Languages Generated by Programmed Grammars with Various Graphs

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Languages Generated by Programmed Grammars with
Various Graphs

• Jurgen Dassow, Benedek Nagy, Cristina Bibire,
  Madalina Barbaiani, Szilard Fazekas, Aidan
  Delaney, Mihai Ionescu, GuangWu Liu, Atif Lodhi

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• A programmed grammar is a quadruple
• G(N, T, S, P) where
• N, T and S are specified as in a context-free
  grammar and
• P is a finite set of triples r (p, s, f) where
  p is a context-free production and s and f are
  subsets of P (s and f are called the success
  field and the failure field, respectively).
• If the failure field of any rule in P is empty
  we say that G is a programmed grammar without
  appearance checking. Otherwise G is a programmed
  grammar with appearance checking.

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• We will consider another interpretation of the
  programmed grammars as graphs. From now on, we
  will consider only programmed grammars without
  appearance checking.
• Let G (N, T, S, P) be a programmed grammar.
  We can construct a graph H(P, E), where P is the
  set of rules from G and (r, r) is in E if and
  only if r is one of the rules from the success
  field of r.

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The corresponding graph is
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• For a particular class of restricted graphs,
  one can determine which class of languages can be
  generated with it.
• Types of graphs that we had in mind
• Complete graphs
• Planar graphs
• Connected graphs
• Eulerian graphs
• Hamiltonian graphs
• Bipartite graphs
• Ring backbones
• Graphs with limited degree (
  )

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Complete Graphs

• Theorem The language obtained by a programmed
  grammar in which the corresponding graph is a
  complete one is context-free.
• Proof We know that the rules in a programmed
  grammar are all context-free. A complete graph is
  simply telling us that each rule of the grammar
  can be followed (in the derivation) by any of the
  other rules. That reduces the power of the given
  programmed grammar to an arbitrary context-free
  grammar.

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Planar Graphs

• Theorem All languages obtained by a programmed
  grammar can be obtained by a programmed grammar
  using a planar graph.
• Proof Planar graphs are graphs which can be
  drawn in such a way that their edges do not
  intersect each other, so our task here is to
  avoid the intersections without altering the
  generated language.
• It suffices to consider a simpler case, with
  only four vertices. We will apply this algorithm
  to all the other intersections from the graph.

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A ? u
B ? v
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A ? u
B ? v
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A ? u
X ? ?
B ? v
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A ? u
B ? BX
X ? ?
A ? AX
B ? v
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A ? u
B ? BXA1
A2 ? ?
X ? ?
A1 ? ?
A ? AXA2
B ? v
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Connected Graphs

• Theorem Any language obtained by a programmed
  grammar can be obtained by a programmed grammar
  with a connected graph.
• Proof Consider the programmed grammar
  associated with the graph .
• We define
• - the union of all maximally
  connected subgraphs of containing at least
  one of the vertices from the set .

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• We introduce a new terminal symbol ,
  which will be the new start symbol for our
  grammar and we define
  .
• The grammar
  is a programmed grammar in which the associated
  graph is connected, and it generates the same
  language as

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Eulerian Graphs

• Theorem Any language obtained by a programmed
  grammar can be obtained by a programmed grammar
  with an eulerian graph.
• Proof Let be a programmed
  grammar and
• the associated graph. We denote the vertices
  by .
• A directed connected graph is eulerian
  if and only if for every vertex, the in-degree
  equals the out-degree.
• We will extend our graph to a new one
  by introducing the vertices
  where . The new
  non-terminals are meant to
  block the undesired
• derivations.
• Our goal is to connect the old nodes to the new
  one such that the in-degree for each node equals
  the out-degree.

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• Put the new edges in the following way
• For each connect the node to
  the node if and only if there was not an
  edge from to . In this way all vertices
  will have out-degree .
• For each connect the node to
  the node if and only if there was not an
  edge from to . In this way all vertices
  will have in-degree .
• In this way in our new graph, for all nodes
  , we have that the in-degree equals the
  out-degree.
• All we have to do now is to make sure that this
  property will hold also for each .

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• The following property is true for any directed
  graph
• But we know that
• From and we obtain
• Connect any node which has higher in-degree
  than out-degree to any node having higher
  out-degree than in-degree. In this way the number
  of in-degrees and out-degrees will increase both,
  and after finitely many steps (the difference for
  any node is not more than n) each node will
  have equal number of in- and out-degree.

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Hamiltonian Graphs

• Theorem Any language obtained by a programmed
  grammar can be obtained by a programmed grammar
  with an Hamiltonian graph.
• Proof Given a programmed grammar
  , let
• ( is the number of rules in
  ) be new non-terminal symbols which were not
  in .
• Let . We introduce
  new nodes
• and we connect each node to the node

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Other Results

• Any language obtained by a programmed grammar can
  be obtained by a programmed grammar with a
  bipartite graph.
• Any language obtained by a programmed grammar can
  be obtained by a programmed grammar with the
  graph having the nodes degree at most 3.

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Conclusions and Further Research

• Combining our constructive proofs one can
  generate any language obtained by a programmed
  grammar without appearance checking with a
  connected planar graph having nodes degree at
  most 3.
• There are some related open questions, for
  example what is the language family generated by
  programmed grammars such a way that there is a
  word which is generated by a Hamiltonian or
  Eulerian trail/cycle.

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Thank you!