1. The Encoding of TM Motivation for encoding of TM
Languages defined by TM
Recursive and recursively enumerable languages
ALAN and MATHISON
UTM --Universal Turing Machine

2. Why Encoding of TM?
It will help us to show that some languages are not accepted by any TM, and other limitations of TM.
One of main purposes of encoding of TMs is to construct languages ALAN and MATHISON.
CWL, Code Word Language, defined by RE: (a+b a+b(a+b)5)*

3. Languages Defined by TM
For every TM, T, which runs on strings from the alphabet S, it will divide the set of all finite strings over S into three disjoint sets: accept (T), loop(T), reject(T).
Recursively enumerable language L = accept(T), and L’ = reject (T) + loop(T)
Recursive language L = accept (T), loop(T)= empty, and L’ = reject (T)

4. ALAN and MATHISON
ALAN -- an example of non-recursively enumerable (r.e.)language, constructed by using CWL. The significance of ALAN is to show that not every language can be recognized by TM.
MATHISON -- an example of r.e.L, that can be recognized by UTM.

5. Universal Turing Machine
A TM is a special purpose computer. Once its transition function is defined, the machine is restricted to carrying out one particular type of computation -- not general purpose or reprogrammable.
A UTM Mw is an automaton that, given as input the description of any TM M and a string w, can simulate the computation of M on w.