1,721,142 research outputs found
Bounding Rationality by Discounting Time
Get PDFConsider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or E-Nash equilibria.Bounded rationality; Discounting; Uniform equilibria; Factoring game
New non-uniform lower bounds for uniform classes
Get PDFWe strengthen the nondeterministic hierarchy theorem for non-deterministic polynomial time to show that the lower bound holds against sub-linear advice. More formally, we show that for any constants d and d' such that 1 <= d < d', and for any time-constructible bound t=o(n^d), there is a language in NTIME(n^d) which is not in NTIME(t)/n^{1/d'}. The best known earlier separation of Fortnow, Santhanam and Trevisan could only handle o(log(n)) bits of advice in the lower bound, and was not tight with respect to the time bounds.
We generalize our hierarchy theorem to work for other syntactic complexity measures between polynomial time and polynomial space, including alternating polynomial time with any fixed number of alternations. We also use our technique to derive an almost-everywhere hierarchy theorem for non-deterministic classes which use a sub-linear amount of non-determinism, i.e., the lower bound holds on all but finitely many input lengths rather than just on infinitely many.
As one application of our main result, we derive a new lower bound for NP against NP-uniform non-deterministic circuits of size O(n^k) for any fixed k. This result is a significant strengthening of a result of Kannan, which states that not all of NP can be solved with P-uniform circuits of size O(n^k) for any fixed k. As another application, we show strong non-uniform lower bounds for the complexity class RE of languages decidable in randomized linear exponential time with one sided error
Secrets
No full textThis chapter analyzes how, in 1976, Whitfield Diffie and Martin Hellman suggested that one could use NP to hide one's own secrets. The field of cryptography, the study of secret messages, changed forever. Diffie and Hellman, building on earlier work of Roger Merkle, proposed a method to get around the problem of network security using what they called “public-key” cryptography. A computer would generate two keys, a public key and a private key. The computer would store the private key, never putting that key on the network. The public key would be sent over the network broadcast to everyone. Diffie and Hellman's idea was to develop a cryptosystem that used the public key for encrypting messages, turning the real message into a coded one. The public key would not be able to decrypt the message. Only the private key could decrypt the message.</p
The Prehistory of P versus NP
No full textThis chapter explores two separate paths that led to the P versus NP question. In the end it was Steve Cook in the West and Leonid Levin in the East who would first ask whether P = NP. Science does not happen in a vacuum, and both sides have a long history leading to the work of Cook and Levin. The chapter covers just a small part of those research agendas, the struggle in the West to understand efficient computation and the struggle in the East to understand the necessity of perebor. Both would lead to P versus NP. Today, with most academic work available over the Internet and with generally open travel around the world, there is now one large research community instead of two separate ones.</p
The Golden Ticket
No full textThis introductory chapter provides an overview of the P versus NP problem. The P versus NP problem asks, among other things, whether one can quickly find the shortest route for a traveling salesman. P and NP are named after their technical definitions, but it is best not to think of them as mathematical objects but as concepts. “NP” is the collection of problems that have a solution that one wants to find. “P” consists of the problems to which one can find a solution quickly. “P = NP” means one can always quickly compute these solutions, like finding the shortest route for a traveling salesman. “P ≠ NP” means one cannot. Ultimately, the P versus NP problem has achieved the status of one of the great open problems in all of mathematics.</p
The Beautiful World
No full textThis chapter examines an efficient algorithm that solves NP problems, the Urbana algorithm. With the Urbana algorithm one can solve all the NP problems quickly, finding the simplest program that classifies data becomes an easy programming exercise. All one needs do is feed in lots of data and the algorithm does the rest. And that lets one learn just about everything. If it turns out that P = NP and the world has efficient algorithms for all NP problems, it will change in ways that will make the Internet seem like a footnote in history. Not only would it be impossible to describe all these changes but the biggest implications of the new technologies would be impossible to predict.</p
Dealing with Hardness
No full textThis chapter demonstrates several approaches for dealing with hard problems. These approaches include brute force, heuristics, and approximation. Typically, no single technique will suffice to handle the difficult NP problems one needs to solve. For moderate-sized problems one can search over all possible solutions with the very fast computers available today. One can use algorithms that might not work for every problem but do work for many of the problems one cares about. Other algorithms may not find the best possible solution but still a solution that's good enough. Other times one just cannot get a solution for an NP-complete problem. One has to try to solve a different problem or just give up.</p
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