I came across following question:

What are maximum number of configuration of Turing Machine after $n$ moves?

The answer given was:

$k^n$, where $k$ is a branching factor.

And that "branching factor" left me confused. So I gave some thoughts: Given $Q$ be total number of states, $\Gamma$ be a tape alphabet and two moves, left and right $\{L,R\}$, for every transition function, we have $2^{Q\times \Gamma \times 2}$ possible transitions at each of those $n$ moves. Thus, $k$ must be $2^{Q\times \Gamma \times 2}$. So, total number of configurations of Turing machine after $n$ moves must be ${(2^{Q\times \Gamma \times 2})}^n$. Am I correct with this?

  • $\begingroup$ The question should probably have been "What is the lowest upper bound on the number of configurations a general Turing machine can be in after $n$ moves?" $\endgroup$ – Peter Taylor Nov 12 '19 at 10:45

Note that you are bounding the maximal number of configurations, that the machine can be in, from above. To see it more clearly, when a TM $M$ runs on a finite word $w$ it induces a configuration tree $G = \langle V, E\rangle$ where:

  • $V$ is the set of configurations that $M$ can be in when it runs on $w$,

  • $q_0w$ is the initial configuration of $M$ on $w$ and it is the root of the configuration tree,

  • and the edges in $E$ are defined such that $u$ is a child of $v$ whenever $u$ a consecutive configuration of $v$ (by consecutive we mean that it is possible to reach $u$ from $v$ without violating the transition function of $M$).

As you have mentioned, $k$ is a branching factor, and we usually mean by that the maximal branching degree in the configuration tree. In this aspect you're not correct, $k$ is at most $|{Q\times\Gamma\times \{L, R\}}| = {2|Q|\cdot |\Gamma|}$, and this follows from the definition of the transition function. Indeed, $k$ bounds from above the number of consecutive configurations (which is a subset of configurations) and not the number of all possible subsets of configurations of the TM.

Considering the configuration tree, it is a tree of height ($n = |w|$) with branching degree $k \leq {2|Q|\cdot |\Gamma|}$. Now its not hard to see that such a tree has at most $k^n$ leaves which bounds from above the number of configurations that $M$ can be in when it runs on a word $w$ of length $n$.

A thing worth mentioning is that, $k$ is a constant that does not depend on the input word $w$ (on $n = |w|$) or on a specific configuration tree, and we know that it exists. This sometimes makes life a bit easier as some algorithms/proofs rely on its existence to conclude some upper bounds.

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