Give implementation-level descriptions of a Turing machine?

A Turing machine (TM) can be formally described as seven tuples −

(Q,X,∑,δ,q0,B,F)

Where,

• Q is a finite set of states.

• X is the tape alphabet.

• ∑ is the input alphabet.

• δ is a transition function:δ𝛿:QxX->QxXx{left shift, right shift}.

• q0 is the initial state.

• B is the blank symbol.

• F is the final state.

A Turing machine T recognises a string x (over ∑) if and only when T starts in the initial position and x is written on the tape,

T halts in a final state.

T is said to recognize a language A, if x is recognised by T and if and only if, x belongs to A.

Be aware that while running a TM, you can also read/write other symbols from/onto a tape, not necessarily only those from the alphabet A.

A Turing machine T does not recognize a string x, if T does not halt in a state that is not final.

There are TMs which for an input don’t halt at all and run forever.

Example 1

Implementation-level descriptions of a Turing machine that decides the following language over the alphabet {0, 1}.

{w|w contains an equal number of 0s and 1s }

M = “On input string w −

• Scan tape and mark the first 0 which has not been marked. If no unmarked 0 is found, go to stage 4. Otherwise, move the head back to the front of the side of the tape.

• Scan tape and mark first 1 which has not been marked. If no unmarked then 1 is found, reject.

• Move the head back to front of the tape and go to stage 1.

• Move the head back to the front of the tape. Scan the tape to see if any unmarked 1s still remain. If none are found, accept; otherwise, reject.”

Example 2

Give a high level description of a TM which accepts the language A, where A consists of all strings that represent undirected graphs are connected.

A = {<G>|G is a connected undirected graph}

A = {<G>|G is a connected undirected graph}

M = “On input string <G> −

• Select the first node of G and mark it.

• Repeat the following stage until no new nodes are marked −

• For each node in G, if it is attached by an edge then, mark to a node that is already marked.

• Scan all nodes of G to determine whether they all are marked or not. If they are, accept; otherwise, reject.”