How to formally prove the dependencies of a computer malware?

I’m in the process of the writing a thesis. A small part of it is to prove that certain malware have certain dependencies which must first be satisfied before they are successful in infecting the host. For instance, a virus must first get on the host and then start executing before infection.

  • Dependency 1: getting on the host
  • Dependency 2: executing

We know these dependencies to be true from experience and common sense, however, how would be go about formally proving these in computer science? I am not asking for all the proofs (I realize that that’s my job!), but just how to approach them since right now I see no way to formally prove it.

Given a DFA M, formally define an NFA N such that L(N) = {x in L(M) | x = reverse(x)}

The english description of the question is (from my understanding) N accepts all strings that are both palindromic (the same forwards as it is backwards) and accepted by M. After a lot of toil and trouble, I’m not convinced that such a definition of N is even possible. My thinking is as follows:

Say M above is the DFA that accepts the language $ (1 + 0)^*$ . Then the desired N would describe all palindromes over the language $ \{1, 0\}$ . However, such a language is not regular (I think). Therefore, is it not the case that such a formal definition is not possible, as NFAs can only describe regular languages?

Perhaps I am missing some subtle nuance in the question, but I believe that such a formal definition is simply not possible. Is my thinking correct?

How does a merger formally impacts an ISO 27001 certification?

Organization A has a service that is ISO 27001 certified. It is acquired by Organization B which does not have any certification.

What are the formal impacts of the acquisition on the ISO 27001 certification?

I am interested in two cases:

  1. right after the acquisition when nothing changed yet in Organization A
    → my understanding is that the certification is intact as i) the scope has not changed and ii) the means to handle the requirements (patch management for instance) has not changed either

  2. Organization B integrates Organization A and the means to handle the requirements have changed. To take the patch management example above, it is now ad-hoc, uncontrolled, in one word not suitable for ISO 27001 requirements.
    → does the ISO certification still holds?

Another way of looking at it is whether the certification is a snapshot checked every year (with the hope that things are correct over the year), or whether any negative change over that year automatically invalidates it.

If the latter: how does this invalidation happens?

How do I formally prove a universal implication?

A textbook I am reading (Discrete Mathematics and its Applications) went from introducing formal propositional and predicate logic (including popular rules of inference like Modus Ponens and Universal Generalization) to introducing direct methods of proof for theorems of the form ∀n(P(n)->Q(n)).

Apparently, most mathematical proofs of any kind of theorem are “informal” and omit many logical rules of inference and argumentative steps for the sake of conciseness. However, because the textbook doesn’t provide even one example of a detailed “tedious” proof that expresses most or all rules of inference and axioms used in the proof, though I have a general idea of the connection between the two, I have been struggling to fully tie together the ideas of formal logic to the ideas of mathematically proving theorems of the form ∀n(P(n)->Q(n)). Can anyone provide an example of a detailed mathematical proof of a simple theorem that omits few (if any) logical steps in the argument? I have personally struggled with (as a personal exercise) meticulously proving the theorem “for all integers, if n is odd then the square of n is odd”, but any logically detailed argument proving a simple theorem similar to that would be very useful.

How do I formally prove a universal implication?

A textbook I am reading (Discrete Mathematics and its Applications) went from introducing formal propositional and predicate logic (including popular rules of inference like Modus Ponens and Universal Generalization) to introducing direct methods of proof for theorems of the form ∀n(P(n)->Q(n)).

Apparently, most mathematical proofs of any kind of theorem are “informal” and omit many logical rules of inference and argumentative steps for the sake of conciseness. However, because the textbook doesn’t provide even one example of a detailed “tedious” proof that expresses most or all rules of inference and axioms used in the proof, though I have a general idea of the connection between the two, I have been struggling to fully tie together the ideas of formal logic to the ideas of mathematically proving theorems of the form ∀n(P(n)->Q(n)). Can anyone provide an example of a detailed mathematical proof of a simple theorem that omits few (if any) logical steps in the argument? I have personally struggled with (as a personal exercise) meticulously proving the theorem “for all integers, if n is odd then the square of n is odd”, but any logically detailed argument proving a simple theorem similar to that would be very useful.

Formally smooth algebra of a field

Let $ k$ be a field of characteristic zero and $ R$ a local $ k$ -algebra. By Stacks \tag 00TX, if we assume that $ R$ is of finite type, then the $ R$ is smooth over $ k$ if and only if $ \Omega_{R/k}$ is free. I’m wondering if the analogous statement is true for formal smoothness, i.e is the formal smoothness of $ R$ over $ k$ equivalent to the freeness of $ \Omega_{R/k}$ ?

Find the Taylor series of this polynomial. How do I formally show radius of convergence?

The question I am asked is this:

Find the Taylor series for $ f(x)$ centered at the given value of a. [Assume that f has a power series expansion. Do not show that $ R_n(x) -> 0.$ Also find the associated radius of convergence.

I’m having trouble finding a general formula of this Taylor series and therefore, also having problems finding the radius of convergence since I can’t perform the ratio test:

$ $ f(x) = x^6 – x^4 + 2$ $ at $ a = -2$

so:

$ $ f'(x) = 6x^5 – 4x^3$ $ $ $ f”(x) = 30x^4 – 12x^2$ $ $ $ f”'(x) = 120x^3 – 24x$ $ $ $ f^{iv}(x) = 360x^2 – 24$ $ $ $ f^{v}(x) = 720x$ $ $ $ f^{vi}(x) = 720$ $

and at a = -2 $ $ f(-2) = 50$ $ $ $ f'(-2) = -160$ $ $ $ f”(-2) = 432$ $ $ $ f”'(-2) = -912$ $ $ $ f^{iv}(-2) = 1416$ $ $ $ f^{v}(-2) = -1440$ $ $ $ f^{vi}(-2) = 720$ $

I’m having trouble finding the general formula for each term. Without it, how am I supposed to find the radius of convergence?

EDIT So the general term I have for the derivative of x is:

$ $ f^n(x) = \frac{6!x^{6-n}}{(6-n)!}$ $

So far the general term I have for the Taylor Series is:

$ $ \sum_{n=0}^{\infty} \frac{6! 2^{6-n}}{(6-n)!n!}(x+2)^n$ $

I can see why the radius of convergence is $ \infty$ , it’s because $ x$ can be anything and it’ll converge. But how do I show this formally? Can I use the ratio test?

Formally proving that a metric is not induced by any norm in $\mathbb{R}^n$

What is the procedure to formally prove that no norm exists in $ \mathbb{R}^n$ , that induces a metric $ d$ ?

My first instinctive idea would be to show that $ d$ is a metric in $ \mathbb{R}^n$ , but after this I don’t know any further. What could I achieve by following this road?

The specific problem I am working on is to prove that no norm $ ||\cdot||$ in $ \mathbb{R}^2$ exists, that induces the metric $ d$ :

$ d((x_\text{1},y_\text{1}),(x_\text{2},y_\text{2})) := \begin{cases} |y_\text{1}-y_\text{2}|, & \text{if }x_\text{1} = x_\text{2},\ |y_\text{1}| + |x_\text{1}-x_\text{2}| + |y_\text{2}|, & \text{if }x_\text{1} \not= x_\text{2}. \end{cases}$

I do not intend to get solutions for my specific problem, but maybe a similar problem exists elsewhere, which I could study. Of course, I have searched MathOverflow, and other sources.