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Lecture 15 Deterministic Push Down Automata COT 4420 Theory of Computation Section 7.3 DPDA • A deterministic pushdown accepter (DPDA) is a pushdown automaton that never has a choice in its move. • Two restrictions on transitions: 1. δ(q, a, b) contains at most one element 2. If δ(q, λ, b) is not empty, then δ(q, c, b) must be empty for every c ∈ Σ. Nondeterministic choices a, b x a, b y Not allowed Nondeterministic choices λ, b x a, b y Not allowed Deterministic context-free language • A language L is said to be a deterministic context-free language if and only if there exists a dpda M such that L = L(M). Deterministic context-free language • The language L = { anbn : n ≥ 0 } is a deterministic context-free language. • Because there exists a dpda that accepts L: a, λ x λ, λ λ b, x λ b, x λ λ, z z Example L = { wwR : w ∈ {a,b}* } a, a aa a, b ab a, z az b, a ba b, b bb b, z bz a, a λ b, b λ λ, a a λ, b b λ, z z Example L = { wwR : w ∈ {a,b}* } a, a aa a, b ab a, z az b, a ba b, b bb b, z bz Not allowed in DPDA a, a λ b, b λ λ, a a λ, b b λ, z z Non-Deterministic language • Having an NPDA for a language does not necessarily mean that the language itself is nondeterministic, since there is the possibility of an equivalent DPDA. • However, it is known that the language {wwR} is not deterministic. NPDA vs DPDA ? Deterministic and nondeterministic push down automata are not equivalent Deterministic context-free languages (DPDA) ⊆ Context-free languages (NPDA) • Every DPDA is also an NPDA We need to show that: • There exists a context-free language L that is not accepted by any DPDA Context-free but not Deterministic CF L = { anbn : n ≥ 0 } ∪ { anb2n : n ≥ 0 } L is context-free, but not deterministic contextfree. L is context-free S S 1 | S2 S1 aS1b | λ S2 aS2bb | λ Context-free grammar Therefore, L is a context-free language. Theorem • Theorem: The language L = { anbn : n ≥ 0 } ∪ { anb2n : n ≥ 0 } is not deterministic context-free. (There is no DPDA that accepts L) We know that {anbncn} is not context-free. Also, L ∪ {anbncn} is not context-free • We will prove these later using pumping lemma for context-free languages. Proof Assume by contradiction that L is deterministic context-free, therefore there is a dpda M that accepts L. Such path exists due to determinism M n n a b b n Proof Using this DPDA M, we construct an NPDA that accepts L ∪ {anbncn}, which is a contradiction! L( M ) = {a nb n } ∪ {a nb 2n } DPDA M n n a b b n Replace b with c L( M ′) = {a nc n } ∪ {a nc 2n } DPDA M’ n n a c c n NPDA that accepts L ∪ {anbncn} Connect the final states of M with the final states of M’ M n n a b b n λ M′ a nc n λ cn Proof – Cont’d Since L ∪ {anbncn} is accepted by an NPDA, it is a context-free language. However, this language is not context-free. Contradiction! Therefore, the assumption that L= { anbn : n ≥ 0 } ∪ { anb2n : n ≥ 0 } is deterministic is false.