Regular Languages Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida Programming Language Translators

Regular Languages We will study: Regular grammars Relation to finite-state automata Regular expressions Equivalence among representations Elimination on non-determinism State minimization

Regular Languages Definition: A grammar G = (Φ, Σ, P, S) is regular iff either (but not both): Every production is of the form A →  or A → B (right linear) Every production is of the form A →  or A → B (left linear), where   Σ*, and A, B  Φ.

Regular Languages Examples: G1: S → a R → abaU → bU → U Regular? Why? → bRU → b → S G2: S → aR → Uaba → UbU → b Regular? Why? → Rb → aS

Regular Languages Let’s devise a machine that accepts L(G1). Observe that S => a bU => bb … bR bS … babaU 1.Every sentential form (except sentences) has exactly one nonterminal. 2.The nonterminal occurs in the right-most position. 3.Applicable productions depend only on that nonterminal. =>

Regular Languages Encode possible derivation sequences with a relation ⊢ on pairs of the form (q, ), where q – current state  – remaining string to accept So, S → bU implies (S, bβ) ⊢ (U, β) State “sentential form ends in S” “moves to” state “sentential form ends in U”

Regular Languages Define a graph, one node per nonterminal, describing the possible actions on each sentential form. So, S → bU implies, R → U implies, S → a implies. SU RU SF b  a

Regular Languages Example: S → a R → abaUU → b → bU → U → aS → bR S U R F aba ε b b a b a

Regular Languages In general, Right-linear grammar → Transition diagram: 1.Nodes: Φ  {f}, f  Φ 2.if A →  B 3. if A →  4. AB S AF α α

Regular Languages Example: Is “babaa” in L(G)? NodeInputDerivation SbabaaS => U abaabU => S baabaS => U aababU => S ababaS => Fbabaa Yes.

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} → 2 Q is a partial function called the transition function, s  Q is called the start state, and F  Q is the set of final states.

Finite-State Automata An FSA is the formal accepting mechanism of a regular language. It requires that each transition be labeled by a string of length < 1. The state diagram is described by the FSA S U R F aba ε b b a b a

Finite-State Automata ({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} RX XY YU a b a

Finite-State Automata TWO “SYMPTOMS” OF NON-DETERMINISM: Note:is not a problem F a X a a X ε a 1. 2.

Finite-State Automata Advantages of FSA’s: Question: What language does the following grammar generate? S → aAA → aBB → aC → ε → E → D C → bDD → bEE → bS Difficult to see. Try FSA.

Finite-State Automata Answer: L*, where L = {ab, aabb, aaabbb} Summary: FSA’s are as powerful as right-linear regular grammars. Are they more powerful? No. Can transform FSA → RG R. F ε SA EDC B b ε b aa ε b a

Finite-State Automata Transition Diagram (FSA) → Right-linear regular grammar 1.Φ = Q 2.A → aBif B  δ (A, a) 3.A → aif f  δ (A, a), and f  F 4.Start symbol = Start state

Finite-State Automata Example: FSA: RGR:A → aBB → bBD → cE → a → bD → c → b E → FF → dGG → H → ε H → A Conclusion: Right-linear regular grammars and FSA’s are equivalent. A GF ε ε ba d H BDE b c ε

Finite-State Automata Relationship between Left-linear regular grammars and FSA’s: Example:F → SaU → SbR → Sb → Ub → RS → Ua → Raba → Derivations: Sbb... F => Ub => Rb... Rabab... Sa => Uaa... a =>

Finite-State Automata Similarities with right-linear grammars: 1.Sentential forms have at most one nonterminal. 2.Sentences have none. 3.Applicable productions depend only on the one nonterminal. Differences with right-linear grammars: 1.Nonterminals appears on left-most position. 2.String generated right-to-left, versus left-to-right for right-linear grammars.

Finite-State Automata Left-linear Regular Grammar → FSA 1. if A → B . 2. if A → , S’ is a new start state. 3.F = {S}, S is the start symbol. BA S’A α α

Finite-State Automata Example: F → SaU → SBS → Ua → Ub → R → ε R → Sb → Raba S U R F aba ε b b a b a S’ ε

Finite-State Automata StateInputDerivation S’babaaa babaaa SbabaaaSbabaaa <= R abaaa Rabaaa <= U aa Uaa <= S a Sa <= F

Finite-State Automata FSA → Left-linear Regular Grammar: 1.A → B  if 2.A →  if 3.S’ → F if BA SA α α F New start symbol

Finite-State Automata Summarizing: RG R RG L REFSA Note: Beware of attempts at direct conversion between left and right-linear grammars. Done Soon