Turing Machines proof notations

In context of “Computability”, I have went over some proofs for Recursion Theorem using Turing Machine description. A TM M stands for a single tape Turing machine and is the description of TM M. Even some semi-decidable machines such as Atm = { < M, w > | TM M accepts w}. My question is simple

What is difference between using < M > ,< w > vs < M , w > for a TM description ?

A simplistic answer I assume is that it is being a convention in writing formal proof that books follows standard of combining all inputs as a single description (all inputs comes under single brackets as opposed to multiple descriptions)?

SharePoint DSC – Build a complete farm with seperate machines?

I’m currently a novice to the Desired State Configuration thing and may not found the “right” source for this. I already created a configuration (ps1) and configuration data file (psd1) and even rolled it out to a SharePoint server.

But how to speak to other machines than the one I’m currently logged on? I was thinking about remote installation of SQL Server, about creating domain users on the domain controller, etc.

Maybe some of you have some sources or experience how to it right?!

Distribute bash script ‘silently’ to multiple macos machines

We want to deploy a script ‘silently’ across all mac machines in organization. We have created a bash script that will run using launchd agent. we need to deploy the launchd agent and script on all macos machines. some instructions are needed during installation:

• The bash script ‘script1’ should be added to folder “/Library”

• Read and execute permissions should be added. Manually it can be done by running the command: “sudo chmod 511 /library/script1”

• The Agent “com.script1.plist” should be added to folder “/Library/LaunchAgents”

• Permission should be updated : “sudo chmod 644 /Library/LaunchAgents/com.script1.plist”

Can you please advise how to package all of this and distribute on all machines silently?

Superset of another language and recognizability of turing machines

$ L_1$ = $ \{\langle M \rangle \mid M$ is a turing machine and $ M$ halts on some string$ \}$

$ L_2$ = $ \{\langle M \rangle \mid M$ is a turing machine and $ M$ halts on all strings $ \}$

a) Is $ L_2$ a superset of $ L_1$ ?

b) Is $ L_2$ not co-recognizable and not recognizable ?

(without formally proving)


my attempt

$ a)$ $ L_2$ is a superset of $ L_1$ since all strings are in $ L_2$ whereas $ L_1$ is some string

$ (b)$ $ L_2$ is not recognizable because, out of the infinite enumerations of strings, a single one could not halt disproving it from being recognizable.

For not co-recognizable: It can be written like this

$ \bar{L_2} = \{ \langle M \rangle \mid M$ is a turing machine and $ M$ loops on some string $ \}$ .

It’s not possible to make a recognizer for loops so its instantly not co-recognizable. This is because looping means it never stops so to come up with a recognizer for anything to prove it loops is impossible since we are working with infinite combinations.

Not sure about it being a superset or not and my explanation for co-recognizable

How to configure, manage and automatically update application built on GCP on a network of 50 odd Ubuntu machines?

I want to manage a network of Ubuntu machines from GCP. These physical devices are at a different location. I will develop and build software using CI/CD with bitbucket, Jenkins, Maven, and Docker on GCP and I need to install and update this software automatically on all Ubuntu machines from GCP.

My questions are:

  1. How to configure, register and manage all Ubuntu machines from GCP?
  2. How to update newly build software docker image everytime on all Ubuntu machines automatically and securely from GCP?

Random Access Machines with only addition, multiplication, equality

The literature is fairly clear that unit-cost RAMs with primitive multiplication are unreasonable, in that they

  1. cannot be simulated by Turing machines in polynomial time
  2. can solve PSPACE-complete problems in polynomial time

However, all of the references I can find on this topic (Simon 1974, Schonhage 1979) also involve boolean operations, integer division, etc.

Do there exist any results for the “reasonableness” of RAMs that only have addition, multiplication, and equality? In other words, which do not have boolean operations, truncated integer division, truncated subtraction, etc?

One would think that such RAMs are still quite “unreasonable.” The main red flag is that they enable the generation of exponentially large integers in linear time, and due to the convolution-ish effects of multiplication, this can get particularly complex. However, I cannot actually find any results showing that this allows for any sort of “unreasonable” result (exponential speedup of Turing machine, unreasonable relationship to PSPACE, etc).

Does the literature have any results on this topic?