Type 3 Grammars and Context-Free Languages Quiz

Administrivia

• Previous topics: regular expressions, properties of regular languages, pumping lemma
• Current topics: context-free languages, pushdown automata, grammars
• Reading assignment: Chapter 2
• Test date poll
• Anonymous feedback requested

Recall

• Automaton language: set of strings accepted by the automaton
• Computation (in this context): determining whether a string belongs to a language (decision problem)
• Finite Automata (FA) variants: deterministic, non-deterministic
• Regular language: a language recognized by some FA
• Regular operations: union, concatenation, Kleene star
• Regular expressions: used to describe regular languages
• Non-regular languages: languages that cannot be described by regular expressions, related to the pumping lemma

Regular Expressions

• Definition R is a regular expression if:
  • R is a for a character a in the alphabet Σ
  • R is ε
  • R is ∅
  • R is (R₁ U R₂) where R₁ and R₂ are regular expressions
  • R is (R₁ R₂) where R₁ and R₂ are regular expressions
  • R is (R₁*) where R₁ is a regular expression

Pumping Lemma (PL) for Regular Languages

• Theorem: If A is a regular language, there exists a number p (pumping length) such that any string s in A with length at least p can be divided into three pieces xyz satisfying these conditions:

  • xy^iz ∈ A, for each i ≥ 0
  • y is greater than 0
  • xy is less than or equal to p

Proving a Language Non-Regular

• Use PL to prove a language isn't regular
• Given language L and p
• Choose a string s of length greater than p in L
• Break down the string into 3 parts xyz to match the PL criteria
• Show a contradiction for every possible breakdown of the string will satisfy the 3 conditions

Examples of Non-Regular Languages

• L = {0ⁿ1ⁿ | n > 0}
• L = {ww | w ∈ {0,1}*}
• L = {0ⁱ1ʲ | i > j}

Context-Free Languages

• Characterized by context-free grammars (CFGs)
• Also recognized by pushdown automata (PDAs)
• PDAs have memory (a stack)
• Example of CFL: L = {0ⁿ1ⁿ | n > 0}
• FAs, DPDAs, and NPDAs aren't equally powerful
• Related to type 2 grammars in Chomsky's hierarchy

Context-Free Grammars (CFGs)

• Definition: A 4-tuple (V, Σ, R, S) where
  • V: finite set of variables
  • Σ: finite set of terminals
  • R: finite set of rules of the form α → β
  • S: start variable (element of V)
• Example grammar: G = ({S}, {0, 1}, R, S)
• R : set of rules: S → 0S1, S → ε
• Language of G: {0ⁿ1ⁿ | n ≥ 0}

Context Free Grammars (Continued)

• Derivation rules:
  • If u, v, and w are strings of variables and terminals, and A → w is a rule, then uAv yields uwv.
  • u ⇒ v if u = v or there exists a sequence u₁ ,u₂, ..., uₖ such that u₁ ⇒ u₂ ⇒ ... ⇒ uₖ = v.
• Language of a CFG: {w ∈ Σ* | S ⇒ w} (all strings w that can be derived from the start variable S)

CFGs (Additional Considerations)

• Parsing and Parse Trees: Compilers and interpreters use parsers to extract program meaning before compilation
• CFGs can be used to build a parser

Context-Free Languages and Grammars

• Examples of strings generated by the specific grammar rules

CFLs and CFGs

• CFGs specify language syntax
• Programming languages like Python and Java use CFGs for parsing
• CYK algorithm (O(n³|G|)) uses dynamic programming to determine if a string can be generated by a CFG
• Context-free language: a language that can be generated by a CFG

Grammars

• Grammars provide another way to describe a language (unlike automata which recognize a language)
• Regular languages can be described using finite automata, regular expressions, or type 3 grammars
• Context-free languages can be described using context-free grammars (also type 2)
• Certain languages can be recognized by either a finite automaton or a pushdown automaton, and can also be generated by grammar type 3 or 2 respectively

Type 3 Grammars - Example

• Converting DFAs to equivalent CFGs
• Create a variable for each DFA state Rᵢ
• If (qᵢ, a) = qⱼ, in the DFA, then Rᵢ → aRⱼ in the CFG
• If qᵢ is an acceptor state, then Rᵢ → ε

Designing Grammars

• Examples of designing grammars for specific languages
• L₁ = {w | w contains at least three 1s}:
  • S₁ → R₁R₁R₁R
  • R → 0R|1R|ε

Designing Grammars - Examples Continued

• L₂ = {w | length of w is odd and middle symbol is 0}
• L₃ = {w | w has more a's than b's}

Ambiguity

• Type 3 grammars and languages are unambiguous (only 1 way)
• Type 2 grammars (CFGs) and context-free languages can be ambiguous (more than 1 way)
• Ambiguity can lead to multiple meanings of programs, undesirable in programming languages
• A grammar is ambiguous if there exists a string with multiple parse trees

Ambiguity (Continued)

• Leftmost derivation: every step replaces the leftmost remaining variable

• Ambiguity definition: A string w is generated ambiguously in a CFG G if it has two or more distinct leftmost derivations

• Ambiguous CFG: CFG that generates strings ambiguously

Pushdown Automata (PDAs)

• Definition: A 6-tuple (Q, Σ, Γ, δ, q₀, F)
  • Q: Finite set of states
  • Σ: Input alphabet
  • Γ: Stack alphabet
  • δ: Transition function

Computation of a PDA

• M accepts w if:
  • Initial state: r₀ = q₀ and s₀ = ε
  • Transition rule (rᵢ₊₁, b) ∈ δ (rᵢ, wᵢ₊₁, a), where sᵢ = a⋅t
  • Final state: rₙ ∈ F Where (rᵢ₊₁, b) ∈ δ(rᵢ, wᵢ₊₁, a) for i ∈ [0, n − 1]
  • a, b ∈ Γ* and t ∈ Γ*

PDA Technicalities

• Empty stack test: adds a special symbol ($) to the stack
• End of input: Acceptor mechanism only activates when the entire input string has been processed
• Understanding PDA transition descriptions: meaning of characters (a, b, c) in transition descriptions (input symbols, stack top symbols, ε, etc.)

Let's Build a PDA for a Language

• Example language L = {aⁱbʲcᵏ | i, j, k > 0 and i = j or i = k}
  • Building a PDA to accept that language(Graphical example)
• Another example L = {wwR | w ∈ {0,1}*}
  • Building a PDA to accept that language (Graphical Example)

Context-Free Grammars and Normal Forms

• Definition: A 4-tuple (V, Σ, R, S) where:

  • V: finite set of variables
  • Σ: finite set of terminals
  • R: finite set of rules (α → β where α is a variable and β is a string of variables and terminals)
  • S: start variable (element of V)

• Example grammar

Normal Forms for Type 2 Grammars

• Chomsky Normal Form (CNF): simplifies CFG rules, string size proportional to number of rules used
  • Polynomial-time algorithm to determine if a string is derivable
• Greibach Normal Form (GNF): makes proving a language can be recognized by a PDA in linear time easier

Chomsky Normal Form

• Definition: CFG in CNF has rules where every rule is either of the form A → BC or A → a, where a is a terminal, and A, B, and C are variables (except that B and C may not be the start variable; S).
• Compare/contrast with right linear grammars.

Any CFL has a CFG in CNF

• Theorem: Every context-free language can be generated by a CFG in CNF
• Proof outline:
  • Start with a CFG for a context-free language
  • Successively replace rules that don't fit CNF criteria with equivalent ones.
    • Add a new start variable
    • Eliminate epsilon rules
    • Eliminate unit rules
    • Put rest of rules into CNF form

CFG to CNF Example

• Step-by-step transformations of a CFG to CNF. (graphical examples in the form of diagrams and rules)

Test your understanding of Type 3 grammars and context-free languages with this quiz. Explore questions regarding the properties of grammars, language definitions, and the characteristics of specific languages. Perfect for students studying formal languages and automata theory.