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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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TL;DR: It is proved directly that the approach presented here is equivalent to the established Turing-machine based approach which dates back to Grzegorczyk and Lacombe, is used by Pour-El & Richards in their foundational work on computable analysis, and is the standard notion of computability among physicists as in the work of Penrose.
81 citations
11 Dec 2013
TL;DR: In this paper, the authors study SMT problems over the reals containing ordinary differential equations, and develop δ-complete algorithms for SMT formulas that are purely existentially quantified, as well as ∃∀-formulas whose universal quantification is restricted to the time variables.
Abstract: We study SMT problems over the reals containing ordinary differential equations,. They are important for formal verification of realistic hybrid systems and embedded software. We develop δ-complete algorithms for SMT formulas that are purely existentially quantified, as well as ∃∀-formulas whose universal quantification is restricted to the time variables. We demonstrate scalability of the algorithms, as implemented in our open-source solver dReal, on SMT benchmarks with several hundred nonlinear ODEs and variables.
81 citations
Book•
01 Jan 1978
TL;DR: The scientific books will also be the best reason to choose, especially for the students, teachers, doctors, businessman, and other professions who are fond of reading.
Abstract: In what case do you like reading so much? What about the type of the machines languages and computation book? The needs to read? Well, everybody has their own reason why should read some books. Mostly, it will relate to their necessity to get knowledge from the book and want to read just to get entertainment. Novels, story book, and other entertaining books become so popular this day. Besides, the scientific books will also be the best reason to choose, especially for the students, teachers, doctors, businessman, and other professions who are fond of reading.
80 citations
Posted Content•
TL;DR: A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed as mentioned in this paper.
Abstract: We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle--that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal--quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.
79 citations
Book•
11 Jan 1989TL;DR: This book discusses Grammatical Basis of Language Translation, Foundations of Recursive Function Theory, and the Scope of Primitive Recursive functions in Partial Recursive Functions.
Abstract: Preliminaries Review of Set Theory Grammatical Basis of Language Translation Historical Background Preview of the Remaining Text Chapter Review Problems Finite Automata and Regular Languages Lexical Analysis Deterministic Finite Automata The Limits of Deterministic Finite Automata Nondeterministic Finite Automata Regular Grammars Regular Expressions Closing Comments Chapter Review Problems Pushdown Automata and Context-Free Languages Pushdown Automata Context-Free Grammars The Limits of Pushdown Automata LL(k) Parsers LR(k) Parsers Turing Machines and Phase-Structure Languages Turing Machines Modular Construction of Turing Machines Turing Machines as Language Accepters Turing-Acceptable Languages Beyond Phrase-Structure Languages Closing Comments Chapter Review Problems Computability Foundations of Recursive Function Theory The Scope of Primitive Recursive Functions Partial Recursive Functions The Power of Programming Languages Closing Comments Chapter Review Problems Complexity Computations The Complexity of Algorithms The Complexity of Problems Time Complexity of Language Recognition Problems Time Complexity of Nondeterministic Machines Closing Comments Chapter Review Problems Appendices A More About Constructing LR(1) Parse Tables B More About Ackerman's Function C Some Important Unsolvable Problems D On the Complexity of the String Comparison Problem E A Sampling of NP Problems Additional Reading 0805301437T04062001
78 citations