1. Finite-state automata
Module 04.2 COP4020 – Programming Language Concepts Manuel E. Bermudez

2. Topics Define Finite-State Automata Non-Determinism
Conversion from FSA to Right-Linear Grammar
Conversion from Left-Linear Grammar to FSA (and back)

3. Finite-state automata
Definition: A (non-deterministic) finite-state automaton
is a 5-tuple M = (Q, Σ, δ, s, F), where
Q is a finite set of states,
Σ is a finite set of transition symbols,
δ: Q x Σ ∪ {ε} → 2Q is the
(partial) transition function,
s  Q is called the start state, and
F ⊆ Q is the set of final states.
An FSA is the formal accepting (parsing)
mechanism of a regular language.

4. Finite-state automataR F
aba ε b a
FSA: ({S, R, U, F, X, Y}, {a, b}, δ, S, {F}),
where
δ (S, a) = {F}
δ (S, b) = {U, R}
δ (R, ε) = {U}
δ (R, a) = {X}
δ (U, a) = {S}
δ (U, b) = {F}
δ (X, b) = {Y}
δ (Y, a) = {U} FSA requires each transition
labeled by a string of length < 1. R X a X Y b Y U a

5. Finite-state automata
TWO “SYMPTOMS” OF NON-DETERMINISM:
NOTE: is not a problem. X a ε 1. 2. F a

6. Finite-state automata
Advantage of FSA’s: What language does this grammar generate?
S → aA A → aB B → aC
→ ε → E → D
C → bD D → bE E → bS
Difficult to see. Try FSA.
Answer: L*, where
L={ab, aabb, aaabbb} F ε S A E D C B b a

7. Finite-state automata
Conversion: FSA to Right-linear regular grammar
Φ = Q
A → aB if B  δ (A, a)
A → a if f  δ (A, a), and f  F
Start symbol = Start state
A → aB B → bB D → cE
→ a → bD → c
E → F → b G → H
H → A F → dG → ε
Conclusion: Right- linear regular grammars and FSA’s are equivalent A G F ε b a d H B D E c

8. Finite-state automata
Conversion between FSAs and Left-linear regular grammars
F → Sa U → Sb R → Sb
→ Ub → R S → Ua
→ Raba →
Sbb ...
F => Ub => Rb ...
Rabab ...
Sa => Uaa ... a
=>
=>
=>
=>

9. Finite-state automata
Similarities with right-linear grammars:
Sentential forms have at most one nonterminal.
Sentences have none.
Applicable productions depend only on the one nonterminal.
Differences with right-linear grammars:
Nonterminal appears on left-most position.
String generated right-to-left, versus
left-to-right for right-linear grammars.

10. Finite-state automata
Conversion: Left-linear regular grammar to FSA:
if A → B.
if A → , S’ is a new start state.
F = {S}, S is the start symbol.
F → Sa U → Sb R → Sb
→ Ub → R S → Ua
→ Raba → B A α S’ A α S U R F
aba ε b a S’

11. Finite-state automataR F
aba ε b a S’
Is “babaaa” in the language ?
State Input Derivation
S’ babaaa babaaa <=
S babaaa Sbabaaa <=
R abaaa Rabaaa <=
U aa Uaa <=
S a Sa <=
F F
F → Sa
→ Ub
U → Sb
→ R
→ Raba
R → Sb
S → Ua →

12. Finite-state automata
Conversion: FSA to Left-linear regular grammar:
A → B if
A →  if
S’ → F if
Conclusion: Left-linear regular grammars and FSA’s are equivalent. B A α S α A F
New start symbol

13. Finite-state automata
Summarizing:
RGR RGL
RE FSA
Note: Beware of attempts at
direct conversion between left
and right-linear grammars.
Done
Soon