The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to time and memory requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
In computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.
In computational complexity theory, NP is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.
In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is not completely determined by its action and the current symbol it sees, unlike a deterministic Turing machine.
In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is a type of automaton that employs a stack.
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). The figure at right illustrates a finite-state machine, which belongs to a well-known type of automaton. This automaton consists of states and transitions. As the automaton sees a symbol of input, it makes a transition to another state, according to its transition function, which takes the previous state and current input symbol as its arguments.
In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turing machine can—unlike a deterministic Turing Machine—have stochastic results; that is, on a given input and instruction state machine, it may have different run times, or it may not halt at all; furthermore, it may accept an input in one execution and reject the same input in another execution.
In complexity theory, UP is the complexity class of decision problems solvable in polynomial time on an unambiguous Turing machine with at most one accepting path for each input. UP contains P and is contained in NP.
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory.
In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. Deterministic refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.
In 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 computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.
In computational complexity theory, the complexity class NEXPTIME is the set of decision problems that can be solved by a non-deterministic Turing machine using time .
In computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length n of the input, but not on the input itself. A decision problem is in the complexity class P/f(n) if there is a polynomial time Turing machine M with the following property: for any n, there is an advice string A of length f(n) such that, for any input x of length n, the machine M correctly decides the problem on the input x, given x and A.
In 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.
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length.
In computer science, a computation history is a sequence of steps taken by an abstract machine in the process of computing its result. Computation histories are frequently used in proofs about the capabilities of certain machines, and particularly about the undecidability of various formal languages.
In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in quantified propositional logic where every variable is quantified, using either existential or universal quantifiers, at the beginning of the sentence. Such a formula is equivalent to either true or false. If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT.
Configuration graphs are a theoretical tool used in computational complexity theory to prove a relation between graph reachability and complexity classes.
Lane A. Hemaspaandra and Jorg Rothe, Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets, SIAM J. Comput., 26(3), 634–653
