COMBINING COMPATIBLE STATES DURING LR(1) PARSER CONSTRUCTION

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COMBINING COMPATIBLE STATES DURING LR(1) PARSER
CONSTRUCTION
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• The LR(0) algorithm for creating compilers is
  one in which contexts are not evaluated, and
  states are considered identical if they consist
  of the same set of marked productions

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• But this algorithm is insufficient for actual
  programming languages, producing parsers with
  numerous conflicts

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• The LR(1) algorithm when applied to creating
  compilers for real computer languages, such as
  those for Java or C, results in a parsing
  machine that is a order or more larger than those
  produced by an LR(0) algorithm for the same
  grammar.

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• On the other hand the LR(1) algorithm, which
  you made use of in your last assignment, produces
  parsers, for the large grammars employed for
  actual computer languages, which are a few orders
  larger than those produced by the LR(0)
  algorithm.

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• As a compromise, various methods, including
  the one employed by Yacc, have been devised for
  subsets of the LR(1) languages, using a hybrid
  approach.

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• This works well for most programming languages,
  but imposes a greater responsibility on the
  compiler writer, to come up with a grammar that
  does not lead to conflicts (i.e. to cases where
  more than one action is defined at a parsing
  machine state for the same next input symbol).
• These methods only work for a subset of the
  LR(1) grammars, and there are applications,
  including ones involving natural language
  processing, for which they are inadequate.

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• However one can employ a definition of
  compatibility between states, which works for all
  LR(1) languages, and which produces parsers of
  the same size as those referred to previously

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• DEFINITION. The nucleus of state consists of the
  configurations in the state in which the marker
  is in a position greater that zero.
• Example
• A configuration in a state of the form
• A ? bc.d, x,y
• would be a member of its nucleus, but a
  configuration such as
• A ? .bcd, x,y
• would not be a member.

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• DEFINITION OF COMPATIBILITY BETWEEN LR(1)
  STATES
• Let S and S? be two states in a LR(1) parsing
  machine whose nuclei consist of the same marked
  productions, which we will denote as P1,,Pn .
• For 1 t n, let Ut denote the set of
  contexts associated with marked production Pt in
  state S, and let Ut? denote the set of contexts
  associated with that marked production in state
  S?.
• Then states S and S? are compatible if, for
  all 1 i lt j n, at least one of the following
  condition holds
• (a) Ui ? Uj? ? and Ui? ? Uj ?
• (? is the empty set, i.e. the
  intersections involved are both empty)
• (b) Ui ? Uj ? ?
• (c) Ui? ? Uj? ? ?

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• Note
• If states S and S? are as described above, and
  their nuclei consist of only a single
  configuration, then according to the above
  definition they are compatible

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• In the case where S and S? as described above are
  compatible, one can combine the states into a
  single state whose nucleus consists of the same
  marked productions listed above, while for 1 t
  n, the set of contexts associated with marked
  production Pt is Ut ? Ut? .

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• One way of looking at the definition is to say
  that every pair of configurations in the nuclei
  must pass a test, and that two states are
  compatible only if they all in fact pass.

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• Fortunately, in grammars for actual
  programming languages such as Java, C, etc.,
  there are at most 6 configurations in the nucleus
  of any state.
• The states may be large, with many immediate
  successors, but the nuclei are all quite small.

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• EXAMPLES
• We show only the nucleus of the states in
  these examples, since, according to the
  definition, states are compatible if and only if
  their nuclei are.

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• S
  S
• The above two states are not compatible
• because the pair consisting of the first and
  last configurations fail the test.
• For this pair condition (a) of the defn. is
  not true, since the context of the first
  configuration of S contains an x, and so does the
  context of the third production of S
• In addition neither of conditions (b) or (c)
  are true.

A ? ab.c x,y B ? b.n s,t C ? rb.ed
u,v
A ? ab.c d B ? b.n s C ? rb.ed
x,v

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• S
  S
• The first and third configurations in this
  case pass the test because condition (b) of the
  defn. applies to the first and third
  configurations of S. Both of these
  configurations contain x in their set of
  contexts. The states in this case are
  compatible.
• Remember, that while every pair of
  configurations in the nucleus must pass the test,
  it only requires that one of conditions (a), (b)
  or (c) be true for a given pair for it to pass.

A ? ab.c x,y B ? b.n s,t C ? rb.ed
x,v
A ? ab.c x,y,d B ? b.n s C ? rb.ed
x,v

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• Since the states are compatible, they can be
  combined to form one whose nucleus is

A ? ab.c x,y,d B ? b.n s,t C ?
rb.ed x,u,v
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• Note.
• In the figure on the next slide, where we omit
  the context set of various configurations (i.e.
  only show the marked production involved), the
  inference involved is that they are irrelevant to
  the assertions being made about the figure.

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States 2 and 8 are not compatible since the first
configuration of state 2 has d as context in
common with the second configuration of state 8.
In fact if we were to combine states 2 and 8, it
would produce a combination of states 3 and 9 as
its u-successor. This state would have a
conflict, in that in had reduce actions, for when
the next input symbol was d, for both Z ? tu and
V ? ?
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• Now consider the altered machine obtained if the
  production X ? aYd where replaced by (say) X ?
  aYa. In this case the first configuration of
  state 2 would be Y ? t.W a. It would then
  follow that states 2 and 8 were compatible and
  could safely be combined to form
• Y ? t.W a, e.
• Z ? t.u c, d
• W ? .uV

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• The Journal paper describing this method of
  combining states contains a formal proof of its
  correctness. But seeing ours is a practically
  oriented course, we will just consider an
  informal justification based on a few examples to
  supply a flavor of the reasoning involved

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• The main argument is that if the parsing machine
  containing the states S and S?, as described in
  the defn. of compatibility, has no conflicts, and
  S and S? are compatible, then the parsing machine
  obtained by combining them will also have no
  conflicts.

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• The argument is by contradiction. Lets consider
  examples of the various ways that two
  configurations in the combination of S and S?
  could have conflicts or lead to conflicts between
  other pairs of configurations in states reachable
  from S. In each case we hope to show that either
  the parsing machine as it was before S and S?
  were combined contained conflicts in the first
  place or that S and S? could not in fact have
  been compatible.

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• Case 1. Let configs 1 and 2 of the combined
  state formed from
• states S and S be
• A ? r B.uv a,b
• C ? t B.uv a,c
• Seeing that the machine as it was before the
• combination contained no conflicts, and
  specifically did not
• contain a conflict in the uv successor of these
  states, either
• state S must have contained the a in its
  version of config1, while state S? contained the
  a in its version of config 2, or
• vice-versa.

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• Case 1 contd.
• A ? r B.uv a,b
• C ? t B.uv a,c
• In either case neither condition (a) nor (b) of
  the defn.
• would then be true for the two configs, and
  since
• condition (c) is also not true, states S and S
  could not
• have been compatible in the first place.

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• Case 2. Let configs 1 and 2 of the combined
  state be
• A ? r B.uv a,b
• D ? t B.Ca
• C ?.uv a
• Either S or S? must contain A ? r B.uv a.. ,
• in which case the original parsing machine would
  have had a conflict at its uv-successor. This is
  in contradiction to our assumption that the
  original parsing machine was conflict-free.

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• Case 3. Let configs 1 and 2 be
• A ? s B.Ea
• E ?.uv a
• D ? t B.Ca
• C ?.uv a
• Here again the original parsing machine would
  have had conflicts in the uv-successors of both S
  and S?

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• Case 4. Let configs 1 and 2 be
• A ? r B.uv
• D ? t B.uvr
• Here too the original parsing machine would have
  had conflicts in the uv-successors of both S and
  S?. In this case the conflict would have been
  between a reduction and a transition.

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• EXERCISE
• Construct an LR(1) parsing machine for the
• grammar on the next slide, combining
  compatible states as you encounter them

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• program ? main statement_list end main
• statement_list ? statement_list statement
• statement
• statement ? assign_statement
• while_statement
• do_statement
• assign_statement ? identifier identifier
• while_statement ? while ( condition )
• 
  statement_list wend
• condition ? identifier identifier
• do_statement ? do identifier number to
• number
  statement_list end do