International Scholarly Research Notices

Research Article

An Application of Iterative Pushdown Automata to Contour Words of Balls and Truncated Balls in Hyperbolic Tessellations

Automaton 3

The 2-pushdown automaton recognizing the contour word of a truncated white sector in the pentagrid or in the heptagrid.
  two states: π‘ž 0 and π‘ž 1 ; input word in { π‘Ÿ , 𝑠 , 𝑏 , 𝑀 } βˆ— ;
   Ξ“ = { 𝑍 , 𝐡 , π‘Š , 𝐡 𝑏 , π‘Š 𝑏 , π‘Š π‘Ÿ , 𝑋 , 𝐹 } ;
  initial state: π‘ž 0 ; initial stack: 𝑍 [ πœ– ] ; transition function 𝛿 :
   𝛿 ( π‘ž 0 , πœ– , 𝑍 ) = { ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 2 ( 𝐹 ) ) , ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( π‘Š π‘Ÿ ) ) }
   𝛿 ( π‘ž 0 , πœ– , 𝑍 𝐹 ) = { ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 2 ( 𝐹 𝐹 ) ) , ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( π‘Š π‘Ÿ ) ) }
   𝛿 ( π‘ž 0 , πœ– , π‘Š 𝐹 ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 2 ) ; 𝛿 ( π‘ž 0 , πœ– , 𝐡 𝐹 ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 2 )
   𝛿 ( π‘ž 0 , π‘Ÿ , π‘Š π‘Ÿ 𝐹 ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 2 ) ; 𝛿 ( π‘ž 0 , π‘Ÿ , π‘Š π‘Ÿ ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 1 )
   𝛿 ( π‘ž 0 , πœ– , π‘Š 𝑏 𝐹 ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 2 )
   𝛿 ( π‘ž 0 , 𝑠 , 𝐡 𝑏 𝐹 ) = ( π‘ž 1 , 𝑝 π‘œ 𝑝 2 ) ; 𝛿 ( π‘ž 0 , 𝑏 , 𝐡 𝑏 ) = ( π‘ž 0 , 𝑝 π‘œ 𝑝 1 ) ; 𝛿 ( π‘ž 0 , 𝑏 , 𝐡 ) = ( π‘ž 0 , 𝑝 π‘œ 𝑝 1 )
   𝛿 ( π‘ž 0 , 𝑀 , π‘Š ) = ( π‘ž 0 , 𝑝 π‘œ 𝑝 1 ) ; 𝛿 ( π‘ž 0 , 𝑀 , π‘Š 𝑏 ) = ( π‘ž 0 , 𝑝 π‘œ 𝑝 1 )
   𝛿 ( π‘ž 0 , 𝑠 , 𝑋 𝐹 ) = ( π‘ž 0 , 𝑝 π‘œ 𝑝 1 )
   𝛿 ( π‘ž 1 , πœ– , π‘Š π‘Ÿ 𝐹 ) = ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( 𝐡 𝑏 π‘Š π‘Š π‘Ÿ ) ) ; 𝛿 ( π‘ž 1 , πœ– , π‘Š 𝑏 𝐹 ) = ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( 𝐡 π‘Š π‘Š π‘Ÿ 𝑋 ) )
   𝛿 ( π‘ž 1 , πœ– , π‘Š 𝐹 ) = ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( 𝐡 π‘Š π‘Š ) )
   𝛿 ( π‘ž 1 , πœ– , 𝐡 𝑏 𝐹 ) = ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( 𝐡 𝑏 π‘Š ) ) ; 𝛿 ( π‘ž 1 , πœ– , 𝐡 𝐹 ) = ( π‘ž 0 , 𝑝 𝑒 𝑠 β„Ž 1 ( 𝐡 π‘Š ) )