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Faster Integer Multiplication
 STOC'07
, 2007
"... For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complex ..."
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For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log ∗ n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.
An overview of computational complexity
 Communications of the ACM
, 1983
"... foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures ..."
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foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NPcompleteness. The ensuing exploration of the boundaries and nature of the NPcomplete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to timespace tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation." During 19701979, Cook did extensive work under grants from the
On Kolmogorov machines and related issues
 Bull. of Euro. Assoc. for Theor. Computer Science
, 1988
"... I felt honored and uncertain when Grzegorsz Rozenberg, the president of EATCS, proposed that I write a continuing column on logic in computer science in this Bulletin. Writing essays wasn’t my favorite subject in high school. After some hesitation, I decided to give it a try. I’ll need all the help ..."
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Cited by 6 (2 self)
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I felt honored and uncertain when Grzegorsz Rozenberg, the president of EATCS, proposed that I write a continuing column on logic in computer science in this Bulletin. Writing essays wasn’t my favorite subject in high school. After some hesitation, I decided to give it a try. I’ll need all the help I can get from you: criticism, comments, queries, suggestions, etc. Andrei Nikolayevich Kolmogorov died a few months ago. In recent years he chaired the Department of Mathematical Logic at the Moscow State University. In a later article or articles, I hope to discuss Kolmogorov’s ideas on randomness and information complexity; here let me take up the issue of Kolmogorov machines and their close relatives, Schönhage machines. I believe, we are a bit too faithful to the Turing model. It is often easier to explain oneself in a dialog. To this end, allow me to introduce my imaginary student Quizani. • Quizani: I think you should introduce yourself too. Don’t assume everyone knows you. • Author: All right. I grew up in the Soviet Union and started my career in the Ural University as an algebraist and selftaught logician. In 1973, I emigrated to Israel where I did logic and taught at BenGurion
Fast OnLine Integer Multiplication
 PROC. 5TH ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1974
"... A Turing machine multiplies binary integers onZine if it receives its inputs loworder digits first and produces the jth digit of the product before reading in the (j+l)st digits of the two inputs. We present a general method for converting any offline multiplication algorithm which forms the prod ..."
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Cited by 5 (1 self)
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A Turing machine multiplies binary integers onZine if it receives its inputs loworder digits first and produces the jth digit of the product before reading in the (j+l)st digits of the two inputs. We present a general method for converting any offline multiplication algorithm which forms the product of two ndigit binary numbers in time F(n) into an online method which uses time only O(F() log ), assuming that F is monotone and satisfies n F() F(2)/2 ! kF() for some constant k. Applying this technique to the fast multiplication algorithm of Schönhage and Strassen gives an upper bound of O(n (log n)² loglog n) for online multiplication of integers. A refinement of the technique yields an optimal method for online multiplication by certain sparse integers. Other applications are to the online computation of products of polynomials, recognition of palindromes, and multiplication by a constant.
MultiHead Finite Automata: Characterizations, Concepts and Open Problems
 EPTCS 1
, 2009
"... Multihead finite automata were introduced in [36] and [38]. Since that time, a vast literature on computational and descriptional complexity issues on multihead finite automata documenting the importance of these devices has been developed. Although multihead finite automata are a simple concept, ..."
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Cited by 2 (0 self)
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Multihead finite automata were introduced in [36] and [38]. Since that time, a vast literature on computational and descriptional complexity issues on multihead finite automata documenting the importance of these devices has been developed. Although multihead finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even nonsemidecidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multihead finite automata for a better understanding of the nature of nonrecursive tradeoffs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature.