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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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19 Jan 2020
TL;DR: Computer science as an academic discipline began in the 1960’s with emphasis on programming languages, compilers, operating systems, and the mathematical theory that supported these areas, but today, a fundamental change is taking place and the focus is more on applications.
Abstract: Computer science as an academic discipline began in the 1960’s. Emphasis was on programming languages, compilers, operating systems, and the mathematical theory that supported these areas. Courses in theoretical computer science covered finite automata, regular expressions, context-free languages, and computability. In the 1970’s, the study of algorithms was added as an important component of theory. The emphasis was on making computers useful. Today, a fundamental change is taking place and the focus is more on applications. There are many reasons for this change. The merging of computing and communications has played an important role. The enhanced ability to observe, collect, and store data in the natural sciences, in commerce, and in other fields calls for a change in our understanding of data and how to handle it in the modern setting. The emergence of the web and social networks as central aspects of daily life presents both opportunities and challenges for theory.
262 citations
Book•
21 Sep 1999
TL;DR: In this article, the authors propose a solution to Post's Problem and Strong Reducibilities for degree theory with jump and many-one and other degree theories, and compare them with degree theory without jump.
Abstract: Recursiveness and Computability. Induction. Systems of Equations. Arithmetical Formal Systems. Turing Machines. Flowcharts. Functions as Rules. Arithmetization. Church's Thesis. Basic Recursion Theory. Partial Recursive Functions. Diagonalization. Partial Recursive Functionals. Effective Operations. Indices and Enumerations. Retraceable and Regressive Sets. Post's Problem and Strong Reducibilities. Post's Problem. Simple Sets and Many-One Degrees. Hypersimple Sets and Truth-Table Degrees. Hyperhypersimple Sets and Q-Degrees. A Solution to Post's Problem. Creative Sets and Completeness. Recursive Isomorphism Types. Variations of Truth-Table Reducibility. The World of Complete Sets. Formal Systems and R.E. Sets. Hierarchies and Weak Reducibilities. The Arithmetical Hierarchy. The Analytical Hierarchy. The Set-Theoretical Hierarchy. The Constructible Hierarchy. Turing Degrees. The Language of Degree Theory. The Finite Extension Method. Baire Category. The Coinfinite Extension Method. The Tree Method. Initial Segments. Global Properties. Degree Theory with Jump. Many-One and Other Degrees. Distributivity. Countable Initial Segments. Uncountable Initial Segments. Global Properties. Comparison of Degree Theories. Structure Inside Degrees. Bibliography. Index.
259 citations
Book•
01 Jan 1991
TL;DR: In this paper, an introduction for undergraduates to the theory of computation is given, which emphasizes formal languages, automata and abstract models of computation, and computability. But it also includes an introduction to computational complexity and NP-completeness.
Abstract: From the Publisher:
This book is an introduction for undergraduates to the theory of computation. It emphasizes formal languages,automata and abstract models of computation,and computability. It also includes an introduction to computational complexity and NP-completeness.
249 citations
TL;DR: A tape hardest deterministic context-free language is described and the best upper bound known on the tape complexity of (deterministic) context- free languages is (log(n) 2).
Abstract: Let DSPACE(L(n)) denote the family of languages recognized by deterministic L(n)-tape bounded Turmg machines The pnnopal result described m this paper is the equivalence of the following statements (l) The determtmsttc context-free language L~ 2) (described m the paper) is m DSPACE(Iog(n)) (2) The simple LL(I) languages are m DSPACE(tog(n)) (3) The simple precedence languages are in DSPACE(Iog(n)). (4) DSPACE(Iog(n)) is identical to the famdy of languages recogmzed by deterministic two-way multlhead pushdown automata m polynomml tmae These results are obtained by constructing a determlmstlc context-free language L~ 2~ which is log(n)-complete for the family of determlmstlc context-free languages In other words, a tape hardest deterministic context-free language is described The best upper bound known on the tape complexity of (deterministic) context-free languages is (log(n)) 2
249 citations
14 Mar 1997
TL;DR: These notes contain some high points from the recent book, emphasising what is different or novel with respect to more traditional treatments of computability and complexity theory, and some new results as well.
Abstract: A programming approach to computability and complexity theory yields proofs of central results that are sometimes more natural than the classical ones; and some new results as well. These notes contain some high points from the recent book [14], emphasising what is different or novel with respect to more traditional treatments. Topics include:
Kleene’s s-m-n theorem applied to compiling and compiler generation.
Proof that constant time factors do matter: for a natural computation model, problems solvable in linear time have a proper hierarchy, ordered by coefficient values. (In contrast to the “linear speedup” property of Turing machines.)
Results on which problems possess optimal algorithms, including Levin’s Search theorem (for the first time in book form).
Characterisations in programming terms of a wide range of complexity classes. These are intrinsic: without externally imposed space or time computation bounds.
Boolean program problems complete for PTIME, NPTIME, PSPACE.
242 citations