By Raihan Al-Ekram, Archana Adma, Olga Baysal

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Proof Construct a Turing machine Mr with k tapes in a similar manner as in the previous theorem, such that each symbol of the tape alphabet of Mr is an r-tuple (which we shall call block) of symbols of the alphabet of M, and some of the states of Mr encode tuples (q, i I , ... , i k ), where q is a state of M and 1 ~ im ~ r. Here a difference arises with the tape compression theorem, since Mr needs other types of states to perform other computations, as described below. Configurations of M correspond to configurations of Mr as in the tape compression theorem.

2. We use the following notation for it. Notation. 2 will be abridged to guess y with Iyl ::; i. 2 Meaning of guess y with Iyl ~ i The computation tree of a nondeterministic machine on input W could be an infinite tree. However, any accepting path must be a finite path, since it ends in a final, accepting configuration. We will use this fact later as follows: if we know that a machine must accept within a given, previously known number of steps, say t, then every computation longer than this number can be aborted, since this computation cannot be the computation accepting in t steps.

As a concrete example, r may be translated into {O,I}*. Then each machine is codified as a string over {O, I}, and the set of all encodings of machines will form a language over {O, I}. Thus, according to the remarks made in the first section of this chapter, to each machine correspond an integer, and to some integers corresponds a machine. Furthermore, the machine can be recovered easily from the integer. This process is known as a godelization of Turing machines, and the number corresponding to each machine is its Godel number.

### diffX - An Algorithm to Detect Changes in Multi-Version XML Documents by Raihan Al-Ekram, Archana Adma, Olga Baysal

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