# Questions tagged [turing-machines]

Questions about Turing machines, a theoretical model of mechanical computation capable of simulating any computer program.

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### Why is this true: “There are countably many Turing Machines” [duplicate]

It is said that there are uncountably many languages but only countably many Turing Machines. Could someone make this clear to me? And this doesn't mean that the set of TM is finite, yes?
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### How do you classify properties as Trivial and Non-trivial? [duplicate]

I understand what Rice's theorem states and what Trivial and Non-trivial properties mean. However, when given some property, I am having a hard time seeing if it is Trivial or Non-trivial. Can someone ...
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### What is the difference between RAM and TM?

In algorithm analysis, we assume a generic one processor Random Access Machine (RAM). As far as I know, the RAM machine is no more efficient than the Turing machine. All algorithms can be implemented ...
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### Can we write algorithms without conditional statements?

Regarding turing completeness, i read that for a language/machine to be turing complete it is required that it has some sort of conditional. Consider the factorial problem, we would typically define ...
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### Quantum Computing and Turing Machines: Are Turing Machines still an Accurate Measure?

In class last week, my professor commented and said that Turing machines are used as a standard measure/model of what is computable and are a helpful basis of discussion for that subject. She also ...
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### Is it decidable whether a TM reaches some position on the tape?

I have these questions from an old exam I'm trying to solve. For each problem, the input is an encoding of some Turing machine $M$. For an integer $c>1$, and the following three problems: ...
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### What is the relationship between Turing Machines with a finite tape and Finite State Automata?

I seem to recall from an undergraduate class that for a Turing Machine with a finite tape there will always exist a corresponding Finite State Automata, but I've been unable to find this confirmed ...
6k views

### The first Turing machine

Does anyone know how efficient was the first Turing machine that Alan Turing made? I mean how many moves did it do per second or so... I'm just curious. Also couldn't find any info about it on the web....
158 views

### Computability of equality to zero for a simple language

Suppose we have a tree in which leaves are labeled with a set of numbers $L$, and internal nodes with a set of operations $O$. In particular $L$ can be $\mathbb{N}, \mathbb{Z}$ or $\mathbb{Q}$, and ...
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### Why is simulation by non deterministic Turing machine faster than a deterministic one?

A deterministic universal Turing machine $U_D$ can simulate a deterministic turing machine $M_D$ in $O(T(n)log(T(n)))$ where $M_D$ runs in $O(T(n))$. But I came across an exercise in Sanjeev Arora and ...
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### Can you recognize or decide if a Turing Machine has an infinite sized language?

That is, can you build a Turing Machine that, if given a Turing Machine as input, can decide (or at least recognize) if the inputted Turing Machine has an infinite number of strings in its language? ...
527 views

### Undecidable problems limit physical theories

Does the existence of undecidable problems immediately imply the non-predictability of physical systems? Let us consider the halting problem, first we construct a physical UTM, say using the usual ...
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### Is the leapfrog automata problem in P?

My question is whether a specific decision problem—finding a computation path through a "leapfrog automaton"—is in P or not. It's straightforwardly in NP, and it resembles the hamiltonian ...
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### What is the difference between a TM accepting and deciding a language?

Frankly I'm very uncomfortable with the material right now. There are some things I can understand, but many I still do not. My first assignment is asking me in one question (which I do know how to ...
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### undecidable problem and its negation is undecidable

A lot of "famous" undecidable problems are nonetheless at least semidecidable, with their complement being undecidable. One example above all can be the halting problem and its complement. However, ...
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### Please explain this formal definition of computation

I am trying to attack TAOCP once again, given the sheer literal heaviness of the volumes I have trouble committing to it seriously. In TAOCP 1 Knuth writes, page 8, basic concepts:: Let $A$ be a ...
567 views

### Is the halting problem always decidable for non-universal programs?

For every non-universal computable program $P$ that takes input of type $D$ does there exist some total computable function $g$ that takes an input $I$ of type $D$ and decides successfully whether $P$ ...
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### Union of R.E. and Non R.E. language

Let \begin{align*} L_1 &=\{\langle M,w\rangle \mid M\text{ halts on }w\}\\ L_2 &=\{\langle M,w\rangle \mid M\text{ does not halt on }w\}\,. \end{align*} Here $M$ represents encoding ...
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### Prove that turing machines and the lambda calculus are equivalent

It is known that a turing machine and the lambda calculus are equivalent in power. I now want to try to prove this myself. I think proving that the lambda calculus is at least as powerful as a turing ...
I have a deep appreciation for formalisms like the Turing Machine and the $\lambda$-Calculus, and enjoy studying them and learning more about how they relate to physical computers. I am now learning ...