Theory of computation

Automata theory (also known as Theory Of Computation) is a theoretical branch of Computer Science and Mathematics, which mainly deals with the logic of computation with respect to simple machines, referred to as automata. Automata* enables scientists to understand how machines compute the functions and solve problems. The main motivation behind developing Automata Theory was to develop methods to describe and analyze the dynamic behavior of discrete systems. String: A string is a finite sequence of symbols from some alphabet. A string is generally denoted as w and the length of a string is denoted as |w|. Empty string is the string with zero occurrence of symbols, represented as ε. Number of Strings (of length 2) that can be generated over the alphabet

Harvard has had a long history of groundbreaking research in the theory of computation (ToC, also known as Theoretical Computer Science). This field addresses the mathematical laws that govern efficient computation, whether by human-made devices or natural phenomena. Today ToC had vastly expanded to touch many problems not just in computer science and engineering, but also pure and applied mathematics, as well as the natural, life and social sciences. Members of the ToC group at Harvard are pursuing both the fundamental questions of computation, as well as its applications to many areas, and are closely collaborating with colleagues across all of these fields.

Electrical Engineering Electrical Engineers design systems that sense, process, and transmit energy and information. We leverage computational, theoretical, and experimental tools to develop groundbreaking sensors and energy transducers, new physical substrates for computation, and the systems that address the shared challenges facing humanity. • Artificial Intelligence + Decision-making Artificial Intelligence and Decision-making combines intellectual traditions from across computer science and electrical engineering to develop techniques for the analysis and synthesis of systems that interact with an external world via perception, communication, and action; while also learning, making decisions and adapting to a changing environment. • • Theory of Computation (TOC) studies the fundamental strengths and limits of computation, how these strengths and limits interact with computer science and mathematics, and how they manifest themselves in society, biology, and the physical world. At its core, TOC investigates tradeoffs among basic computational resources. These resources include computation time, space, communication, parallelization, randomness, quantum entanglement, and more. As computational systems come in many forms and the goals of computation are diverse, TOC studies the limits of computation in its many manifestations. These are determined by what access we have to the computation’s input: do we have access to it as a whole, or does it come as a stream; as samples...

Main article: Grammar Languages Automaton Production rules (constraints) Type-0 α → β 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 In order to analyze how much time and space a given 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 grows linearly in the size of the problem. To simplify this problem, computer scientists have adopted O ( n ) steps to solve. Perhaps the most important open problem in all of Main article: Aside from a A computation consists of an initial lambda expression (or two if you want to separate the function and its input) plus a finite sequence of lambda terms, each deduced from the preceding term by one application of is a concept which has many similarities to λ , then for 'f(5)=3' to appear, terms like 'g(5)=6' and 'h(5,6)=3' must occur above. The computation terminates only if the final term gives the value of the recursive function applied to the inputs. a is a theoretically interesting...

This course emphasizes computability and computational complexity theory. Topics include regular and context-free languages, decidable and undecidable problems, reducibility, recursive function theory, time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, … Show more This course emphasizes computability and computational complexity theory. Topics include regular and context-free languages, decidable and undecidable problems, reducibility, recursive function theory, time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Show less

Introduction The notion that mathematical statements should be ‘well-defined’ had been argued by mathematicians since at least the 1-definability. Today, any formal statement or calculation that exhibits this quality of well-definedness is termed computable, while the statement or calculation itself is referred to as a computation. Turing’s definition apportioned “well-definedness” to a very large class of mathematical statements, including all well-formed Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes Some examples of mathematical statements that are computable include: • All statements characterised in modern programming languages, including • All calculations carried by an electronic • All calculations carried out on an • All calculations carried out on a • The majority of mathematical statements and calculations given in maths textbooks. Some examples of mathematical statements that are not computable include: • Calculations or statements which are ill-defined, such that they cannot be unambiguously encoded into a Turing machine: (“Paul loves me twice as much as Joe”). • Problem statements which do appear to be well-defined, but for which it can be proved that no Turing machine exists to solve them (such as The Physical process of computation Computation can be seen as a purely physical process occurring inside a closed Alternative accounts of computa...

Automata Tutorial Theory of automata is a theoretical branch of computer science and mathematical. It is the study of abstract machines and the computation problems that can be solved using these machines. The abstract machine is called the automata. An automaton with a finite number of states is called a Finite automaton. In this tutorial, we are going to learn how to construct deterministic finite automata, non-deterministic finite automata, Regular expression, context-free grammar, context-free language, Push down automata, Turning machines, etc. Prerequisite Before learning Automata, you should have a basic understanding of string, language, alphabets, symbols. Audience Our Automata Tutorial is designed to help beginners and professionals. Problems We assure that you will not find any problem in this Automata Tutorial. But if there is any mistake, please post the problem in contact form.

Lecture Notes LEC # TOPICS PDF PPT 1 Introduction, Finite Automata, Regular Expressions ( ( 2 Nondeterminism, Closure Properties, Regular Expressions → Finite Automata ( ( 3 The Regular Pumping Lemma, Finite Automata → Regular Expressions, CFGs ( ( 4 Pushdown Automata, CFG ↔ PDA ( ( 5 The CF Pumping Lemma, Turing Machines ( ( 6 TM Variants, the Church-Turing Thesis ( ( 7 Decision Problems for Automata and Grammars ( ( 8 Undecidability ( ( 9 Reducibility ( ( 10 The Computation History Method ( ( 11 The Recursion Theorem and Logic ( ( 12 Time Complexity ( ( 13 Midterm Exam [no lecture] 14 P and NP, SAT, Poly-time Reducibility ( ( 15 NP-Completeness ( ( 16 Cook-Levin Theorem ( ( 17 Space Complexity, PSPACE, Savitch’s Theorem ( ( 18 PSPACE-Completeness ( ( 19 Games, Generalized Geography ( ( 20 L and NL, NL = coNL ( ( 21 Hierarchy Theorems ( ( 22 Provably Intractable Problems, Oracles ( ( 23 Probabilistic Computation, BPP ( ( 24 Probabilistic Computation (cont.) ( ( 25 Interactive Proof Systems, IP ( ( 26 coNP ⊆ IP ( (

CS 332: Elements of the Theory of Computation, Fall 2021 Course Overview This course is an introduction to the theory of computation. This is the branch of computer science that aims to understand which problems can be solved using computational devices and how efficiently those problems can be solved. To be able to make precise statements and rigorous arguments, computational devices are modeled using abstract mathematical "models of computation." The learning objectives of the course are to: • Foremost, understand how to rigorously reason about computation through the use of abstract, formal models. • Learn the definitions of several specific models of computation including finite automata, context-free grammars, and Turing machines, learn tools for analyzing their power and limitations, and understand how they are used in other areas of computer science. • Learn how fundamental philosophical questions about the nature of computation (Are there problems which cannot be solved by computers? Can every problem for which we can quickly verify a solution also be solved efficiently?) can be formalized as precise mathematical problems. • Gain experience with creative mathematical problem solving and develop the ability to write correct, clear, and concise mathematical proofs. Instructor: Instr. Office Hours: Mon 5:00-6:00 PM (MCS 114) Fri 5:00-6:00 PM (MCS 114) Teaching Fellow: TF Office Hours: Tue 5:00-6:00 PM (MCS B21) Thu 4:00-5:00 PM (MCS B21) Class Times: Tue, Thu 2:00-3:1...

Could a machine think? Could the mind itself be a thinking machine? The computer revolution transformed discussion of these questions, offering our best prospects yet for machines that emulate reasoning, decision-making, problem solving, perception, linguistic comprehension, and other mental processes. Advances in computing raise the prospect that the mind itself is a computational system—a position known as the computational theory of mind (CTM). Computationalists are researchers who endorse CTM, at least as applied to certain important mental processes. CTM played a central role within cognitive science during the 1960s and 1970s. For many years, it enjoyed orthodox status. More recently, it has come under pressure from various rival paradigms. A key task facing computationalists is to explain what one means when one says that the mind “computes”. A second task is to argue that the mind “computes” in the relevant sense. A third task is to elucidate how computational description relates to other common types of description, especially neurophysiological description (which cites neurophysiological properties of the organism’s brain or body) and intentional description (which cites representational properties of mental states). The intuitive notions of computation and algorithm are central to mathematics. Roughly speaking, an algorithm is an explicit, step-by-step procedure for answering some question or solving some problem. An algorithm provides routine mechanical instruc...