International Scholarly Research Notices

Research Article

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

Automaton 1

The 2-pushdown automaton recognizing the Fibonacci sequence.
three states: 𝑞 0 , 𝑞 1 and 𝑞 2 ; input word in { 𝑎 } ∗ ; Γ = { 𝑍 , X 1 , 𝑋 2 , 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 ( 𝑋 2 ) ) }
    𝛿 ( 𝑞 0 , 𝜖 , 𝑍 𝐹 ) = { ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 2 ( 𝐹 𝐹 ) ) , ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑋 2 ) ) }
    𝛿 ( 𝑞 0 , 𝜖 , 𝑋 1 𝐹 ) = ( 𝑞 1 , 𝑝 𝑜 𝑝 2 )
    𝛿 ( 𝑞 0 , 𝜖 , 𝑋 2 𝐹 ) = ( 𝑞 2 , 𝑝 𝑜 𝑝 2 )
    𝛿 ( 𝑞 0 , 𝑎 , 𝑋 1 ) = ( 𝑞 0 , 𝑝 𝑜 𝑝 1 )
    𝛿 ( 𝑞 0 , 𝑎 , 𝑋 2 ) = ( 𝑞 0 , 𝑝 𝑜 𝑝 1 )
    𝛿 ( 𝑞 1 , 𝜖 , 𝑋 1 𝐹 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑋 1 𝑋 2 ) )
    𝛿 ( 𝑞 2 , 𝜖 , 𝑋 2 𝐹 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑋 1 ) )
    𝛿 ( 𝑞 1 , 𝜖 , 𝑋 1 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑋 1 𝑋 2 ) )
    𝛿 ( 𝑞 2 , 𝜖 , 𝑋 2 ) = ( 𝑞 0 , 𝑝 𝑢 𝑠 ℎ 1 ( 𝑋 1 ) )