### Computability And Complexity From A Programming Perspective

Goerdt, A. Characterizing complexity classes by general recursive definitions in higher types. Information and Computation , — Characterizing complexity classes by higher type primitive recursive definitions. Theoretical Computer Science , 45— Characterizing complexity classes by higher type primitive recursive definitions, Part II. Hutton, G. A tutorial on the universality and expressiveness of fold. Journal of Functional Programming 1 1 :1—17 Jones, N. CrossRef Google Scholar. Symposium on Theory of Computing , pages — ACM Press, The MIT Press, Partial Evaluation and Automatic Program Generation.

Prentice-Hall International, Theoretical Computer Science , Even simple programs are hard to analyze. Journal of the Association for Computing Machinery , 24 2 —, For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm.

Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication. This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X , since an algorithm for X allows us to solve any problem in C. The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

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This means that X is the hardest problem in C. Since many problems could be equally hard, one might say that X is one of the hardest problems in C. Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm.

This hypothesis is called the Cobham—Edmonds thesis. The complexity class NP , on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem , the Hamiltonian path problem and the vertex cover problem.

Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP. The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. These include various types of integer programming problems in operations research , many problems in logistics , protein structure prediction in biology ,  and the ability to find formal proofs of pure mathematics theorems.

The graph isomorphism problem , the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P , NP-complete , or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. The integer factorization problem is the computational problem of determining the prime factorization of a given integer.

Phrased as a decision problem, it is the problem of deciding whether the input has a prime factor less than k.

No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. If the problem is NP-complete , the polynomial time hierarchy will collapse to its first level i. However, the best known quantum algorithm for this problem, Shor's algorithm , does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. Many known complexity classes are suspected to be unequal, but this has not been proved. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Along the same lines, co-NP is the class containing the complement problems i. It is believed  that NP is not equal to co-NP ; however, it has not yet been proven. Similarly, it is not known if L the set of all problems that can be solved in logarithmic space is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC , and it is not known if they are distinct or equal classes. It is suspected that P and BPP are equal. A problem that can be solved in theory e.

The term infeasible literally "cannot be done" is sometimes used interchangeably with intractable ,  though this risks confusion with a feasible solution in mathematical optimization. However, this identification is inexact: a polynomial-time solution with large degree or large leading coefficient grows quickly, and may be impractical for practical size problems; conversely, an exponential-time solution that grows slowly may be practical on realistic input, or a solution that takes a long time in the worst case may take a short time in most cases or the average case, and thus still be practical.

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Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem. To see why exponential-time algorithms are generally unusable in practice, consider a program that makes 2 n operations before halting.

Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress.

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## Computability and Complexity: From a Programming Perspective by Neil D. Jones

However, an exponential-time algorithm that takes 1. Similarly, a polynomial time algorithm is not always practical. If its running time is, say, n 15 , it is unreasonable to consider it efficient and it is still useless except on small instances. Indeed, in practice even n 3 or n 2 algorithms are often impractical on realistic sizes of problems. Continuous complexity theory can refer to complexity theory of problems that involve continuous functions that are approximated by discretizations, as studied in numerical analysis.

One approach to complexity theory of numerical analysis  is information based complexity. Continuous complexity theory can also refer to complexity theory of the use of analog computation , which uses continuous dynamical systems and differential equations. Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers.

## Useful Books for Theory of Computation

Most influential among these was the definition of Turing machines by Alan Turing in , which turned out to be a very robust and flexible simplification of a computer. The beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard E.

Stearns , which laid out the definitions of time complexity and space complexity , and proved the hierarchy theorems. Earlier papers studying problems solvable by Turing machines with specific bounded resources include  John Myhill 's definition of linear bounded automata Myhill , Raymond Smullyan 's study of rudimentary sets , as well as Hisao Yamada 's paper  on real-time computations Somewhat earlier, Boris Trakhtenbrot , a pioneer in the field from the USSR, studied another specific complexity measure.

However, [my] initial interest [in automata theory] was increasingly set aside in favor of computational complexity, an exciting fusion of combinatorial methods, inherited from switching theory , with the conceptual arsenal of the theory of algorithms. These ideas had occurred to me earlier in when I coined the term "signalizing function", which is nowadays commonly known as "complexity measure". In , Manuel Blum formulated a set of axioms now known as Blum axioms specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called speed-up theorem.

The field began to flourish in when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In , Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

In the s, much work was done on the average difficulty of solving NP-complete problems—both exactly and approximately. At that time, computational complexity theory was at its height, and it was widely believed that if a problem turned out to be NP-complete, then there was little chance of being able to work with the problem in a practical situation.

However, it became increasingly clear that this is not always the case [ citation needed ] , and some authors claimed that general asymptotic results are often unimportant for typical problems arising in practice. From Wikipedia, the free encyclopedia. Theory classifying computational problems by their inherent difficulty. Main article: Turing machine.

Main article: Complexity class. Main articles: time hierarchy theorem and space hierarchy theorem. Main article: Reduction complexity. Main article: P versus NP problem. See also: Combinatorial explosion. Context of computational complexity Descriptive complexity theory Game complexity List of complexity classes List of computability and complexity topics List of important publications in theoretical computer science List of unsolved problems in computer science Parameterized complexity Proof complexity Quantum complexity theory Structural complexity theory Transcomputational problem Computational complexity of mathematical operations.

Graph isomorphism is in the low hierarchy. Lecture Notes in Computer Science. September 13, Algorithms and Complexity. Writing for Computer Science. Acta Numerica. Cambridge Univ Press. Changing Conceptions of What is Computable. Cooper, S. Theoretical Computer Science Stack Exchange is a question and answer site for theoretical computer scientists and researchers in related fields. It only takes a minute to sign up. Currently, our ToC Theory of Computation courses are designed with the following progression of topics:.

The kind of approach to theory of computation you describe is what I like to call an abstract machine based computability theory : i. Computability and Complexity from a Programming Perspective. It also spends a significant number of pages on complexity.

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This is also unlike most other classical computability texts that are based on models other than the Turing model eg Lambda-calculus. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 4 years, 7 months ago. Active 3 years, 11 months ago. Viewed times.