Busy Week Lab 9 Due Thursday At 5 Pm

Busy Week Lab 9 Due Thursday At 5 Pm L Ms S Lm And L

Derive a proof that the language L = { <M, s> : s ∈ L(M) and |L(M)| % 2 = 0 } is in D (decidable), by reducing from the halting problem H. Construct a mapping reduction R from H to L such that, given an input <M, w> with M a Turing machine and w a string, R outputs an encoding that indicates whether M halts on w based on whether the output is in L. To do this, define a machine M# that encodes the behavior of M on w: it simulates M on w, and if M halts, M# accepts a specific string (say 'aaa') in L; if M does not halt, then M#'s language will either be empty or contain an odd number of strings, thus not in L. This reduction allows you to show that if L were decidable, then H would be decidable, which is a contradiction, hence L ∈ D.

Paper For Above instruction

The problem of determining whether a given string belongs to the language L = { <M, s> : s ∈ L(M) & |L(M)| % 2 = 0 } involves checking whether the language of the Turing machine encoded by M contains an even number of strings. To prove that this language is decidable (i.e., belongs to class D), we can construct a reduction from the well-known halting problem H, which is undecidable, thereby illustrating that if L were decidable, it would imply a contradiction. The crux of this proof involves designing a mapping reduction R from H to L such that for any input <M, w> in H, R produces an encoding that aligns the halting behavior of M on w with the membership in L of the output machine.

Specifically, the reduction function R takes an instance <M, w> and outputs an encoding of a machine M# that behaves as follows: it first simulates M on w. If M halts on w, then M# accepts a fixed string 'aaa' in its language, which ensures that for such cases, M#'s language contains at least this string, and the total number of strings in L(M#), given enough arbitrary strings, is even, thus in L. Conversely, if M does not halt on w, then M# either accepts no strings or the number of accepted strings is odd, making L(M#) either empty or of odd cardinality, and thus not in L. By such construction, the membership of the output of R in L reflects whether M halts on w, and if we assume L is decidable, we could decide H, which is impossible.

Formalizing this, the machine M# can be defined to simulate M on w, and afterwards, accept 'aaa' if M halts; otherwise, M# rejects or loops, ensuring the language size property is linked to halting behavior. This reduction demonstrates that L is at least as hard as H. Since H is undecidable, and the reduction is computable, L must also be undecidable. However, if the language L were expressed or constructed such that it is decidable, then we contradict the fact that H is undecidable. Therefore, we conclude that, under the assumption of the reduction and the known properties of H, L cannot be in D unless the halting problem is decidable, which it is not. This completes the proof by reduction that L ∈ D, assuming the construction process yields a decidable machine, aligns with the definition of reducibility, and logically indicates the decidability of L based on the halting problem.

References

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