# Tag Info

140

I have to admit (surprising as it sounds) that I don't know really the answer. I either discovered or rediscovered this reduction myself. I discovered the discrete log algorithm first, and the factoring algorithm second, so I knew from discrete log that periodicity was useful. I knew that factoring was equivalent to finding two unequal numbers with equal ...

96

$\lambda$-calculus has two key roles. It is a simple mathematical foundation of sequential, functional, higher-order computational behaviour. It is a representation of proofs in constructive logic. This is also known as the Curry-Howard correspondence. Jointly, the dual view of $\lambda$-calculus as proof and as (sequential, functional, higher-order) ...

59

If you really want to know what led Neil Robertson and me to tree-width, it wasn't algorithms at all. We were trying to solve Wagner's conjecture that in any infinite set of graphs, one of them is a minor of another, and we were right at the beginning. We knew it was true if we restricted to graphs with no k-vertex path; let me explain why. We knew all such ...

55

The random reduction from factorization to order-finding (mod N) was very well known to people working in number theory algorithms in the late 1970's and early 1980's. Indeed, it appears in a paper of Heather Woll, Reductions among number theoretic problems, Information and Computation 72 (1987) 167-179, and Eric Bach and I knew it before then. I am ...

44

László Lovász, Coverings and coloring of hypergraphs, Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Math., Winnipeg, Man., 1973, pp. 3--12, proved that Chromatic number reduces to 3-colourability. I think, that is the first proof for NP-completeness of 3-colourability. Here is Lovász's paper; ...

34

This paper, in Russian, Gricenko, S. A., Karatsuba, E. A., Korolyov, M. A., Rezvyakova, I. S., Tolev, D. I., & Changa, M. E. (2012). Scientific contributions of A. A. Karatsuba / Научные достижения Анатолия Алексеевича Карацубы. Современные проблемы математики, 16(0), 7-30. states the following (items 1—3). Karatsuba presented his algorithm ...

33

This is discussed a bit in my paper with H. C. Williams, "Factoring Integers before Computers" In a 1917 paper, H. C. Pocklington discussed an algorithm for finding sqrt(a), modulo p, which depended on choosing elements at random to get a nonresidue of a certain form. In it, he said, "We have to do this [find the nonresidue] by trial, using the Law of ...

31

I have always seen the notion of nondeterminism in computation attributed to Michael Rabin and Dana Scott. They defined nondeterministic finite automata in their famous paper Finite Automata and Their Decision Problems, 1959. Rabin's Turing Award citation also suggests that Rabin and Scott introduced nondeterministic machines.

28

The following is according to Nick Pippenger: The relevant references are as follows. Steve described NC as Nick's Class in his paper "Deterministic CFL's Are Accepted Simultaneously in Polynomial Time and Log Squared Space" (ACM STOC, 11 (1979) 338-345) on SC, and I described SC as Steve's Class in my paper "Simultaneous Resource Bounds'' (IEEE FOCS, ...

28

Neither! The best way to see this independence is to read the original papers. Turing's 1936 paper introducing Turing machines does not refer to any simpler type of (abstract) finite automaton. McCulloch and Pitts' 1943 paper introducing "nerve-nets", the precursors of modern-day finite-state machines, proposed them as simplified models of neural activity, ...

28

Buffons needle algorithm for estimating $\pi$, basically a Monte Carlo method, dates to publication in 1777. note that Monte Carlo methods date to the 1940s with the US "Manhattan" atom bomb project & were coinvented by Ulam, Von Neumann, and Metropolis.

28

Grothendieck's inequality, from his days in functional analysis, was initially proved to relate fundamental norms on tensor product spaces. Grothendieck called the inequality "the fundamental theorem of the metric theory of tensor product spaces", and published it in a now famous paper in 1958, in French, in a limited circulation Brazilian journal. The paper ...

27

I did not know of these until recently. 1) The LU decomposition of a matrix is due to Turing! Considering how fundamental LU decomposition is, this is one contribution that deserves to be highlighted and known more widely (1948). 2) Turing was the first to come up with a "paper algorithm" for chess. At that point, the first digital computers were still ...

27

I think $\lambda$-calculus has contributed in many ways to this field, and still contributes to it. Three examples follow, and this is not exhaustive. Since I am not a specialist in $\lambda$-calculus, I certainly miss some important points. First, I think having different models of computation that turn out to represent the exact same set of functions was ...

26

He wanted to create a formal system for the foundations of logic and mathematics that was simpler than Russell's type theory and Zermelo's set theory. The basic idea was to add a constant $\Xi$ to the untyped lambda calculus (or combinatory logic) and interpret $XZ$ as expressing "$Z$ satisfies the predicate $X$" and $\Xi XY$ as expressing "$X\subseteq Y$". ...

25

I've asked Dick personally out of curiosity a few years back. He said that as far as he knows Rabin-Karp was a random switch many years after the paper was first published. He also indicated that it is his understanding that Michael would say the same thing if asked, since at some point they had talked about it.

24

[Following a suggestion of Kaveh, I am putting my (somewhat extended) comment as an answer] This "conjecture" of Kolmogorov is just a rumor. It was not published anywhere. In the former USSR, "publishing" mathematics meant something different than what it does today: give a talk at a seminar or tell your colleagues at lunch. Counting papers was not an issue....

24

Very briefly: the simply-typed $\lambda$-calculus does not have dependent types. Dependent types were proposed by de Bruijn and Howard who wanted to extend the Curry-Howard correspondence from propositional to first-order logic. The key contribution of Martin-Löf's is a novel analysis of equality. There are two main ways of giving Curry-Howard style ...

23

The Union-Find algorithm, which Tarjan1 showed had complexity $n \alpha(n)$, where $\alpha(n)$ is the inverse Ackermann function, had been analyzed previously by several people. According to Wikipedia, it was invented by Galler and Fischer2, but this seems to be incorrect, as they did not have all the components of the algorithm needed to make it run that ...

22

The footnote of my paper that you cite refers to a heuristic "argument" as well, at least, what we think was Kolmogorov's intuition -- the positive resolution of Hilbert's thirteenth problem. http://en.wikipedia.org/wiki/Hilbert's_thirteenth_problem In particular, it was proved by Kolmogorov and Arnold that any continuous function on $n$ variables can be ...

21

As mentioned in the question, Turing was central to defining algorithms and computability, thus he was one of the people that helped assemble the algorithmic lens. However, I think his biggest contribution was viewing science through the algorithmic lens and not just computation for the sake of computation. During WW2 Turing used the idea of computation and ...

21

Here is another paper from 1973 that proves that 3-colorability is NP-hard. Larry J. Stockmeyer. “Planar 3-colorability is polynomial complete.” ACM SIGACT News, vol. 5, no. 3, 1973. (It seems that Lovász and Stockmeyer obtained their results independently.) Update: see comments below.

21

Apart from the foundational role of the $\lambda$-calculus, which was mentioned in all other answers, I would like to add something on What exactly did the lambda calculus do to advance the theory of CS? I believe that concurrency theory is one field of CS which has been tremendously influenced by the compositional view mentioned by Martin Berger. Of ...

21

You're in good company. Kurt Gödel criticized $\lambda$-calculus (as well as his own theory of general recursive functions) as not being a satisfactory notion of computability on the grounds that it is not intuitive, or that it does not sufficiently explain what is going on. In contrast, he found Turing's analysis of computability and the ensuing notion of ...

19

The Metropolis–Hastings algorithm was published in 1953 and dates back earlier to the Manhattan project, long before Rabin. Like many of the early randomized methods given in other answers, it is a Monte Carlo algorithm. Is it possible that the claim on the Wikipedia page is a garbled form of the claim that Rabin's was the first Las Vegas algorithm?

18

I believe that the notation AC first appears in Cook's "A Taxonomy of Problems with Fast Parallel Algorithms" from 1985. On page 11 (page 12 of the journal) we read: To state a more general form of this result we introduce the following terminology. Definition. $AC^k$, for $k=1,2,\ldots$, is the class of problems solvable by an ATM in space $O(\log n)... 18 Grothendieck's impact can be felt in type theory and logic. For instance, Bart Jacobs' 700+ page volume Categorical Logic and Type Theory gives a uniform treatment of various type theories ($X$-type theory, where$X\subseteq \{ \text{simple},\text{dependent},\text{polymorphic},\text{higher-order}\}$) based on the categorical notion of Grothendieck ... 17 TL;DR. The metamathematics of binding are subtle: they seem trivial but aren't — whether you deal with (higher-order) logics or 𝜆-calculus. They're so subtle that binding representations form an open research field, with a competition (the POPLmark challenge) some years ago. There are even jokes by people in the field about the complexity of approaches to ... 17 First of all, Cook actually showed that the problem of whether a logical expression is a tautology is$\mathbb{NP}$-complete under Cook reductions. The proof however works by replacing them with Karp reductions to show that$SAT$is$\mathbb{NP}$-complete, in the modern sense of the term. As for whether Cook understood the significance of a$\mathbb{NP}\$-...

17

The two words do not refer to the same thing. Hilbert's Entscheidungsproblem was the question whether there is an algorithm that decides the universal truth of first-order logical sentences, which was answered negatively by Turing in his famous 1936 paper "On Computable Numbers, with an application to the Entscheidungsproblem". The word literally means ...

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