I stuck on this question for a long time and cannot figure out how to prove it?
Tag: prove
a^ib^jc^k, i < j < k is a contextsensitive 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 contextsensitive 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 sideeffects 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 NPcomplete (or NPHard)
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, .., T1$ , 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,T1] 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 pseudpolynomial. 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 worstcase 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, .., T1$ , 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,
Verifying passwords = Polynomial (linear)
Guessing passwords = Exponential
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 maxflow linear program indeed is a mincut linear program?
So the wikipedia page gives the following linear programs for maxflow, and the dual program :
While it is quite straight forward to see that the maxflow 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 maxflow linear program indeed is the LP of the mincut 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 maxflow LP indeed is the LP formulation for mincut ?
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.