are equal to those of  in the test and the following answers are equal to , 
will pass the test ,ℌ,∗ . That completes the proof of Items 1 and 2 of Theorem 5.1.

Finally, generator is a communicable TM and hence Item 3 of Theorem 5.1 is the corollary of Item 2 of Theorem 5.1. □ 5.3. Lemmas To prove Item 4 of Theorem 5.1 let us:
 Fix the oracle  from the hypothesis of this item. Assume the existence of TM  that was specified in Item 4 of the theorem:
(ℑ) ⊵ ℑ
,  .

 Prove the following lemmas.

LEMMA 5.1. For each SP  and each TM , the following values *are equal for* 
,, ℓ and 
,,  :
1. The numbers of tests' *steps.*
2. The numbers *(maybe infinite) and contents of answers* of .

3. The numbers (maybe *infinite) and contents of answers of* .

PROOF. By definition of interrogator ℑ
, it treats the answers of the subjects of the test symmetrically, and that proves Item 1 of the lemma. Items 2 and 3 of the lemma follow from Item 1 of this lemma and from the dumbness of ℑ
. □
Taking Lemma 5.1 into account for the strict tests with a tester that has a form of 
, we shall discuss the number of the steps in the test and the answers of the subjects of the test without specifying the orientation of the test (the left one or the right one).

To formulate and prove Lemmas 5.2, 5.3, 5.4, and 5.5 we need the following definitions and notations.

 Denote by  the number of words in the sequence  of words over ; we say that sequences of words  and  satisfy the relation ⥽ if any of the following cases holds:
 0 =  < μ . ∞ ≠  < μ and  is the beginning of .

  = μ = ∞ and  = .