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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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Book
02 Oct 1997
TL;DR: The Bounds of Computability Part II: A Hierarchy of Automata and Formal Languages and the Chomsky Hierarchy Epilogue are presented.
Abstract: Preface Chapter 0 - Mathematical Preliminaries Part I: Models of Computation Chapter 1 - Turning Machines Chapter 2 - Additional Varieties of Turning Machines Chapter 3 - An Introduction to Recursion Theory Chapter 4 - Markov Algorithms Chapter 5 - Register Machines Chapter 6 - Post Systems (Optional) Chapter 7 - The Vector Machine Model of Parallel Computation (Optional) Chapter 8 - The Bounds of Computability Part II: A Hierarchy of Automata and Formal Languages Chapter 9 - Regular Languages and Finite-State Automata Chapter 10 - Context-Free Languages and Pushdown-Stock Automata Chapter 11 - Context-Free Languages and Compiler Design Theory (Optional) Chapter 12 - Context-Sensitive Languages and Linear Bounded Automata Chapter 13 - Generative Grammars an the Chomsky Hierarchy Epilogue

71 citations

Proceedings ArticleDOI
23 May 1994
TL;DR: It is shown that a synchronization problem has a wait-free solution if and only if its input complex can be continuously “stretched and folded” to cover its output complex.
Abstract: A Simple (constructive Computability Theorem for wait-free ~omputation In modern shared-memory multiprocessors, processes can be halted or delayed without warning by interrupts, pre-emption, or cache misses. In such environments, it is desirable to design synchronization protocols that are wait-free: any processes that continues to run will finish the protocol in a fixed number of steps, regardless of delays or failures by other processes. Not all synchronization problems have wait-free solutions. In this paper, we give a new, remarkably simple necessary and sufficient combinatorial condition characterizing the problems that have wait-free solutions using shared read/write memory. We associate the range of possible input and output values for any synchronization problem with a high-dimensional geometric structure called a simplicial complex. We show that a synchronization problem has a wait-free solution if and only if its input complex can be continuously “stretched and folded” to cover its output complex. The key to the new theorem is a novel “simplex agreement” protocol, allowing processes to converge asynchronously to a common simplex of a simplicial complex. The proof exploits a number of classical results from algebraic and combinatorial topology. Permission to copy without fee all or part of thk material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. STOC 945/94 Montreal, Quebec, Canada @ 1984 ACM 0-89791 -663-8/84/0005..$3.50 Nir Shavit Computer Science Department Tel-Aviv University shanir@math. tau. ac.il

71 citations

Journal ArticleDOI
TL;DR: An overview of quantum cellular automata theory is given, with particular focus on structure results; computability and universality results; and quantum simulation results.
Abstract: Quantum cellular automata are arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.

71 citations

Book ChapterDOI
01 Jan 2014
TL;DR: The notion of degree of unsolvability was introduced by Post in [Post, 1944] and has been used extensively in computability theory as mentioned in this paper, where a set A is computable relative to a set B, and B is Turing reducible to A.
Abstract: Modern computability theory began with Turing [Turing, 1936], where he introduced the notion of a function computable by a Turing machine. Soon after, it was shown that this definition was equivalent to several others that had been proposed previously and the Church-Turing thesis that Turing computability captured precisely the informal notion of computability was commonly accepted. This isolation of the concept of computable function was one of the greatest advances of twentieth century mathematics and gave rise to the field of computability theory. Among the first results in computability theory was Church and Turing’s work on the unsolvability of the decision problem for first-order logic. Computability theory to a great extent deals with noncomputable problems. Relativized computation, which also originated with Turing, in [Turing, 1939], allows the comparison of the complexity of unsolvable problems. Turing formalized relative computation with oracle Turing machines. If a set A is computable relative to a set B, we say that A is Turing reducible to B. By identifying sets that are reducible to each other, we are led to the notion of degree of unsolvability first introduced by Post in [Post, 1944]. The degrees form a partially ordered set whose study is called degree theory. Most of the unsolvable problems that have arisen outside of computability theory are computably enumerable (c.e.). The c.e. sets can intuitively be viewed as unbounded search problems, a typical example being those formulas provable in some effectively given formal system. Reducibility allows us to isolate the most difficult c.e. problems, the complete problems. The standard method for showing that a c.e. problem is undecidable is to show that it is complete. Post [Post, 1944] asked if this technique always works, i.e., whether there is a noncomputable, incomplete c.e. set. This problem came to be known as Post’s Problem and it was origin of degree theory. Degree theory became one of the core areas of computability theory and attracted some of the most brilliant logicians of the second half of the twentieth century. The fascination with the field stems from the quite sophisticated techniques needed to solve the problems that arose, many of which are quite easy to state. The hallmark of the field is the priority method introduced by

71 citations

Proceedings ArticleDOI
03 Nov 1997
TL;DR: Given the organization of the proposed SAT algorithm, the resulting ILP procedures implement powerful search pruning techniques, including a non-chronological backtracking search strategy, clause recording procedures and identification of necessary assignments.
Abstract: The computation of prime implicants has several and significant applications in different areas, including automated reasoning, non-monotonic reasoning, electronic design automation, among others. The authors describe a new model and algorithm for computing minimum-size prime implicants of propositional formulas. The proposed approach is based on creating an integer linear program (ILP) formulation for computing the minimum-size prime implicant, which simplifies existing formulations. In addition, they introduce two new algorithms for solving ILPs, both of which are built on top of an algorithm for propositional satisfiability (SAT). Given the organization of the proposed SAT algorithm, the resulting ILP procedures implement powerful search pruning techniques, including a non-chronological backtracking search strategy, clause recording procedures and identification of necessary assignments. Experimental results, obtained on several benchmark examples, indicate that the proposed model and algorithms are significantly more efficient than other existing solutions.

71 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202344
2022119
202189
202098
2019111
201897