## Show {𝑎^i𝑏^j𝑐^k, i!=j!=k} is context free or not, how can we prove it?

I stuck on this question for a long time and cannot figure out how to prove it?

## a^ib^jc^k, i < j < k is a context-sensitive language, how can prove it as a context sensitive

I am thinking this question for a long time, that a^ib^jc^k, i < j < k is a context-sensitive language, how we can prove it as a context sensitive or which grammar can generate such a language. Thanks for all of your help

## Methods to Prove Data Authenticity from Potentially Compromised Sources?

I’ve been thinking about this problem for some time and I wanted to ask if there are any known methods, or research papers, about how to prove "authenticity" or correctness of data originating from a potentially compromised source (remote server, process, etc). Specifically what I’ve been imagining is say you have service A and service B, service B sources data from A but is worried that A has been compromised such that even if data is signed by A, B can’t trust that it was generated by code written by A‘s developers. Is it possible for B to prove to itself that data from A is authentic, that it was indeed generated by the expected code and not injected or generated by an attacker who has compromised A?

One solution I’ve been thinking about is using a sort of distributed ledger or blockchain so that multiple nodes compute the same data, and in doing so raises the bar such that an attacker would have to compromise N% of the services producing the needed data, this provides naturally replication and I can use an appropriate consensus protocol, but ofc introduces some overhead, efficiency concerns, and I would need to think hard about side-effects being performed more than once.

If there is only one node possible of generating data, such as a sensor node, and it is compromised, I’d imagine all hope is lost, but I also wouldn’t be surprised if there is some clever crypto scheme that attempts to solve this problem as well.

I hope it’s clear as to what the question is, thank you.

## Prove following statement about Kruskal Algorithm

Let G be undirected graph, G=(V,E). Consider an edge e=(u,v)∈E that wasn’t included in the solution obtained from applying Kruskal Algorithm to G. Prove that this edge isn’t in any Minimimum Spanning Tree of G.

## prove: max (w(E), w(E)) is a 1/2 approximation to the value OPT

Hey I would like to find a answer for b. for a look to the picture that is my answer for it. But I dont habe any Idea how i can solve this. Thank you guys. (I had to translate it to english maybe it will be hard to understand it sorry about that).

## How to prove that this problem is NP-complete (or NP-Hard)

I have the following data:

• A set $$V$$ of tasks, the starting time $$s_j$$ of each task and the duration $$p_j$$ of each task.

• A set $$K$$ of resource, each resource has an availability function $$R_{k}$$ that is piecewise constant.That is, for each $$t = 0, .., T-1$$, we precise $$R_{k}(t)$$ the number of units available at $$t$$. $$R_k$$ is an array of length $$T$$.

• Each task $$j$$ needs $$r_{j,k}$$ resources to be processed (it could be zero). This quantity needs to be available during all the processing time starting from $$s_j$$.

For example consider :

• Task$$A$$ has processing time $$3$$ and starts at time period $$t=2$$ and needs 2 units of some resource $$k$$
• Task $$B$$ has processing time $$4$$ starts at time $$t=3$$ and needs 3 units of the same resource $$k$$.

Then if $$R_{k}(t) = [*,*,2,6,6,3,*,*]$$ then we are ok since at time $$t=2$$ only task $$A$$ is active and it requires $$2$$ units, at time $$t=3$$, both tasks are active and the sum of their utilization is $$2+3 = 5 \leq 6$$; same at time $$t=4$$. At time $$t=4$$, only task $$B$$ is active and it requires $$3$$ units.

However, if $$R_{k}(t) = [*,*,2,4,6,3,*,*]$$, is not ok since at time $$t=3$$, both tasks $$A$$ and $$B$$ are active and their total use is equal to $$5$$ wheras only $$4$$ units are available.

Here is my attempt to verify that the resource utilization at each $$t$$ is no larger than the availability function. So the answer is yes or no (we can say that this is a decision problem).

For each time t in [0,T-1]   For each resource k in K      total_use = 0, active_set = A     for each task j in V       if s_j<=t and s_j+p_j > t and r_{j,k}>0 \if the task is active at time t and it requires positive amount of resource k in order to be processed)         total_use += r_{j,k}         active_set := active_set U {j}        if total_use > R_{k}(t)         print(at time t the usage of resource k exceeds its capacity, active_set)         return False return True 

The algorithm here is pseud-polynomial. Unfortunately, I need to find a polynomial one in order to say that the problem is in $$\mathcal{NP}$$.

## How to prove that there is no algorithm with worst-case running time better than this one?

I have the following data:

• A set $$V$$ of tasks, the starting time $$s_j$$ of each task and the duration $$p_j$$ of each task.

• A set $$K$$ of resource, each resource has an availability function $$R_{k}$$ that is piecewise constant.That is, for each $$t = 0, .., T-1$$, we precise $$R_{k}(t)$$ the number of units available at $$t$$. $$R_k$$ is an array of length $$T$$.

• Each task $$j$$ needs $$r_{j,k}$$ resources to be processed (it could be zero). This quantity needs to be available during all the processing time starting from $$s_j$$.

Here is my attempt to verify that the resource utilization at each $$t$$ is no larger than the availability function.

Algorithm

## Why don’t passwords prove P != NP?

Pardon my ignorance on the matter but,

Since each guess has nothing to do with one another, exponential time is best possible time (but verifiable in linear time).

## How to prove that the dual linear program of the max-flow linear program indeed is a min-cut linear program?

So the wikipedia page gives the following linear programs for max-flow, and the dual program :

While it is quite straight forward to see that the max-flow linear program indeed computes a maximum flow (every feasable solution is a flow, and every flow is a feasable solution), i couldn’t find convincing proof that the dual of the max-flow linear program indeed is the LP of the min-cut problem.

An ‘intuitive’ proof is given on wikipedia, namely : $$d_{uv}$$ is 1 if the edge $$(u,v)$$ is counted in the cut and else $$0$$, $$z_u$$ is $$1$$ if $$u$$ is in the same side than $$s$$ in the cut, and $$0$$ if $$u$$ is in the same side of the cut than $$t$$

But that doesn’t convince me a lot, mainly why should all the variables be integers, while we don’t have integer conditions ?

And in general, do you have a convincing proof that the dual of the max-flow LP indeed is the LP formulation for min-cut ?

## How to prove νX. A × X ≅ (μX. 1 + X) -> A?

How can we prove Stream A = νX. A × X is isomorphic to Nat -> A = (μX. 1 + X) -> A ?

In programming sense, Stream A can be seen as a function from Nat to A, and I can write isomorphisms between them. But how can this be proven mathematically?

I would also like to know the conversion mechanism from μ to ν, and vice versa.