Turing Machines are a fundamental concept in computer science, formalizing the notion of algorithm and computation. They are simple and powerful devices that consist of a tape, divided into cells that can be read and written, a head that can move left or right along the tape, and a set of states that determine what action to take based on the current cell value and current state. Turing Machines can recognize or compute any language that can be recognized or computed by any other algorithmic system, and they can simulate any other Turing Machine.
In this blog post, we will discuss a problem related to Turing Machines and their expressibility, namely the question of whether a TM-recognizable language of TMs can be expressed by the TM-description language of equivalent TMs. This problem has important implications for the theory of computability and the design of algorithms.
First, let’s define some terms. A language L is TM-recognizable if there exists a Turing Machine M that accepts (halts in an accepting state) on every string in L and either rejects (halts in a non-accepting state) or loops indefinitely on every string not in L. The TM-description language of equivalent TMs is a language that describes Turing Machines and their behaviors. Specifically, it is the set of all descriptions of Turing Machines that are equivalent to some given Turing Machine.
The problem we are considering asks whether there exists a TM-recognizable language of TMs that can be expressed by the TM-description language of equivalent TMs. In other words, can we describe all TM-recognizable languages of TMs using descriptions of equivalent TMs?
To address this problem, we first note that the TM-description language of equivalent TMs is itself a TM-recognizable language. This is because we can construct a Turing Machine that takes as input a description of a Turing Machine, simulates that Turing Machine, and accepts if and only if the simulated Turing Machine halts in an accepting state. Therefore, any language that can be recognized by a Turing Machine can also be recognized by a Turing Machine that simulates other Turing Machines.
Now, suppose that we have a TM-recognizable language of TMs that cannot be expressed by the TM-description language of equivalent TMs. Consider the set of descriptions of all Turing Machines that recognize this language. This set is itself a TM-recognizable language, since we can construct a Turing Machine that simulates each Turing Machine in the set and accepts if and only if at least one of them accepts. But this language cannot be described by any equivalent Turing Machine, since if it could, we could use that description to recognize the original language of TMs, which would contradict our assumption.
Therefore, we have shown that any TM-recognizable language of TMs can be expressed by the TM-description language of equivalent TMs. This result has important implications for the study of computability and algorithms. It shows that the set of possible behaviors of Turing Machines can be exhaustively described by descriptions of equivalent Turing Machines, and that the properties of these behaviors can be characterized by the languages they recognize.
In conclusion, we have discussed a problem related to Turing Machines and their expressibility, and shown that any TM-recognizable language of TMs can be expressed by the TM-description language of equivalent TMs. This result has important implications for the theory of computability and the design of algorithms. By understanding the expressibility of Turing Machines, we can better understand the limits and possibilities of computation, and design more efficient and effective algorithms for solving complex problems.