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\section{Honors Theory: Hints for HW1}

All machines here are deterministic.

\paragraph{(1) Using PM to simulate TM.}

	This is relatively easy.

	Let M be any k-tape TM.

	We construct a PM P to simulate M.

	Let
		$$\Delta=\{c, q, \wh{q},
		t_0,t_1\dd t_k, h_0\dd h_k,\ldots\}.$$
	The meanings of these labels will be explained.

	If $w\in\Delta^*$, then let $[w]$ denote the
	node pointed to by $w$ in the current $\Delta$-structure.
	
	The contiguous cells containing non-blank symbols
	on the $i$th tape of M corresponds
	to the nodes $[t_i^+]$.  (Note: $t_i^+$ is the set
	$\{t_i, t_i^2, t_i^3 ,\ldots\}$ and $[t_i^+]$ is the set of nodes
	$\{[t_i], [t_i^2], [t_i^3] ,\ldots\}$.)

	If M has $m$ tape symbols, then these are represented
	by the nodes in the set $[c^+]$.  This set has exactly
	$m$ nodes (we arrange these nodes in a linear list
	with the last node have its $c$-edge pointing back to itself).

	The symbol stored in cell $[t_i^m]$ is given by $[t_i^m c]$.

	The cell being scanned by the $i$th tape head is given by $[h_i]$.

	It is assume that P is initialized with the correct
	input tape contents.  The work tapes are initially
	empty: that is, for $i=1\dd k$,
	we have 
		$$[t_i^+]=[\ess]$$
	(where $[\ess]$ is the center.

	If M has $n$ states, then $[q^+]$ will have $n$ nodes
	arranged in a linear list, as usual.
	The label $\wh{q}$ indicates the current state of M:
	if the current state is $[q^m]$, then we have $[\wh{q}]=[q^m]$.
	Note: the first thing that P does is to initialize
	$[q^+]$ and $[\wh{q}]$ correctly.

	We leave the rest of the details to the student.


\paragraph{(2) Using a TM to simulate a DPM.}

	This is harder.

	Let P be any PM and M be a TM we will construct
	to simulate P.

	We assume the input to P follows the convention in part (1)
	above.  Thus, we initially have the input symbols
	$[c^+]$ and the input word $[t_0^+]$.

	
	Moreover, we assume that P never modify the initial
	input (to conform to our conventions for TMs).
	That is, assignments of the form
		
	$$t_0^m \leftarrow w,  t_0^m c \leftarrow w, c^m \leftarrow w$$

	is illegal for all $m\in\NN$.
	

	The machine M has several tapes with the following functions.
	Basically, we need to store some representation of the
	$\Delta$-structure used by P.  Assume $|\Delta|=d$.

	Tape 1: stores in binary notation the value $n$ equal to
		current number of nodes in the current
		$\Delta$-structure.  The length of
		$n$ is $\floor{\log_2 n}+1$ is denoted by $s$.

		NOTE: it is assume that the last bit in the
		binary notation has some additional mark
		to indicate that it is the last bit.  This
		self-limiting notation is assumed
		in all our binary notations.

	Tape 2: stores a representation of all the nodes in
		the $\Delta$-structure.
		If $u$ is the $i$th node represented on tape 2,
		let $#(u)$ be the binary notation for $i$
		(padded out to exactly $s$ bits).
	
		If the $d$ successors of a node $u$ are
		$u_1\dd u_d$, then the node $u$ is represented
		by the binary sequence
			$$#(u_1), #(u_2) \dd #(u_d).$$
		This sequence can be stored using exactly
		$ds$ cells.  There are $n$ nodes, and hence
		the space used in tape 2 is $nds$.

	Tape 3: There are finitely many words that are
		referenced in the instruction set of P.  Note:
		in the instructions
		
		$$w.i\leftarrow w',
			w.i\leftarow \NEW,
			\IF\ (w=w') \THEN\ \GOTO\ L$$

		the words $w$ and $w'$ are said to be referenced.

		Suppose $w_1\dd w_\ell$ are referenced in P.
		Then in tape 3, we store the sequence

			$$#([w_1]), #([w_2])\dd #([w_\ell]).$$
			
		This takes space $\ell s$.

	Tape 4, 5: This provides scratch space in our computation.

	Our machine M has various states, but certain states
	are called DISTINGUISHED.  There is a 1-1 correspondence
	between the distinguished states and the instructions of P.
	Whenever we enter a DISTINGUISHED state, the contents
	of tapes 1,2 and 3 will conform to the specifications
	above.  In the non-distinguished states, the tape contents
	are in transition and may not conform to the specifications.

	Each time a new node is created in the $\Delta$-structure,
	we need to increment the value $n$ on tape 1.  If this
	causes the value $s$ to increase, then we must
	first update the contents of tapes 2 and 3 in the natural
	way before proceeding.

	We leave the remaining details to the reader.


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