Introduction to the Theory of Computing

• The course "Introduction to the Theory of Computing" (18.404/6.840) will explore the capabilities and limitations of computer algorithms and computation, divided into two halves: computability theory and complexity theory.

• Computability theory, studied in the first half of the course, examines what can be computed with an algorithm in principle, a research area that was active in the early 20th century but is now largely closed off.

• The course will cover models of computation, including finite automata, Turing machines, and others, to understand what kinds of problems can be solved with an algorithm.

• One key concept is that there are problems that cannot be solved with an algorithm, such as determining whether a program meets a certain specification or verifying mathematical truths.

• The second half of the course will focus on complexity theory, which explores what can be computed in practice, considering the time and space requirements of algorithms.

• Topics in complexity theory include the factoring problem, cryptography, and the P vs. NP problem, a famous problem in computer science and mathematics.

• The course has prerequisites, including prior experience with mathematical theorems and proofs, and will require students to produce proofs and understand mathematical statements.

• The instructor, Mike Sipser, emphasizes the importance of theoretical computer science in understanding computation and its limitations, citing the fact that we still don't understand how the brain works or how to create a computer program that can do mathematics or creative tasks.

• The course will start with simple models of computation, such as finite automata, to understand the fundamental principles of computation, and then move on to more complex models.

• Finite automata, a simple model of computation, will be explored in detail, including their capabilities and limitations, and how they approximate real computers.

• The course aims to provide a comprehensive theory of computation, covering the basics of computability and complexity, and their applications in computer science.Here is a summary of the text in detailed bullet points:

• Finite automata are computational models that process input strings and accept or reject them based on their structure.

• A finite automaton has a starting state, accepts input strings one symbol at a time, and moves to a new state based on the current state and input symbol.

• The automaton accepts a string if it ends in an accepting state after processing the entire string.

• A language is a set of strings, and a finite automaton recognizes a language if it accepts all strings in the language and rejects all strings not in the language.

• The language of a machine is the set of strings it accepts.

• A string is a finite sequence of symbols from an alphabet, and a language is a set of strings.

• The empty string is a special string of length 0, and the empty language is a language with no strings.

• A regular language is a language that can be recognized by a finite automaton.

• An automaton accepts a string if there is a sequence of states that satisfies three properties: it starts at the starting state, each state legally follows the previous state according to the transition function, and the final state is an accepting state.

• The language of a machine is the set of strings it accepts, and a machine recognizes a language if it accepts all strings in the language and rejects all strings not in the language.

• A language is regular if there is a finite automaton that recognizes it.

• Examples of regular languages include the set of strings with the substring 11, and the set of strings with an even number of 1's.

• An example of a non-regular language is the set of strings with an equal number of 0's and 1's.

• Regular operations are operations that apply to languages, such as union, concatenation, and star.

• The union of two languages is the set of strings that are in either language.

• The concatenation of two languages is the set of strings formed by concatenating a string from the first language with a string from the second language.

• The star operation applies to a single language and forms the set of strings that can be formed by concatenating zero or more strings from the original language.

• Examples of regular operations include the union of two languages, the concatenation of two languages, and the star operation on a single language.Here are the detailed bullet points summarizing the text:

• The empty language or a language containing only the empty string has an infinite language as its star (A*).

• Regular expressions are built using regular operations (union, concatenation, and star) on atomic elements (elements of σ, σ itself, the empty language, and the empty string).

• Sigma star (σ*) represents all possible strings over the alphabet σ.

• Sigma star 1 represents all strings that end with 1.

• Sigma star 11 sigma star represents all strings that contain 11.

• Regular languages are equivalent in power to finite automata, meaning anything that can be described by a regular expression can also be recognized by a finite automaton.

• The course will prove that regular languages are closed under union, meaning that the union of two regular languages is also a regular language.

• To prove this, we assume that A1 and A2 are regular languages recognized by finite automata M1 and M2, and we need to show that A1 union A2 is also a regular language.

• We can build a new finite automaton M that recognizes A1 union A2 by combining M1 and M2.

• M has pairs of states from M1 and M2, and the transition function is built from the transition functions of M1 and M2.

• The accepting states of M are pairs where either the M1 state or the M2 state is an accepting state.

• The number of states in M is the product of the number of states in M1 and M2.

• We started discussing closure under concatenation, where we want to build a machine that recognizes the concatenation language A1A2.

• The proposed strategy is to simulate M1 and M2, but this approach is flawed because it doesn't consider all possible ways to divide the input string w.