Alternating Turing machineIn computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981. The definition of NP uses the existential mode of computation: if any choice leads to an accepting state, then the whole computation accepts.
Computational resourceIn computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems. The simplest computational resources are computation time, the number of steps necessary to solve a problem, and memory space, the amount of storage needed while solving the problem, but many more complicated resources have been defined. A computational problem is generally defined in terms of its action on any valid input.
Interactive proof systemIn computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties: a prover and a verifier. The parties interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover possesses unlimited computational resources but cannot be trusted, while the verifier has bounded computation power but is assumed to be always honest.
Knapsack problemThe knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.
Complement (complexity)In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem. For example, one important problem is whether a number is a prime number. Its complement is to determine whether a number is a composite number (a number which is not prime). Here the domain of the complement is the set of all integers exceeding one.
Savitch's theoremIn computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic space complexity. It states that for any function , In other words, if a nondeterministic Turing machine can solve a problem using space, a deterministic Turing machine can solve the same problem in the square of that space bound. Although it seems that nondeterminism may produce exponential gains in time (as formalized in the unproven exponential time hypothesis), Savitch's theorem shows that it has a markedly more limited effect on space requirements.
Space hierarchy theoremIn computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in (asymptotically) more space, subject to certain conditions. For example, a deterministic Turing machine can solve more decision problems in space n log n than in space n. The somewhat weaker analogous theorems for time are the time hierarchy theorems.
Nondeterministic algorithmIn computer programming, a nondeterministic algorithm is an algorithm that, even for the same input, can exhibit different behaviors on different runs, as opposed to a deterministic algorithm. There are several ways an algorithm may behave differently from run to run. A concurrent algorithm can perform differently on different runs due to a race condition. A probabilistic algorithm's behaviors depends on a random number generator.
Average-case complexityIn computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the algorithm, averaged over all possible inputs. It is frequently contrasted with worst-case complexity which considers the maximal complexity of the algorithm over all possible inputs. There are three primary motivations for studying average-case complexity.
Log-space reductionIn computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means it can keep a constant number of pointers into the input, along with a logarithmic number of fixed-size integers. It is possible that such a machine may not have space to write down its own output, so the only requirement is that any given bit of the output be computable in log-space. Formally, this reduction is executed via a log-space transducer.
