International Scholarly Research Notices

Research Article

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

Automaton 2

The 2-pushdown automaton recognizing the contour word of a ball in the pentagrid or in the heptagrid.
two states: 𝑞 0 and 𝑞 1 ; input word in { 𝑏 , 𝑤 } ∗ ; Γ = { 𝑍 , 𝐵 , 𝑊 , 𝐹 } ;
initial state: 𝑞 0 ; initial stack: 𝑍 [ 𝜖 ] ; transition function 𝛿 :
    𝛿 ( 𝑞 0 , 𝜖 , 𝑍 ) = { ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 2 ( 𝐹 ) ) , ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑊 𝛼 ) ) }
    𝛿 ( 𝑞 0 , 𝜖 , 𝑍 𝐹 ) = { ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 2 ( 𝐹 𝐹 ) ) , ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑊 𝛼 ) ) }
    𝛿 ( 𝑞 0 , 𝜖 , 𝑊 𝐹 ) = ( 𝑞 1 , 𝑝 𝑜 𝑝 2 )
    𝛿 ( 𝑞 0 , 𝜖 , 𝐵 𝐹 ) = ( 𝑞 1 , 𝑝 𝑜 𝑝 2 )
    𝛿 ( 𝑞 0 , 𝑏 , 𝐵 ) = ( 𝑞 0 , 𝑝 𝑜 𝑝 1 )
    𝛿 ( 𝑞 0 , 𝑤 , 𝑊 ) = ( 𝑞 0 , 𝑝 𝑜 𝑝 1 )
    𝛿 ( 𝑞 1 , 𝜖 , 𝑊 𝐹 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝐵 𝑊 𝑊 ) )
    𝛿 ( 𝑞 1 , 𝜖 , 𝐵 𝐹 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝐵 𝑊 ) )
    𝛿 ( 𝑞 1 , 𝜖 , 𝑊 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝐵 𝑊 𝑊 ) )
    𝛿 ( 𝑞 1 , 𝜖 , 𝐵 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝐵 𝑊 ) )