1 / 12

Exploring Context-Free Grammars and Turing Machines in Computability Theory

This text delves into the concepts of context-free grammars (CFGs) and Turing machines, focusing on the language {anbncn | n ≥ 0}. We discuss the Pumping Lemma for context-free languages, the Chomsky hierarchy, and prove whether specific languages are context-free. Moreover, we introduce Turing machines, their structure (7-tuple description), and illustrate configurations and the process of recognizing languages. This foundational framework is essential for understanding formal languages and automata theory in computer science.

camilla-cardenas
Télécharger la présentation

Exploring Context-Free Grammars and Turing Machines in Computability Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Turing machines Sipser 2.3 and 3.1 (pages 123-144)

  2. A Context-free Grammar for {anbncn| n ≥ 0}? • Theorem 2.34 (Pumping lemma for CFLs): If A is a CFL, then there is a number p where, if s is any string in A of length ≥ p, then s = uvxyzsuch that: • For each i ≥ 0, uvixyiz ∈ A, • |vy| > 0, and • |vxy| ≤ p

  3. Proof idea • Surgery on parse trees

  4. So… • Is {anbncn| n ≥ 0} a CFL?

  5. Chomsky hierarchy anbncn Context-free languages Regular languages 0n1n

  6. Introducing… Turing machines Infinite tape a b a b ⨆ ⨆ ⨆ Bi-directional read/write head Finite control

  7. Formally… • A Turing machine is a 7-tuple (Q, Σ, Γ, δ, q0, qaccept, qreject), where • Q is a finite set called the states • Σis a finite set not containing the blank symbol ⨆ called the input alphabet • Γ is a finite set called the tape alphabet with ⨆∈Γand Σ⊆Γ • δ:Q ×Γ→ Q ×Γ ×{L,R} • q0∈Q is the start state • qaccept∈Q is the accept state • qaccept∈Qis the reject state

  8. Recognizing {anbncn| n ≥ 0} Infinite tape a a b b c c ⨆ ⨆ ⨆ Finite control

  9. Configurations • A configurationis • Current state • Current tape contents • Current head location • uqvmeans • Current state is q • Current tape contents isuv • Current head points at first symbol of v • Example • âaq1bbcc • In state q1 • Tape contents are âabbcc • Tape head is on first b

  10. Yields • A configuration C1 yields configuration C2 if the Turing machine can legally go from C1 to C2 in a single step • yields • Written ⊢

  11. Turing-recognizable languages • A Turing machine accepts input w if a sequence of configurations C1,C2,...,Ck exists where • C1 is the start configuration of M on input w • Each Ci yields Ci+1 • Ck is an accepting configuration • Defn 3.5: A language is Turing-recognizable if it is accepted by some Turing machine.

  12. Recognizing .

More Related