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En provenance du cours de Princeton University

Computer Science: Algorithms, Theory, and Machines

84 notes

Princeton University

Computer Science: Algorithms, Theory, and Machines

84 notes

This course introduces the broader discipline of computer science to people having basic familiarity with Java programming. It covers the second half of our book Computer Science: An Interdisciplinary Approach (the first half is covered in our Coursera course Computer Science: Programming with a Purpose, to be released in the fall of 2018). Our intent is to demystify computation and to build awareness about the substantial intellectual underpinnings and rich history of the field of computer science. First, we introduce classic algorithms along with scientific techniques for evaluating performance, in the context of modern applications. Next, we introduce classic theoretical models that allow us to address fundamental questions about computation, such as computability, universality, and intractability. We conclude with machine architecture (including machine-language programming and its relationship to coding in Java) and logic design (including a full CPU design built from the ground up). The course emphasizes the relationships between applications programming, the theory of computation, real computers, and the field's history and evolution, including the nature of the contributions of Boole, Shannon, Turing, von Neumann, and others. All the features of this course are available for free. No certificate will be offered upon completion.

À partir de la leçon

TURING MACHINES

In 1936, Alan Turing published a paper that is widely hailed as one of the most important scientific papers of the 20th century. This lecture is devoted to the two far-reaching central ideas of the paper: All computational devices have equivalent computational power, and there are limitations to that power.

A simple model of computation26:32

Universality15:51

Computability14:08

Implications11:34

Rencontrer les enseignants

Robert Sedgewick
William O. Baker *39 Professor of Computer Science
Computer Science
Kevin Wayne
Phillip Y. Goldman '86 Senior Lecturer
Computer Science

Today we're going to talk about a subject that's at the heart of theoretical

computer science, a formal basis for the study of computation that was developed by

the English mathematician Alan Turing in the 1930s.

We'll begin with some context.

Just briefly to put this material within the context of mathematics.

What we're talking about is the idea of universality and computability,

which are fundamental questions at the heart of understanding computation.

We want to know what is a general-purpose computer, really?

Are there some limits on what we can do with digital computers?

Are there limits on the power of machines we can build?

In the 1930s, there was pioneering work, a lot of it done at Princeton.

Mathematician David Hilbert, around the turn of the 19th to

20th century, asked some basic questions.

He laid out ten problems at a famous mathematical congress.

And then in the early 30s, the Princeton

mathematician Kurt Godel solved one of the important math problems laid

out by Hilbert on a result that really stunned the mathematical world.

And there was a similar problem, related,

but closer to computing, that was laid out by Alonzo Church,

who was a professor of mathematics at Princeton around that time.

And then a young British student came to Princeton and

presented work that he had done in England that really

provided the answers that have kept people studying

computation from a theoretical basis ever since.

This is just a basic overview of the concept from a mathematical point of view.

So we can think of mathematics as

any formal system that's powerful enough to express arithmetic.

It seems like this should be an easy thing to do, but

it really gets right at the heart of the matter.

And lots and lots of different formal systems have been developed for

this purpose.

So we say that a formal system is complete if you can prove the truth or

falsity of every statement that you can make within the system.

And you want it to be consistent.

You want to say that within your system, you can't prove contradictions,

like 2 + 2 = five when you know that 2 + 2 should equal 4.

And you want it to be decidable,

you want to know that an algorithm exists to determine the truth of every statement.

So what Hilbert asks, around 1900, was the question,

is mathematics complete and consistent?

And people thought about this as something that really

people thought we should be able to know.

And Godel proved in 1931 that actually, no, it's not possible,

to have both, that you can prove the truth or falsity of every arithmetic statement.

And that you cannot prove contradictions.

Now, it's quite an amazing result that was very surprising to people,

and shook the foundations of the mathematical world.

And Hilbert also asked the question known as the Entscheidungsproblem.

Is mathematics decidable?

Is there an algorithm to determine the truth of every statement?

And Church and Turing, in results like we're going to describe today,

also answered that question in the negative.

And again,

a quite surprising result that shook the foundations of the mathematical world.

And really gets at the heart of our understanding of computation.

Now, we're going to look at this at a level much more closely related to

the simple machines that we talked about in the last lecture.

But it's important to be able to come back to this context to

understand why people were so interested in these kinds of results.

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