Finite State Machine (Prerequisites)

Summary

In this lecture on the Theory of Computation, we explore fundamental concepts like symbols, alphabets, strings, and languages, which are essential for understanding Finite State Machines. The lecture covers the definitions and examples of each concept, explaining how alphabets form the basis of string construction, and how languages are sets of strings. It also introduces the powers of Sigma, cardinality, and the concept of Sigma star, detailing their significance in computational theory. This foundational knowledge is crucial for students to grasp the complexities of automata and computational models in subsequent lectures.

Takeaways

• 😀 A symbol is a basic unit in the theory of computation and can be any representation like a, b, 0, 1, etc.
• 😀 An alphabet (denoted by Σ) is a collection of symbols used to form strings. Examples include Σ = {a, b} or Σ = {0, 1}.
• 😀 A string is a sequence of symbols. For example, 'a', 'ab', '011' are all valid strings.
• 😀 A language is a set of strings formed from an alphabet. For example, L1 = {00, 01, 10, 11} is a language over the alphabet {0, 1}.
• 😀 The set of all strings of a given length is finite, but some languages, such as those starting with a particular symbol, can be infinite.
• 😀 Powers of Σ define sets of strings of varying lengths, such as Σ^0 for strings of length 0 (empty string), Σ^1 for strings of length 1, and so on.
• 😀 The cardinality of a set refers to the number of elements it contains. For example, the cardinality of Σ^2 (strings of length 2) is 4.
• 😀 The cardinality of Σ^n, where n is the length of strings, is calculated as 2^n for binary alphabets like {0, 1}.
• 😀 Sigma star (Σ*) represents the set of all possible strings (including the empty string) of any length formed from the alphabet Σ.
• 😀 Sigma star (Σ*) is an infinite set because it includes strings of all possible lengths, from length 0 (ε) to any larger length.

Q & A

What is a symbol in the context of theory of computation?

A symbol is any individual character used to represent something in the theory of computation. Examples include 'a', 'b', '0', '1', etc.

How is an alphabet defined in theory of computation?

An alphabet is a collection of symbols, often denoted by Sigma (Σ). It can include characters, numbers, or other symbols, such as Σ = {0, 1}.

What is a string in computational theory?

A string is a sequence of symbols from an alphabet. For example, 'ab', '101', and 'aabb' are strings formed from symbols of a given alphabet.

What is a language in the theory of computation?

A language is a set of strings that are formed from an alphabet. A language is defined by specific rules or conditions that the strings must satisfy.

What does the term 'Sigma power' (Σ^n) mean?

Sigma power (Σ^n) refers to the set of all strings of length 'n' formed from the alphabet Σ. For example, Σ^2 would contain all strings of length 2 formed from the symbols of Σ.

Can you provide an example of strings in Sigma power 2?

For an alphabet Σ = {0, 1}, the strings in Σ^2 (strings of length 2) would be: '00', '01', '10', and '11'.

What is the cardinality of a set in the context of finite automata?

Cardinality refers to the number of elements in a set. For instance, the cardinality of Σ^2 (with Σ = {0, 1}) is 4 because there are four strings of length 2: '00', '01', '10', '11'.

What is the significance of Sigma star (Σ*) in formal language theory?

Sigma star (Σ*) represents the set of all possible strings of all lengths (including the empty string) formed from an alphabet. It is an infinite set because it includes strings of any length.

Why is Sigma power 0 (Σ⁰) important in the theory of computation?

Sigma power 0 (Σ⁰) represents the set of all strings of length zero, which is denoted by the empty string 'ε'. It is important because it serves as the base case in many computations and language definitions.

What does the union of Sigma powers (Σ⁰ ∪ Σ¹ ∪ Σ² ...) represent?

The union of Sigma powers (Σ⁰ ∪ Σ¹ ∪ Σ² ...) represents Sigma star (Σ*), the set of all possible strings of all lengths over the alphabet. This includes strings of length zero, one, two, three, and so on.