# A Formalization of the Turing Test Evgeny Chutchev

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1 A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant of the test (a person), each of which tries to appear human. The interrogator does not know from whom exactly he receives answers and has the objective to tell the Turing machine from the person (for more details, see, for example, [3]). In this paper, we consider a formalization of the Turing test and obtain the following results (Figure 1.1): Second participant does not follow an algorithm Every Turing machine fails a certain test from this area (Theorem 3.2) Second participant follows some algorithm Every Turing machine that answers any questions and is limited in response time or memory fails certain tests from this area (Theorems 4.2 and 4.3) Every Turing machine that answers any questions fails a certain test from this area (Theorem 4.1) For every test in the shaded area, an appropriate Turing machine passes the test (Theorems 3.1 and 5.1) Interrogator follows some algorithm Figure 1.1. Summary of results Interrogator does not follow an algorithm The rest of the paper is organized as follows: Section 2 gives definitions and introduces notations. Section 3 considers tests with arbitrary Turing machines. Section 4 considers tests with Turing machines of some special classes. Section 5 considers the strict Turing test. Section 6 presents the conclusion. Section 7 contains a list of definitions and notations. Section 8 contains references. Section 9 is an appendix, where we consider a random answers model for the second participant of the test that does not follow an algorithm. 2. General definitions and notations 2.1. Words and numbers 1. Suppose A is a finite alphabet and symbol θ does not belong to A. By A denote all words over A, including the empty word λ. By λ n denote the sequence of n empty words λ. 2. Denote by N the set of all natural numbers (excluding zero), and put N 0 N {0}. We will not make any distinction between the elements of N 0 and their notations over the alphabet A, assuming N 0 = A \*λ+. 3. Fix some letter a A, and for each word ω A denote by ω the word aω.

4 work tape equal to the question (see [2]). Then note that in accordance with 42 [1], there is an elementary arithmetic formula f(x, y) with free occurrence of variables x and y, which is true for x M and y M if and only if x and y are on the tape Π. 2. We consider SP as the tester constituent and explain this as follows. The original description of the test offered by A. Turing assumes the loyalty of SP to interrogator. Now if interrogator gives in his zero numbered test question the instructions on how SP should operate (for example, instructions in the form Mf for OTM M Π ), then SP, due to his loyalty, will fulfill these instructions. The instructions to simulate the OTM equipped with computable oracle are executable by a human. The problem of simulation of OTM equipped with non-computable oracle is beyond the scope of this paper. 3. Tests with arbitrary Turing machines We shall consider tests under various conditions of reducibility of interrogator and SP to TM (Theorems 3.1 and 3.2) and prove at first the following lemma. LEMMA For each enumerator E that enumerates some communicable TMs 3 and for each oracle Φ, the tester T E Φ is successful for E* Tester T Φ E will finish the test τ = T Φ E, E(n), not later than at the nth step. PROOF. The interrogator I Φ E is mute, and due to inequality E,n- E(n) (λ n ), this interrogator successfully completes the test τ not later than at the nth step. THEOREM For each tester with SP that reduces to TM, the specified TM can pass the test; hence, any such tester is not successful for M For each tester T of the type ( M, M), some generator H can pass the test,t, H-. 3. The problem of constructing the generator H that can pass the test, I, Q, H-, where I and Q are arbitrary TMs, is algorithmically unsolvable. PROOF. The proof of Item 1 of the theorem arises from the definition of TM s ability to pass the test. To prove Item 2 of the theorem consider H = M, where TM M is defined as follows: If J and Q are TMs, to which interrogator and SP are reduced, then M is an autonomous TM that simulates, I, Q, Q, - and gives the answers that are the same as those of Q in this test. In order to prove Item 3 of the theorem assume that there exists TM B such that generator H = B(IθQ) can pass the test, I, Q, H-. Then consider the following enumerator E that enumerates generators: E(n) B I An θa n. According to the definition of E, we Θ have n N E(n) T An and thus T T Θ n (E(n) T). But Item 1 of Lemma 3.1 implies that T T Θ n (E(n) T). for M. 4 THEOREM 3.2. Some tester of the type ( M, M) with dumb interrogator is successful PROOF 5. Let us describe the desired tester T = I, Q, where Q is equipped with oracle Π: SP Q, after receiving the nth test question, acts as follows: Q calculates A n and queries its oracle Π whether A n recognizes itself. If the oracle answers in the affirmative, Q simulates the work of A n with question A n, calculates the number t of cycles that is necessary for A n to compute A n (A n ), and gives the answer t. 3 Note that if there is an enumerator E that enumerates all communicable TMs, then E* + would include all generators, and then any generator would be similar to some element in R* +, where R(n) E(n), but R R* +. 4 Thus, for the successfulness of such tester either SP should not reduce to TM (see Theorem 3.2) or SP, which reduces to TM, should not belong to the class of TMs that are subjects of the test (see Section 4). 5 The idea of the proof is based on Subsection 3 of Section 6 of [4].

6 Now construct an enumerator E that enumerates M t for all M M and all t 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 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 ( M, M) with dumb interrogator is successful for all communicable limited in memory TMs. 6 PROOF. We need the following definitions: Denote by L 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 K the set of all TMs from L 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 N that L A*N+ and put L N *M L: M(λ N ) + (note that K L N ). Observing the work of M L that calculates M N, it is possible to reject those M for which: The length of the work tape segment that M makes use of indicates that M K. At the processing of current question, the configuration of M (i.e. the combination (state of M, position and content of the work tape)) was repeated, whence it follows that M L N. Thus we can reject all TMs from L\L N and some TMs from L N \K. Then it is possible to construct an enumerator R, based on A, on the definition of L, and on the described above rejection, such that K R*N+ L N. Finally, it is possible to construct the following enumerator E that enumerates generators: If n N, then E(n),k- R(n) λk if k N, λ if k > N; and if n > N, then k N (E(n),k- λ). For each C K there is an arbitrary TM in E*N+ that is N-similar to C, whereby the theorem statement follows from Item 2 of Lemma 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 T, M, l, T, M, r, T, M,, T, M, M T, and M T by analogy with the notations,t, M, l-,,t, M, r-,,t, M, -,,T, M-, M T, and M T for the ordinary tests. It is clear that if M T, then M T. 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 T reduces to TM Q, then Q may fail the strict test T, Q (take for example TM Q that 8 Note that Item 1 of Theorem 3.1 implies that SP of the described tester is not limited in time, and this is also clear from the principle of how this SP works. 9 Note that from Item 1 of Theorem 3.1 it follows that SP of the described tester does not belong to K, and this is also clear from the principle of how this SP works.

7 does not answer any question). However, the modified Theorem 3.1, given in following subsection, is valid for the strict test Successfulness of the testers for the strict Turing test THEOREM For each tester, which SP is reducible to TM, an arbitrary TM can pass the strict test with this tester For each tester T of the type ( M, M) some generator H can pass the strict test T, H For each tester T of the type ( M, M) some communicable TM C can pass the strict test T, C For a given oracle Φ and for any TMs I and Q, the problem of constructing the TM M that can pass the strict test I Φ, Q, M is algorithmically unsolvable 13. PROOF OF ITEMS 1, 2, AND 3 OF THEOREM 5.1. For an arbitrary tester T, denote by Q the TM to which SP of this tester is reduced, and consider the strict test τ = T, Q,. If Q 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 Q does not answer the nth test question in the test τ, where n N. Then Q fails the test τ, but generator H, which first n 1 answers are equal to those of Q in the test and the following answers are equal to, will pass the test T, 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 Lemmas To prove Item 4 of Theorem 5.1 let us: Fix the oracle Φ from the hypothesis of this item. Assume the existence of TM F that was specified in Item 4 of the theorem: F(IQ) I Φ, Q. Prove the following lemmas. LEMMA 5.1. For each SP Q and each TM M, the following values are equal for T Q Φ, M, l and T Q Φ, M, r : 1. The numbers of tests' steps. 2. The numbers (maybe infinite) and contents of answers of Q. 3. The numbers (maybe infinite) and contents of answers of M. PROOF. By definition of interrogator I Φ Q, 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 Φ Q. Taking Lemma 5.1 into account for the strict tests with a tester that has a form of T Φ Q, 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 A; 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 = An analogue of Item 1 of Theorem An analogue of Item 2 of Theorem An analogue of Item 1 of Theorem That accentuates the difference between the strict and the ordinary tests: See Item 1 of Theorem 3.1.

8 8 Fix some SP Q and put Q 0 Q, Q n+1 F I Qn Q n, I n I Φ Qn, T n T Φ Qn, where n N 0. For n > 0 and the strict test T n 1, Q n : Denote by (n) the sequence of answers of SP Q n 1 in this test. Denote by (n) = γ 1 (n), γ2 (n), the sequence of answers of TM Qn that is the subject of this test. LEMMA 5.2. n N (n) (n). PROOF. For n > 0, Q n T n 1. Thus, owing to the specificity of how interrogator I n operates, only the two following cases are possible: 1. The number of test steps is infinite, and moreover, (n) = (n). 2. SP Q n 1 has not answered to the jth (j 1) test question that has been answered by TM Q n. Thus γ (n) = (n) + 1 and this implies that for j = 1, the equation (n) = 0 is satisfied, (n). and for j > 1, (n) = γ 1 (n), γ2 (n),, γj 1 LEMMA 5.3. n N (n) (n+1). PROOF. Consider the tests T n 1, Q n, and T n, Q n+1,. The interrogators I n 1 and I n are dump, and Q n+1 T n. Thus (n) = (n+1) or (n) (n+1), due to the definition of I n. Then the proof follows from Lemma 5.2. LEMMA 5.4. n N ( γ (n) n). PROOF. Lemma 5.2 implies that γ (1) > 0, and for n > 1 the statement of the lemma follows from Lemma 5.3. LEMMA 5.5. For each n N and for every k, where 0 < k γ (n), the word γ (n) k is calculated algorithmically. PROOF. Owing to definition of I M, this TM can be algorithmically constructed from TM M. Then, due to definition of Q n, every I Qn and thus every Q n can be algorithmically constructed from TM Q. Finally, γ k (n) = Qn λ k Proof of Item 4 of Theorem 5.1 (n) To prove Item 4 of Theorem 5.1 assign to each Q M the following generator H Q : H Q,n- γ n (H Q is well defined due to Lemmas 5.4 and 5.5). Now, if is the sequence of all answers of H Q, then according to Lemmas 5.3 and 5.4, (1), hence Q 1 T Φ Q H Q T Φ Q H Q T Φ Q. Now construct the following enumerator E that enumerates generators: E(n) H An. We Φ have n N E(n) T An and thus T T Φ n (E(n) T). However, according to Item 1 of Lemma 3.1, T T Φ n (E(n) T). This contradiction proves that F does not exist. 6. Conclusion The summary of results is given in Figure 1.1 (see Section 1). Note that some results can be extended to the case when each participant of the test can be an oracle Turing machine. 7. Definitions and notations In tables given below, the global definitions and notations (terms) are shown with specifying the number of the subsection, where the corresponding definition or notation was denoted Definitions and notations Sets Symbols, words, and related concepts Special TMs Testers, interrogators, tests A, A * θ M I, Q B λ A k T Q Φ 2.6.4

9 N, N λ n E*N+ (E* +) T n 5.3 M ω A I Q C (n) 5.3 E I n 5.3 T Φ (n), γ k (n) Q n 5.3,T, M, l-,,t, M, r-,,t, M, Relations 5.3 F 5.3,T, M T, M, l, T, M, r, Oracles, OTM Formula T, M,, T, M 5.1, Π f Answers of (O)TM to questions, Θ V(), V() = 2.2.4, 2.4.2, 5.1 M Φ H,n Terms answer of (O)TM 2.2.3, left (right) test reducible to TM answers to the questions, that, (O)TM 2.2.4, limited in memory, TM 4.3 similar OTMs autonomous TM limited in time, TM 4.2 SP communicable TM N-similar OTMs strict test 5.1 dumb interrogator oracle successful tester enumerator oracle interface of (O)TM 2.2.1, test question fails the test, that, TM OTM tester generator output of (O)TM 2.2.1, TM input of (O)TM 2.2.1, question to (O)TM 2.2.3, type of OTM interrogator recognition of TM type of tester References [1] Kleene S.C. (1967), Mathematical Logic, John Wiley and Sons, New York London Sydney.,2- Papadimitriou, Christos H. (1994), Computational Complexity, Addison-Wesley. [3] Saygin, A.P., Cicekli, I. and Akman, V. (2000), Turing Test: 50 Years Later, Minds and Machines 10, pp [4] Turing, A. (1950), Computing Machinery and Intelligence, Mind 59(236), pp Appendix. Probabilistic test Consider some variation of Theorem 3.2: Let SP be equipped with the oracle that creates physically, namely, can be replaced with a random number generator. There are at least two symbols in the alphabet B: and λ, and in order to use the notations that are conventional to binary random number generators we replace these symbols with 0 and 1. Then consider the following OTM Z = Z Ξ : Oracle Ξ is filled up with symbols 0 and 1 randomly and independently; where 0 appears with the probability p 0 and 1 appears with the probability p 1 (hence Ξ is randomly chosen from the set of all oracles). TM Z satisfies the following condition: If Ξ = ξ 1 ξ 2, then Z( n ) ξ n. THEOREM 9.1. For some dumb interrogator I of the type M, which depends on m N, if p = max(p 0, p 1 ) < 1, then C C.P*C I, Z + 1 pm 1 p /. PROOF. Without loss of generality, we equate each nonzero answer of any TM to 1. Now fix m N and construct the dumb interrogator I by the following algorithm. Interrogator I consists of supervisor S and two assistants: The left assistant L and the right assistant R. The left (the right) assistant processes the answers of the left (the right) subject of the test and transmits the results of the processing to S.

10 After the nth answer has been received, an assistant carries out the following instructions: 1. Save the answer in memory and put = For k = 1,, n, start up A k ( n ) for n cycles of calculation. 3. If A k has provided not less than m + k 1 answers, and these answers are equal to the received answers, put = Report the value of to S. Supervisor S treats the messages δ L and δ R of L and R according to the following table. δ R = δ L = Continue the test Complete the test with result SP is on the left 1 Complete the test with result SP is on the right If some communicable TM C has passed the test, then symbol 1 appears in the sequence of messages from the assistant that processes the answers of Z, and then for some k the first m + k 1 m +k 1 answers of Z are equal to the answers of A k. As a result, P{C I, Z } p. 10 k=1

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