Chapter Two - Regular Expression and Regular languages.pptx
- 1. Regular Expression and Regular languages Chapter 2
- 2. Outline 2.1. Regular expressions 2.2. Connection between regular expression and regular languages 2.3. Regular grammar 2.4. Pumping lemma
- 4. Introduction • A Regular Expression (RE) is a symbolic method to describe patterns of strings in a language. • It represents a regular language, which is a language accepted by finite automata (either DFA or NFA). • Regular expressions are essential in: • Lexical analysis in compilers • Pattern matching (e.g., grep, regex in Python) • Describing tokens in programming languages
- 5. Formal Definition of Regular Expressions I. Base Cases: are the simplest regular expressions — the building blocks: 1. (phi): ϕ represents the empty language: L( ) = ϕ ∅ This language contains no strings at all. 2.ε (epsilon): represents the language containing only the empty string: L(ε) = {ε} The empty string is a string of length 0. 3.a, for any symbol a Σ: represents the language containing ∈ just the string "a": L(a) = {a}
- 6. Cont … II. Recursive Rules (Building Larger Expressions) • If r and s are regular expressions representing languages L(r) and L(s), • Then the following are also regular expressions: 1.Union: r s ∪ The expression r s ∪ denotes the language: L(r s) = L(r) L(s) ∪ ∪ That is, any string in either L(r) or L(s). 2.Concatenation: rs The expression rs denotes the concatenation of L(r) and L(s): L(rs) = { xy | x L(r), y L(s) } ∈ ∈ 3.Kleene Star: r* The expression r* denotes the language containing zero or more concatenations of strings from L(r): L(r) = {ε, w , w w , w w w , ... | each w L(r)} ₁ ₁ ₂ ₁ ₂ ₃ ᵢ ∈ *
- 7. Precedence of Operators • To interpret expressions correctly without excessive parentheses, operator precedence is used: • Kleene Star * – highest precedence • Concatenation • Union | or ∪ – lowest precedence • Example: • a|bc* is interpreted as a (b(c*)) ∪ • Use parentheses to control grouping: • (a|b)*abb means: any number of a’s and b’s followed by a, then b, then b.
- 8. Cont … • To interpret regular expressions without excessive parentheses, we rely on a standard precedence of operators: 1.Kleene Star (*) – Highest Precedence •Applies to the symbol or group directly before it. •Example: a* means zero or more occurrences of a. 2.Concatenation – Medium Precedence •Joins two patterns end to end. •Example: ab* is interpreted as a(b*), not (ab)*. 3.Union (| or ) ∪ – Lowest Precedence •Represents choice between alternatives. •Example: a|b* is interpreted as a | (b*), not (a | b)*.
- 9. Algebraic Laws of Regular Expressions • Regular expressions follow certain algebraic identities, helping in simplification: • Union is commutative: r s = s r ∪ ∪ • Concatenation is associative: (rs)t = r(st) • Distributive property: r(s t) = rs rt ∪ ∪ • Identity element for union: r ∪ = ϕ r • Identity for concatenation: rε = εr = r • Kleene star properties: • r* = ε ∪ r rr rrr ... ∪ ∪ ∪ • (r*)* = r*
- 10. Excersice • Write a regular expression for the set of all strings over the alphabet {a, b} that start with 'a’. • Give a brief explanation of how your regular expression works. • Answer: Regular Expression: a(a|b)* • Explanation: • The string must start with an a. • The expression (a|b)* means zero or more occurrences of either a or b. • So the string starts with a and is followed by any sequence (including none) of a’s and b’s. • Valid examples: a, ab, aabbb, abab, aaaa. • Write a regular expression for all strings over {a, b} that end with 'b’. • Explain how the expression ensures the last character is 'b’. • Answer: Regular Expression: (a|b)*b • Explanation: • (a|b)* matches any sequence of a's and b’s. • The final b ensures that the string ends with b. • Valid examples: b, ab, aab, bbab, bbbb.
- 11. Cont … • Write a regular expression for strings that contain the substring "ab". • Explain how your regex ensures "ab" appears. • Answer: Regular Expression: (a|b)*ab(a| b)* • Explanation: • (a|b)* before and after means any characters can come before and after ab. • The required ab substring must appear at least once in the string. • Examples: ab, aab, babab, aaabba, bbaab. • Write a regular expression for strings over {a, b} that contain exactly two a’s. • Describe how this restricts the count of 'a’. • Answer: Regular Expression: (b*)a(b*)a(b*) • Explanation: • a appears exactly twice. • Between and around the as, there can be any number of bs (even zero). • No more than two as are allowed. • Examples: aab, baab, babab, bbabb.
- 12. 2.2. Connection between regular expression & regular languages
- 13. Introduction Regular Expressions (RE): • Symbolic notation to describe patterns in strings. • Built from basic symbols using operations: union ( ), concatenation, and ∪ Kleene star (*). • Example: (a|b)*abb Regular Languages (RL): • A class of languages that can be recognized by finite automata (DFA/NFA). • These are exactly the languages that can be described by regular expressions.
- 14. From Regular Languages to Regular Expressions • Every regular language can also be represented by a regular expression. • Why? • Regular languages are recognized by DFA/NFA. • Kleene's Theorem: If a language is recognized by a finite automaton, then there exists a regular expression that generates it. • Conversion: Convert NFA Regular Expression →
- 16. Regular Grammar • Regular grammar is a formal grammar used to describe regular languages, which are the languages that can be recognized by finite automata. • There are two standard forms of regular grammar: • Right-Linear Grammar • Left-Linear Grammar
- 17. Cont … • A regular grammar is a restricted type of context- free grammar (CFG) where all production rules follow specific patterns. • Formal Definition • A regular grammar G is a 4-tuple (V,Σ,P,S): • V: Finite set of non-terminal symbols (e.g., S,A,B) • Σ: Finite set of terminal symbols (e.g., a,b) • P: Production rules of specific forms • S: Start symbol (S V) ∈
- 18. Types of Regular Grammars I. Right-Linear Grammars • Rule Forms: • A aB(Non-terminal → → terminal + non-terminal) • A a (Non-terminal → → terminal) • A→ε (Only if A is the start symbol) II. Left-Linear Grammars • Rule Forms: • A Ba(Non-terminal → → non-terminal + terminal) • A a (Non-terminal → → terminal) • A→ε (Only if A is the start symbol) • A language is regular if and only if it can be generated by a right-linear or left-linear grammar.
- 19. I. Right Linear Grammar • Right Linear Grammars are special type of CFGs, where each production rule has at most 1 variable on RHS & that variable is on right most position. • A xB ⇢ • A x ⇢ • where A,B V and x T* ∈ ∈ • Example 1: • S -> aA | B • A -> aaB • B -> bB | a • Grammar G is right-linear
- 20. Cont … • Example 2: FA for accepting strings that start with b • ∑ = {a, b} • Initial state(q0) = A • Final state(F) = B • The RLG corresponding to FA is • A bB ⇢ • B /aB/bB ⇢ ∈ • The above grammar is RLG, which can be written directly through FA.
- 21. II. Left Linear Grammar • Left Linear Grammars are Special type of CFGs, where Each Production Rule has At Most 1 Variable on RHS & that variable is on Left Most position. • A Bx ⇢ • A x ⇢ • where A,B V and x T* ∈ ∈ • Example: • A -> Da | Bc | b • B -> Bf | Ca | a • C -> Ca | D • D -> 𝛆
- 22. 2.4. Pumping lemma and non-regular language grammars
- 23. Introduction • pumping lemma is used to prove that a language is not regular. • what are regular languages and what are non-regular languages? • In previous lessons, we learned that regular languages are exactly those that can be described by regular expressions or recognized by finite automata (DFA/NFA). • All regular languages can be described using regular expressions. • However, not all languages are regular. • To prove that a language is not regular, we use a powerful tool called the Pumping Lemma.
- 24. What is the Pumping Lemma? • If a language A is regular, then there exists a pumping length p≥1 such that any string s A, where ∈ s ≥p, can be split into three parts: ∣ ∣ • s=xyz • such that the following three conditions are true: • xyi z A for all ≥0 ∈ 𝑖 • ∣y ≥1 ∣ • ∣xy ≤p ∣
- 25. Cont … • Let’s break them down: • xyz is the original string. • Y is the part that can be repeated (pumped). • Condition 1 ensures that no matter how many times we repeat y (including zero times), the new string stays in the language. • Condition 2 ensures that we are actually repeating something (not the empty string). • Condition 3 limits the position of the pumpable section y — it must be within the first p characters.
- 26. Proof by contradiction • How to Use the Pumping Lemma to Prove a Language is Not Regular? • We use a proof by contradiction: • Assume the language A is regular. • Then there must exist a pumping length p. • Choose a string s A such that s ≥p. ∈ ∣ ∣ • split s into xyz • Find a value of i such that the pumped string xyi z A. ∉ • This contradiction means that the language cannot be regular.
- 27. Summary Table Concept Meaning Pumping Lemma A property that all regular languages must satisfy Pumping Length (p) The length beyond which strings can be pumped Goal Show that no matter how a long string is split, the conditions will fail Outcome If conditions fail contradiction language is → → not regular
- 28. Example • Example 1: Language 1={a 𝐿 n bn ≥0}L 1={a ∣𝑛 n bn n≥0}This ∣ language contains strings with equal number of a’s followed by equal number of b’s • (e.g., ab, aabb, aaabbb, etc.).
- 29. End Of Chapter Instructor: Biniyam E.