(a) Show that if a partial function f ∈ℱ1^* is T-computable, then it is computable by a Turing m

(a) Show that if a partial function f ∈ℱ1^* is T-computable,...

(a) Show that if a partial function $f \in \mathcal{F}_{1}^{*}$ is $T$-computable, then it is computable by a Turing machine that has exactly three bands. (b) Consider the set $\mathcal{M}_{n}$ of Turing machines that have three bands and $n$ states. Set these machines in operation with an initial configuration in which all bands are clean. If machine $\mathcal{M}$ halts, we let $\sigma(\mathcal{M})$ be the number of strokes written on its second band at the instant it halts; otherwise, we set $\sigma(\mathcal{M})=0$. Show that the set $$ \left\{\sigma(\mathcal{M}): \mathcal{M} \in \mathcal{M}_{n}\right\} $$ is bounded. We will denote the upper bound of this set by $\Sigma(n)$. (c) Let $f$ be a partial function of one variable that is computable by a machine $\mathcal{M}$ in $\mathcal{M}_{n} .$ For every integer $p$, construct a machine $\mathcal{N}_{p}$ with three bands which, when started in an initial configuration in which all bands are clean, begins by writing $p$ strokes on its first band, then returns its head to the beginning of the tape, and continues to behave exactly as $\mathcal{M}$ would. How many states does $\mathcal{N}_{p}$ have? (d) Show that the function $\Sigma$ is not $T$-computable.

(a) Show that if a partial function $f \in \mathcal{F}_{1}^{*}$ is $T$-computable, then it is computable by a Turing machine that has exactly three bands.
(b) Consider the set $\mathcal{M}_{n}$ of Turing machines that have three bands and $n$ states. Set these machines in operation with an initial configuration in which all bands are clean. If machine $\mathcal{M}$ halts, we let $\sigma(\mathcal{M})$ be the number of strokes written on its second band at the instant it halts; otherwise, we set $\sigma(\mathcal{M})=0$. Show that the set
$$
\left\{\sigma(\mathcal{M}): \mathcal{M} \in \mathcal{M}_{n}\right\}
$$
is bounded. We will denote the upper bound of this set by $\Sigma(n)$.
(c) Let $f$ be a partial function of one variable that is computable by a machine $\mathcal{M}$ in $\mathcal{M}_{n} .$ For every integer $p$, construct a machine $\mathcal{N}_{p}$ with three bands which, when started in an initial configuration in which all bands are clean, begins by writing $p$ strokes on its first band, then returns its head to the beginning of the tape, and continues to behave exactly as $\mathcal{M}$ would.
How many states does $\mathcal{N}_{p}$ have?
(d) Show that the function $\Sigma$ is not $T$-computable.

Key Concepts

Turing Computability

Turing computability is a foundational concept in computability theory that formalizes what it means for a function to be effectively computed. It underpins the idea that if a function can be computed by some abstract algorithmic process (or Turing machine), then there exists a precise mechanical procedure that, step by step, produces the result. This concept establishes Turing machines as the standard model for what is computable and is central to proving equivalence between different models of computation.

Three?Band Turing Machines

The notion of a three?band Turing machine is an extension of the standard single-tape Turing machine model, where the computation is conducted on three separate tapes (or bands). This multi-band model allows more structured manipulations by dedicating specific tapes for particular purposes. As shown in the problem, any function that is Turing computable can be simulated by such a machine with exactly three bands, thereby exploring the sufficiency and efficiency of limited multi-tape architectures in computational devices.

Simulation of Turing Machines

Simulation of one Turing machine by another is a key concept where one constructs a machine that replicates the behavior of another machine, often with modifications such as additional initial input or resource constraints. In the problem, a new machine is constructed to simulate another machine's behavior after performing an initial operation (writing a certain number of strokes), thereby demonstrating how computational processes can be emulated or transformed by machines with different configurations while preserving their computational properties.

State Complexity in Turing Machines

State complexity refers to the number of states used in a Turing machine and is a measure of the machine's descriptive compactness or efficiency. The problem touches on this by asking how modifications to a machine (like writing extra strokes on the tape) affect the overall number of states. Analyzing state complexity is important in understanding trade-offs in the design of machines, especially when considering optimizations and resource bounds in algorithm design.

Boundedness and Non-computability of Resource Functions

This concept involves analyzing functions that measure certain computational resources (such as the number of strokes written on a tape) and determining their computability properties. In the given problem, the function ?(n) represents an upper bound on a resource (strokes written) for machines with n states. Showing that such functions are not Turing computable highlights that there are inherent limitations in algorithmically predicting or optimizing certain computational resources, which is a recurring theme in the study of algorithmic complexity and undecidability.

Transcript

00:02 We're asked to prove that s is 0 related to t if and only if s and t are both final states, or neither are final states.

00:19 So, first, let s be 0 related to t.

00:30 This is part a, by the way.

00:38 This implies that f of s x and if tx are both final or non -final.

00:57 States for all strings x such that the length of x is less than or equal to zero.

01:15 In other words, since lambda, the empty string is the only string of length zero, it follows that f of s lambda and f of t lambda are either both final or both non -final states, where lambda is the empty string.

01:43 Now we have that by definition f of s lambda is s and f of t lambda is t.

02:02 So we have that s and t are both final states or both non -final states.

02:18 Now that was one direction.

02:21 We want to prove the other direction.

02:23 So let's suppose that s and t are both final states or neither are final states.

02:27 This implies that f of s lambda and f t lambda are both final or are both non -final states, since lambda is the empty string, and this is true by definition.

03:19 Now, lambda is the only string of length 0, so it follows that f of s x and f of t x are both final or non -final states for all strings x of length less than are equal to zero.

04:07 Therefore, it follows that s is zero related to t.

04:12 This is what we wanted to prove.

04:16 So we've shown s is zero related to if only if t and s are both final states, and neither are final states.

04:34 Now we're asked to conclude that each final state of 1 over m, which is an r star equivalence class, contains only final states of m.

04:48 So let a be an equivalence class of m bar.

05:08 Now this implies that a is an r -star equivalence class of m.

05:31 So all states in a are either all final states or are all non -final states in m by a previous exercise.

06:08 And this implies that all final states of m bar are also final states in m.

06:43 Now in part b, we're asked to prove that if k is a positive integer, then it follows that s.

06:52 S is k related to t if and only if s is k minus 1 related to t and f of s a are k minus 1 f of t a for every a in the language.

07:23 So let k be a positive integer.

07:34 Now suppose that s is k related to t then it follows that f of s x and f of tx are both final states or both not final states.

07:52 For all strings x of length less than or equal to k.

08:22 Now let y be an input string of length k minus 1 and let a be an element of the input set.

08:49 So since y is an input string of length at most k, it follows that s is k minus 1 related to t.

09:07 Now, ya is the strength of length at most k.

09:19 Well, we have that f of s, a .y is equal to f of s...

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