# 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, Class 10 Nancy Lynch

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1 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 10 Nancy Lynch

2 Today Final topic in computability theory: Self-Reference and the Recursion Theorem Consider adding to TMs (or programs) a new, powerful capability to know and use their own descriptions. The Recursion Theorem says that this apparent extra power does not add anything to the basic computability model: these self-referencing machines can be transformed into ordinary nonself-referencing TMs.

3 Today Self-Reference and the Recursion Theorem Topics: Self-referencing machines and programs Statement of the Recursion Theorem Applications of the Recursion Theorem Proof of the Recursion Theorem: Special case Proof of the Recursion Theorem: General case Reading: Sipser, Section 6.1

4 Self-referencing machines and programs

5 Self-referencing machines/programs Consider the following program P 1. P 1 : Obtain < P 1 > Output < P 1 > P 1 simply outputs its own representation, as a string. Simplest example of a machine/program that uses its own description.

6 Self-referencing machines/programs A more interesting example: P 2 : On input w: If w = ε then output 0 Else Obtain < P 2 > Run P 2 on tail(w) If P 2 on tail(w) outputs a number n then output n+1. What does P 2 compute? It computes w, the length of its input. Uses the recursive style common in LISP, Scheme, other recursive programming languages. We assume that, once we have the representation of a machine, we can simulate it on a given input. E.g., if P 2 gets < P 2 >, it can simulate P 2 on any input.

7 Self-referencing machines/programs One more example: P 3 : On input w: Obtain < P 3 > Run P 3 on w If P 3 on w outputs a number n then output n+1. A valid self-referencing program. What does P 3 compute? Seems contradictory: if P 3 on w outputs n then P 3 on w outputs n+1. But according to the usual semantics of recursive calls, it never halts, so there s no contradiction. P 3 computes a partial function that isn t defined anywhere.

8 Statement of the Recursion Theorem

9 The Recursion Theorem Used to justify self-referential programs like P 1, P 2, P 3, by asserting that they have corresponding (equivalent) basic TMs. Recursion Theorem (Sipser Theorem 6.3): Let T be a TM that computes a (possibly partial) 2- argument function t: Σ* Σ* Σ*. Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w).

10 The Recursion Theorem Recursion Theorem: Let T be a TM that computes a (possibly partial) 2-argument function t: Σ* Σ* Σ*. Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w). Thus, T is a TM that takes 2 inputs. Think of the first as the description of some arbitrary 1-input TM M. <M> T w t(<m>, w) Then R behaves like T, but with the first input set to <R>, the description of R itself. Thus, R uses its own representation. R w t(<r>, w)

11 The Recursion Theorem Recursion Theorem: Let T be a TM that computes a (possibly partial) 2-argument function t: Σ* Σ* Σ*. Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w). Example: P 2, revisited Computes length of input. What are T and R? Here is a version of P 2 with an extra input <M>: T 2 : On inputs <M> and w: If w = ε then output 0 Else run M on tail(w); if it outputs n then output n+1. <M> T R w t(<m>, w) w t(<r>, w)

12 The Recursion Theorem Example: P 2, revisited T 2 : On inputs <M> and w: If w = ε then output 0 Else run M on tail(w); if it outputs n then output n+1. T 2 produces different results, depending on what M does. E.g., if M always loops: T 2 outputs 0 on input w = ε and loops on every other input. E.g., if M always halts and outputs 1: T 2 outputs 0 on input w = ε and outputs 2 on every other input. <M> T R w t(<m>, w) w t(<r>, w)

13 The Recursion Theorem Example: P 2, revisited T 2 : On inputs <M> and w: If w = ε then output 0 Else run M on tail(w); if it outputs n then output n+1. Recursion Theorem says there is a TM R computing t(<r>, w)---just like T 2 but with input <M> set to <R> for the same R. This R is just P 2 as defined earlier. <M> T R w t(<m>, w) w t(<r>, w)

14 The Recursion Theorem Recursion Theorem (Sipser Theorem 6.3): Let T be a TM that computes a (possibly partial) 2-argument function t: Σ* Σ* Σ*. <M> T w t(<m>, w) Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w). R w t(<r>, w)

15 Applications of the Recursion Theorem

16 Applications of Recursion Theorem The Recursion Theorem can be used to show various negative results, e.g., undecidability results. Application 1: Acc TM is undecidable We already know this, but the Recursion Theorem provides a new proof. Suppose for contradiction that D is a TM that decides Acc TM. Construct another machine R using self-reference (justified by the Recursion Theorem): R: On input w: Obtain < R > (using Recursion Theorem) Run D on input <R, w> (we can construct <R, w> from <R> and w) Do the opposite of what D does: If D accepts <R, w> then reject. If D rejects <R, w> then accept.

17 Application 1: Acc TM is undecidable Suppose for contradiction that D decides Acc TM. R: On input w: Obtain < R > Run D on input <R, w> Do the opposite of what D does: If D accepts <R, w> then reject. If D rejects <R, w> then accept. RT says that TM R exists, assuming decider D exists. Formally, to apply RT, use the 2-input machine T: T: On inputs <M> and w: Run D on input <M, w> Do the opposite of what D does: If D accepts <M, w> then reject. If D rejects <M, w> then accept.

18 Application 1: Acc TM is undecidable Suppose for contradiction that D decides Acc TM. R: On input w: Obtain < R > Run D on input <R, w> Do the opposite of what D does: If D accepts <R, w> then reject. If D rejects <R, w> then accept. Now get a contradiction: If R accepts w, then D accepts <R, w> since D is a decider for Acc TM, so R rejects w by definition of R. If R does not accept w, then D rejects <R, w> since D is a decider for Acc TM, so R accepts w by definition of R. Contradiction. So D can t exist, so Acc TM is undecidable.

19 Applications of Recursion Theorem Application 2: Acc01 TM is undecidable Similar to the previous example. Suppose for contradiction that D is a TM that decides Acc01 TM. Construct another machine R using the Recursion Theorem: R: On input w: (ignores its input) Obtain < R > (using RT) Run D on input <R> Do the opposite of what D does: If D accepts <R> then reject. If D rejects <R> then accept. RT says that R exists, assuming decider D exists.

20 Application 2: Acc01 TM is undecidable Suppose for contradiction that D decides Acc01 TM. R: On input w: Obtain < R > Run D on input <R> Do the opposite of what D does: If D accepts <R> then reject. If D rejects <R> then accept. Now get a contradiction, based on what R does on input 01: If R accepts 01, then D accepts <R> since D is a decider for Acc01 TM, so R rejects 01 (and everything else), by definition of R. If R does not accept 01, then D rejects <R> since D is a decider for Acc01 TM, so R accepts 01 (and everything else), by definition of R. Contradiction. So D can t exist, so Acc01 TM is undecidable.

21 Applications of Recursion Theorem Application 3: Using Recursion Theorem to prove Rice s Theorem Rice s Theorem: Let P be a nontrivial property of Turing-recognizable languages. Let M P = { < M > L(M) P }. Then M P is undecidable. Nontriviality: There is some M 1 with L(M 1 ) P, and some M 2 with L(M 2 ) P. Implies lots of things are undecidable. We already proved this; now, a new proof using the Recursion Theorem. Suppose for contradiction that D is a TM that decides M P. Construct machine R using the Recursion Theorem:

22 Application 3: Using Recursion Theorem to prove Rice s Theorem Rice s Theorem: Let P be a nontrivial property of Turingrecognizable languages. Let M P = { < M > L(M) P }. Then M P is undecidable. Nontriviality: L(M 1 ) P, L(M 2 ) P. D decides M P. R: On input w: Obtain < R > Run D on input <R> If D accepts <R> then run M 2 on input w and do the same thing. If D rejects <R> then run M 1 on input w and do the same thing. M 1 and M 2 are as above, in the nontriviality definition. R exists, by the Recursion Theorem. Get contradiction by considering whether or not L(R) P:

23 Application 3: Using Recursion Theorem to prove Rice s Theorem Rice s Theorem: Let P be a nontrivial property of Turingrecognizable languages. Let M P = { < M > L(M) P }. Then M P is undecidable. L(M 1 ) P, L(M 2 ) P. D decides M P. R: On input w: Obtain < R > Run D on input <R> If D accepts <R> then run M 2 on input w and do the same thing. If D rejects <R> then run M 1 on input w and do the same thing. Get contradiction by considering whether or not L(R) P: If L(R) P, then D accepts <R>, since D decides M P, so L(R) = L(M 2 ) by definition of R, so L(R) P.

24 Application 3: Using Recursion Theorem to prove Rice s Theorem Rice s Theorem: Let P be a nontrivial property of Turingrecognizable languages. Let M P = { < M > L(M) P }. Then M P is undecidable. L(M 1 ) P, L(M 2 ) P. D decides M P. R: On input w: Obtain < R > Run D on input <R> If D accepts <R> then run M 2 on input w and do the same thing. If D rejects <R> then run M 1 on input w and do the same thing. Get contradiction by considering whether or not L(R) P: If L(R) P, then D rejects <R>, since D decides M P, so L(R) = L(M 1 ) by definition of R, so L(R) P. Contradiction!

25 Applications of Recursion Theorem Application 4: Showing non-turing-recognizability DefineMIN TM = { < M > M is a minimal TM, that is, no TM with a shorter encoding recognizes the same language }. Theorem: MIN TM is not Turing-recognizable. Note: This doesn t follow from Rice: Requires non-t-recognizability, not just undecidability. Besides, it s not a language property. Proof: Assume for contradiction that MIN TM is Turing-recognizable. Then it s enumerable, say by enumerator TM E. Define TM R, using the Recursion Theorem: R: On input w:

26 Application 4: Non-Turing-recognizability MIN TM = { < M > M is a minimal TM }. Theorem: MIN TM is not Turing-recognizable. Proof: Assume that MIN TM is Turing-recognizable. Then it s enumerable, say by enumerator TM E. R: On input w: Obtain <R>. Run E, producing list < M 1 >, < M 2 >, of all minimal TMs, until you find some < M i > with < M i > strictly greater than < R >. That is, until you find a TM with a rep bigger than yours. Run M i (w) and do the same thing. Contradiction: L(R) = L(M i ) < R > less than < M i > Therefore, M i is not minimal, and should not be in the list.

27 Proof of the Recursion Theorem: Special case

28 Proof of Recursion Theorem: Special Case Start with easier first step: Produce a TM corresponding to P 1 : P 1 : Obtain < P 1 > Output < P 1 > P 1 outputs its own description. Lemma: (Sipser Lemma 6.1): There is a computable function q: Σ* Σ* such that, for any string w, q(w) is the description of a TM P w that just prints out w and halts. Proof: Straightforward construction. Can hard-wire w in the FSC of P w. Q w q(w) = < P w >

29 Proof of RT: Special Case Lemma: (Sipser Lemma 6.1): There is a computable function q: Σ* Σ* such that, for any string w, q(w) is the description of a TM P w that just prints out w and halts. Now, back to the machine that outputs its own description Consists of 2 sub-machines, A and B. Q w q(w) = < P w > A B Output of A feeds into B. Write as A B.

30 Construction of B <M> B < P <M> M > B expects its input to be the representation <M> of a 1- input TM (a function-computing TM, not a language recognizer). If not, we don t care what B does. B outputs the encoding of the combination of two machines, P <M> and M. The first machine is P <M>, which simply outputs <M>. The second is the input machine M. P <M> M: P <M> <M> M Some output

31 Construction of B <M> B < P <M> M > How can B generate < P <M> M >? B can generate a description of P <M>, that is, <P <M> >, by Lemma 6.1. B can generate a description of M, that is, <M>, since it already has <M> as its input. Once B has descriptions of P <M> and M, it can combine them into a single description of the combined machine P <M> M, that is, < P <M> M >. P <M> <M> M Some output

32 Construction of A A <B> A is P <B>, the machine that just outputs <B>, where B is the complicated machine constructed above. A has no input, just outputs <B>.

33 Combining the Pieces A B: A B Claim A B outputs its own description, which is < A B >. Check this A is P <B>, so the output from A to B is <B>: A = P <B> <B> B Substituting B for M in B s output: A = P <B> <B> < P <B> B > B

34 Combining the Pieces A B: A B Claim A B outputs its own description, which is < A B >. A = P <B> <B> < P <B> B > B The output of A B is, therefore, < P <B> B > = < A B >. As needed! A B outputs its own description, < A B >.

35 Proof of the Recursion Theorem: General case

36 Proof of the RT: General case So, we have a machine that outputs its own description. A curiosity---this is not the general RT. RT says not just that: There is a TM that outputs its own description. But that: There are TMs that can use their own descriptions, in arbitrary ways. The arbitrary ways are captured by the machine T in the RT statement. <M> w T t(<m>, w)

37 The Recursion Theorem Recursion Theorem: Let T be a TM that computes a (possibly partial) 2-argument function t: Σ* Σ* Σ*. <M> T w Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w). t(<m>, w) w R t(<r>, w)

38 The Recursion Theorem Recursion Theorem: Let T be a TM that computes a (possibly partial) 2-argument function t: Σ* Σ* Σ*. <M> T w Then there is another TM R that computes the function r: Σ* Σ*, where for any w, r(w) = t(<r>, w). t(<m>, w) Construct R from: w The given T, and R Variants of A and B from the specialcase proof. t(<r>, w)

39 Proof of RT: General Case R looks like: A B T Write this as (A B) 1 T The 1 means that the output from (A B) connects to the first (top) input line of T.

40 Proof of RT: General Case R = (A B) 1 T A B T New A: P <B 1 T>, where B 1 T means: B T

41 Proof of RT: General Case New B: <M> B < P <M> 1 M > Like B in the special case, but now M is a 2- input TM. P <M> 1 M: 1-input TM, which uses output of P <M> as first input of M. P <M> M

42 Combining the Pieces R = (A B) 1 T A B T Claim R outputs t(<r>, w): A is P <B 1 T>, so the output from A to B is <B 1 T >: w A = P <B 1 T> <B 1 T > B < P <B 1 T> 1 (B 1 T) > Now recall definition of B: < P <M> <M> 1 M > B Plug in B 1 T for M in B s input, and obtain output for B.

43 Combining the Pieces A = P <B 1 T> <B 1 T > < P 1 <B T> 1 (B 1 T) > B B s output = < A 1 (B 1 T) > = < R >: A <B 1 T > < R> B Now combine with T, plugging in R for M in T s input: A <B 1 T > < R> B T t(<r>,w) w

44 Combining the Pieces A <B 1 T > < R> B T t(<r>,w) w Thus, R = (A B) 1 T, on input w, produces t(<r>,w), as needed for the Recursion Theorem. w R t(<r>, w)

45 Next time More on computabilty theory Reading: "Computing Machinery and Intelligence" by Alan Turing:

46 MIT OpenCourseWare J / J Automata, Computability, and Complexity Spring 2011 For information about citing these materials or our Terms of Use, visit:

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