Mit 18.404j Theory Of Computation

[Lina Drury]

Intheoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".[1]

The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used. In the last century, it separated from mathematics and became an independent academic discipline with its own conferences such as FOCS in 1960 and STOC in 1969, and its own awards such as the IMU Abacus Medal (established in 1981 as the Rolf Nevanlinna Prize), the Gdel Prize, established in 1993, and the Knuth Prize, established in 1996.

Automata theory is the study of abstract machines (or more appropriately, abstract 'mathematical' machines or systems) and the computational problems that can be solved using these machines. These abstract machines are called automata. Automata comes from the Greek word (Αυτόματα) which means that something is doing something by itself.Automata theory is also closely related to formal language theory,[5] as the automata are often classified by the class of formal languages they are able to recognize. An automaton can be a finite representation of a formal language that may be an infinite set. Automata are used as theoretical models for computing machines, and are used for proofs about computability.

Language theory is a branch of mathematics concerned with describing languages as a set of operations over an alphabet. It is closely linked with automata theory, as automata are used to generate and recognize formal languages. There are several classes of formal languages, each allowing more complex language specification than the one before it, i.e. Chomsky hierarchy,[6] and each corresponding to a class of automata which recognizes it. Because automata are used as models for computation, formal languages are the preferred mode of specification for any problem that must be computed.

Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer. The statement that the halting problem cannot be solved by a Turing machine[7] is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine. Much of computability theory builds on the halting problem result.

Another important step in computability theory was Rice's theorem, which states that for all non-trivial properties of partial functions, it is undecidable whether a Turing machine computes a partial function with that property.[8]

Computability theory is closely related to the branch of mathematical logic called recursion theory, which removes the restriction of studying only models of computation which are reducible to the Turing model.[9] Many mathematicians and computational theorists who study recursion theory will refer to it as computability theory.

Complexity theory considers not only whether a problem can be solved at all on a computer, but also how efficiently the problem can be solved. Two major aspects are considered: time complexity and space complexity, which are respectively how many steps it takes to perform a computation, and how much memory is required to perform that computation.

In order to analyze how much time and space a given algorithm requires, computer scientists express the time or space required to solve the problem as a function of the size of the input problem. For example, finding a particular number in a long list of numbers becomes harder as the list of numbers grows larger. If we say there are n numbers in the list, then if the list is not sorted or indexed in any way we may have to look at every number in order to find the number we're seeking. We thus say that in order to solve this problem, the computer needs to perform a number of steps that grow linearly in the size of the problem.

To simplify this problem, computer scientists have adopted Big O notation, which allows functions to be compared in a way that ensures that particular aspects of a machine's construction do not need to be considered, but rather only the asymptotic behavior as problems become large. So in our previous example, we might say that the problem requires O ( n ) \displaystyle O(n) steps to solve.

Perhaps the most important open problem in all of computer science is the question of whether a certain broad class of problems denoted NP can be solved efficiently. This is discussed further at Complexity classes P and NP, and P versus NP problem is one of the seven Millennium Prize Problems stated by the Clay Mathematics Institute in 2000. The Official Problem Description was given by Turing Award winner Stephen Cook.

In addition to the general computational models, some simpler computational models are useful for special, restricted applications. Regular expressions, for example, specify string patterns in many contexts, from office productivity software to programming languages. Another formalism mathematically equivalent to regular expressions, Finite automata are used in circuit design and in some kinds of problem-solving. Context-free grammars specify programming language syntax. Non-deterministic pushdown automata are another formalism equivalent to context-free grammars. Primitive recursive functions are a defined subclass of the recursive functions.

Different models of computation have the ability to do different tasks. One way to measure the power of a computational model is to study the class of formal languages that the model can generate; in such a way to the Chomsky hierarchy of languages is obtained.

Working with experimental biologists, we use these computational analyses and theoretical frameworks to design new experiments that refine and test our models, filling in the biggest gaps in our understanding of biological processes.

06/30/22 Cosmological thinking meets neuroscience in new theory about brain connections: A collaboration between a former cosmologist and a computational neuroscientist at Janelia generates a new way to identify essential connections between brain cells.

The mission of the ITC is to advance our knowledge and understanding of the universe through computational and analytical means, to create a forum for exploration and discoveries in theoretical astrophysics, and to train the next generation of astrophysicists. Read More about the ITC Mission.

Harvard University astronomers and Smithsonian Institution astronomers explore the cosmos together. The combination of the Harvard College Observatory (HCO) and the Smithsonian Astrophysical Observatory (SAO) is known as the Harvard-Smithsonian Center for Astrophysics (CfA).

TOC has undergone a number of evolutions in a short span of time. From its beginning in the 1960s as an outgrowth of mathematical logic and information theory, it evolved into a branch of mathematics where one looks at classical problems with the aesthetics of computational complexity and asks new questions concerning non-determinism, randomness, approximation, interaction, and locality. It then took a foundational role in addressing challenges arising in computer systems and networks, such as error-free communication, cryptography, routing, and search, and is now a rising force in the sciences: exact, life, and social. The TOC group at MIT has played a leadership role in theoretical computer science since its very beginning. Today, research done at the TOC group covers an unusually broad spectrum of research topics.

Theory of Computation (TOC) is the study of the inherent capabilities and limitations of computers: not just the computers of today, but any computers that could ever be built. By its nature, the subject is close to mathematics, with progress made by conjectures, theorems, and proofs. What sets TOC apart, however, is its goal of understanding computation -- not only as a tool but as a fundamental phenomenon in its own right.

At MIT, we are interested in a broad range of TOC topics, including algorithms, complexity theory, cryptography, distributed computing, computational geometry, computational biology, and quantum computing.

The Mathematics Theory and Computation B.S. provides an excellent scientific background from which to pursue a variety of career opportunities. UC Santa Cruz graduates with degrees in mathematics hold teaching posts at all levels, as well as positions in law, government, civil service, insurance, software development, business, banking, actuarial science, forensics, and other professions where skills in logic, numerical analysis, and computing are required. In particular, students of mathematics are trained in the art of problem-solving, an essential skill in all professions.

The undergraduate adviser may be contacted via email at mathad...@ucsc.edu. The adviser provides information about requirements, prerequisites, policies and procedures, learning support, scholarships, and special opportunities for undergraduate research. In addition, the adviser assists with the drafting of study plans, as well as certifying degrees and minors. Students are urged to stay informed and involved with their major, as well as to seek advice should problems arise.

The Mathematics Department website is a critical resource for students. Here you will find a link to the undergraduate program; the materials at that link constitute the undergraduate handbook. Students should visit this first to seek answers to their questions, because it hosts a wealth of information. Each student in the major is encouraged to regularly review the materials posted to stay current with requirements, course curriculum, and departmental policy. Transfer students should consult the Transfer Information and Policy section.

3a8082e126