Context Sensitive Languages and Linear Bounded Automata

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Context Sensitive Languages and Linear Bounded
Automata

• Benjamin Mayne
• CPSC 627

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Introduction

• Context Sensitive languages/grammars
• Linear Bounded Automata (LBA)
• Equivalence of CSL and LBA
• Complexity of CSLs / Variants
• Closure properties of CSLs
• Decidability of CSLs

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Context Sensitive Grammars

• G (V, ?, R, S)
• V is set of variables
• ? is set of terminals
• R are rules of the form
• aAß -gt a?ß
• (A goes to ? in the context of a and ß)
• where A ? V, a, ß ? (V U S), and
• ? ? (V U S)
• plus the rule S ? e if S is not on right side
  of any rule.

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Continued

• aAß a?ß
• CSGs are called noncontracting grammars because
  no rule decreases the size of the string being
  generated.
• For example
• S ? aSBc ? aaSBcBc ? aaabcBcBc ?
• aaabBcBcc ? aaabbBccc ? aaabbbccc

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CSL example

• Consider the following CSG
• S ? aSBc
• S ? abc
• cB ? Bc
• bB ? bb
• The language generated is L(G) anbncn n?1

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Chomsky Hierarchy
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Chomsky Hierarchy
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Context-sensitive languages

• Clearly, context-sensitive rules give a grammar
  more power than context-free grammars. A
  context-sensitive grammar can use the surrounding
  characters to decide to do different things with
  a variable, instead of always having to do the
  same thing every time.
• All productions in context-sensitive grammars are
  non-decreasing or non-contracting that is, they
  never result in the length of the intermediate
  string being reduced.

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Linear Bounded Automata

• A Turing machine that has the length of its tape
  limited to the length of the input string is
  called a linear-bounded automaton (LBA).
• A linear bounded automaton is a 7-tuple
  nondeterministic Turing machine M (Q, S, G, d,
  q0,qaccept, qreject) except that
• 1. There are two extra tape symbols lt and gt,
  which are not elements of G.
• 2. The TM begins in the configuration (q0ltxgt),
  with its tape head scanning the symbol lt in cell
  0. The gt symbol is in the cell immediately to
  the right of the input string x.
• 3. The TM cannot replace lt or gt with anything
  else, nor move the tape head left of lt or right
  of gt.

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Linear Bounded Automata
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• L anbncn n ? 0
• Q s,t,u,v,w ? a,b,c
• ? a,b,c,x q0 s
• ?
• ((s, lt), (t, lt, R)), ((t, gt), (t, gt, L )),
• ((t, x), (t, x, R)), ((t, a), (u, x, R)),
• ((u, a), (u, a, R)), ((u, x), (u, x, R)),
• ((u, b), (v, x, R)), ((v, b), (v, b, R)),
• ((v, x), (v, x, R)), ((v, c), (w, x, L)),
• ((w, c), (w, c, L)), ((w, b), (w, b, L)),
• ((w, a), (w, a, L)), ((w, x), (w, x, L)),
• ((w, lt), (t, lt, R))

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• The intuition behind the previous example is
• that on each pass through the input string, we
• match one a, one b and one c and replace each
• of them with an x until there are no a's, b's or
  c's
• left.
• Each of the states can be explained as follows
• State t looks for the leftmost a, changes this to
  an x, and moves into state u. If no symbol from
  the input alphabet can be found, then the input
  string is accepted.
• State u moves right past any a's or x's until it
  finds a b. It changes this b to an x, and moves
  into state v.
• State v moves right past any b's or x's until it
  finds a c. It changes this c to an x, and moves
  into state w.
• State w moves left past any a's, b's, c's or x's
  until it reaches the start boundary, and moves
  into state t.

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CSG LBA

• A language is accepted by an LBA iff it is
  generated by a CSG.
• Just like equivalence between CFG and PDA
• Given an x ? CSG G, you can intuitively see that
  and LBA can start with S, and nondeterministically
  choose all derivations from S and see if they
  are equal to the input string x. Because CSLs
  are non-contracting, the LBA only needs to
  generate derivations of length ? x. This is
  because if it generates a derivation longer than
  x, it will never be able to shrink to the size
  of x.

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Complexity of CSL/Variants

• Since a context-sensitive language is equivalent
  to the languages recognized by an LBA,
  context-sensitive languages are exactly
  ?NSPACE(cn).
• Can be solved by nondeterministic TM using cn
  space
• In complexity theory is thought to lie outside of
  NP. Recall that
• P ? NP ? NPSPACE
• The degree of complexity of context-sensitive
  languages is too high for practical applications.

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• On the other hand, the context-free languages
  (CFL) are not powerful enough to completely
  describe all the syntactical aspects of a
  programming language like PASCAL, since some of
  them are inherently context dependent.
• So, there are classes of languages that are
  strictly in between CFL and CSL.
• Can make CFGs more powerful or restrict the
  power of CSGs.

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Growing context-sensitive languages

• The start symbol occurs only in the left-hand
  side of a rule
• All rules are of the form that either the
  left-hand side consists of the start symbol or
  the right-hand side is strictly longer than the
  left-hand side
• Membership problem is NP-complete

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Closure Properties

• Closed under
• Union, Concatenation,
• Closed under Intersection
• But what about
• Complementation

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Closed under Complementation?

• Up until 1988, context-sensitive languages were
  not known to be closed under complementation.

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Complementation (continued)

• Show That NSPACE(n) co-NSPACE(n)
• This means that all problems in NSPACE(n) are in
  co-NSPACE(n) and vice versa which means NSPACE(n)
  is closed under complementation.
• It immediately follows that context-sensitive
  languages are closed under complementation.

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Decidability

• ALBA ltM,wgt M is an LBA and M accepts w
• Unlike ATM, ALBA is decidable.
• Proof
• The ID of an LBA (like a TM) consists of the
  current tape contents (wi), the current state
  (q), and the current head position. (w1q0w2w3w4)
• For a turing machine, there are infinitely many
  IDs
• However, for an LBA, there are a finite number.
  Precisely, there are nQ?n possible IDs
  where n is the length of the input string. ?n
  is the number of possible tape strings. Q is
  the number of possible states. And n is the
  number of head positions.

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• Recall that computation of a Turing Machine was
  defined as a chain of IDs
• ID0 ? ID1 ? ? IDk, where ID0 is an initial
  configuration
• If an ID appears twice, then the machine is in a
  loop.
• On input M,w, where M is an LBA and w is an
  input word,
• 1. Simulate machine M for at most nQ?n
  steps of computation.
• 2. If M accepted, accept. If M rejected, reject.
  Otherwise, M must be in a loop reject.

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Decidability (continued)

• Theorem
• ACSG ltG,wgt G is a CSG that accepts w is
  decidable.
• Theorem Every context-sensitive language is
  decidable
• Like context-free languages

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• Theorem ELBA ltMgt M is an LBA and L(M) ?
  
• is undecidable
• (This differs from context-free languages)

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Sources

• Brainerd, Walter S. and Lawrence Landweber.
  Theory of Computation. New York John Wiley
  Sons, 1974.
• Immerman, Neil. Nondeterministic Space is Closed
  Under Complementation. Yale University,
  http//ieeexplore.ieee.org/iel2/209/274/00005270.p
  df?isNumber274prodIEEE20CNFarnumber5270arSt
  112ared115arAuthorImmerman2CN.3B

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Questions?