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Unformatted text preview: ECS 120 Lesson 20 The Post Correspondence Problem Oliver Kreylos Wednesday, May 16th, 2001 Today we will continue yesterdays discussion of reduction techniques by looking at another common strategy, reduction by computation chains . We will apply this method to prove the undecidabiliy of the emptiness problem for LBAs, and later to prove the undecidability of a problem not directly related to language theory, the Post Correspondence Problem (PCP) . 1 Reduction Using Computation Chains Another reduction strategy is based on computation chains for Turing Ma- chines. If M is a Turing Machine, and w is an input word, then the computa- tion of M on w can be simulated by a UTM. This UTM will typically store the current configuration of M on its tape, and will update it for every computa- tion step. We can now modify the Universal Turing Machine to write a new configuration for each step instead of modifying the old one. The simulation will work the same way, but after simulation is complete, the UTMs tape will contain a valid chain of IDs for machine M and input word w . Technically, the modified UTM will write the chain as a string # ID # ID 1 # ID 2 # . . . # ID k # . If a computation chain leads to a halting configuration, it must be of finite length; if it does not, we will not consider it being valid. We will then use the fact that a computation chain can be checked for validity by an LBA: A computation chain can be invalid for three reasons: 1. The first configuration ID is not an initial configuration, 2. the last configuration ID k is not a halting configuration, or 1 3. there is a pair of configurations ID i , ID i +1 that is not related by the turnstile relation: ID i 6` ID i +1 . All three of these conditions can be checked by a Turing Machine that does not write past its input, i. e., an LBA. Since the acceptance problem for LBAs is decidable, so is the validity of a computation chain for any Turing Machine. 1.1 The Emptiness Problem for LBAs Though the acceptance problem for LBAs is decidable, not all problems about LBAs are. One undecidable problem is the emptiness of the language accepted by an LBA. To prove this, we will use a reduction from A TM using computation chains. We will assume that the emptiness problem is decidable, and will construct a helper LBA whose language is the set of all accepting computation chains of M on input word w . If this LBA has a non-empty language L ( B ), there exists a valid computation chain in M on w , which means that w is accepted by M . Otherwise, L ( B ) is empty, there are no valid computation chains, and M does not accept w . This means we have reduced the problem of deciding whether a Turing Machine M accepts a word w to the problem of deciding whether an LBA B s language L ( B ) is empty....
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