complexity theory

complexity theory. In computer science and mathematics, complexity theory is the study of the resources required to solve computational problems. It classifies problems based on their inherent difficulty and the efficiency of algorithms designed to solve them. The field provides a rigorous framework for understanding the limits of computation, most famously crystallized in the P versus NP problem.

Overview and basic concepts

The foundational work in the field is often traced to the 1965 paper by Juris Hartmanis and Richard Stearns, which introduced time hierarchy theorems. Central to the discipline is the Turing machine, the abstract model of computation defined by Alan Turing. Problems are analyzed by the amount of resources—such as time and memory—an algorithm needs, typically expressed using Big O notation. Key concepts include decision problems, the distinction between deterministic and nondeterministic models, and reductions that allow the comparison of problem difficulty. The Church–Turing thesis underpins the notion that all reasonable models of computation are equivalent in power.

Computational complexity classes

Complexity classes group problems that can be solved with similar amounts of resources. The class P contains problems solvable by a deterministic Turing machine in polynomial time. Its famous counterpart, NP, contains problems whose solutions can be verified quickly, and includes many practical challenges from operations research and logic. Within NP lie the NP-complete problems, such as the Boolean satisfiability problem identified by Stephen Cook in the Cook–Levin theorem; a polynomial-time algorithm for one would solve all in NP. Other critical classes include PSPACE for problems solvable with polynomial memory, EXPTIME for problems requiring exponential time, and the class BPP for problems solvable efficiently with a randomized algorithm. The relationships between these classes, such as whether P equals NP, remain profound open questions.

Major results and theorems

Landmark results provide structural insights into these classes. The time hierarchy theorem and space hierarchy theorem, proven by Hartmanis and Stearns, establish that more time or space allows more problems to be solved. Richard Karp's 1972 work demonstrated the ubiquity of NP-completeness by cataloging 21 classic problems, including the knapsack problem and the Hamiltonian path problem. The PCP theorem, a culmination of work by researchers like Sanjeev Arora and Shafi Goldwasser, revolutionized the understanding of probabilistically checkable proofs and hardness of approximation. Other pivotal theorems include Ladner's theorem, which shows the existence of intermediate problems if P ≠ NP, and Savitch's theorem, which relates deterministic and nondeterministic space.

Applications

The implications of complexity theory extend far beyond pure theory. In cryptography, the security of protocols like RSA relies on the conjectured hardness of problems such as integer factorization, which is in NP but not believed to be in P. The theory of NP-completeness guides algorithm design in fields like bioinformatics and VLSI design, helping researchers identify when to seek heuristic or approximate solutions, as seen with the traveling salesman problem. Concepts from algorithmic game theory and quantum complexity theory, such as BQP, are essential for understanding the power of quantum computers like those explored by IBM and Google.

Relationships to other fields

Complexity theory has deep and symbiotic connections with numerous disciplines. In mathematical logic, it interacts with descriptive complexity theory, which links complexity classes to logical formalisms like first-order logic. Work in combinatorics and graph theory often involves classifying the complexity of problems on structures like the Petersen graph. The field informs philosophy of mind through debates on computationalism and the nature of artificial intelligence. It also overlaps with statistical mechanics in the study of phase transitions in random instances of constraint satisfaction problems, and with economics through mechanism design and computational social choice.

Category:Theoretical computer science Category:Computational complexity theory Category:Mathematical logic