ContextSensitive Languages and LinearBounded Automata

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Context-Sensitive Languages and Linear-Bounded
Automata

• Another Intermediate-Strength Model of Computation

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Context-Sensitive Grammars

• Language anbncnn gt 0 not context-free
• Application of Pumping Lemma for Context-Free
  Grammars
• Generative grammars exist that generate L
• Example 9.8.3

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Example 11.1.1
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Context-Sensitivity

• Context-sensitive production any production a b
  satisfying a ? b.
• Context-sensitive grammar any generative grammar
  G ?S, G, ?, s? such that every production in ?
  context-sensitive.
• No empty productions.

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Context-Sensitive Language

• Language L context-sensitive if there exists
  context-sensitive grammar G such that either L
  L(G) or L L(G) ? e.

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Context-Free and Context-Sensitive Languages

• Any context-free language context-sensitive
  despite fact that there exist context-free
  grammars that are not context-sensitive grammars.
• A bit of engineering.

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Generates aibjcidji, j ? 1
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Generates aibjck1 ? i ? j ? k
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Linear-Bounded Automaton

• Special sort of nondeterministic, single-tape
  Turing machine
• Language acceptor only
• Left endmarker l and right endmarker r flanking
  input
• Multiple tapes possible (also with l and r in
  same positions)

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Summary of Formal Definition

• Upon encountering l, either M halts or Ms
  read/write head must move to right
• Upon encountering r, either M halts or Ms
  read/write head must move to the left.
• M nevers writes either l or r.
• l and r can never be moved.
• Mark bounds of memory.

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Word Acceptance

• Input tape contains w immediately preceded by l
  and immediately followed by r and otherwise
  blank.
• M (is able to) halt scanning symbol 1 on a tape
  that contains string l1Bw-1r and that is blank
  otherwise.
• Input w ? a special case.

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LBA-Acceptable Language

• LBA M accepts language L if M accepts all and
  only words of L.
• L is LBA-acceptable if there exists LBA that
  accepts it.

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Powerful Model of Computation

• N. Savoius Deus ex machina (software simulation)
  
• Example 11.3.1 accepts anbncnn ? 1.
• Example 11.3.2 accepts aii prime
• Example 11.3.3 accepts ai2i ? 0

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Context-Sensitive Languages and Linear-Bounded
Automata Related

• L is context-sensitive language if and only if L
  is LBA-acceptable.
• Tool (equivalent) variant notion whereby M
  accepts by terminal state