In theoretical computer science, the terms "oracle" and "adversary" are often used in the context of complexity theory, where they play important roles in proving lower bounds on the computational complexity of problems.
An oracle is an abstract computational device that is used to represent a hypothetical source of information that can be queried by an algorithm. In other words, an oracle is an idealized black box that provides the answer to a particular decision problem when queried. For example, an oracle for the graph isomorphism problem could provide the answer to the question "Are these two graphs isomorphic?" when given the two graphs as input. The idea of using oracles is a way to abstract away the complexity of certain problems and focus on the complexity of the algorithm itself.
An adversary, on the other hand, is a hypothetical entity that is used to model an attacker in a computational security context. The adversary is assumed to have unlimited computational power, and its goal is to break a security protocol or algorithm by finding a weakness or exploiting a vulnerability. The use of an adversary is a way to analyze the security of a system or algorithm in a worst-case scenario, where an attacker has the maximum amount of resources and knowledge available to them.
In complexity theory, oracles and adversaries are often used in conjunction to prove lower bounds on the computational complexity of problems. An adversary argument involves proving that no algorithm can solve a given problem with a certain amount of resources, even when given access to an oracle that provides additional information about the problem. An oracle argument, on the other hand, involves proving that no algorithm can solve a given problem with a certain amount of resources, even when given access to an oracle that provides additional computational power.
For example, to prove a lower bound on the time complexity of a problem, one might use an adversary argument to show that an adversary can construct a worst-case input that requires a certain amount of time to solve. To prove a lower bound on the space complexity of a problem, one might use an oracle argument to show that no algorithm can solve the problem with a certain amount of space, even when given access to an oracle that can solve a related problem in constant time.
Overall, oracles and adversaries are powerful tools for analyzing the computational complexity of problems and proving lower bounds on the resources required to solve them. They are essential concepts in complexity theory and are used extensively in a wide range of research areas, including cryptography, computational geometry, and graph algorithms.
you want simple answer?
Sure! In computer science, we use the terms "oracle" and "adversary" to help us study how difficult it is to solve certain problems. An oracle is like a magic genie that can answer any question we ask it, but we don't know how it works or what it does behind the scenes. An adversary is like a really smart and sneaky hacker who is trying to break into a system, and we want to make sure our system is strong enough to withstand their attacks.
We use oracles and adversaries to help us understand how much time or space it takes to solve certain problems. For example, imagine we have a problem where we want to figure out if two graphs are the same or different. We might use an oracle to help us solve this problem - we can ask the oracle if the two graphs are isomorphic (meaning they have the same structure), and the oracle will tell us yes or no. But we don't know how the oracle does this - it might be really fast or really slow, and we can't control it.
On the other hand, an adversary is like an opponent who is trying to make our problem harder to solve. For example, if we want to sort a list of numbers, an adversary might give us a really tricky list that is hard to sort quickly. We want to make sure that our sorting algorithm can handle even the trickiest lists an adversary can throw at us.
So, oracles and adversaries are important tools that help us study how difficult it is to solve problems in computer science. We use them to prove that certain problems are really hard to solve, even with the best algorithms and the most powerful computers.
example which is in detail but simple to understand
Sure, I can provide a simple example to explain oracles and adversaries.
Let's say you want to find the square root of a large number, say 100 digits long. One way to do this is to use an algorithm that repeatedly tries different values until it finds one that, when squared, is close to the target number. This is a difficult problem, and the algorithm might take a long time to find the answer.
Now, let's introduce an oracle into the problem. Imagine you have access to an oracle that can quickly tell you whether a given number is a perfect square or not. In other words, you can ask the oracle whether a number has an integer square root or not, and the oracle will tell you the answer in a single operation.
With this oracle, you can significantly speed up the process of finding the square root of a number. You can use the oracle to eliminate a lot of candidate values that won't work, since you know they are not perfect squares. This reduces the amount of work the algorithm has to do, and makes the problem easier to solve.
However, the problem is not trivial yet. An adversary can still make the problem difficult by giving you a target number that is not a perfect square, but is very close to one. In this case, the oracle is not very helpful, since you will need to try many values before finding one that is close enough to the square root. The adversary has found a weakness in your algorithm, and you need to come up with a better one that can handle these tricky cases.
So, in summary, an oracle is a hypothetical tool that can provide information that makes a problem easier to solve, while an adversary is a hypothetical opponent who can make the problem more difficult to solve. In practice, we don't have perfect oracles or adversaries, but the concepts are useful for understanding the difficulty of different problems in computer science.
one more example with proof of lower bound?
Sure, I can provide an example and prove its lower bound using the concepts of oracles and adversaries.
Consider the problem of finding the maximum element in an unsorted list of n numbers. One algorithm for solving this problem is to simply scan the list and keep track of the maximum element seen so far. This algorithm takes O(n) time, which is the best possible running time if we only consider the input size n.
However, suppose we introduce an oracle that can quickly tell us whether a given number is the maximum element in the list or not. In other words, we can ask the oracle whether a given number is greater than all the other numbers in the list or not, and the oracle will tell us the answer in a single operation.
With this oracle, we can significantly speed up the algorithm for finding the maximum element. We can start by asking the oracle whether the first element is the maximum element or not. If it is, we are done. Otherwise, we can eliminate all the elements that are less than the first element, since we know they cannot be the maximum element. We can then repeat this process with the remaining elements until we find the maximum element.
This algorithm takes advantage of the oracle to reduce the number of comparisons we need to make, and can be shown to take O(log n) time. However, this is not the best possible running time.
An adversary can make the problem more difficult by giving us a list where the maximum element is not significantly larger than the other elements. For example, if the maximum element is only 1% larger than the other elements, we will need to make many comparisons before finding it, even with the help of the oracle. In fact, it can be shown that any algorithm that uses an oracle to find the maximum element must make at least log n comparisons in the worst case, regardless of the quality of the oracle. This is known as the "oracle lower bound" for this problem.
To prove this lower bound, we can construct an adversary that gives us a list where the maximum element is always at the end of the list, and all the other elements are randomly permuted. This adversary can then swap adjacent pairs of elements in the list, making it difficult to eliminate any elements using the oracle. It can be shown that this adversary requires us to make at least log n comparisons in the worst case, even with the help of the oracle.
So, in summary, the use of an oracle can reduce the number of comparisons required to solve the problem of finding the maximum element in an unsorted list, but there is a lower bound on the number of comparisons that any algorithm can make, even with the help of the best possible oracle.
No comments:
Post a Comment