Four-Dimensional Homogeneous Systolic Pyramid Automata

Cellular automaton is famous as a kind of the parallel automaton. Cellular automata were investigated not only in the viewpoint of formal language theory, but also in the viewpoint of pattern recognition. Cellular automata can be classified into some types. A systolic pyramid automata is also one parallel model of various cellular automata. A homogeneous systolic pyramid automaton with four-dimensional layers (4HSPA) is a pyramid stack of four-dimensional arrays of cells in which the bottom four-dimensional layer (level 0) has size an (a≥1), the next lowest 4(a-1), and so forth, the (a-1)st fourdimensional layer (level (a-1)) consisting of a single cell, called the root. Each cell means an identical finite-state machine. The input is accepted if and only if the root cell ever enters an accepting state. A 4-HSPA is said to be a real-time 4-HSPA if for every four-dimensional tape of size 4a (a≥1), it accepts the fourdimensional tape in time a-1. Moreover, a 1way fourdimensional cellular automaton (1-4CA) can be considered as a natural extension of the 1-way two-dimensional cellular automaton to four-dimension. The initial configuration is accepted if the last special cell reaches a final state. A 1-4CA is said to be a realtime 1-4CA if when started with fourdimensional array of cells in nonquiescent state, the special cell reaches a final state. In this paper, we proposed a homogeneous systolic automaton with four-dimensional layers (4-HSPA), and investigated some properties of real-time 4-HSPA. Specifically, we first investigated the relationship between the accepting powers of real-time 4-HSPA’s and real-time 1-4CA’s. We next showed the recognizability of four-dimensional connected tapes by real-time 4-HSPA’s.


Introduction
In recent years, due to the advances in many application areas such as dynamic image processing, computer animation, VR(virtual reality), AR (augmented reality), and so on, the study of four-dimensional pattern processing has been of crucial importance.And the question of whether processing four-dimensional digital P -670 patterns is much more difficult than three-dimensional ones is of great interest from the theoretical and practical standpoints.Thus, the study of four-dimensional automata as a computational model of four-dimensional pattern processing has been meaningful [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].On the other hand, cellular automata were investigated not only in the viewpoint of formal language theory, but also in the viewpoint of pattern recognition.Cellular automata can be classified into some types [2].A systolic pyramid automaton is also one parallel model of various cellular automata.In this paper, we propose a homogeneous systolic pyramid automaton with four-dimensional layers (4-HSPA) as a new four-dimensional parallel computational model, and investigate some properties of real-time 4-HSPA.
Let Σ be a finite set of symbols.A four-dimensional tape over Σ is a four-dimensional rectangular array of elements of Σ.The set of all four-dimensional tapes over ∑ is denoted by Σ (4) .Given a tape x ∈ Σ (4) , for each integer j (1≤ j ≤4), we let lj(x) be the length of x along the jth axis.The set of all x ∈ Σ (4) with l1(x) = n1, l2(x) = n2, l3(x) = n3, and l4(x) = n4 is denoted by Σ ( , , , ) .When 1≤ ij≤ lj(x) for each j(1≤ j ≤4), let x(i1, i2, i3, i4) denote the symbol in x with coordinates (i1, i2, i3, i4).Furthermore, we define when 1 ≤ ij ≤ i'j ≤ lj(x) for each integer j(1≤ j ≤4), as the four-dimensional input tape y satisfying the following conditions : (i)for each j(1≤ j ≤4), lj(y)= ij -ij+1; (ii)for each r1, r2, r3, r4 For each x ∈ Σ ( , , , ) and for each 1  A four-dimensional homogeneous systolic pyramid automaton (4-HSPA) is a pyramidal stack of fourdimensional arrays of cells in which the bottom fourdimensional layer (level 0) has size 4a (a ≥ 1), the next lowest 4(a − 1), and so forth, the (a−1)st fourdimensional layer (level (a − 1)) consisting of a single cell, called the root .Each cell means an identical finitestate machine, M = (Q, Σ, , #, F), where Q is a finite set of states, Σ ⊆ Q is a finite set of input states, # ∈ −Σ is the quiescent state, F ⊆ Q is the set of accepting states, and : 17 → Q is the state transition function, mapping the current states of M and its 16 son cells in a 2 × 2 × 2× 2 block on the four-dimensional layer below into M's next state.The input is accepted if and only if the root cell ever enters an accepting state.A 4-HSPA is said to be a real-time 4-HSPA if for every four-dimensional tape of size 4a (a ≥ 1) it accepts the four-dimensional tape in time a − 1.By £ R [4-HSPA] we denote the class of the sets of all the four-dimensional tapes accepted by a real-time 4-HSPA [1].A 1-way four-dimensional cellular automaton (1-4CA) can be considered as a natural extension of the 1-way two-dimensional cellular automaton to four dimensions [3].The initial configuration of the cellular automaton is taken to be an l1(x) × l2(x) ×l3(x) × l4(x) array of cells in the nonquiescent state.The initial configuration is accepted if the last special cell reaches a final state.A 1-4CA is said to be a real-time 1-4CA if when started with an l1(x) × l2(x) ×l3(x) × l4(x) array of cells in the nonquiescent state, the special cell reaches a final state in time l1(x) + l2(x) + l3(x) × l4(x) − 1.By £R[1-4CA] we denote the class of the sets of all the four-dimensional tapes accepted by a real-time 1-4CA [3].
[Proof : Suppose to the contrary that ( = ( ).We consider two tapes x', y' ∈ W(t) satisfying the following : (i) 1st cube of x and nth cube of x are equal to 1st cube of x', respectively (ii) 1st cube of y' is equal to 1st cube of y, and nth cube of y′ is equal to 1st cube of x.
As is easily seen, x′ ∈ V and so x′ is accepted by M. On the other hand, from Proposition 2.1(ii), ( = ( ′).From Proposition 2.1(i), ( = ( ( ) = ( ′).It follows that y′ must be also accepted by M. This contradicts the fact that y′ is not in V .□] Proof of Theorem 2.1 (continued) : Let p(t) be the number of tapes in W(t) whose 1st cubes are different, and let where k is the number of states of each cell of M.Then, p(t) =2 , and Q(t) ≤ k 4 .It follows that p(n) > Q(t) for large t.Therefore, it follows that for large t, there must be two tapes x,y in W(t) such that their 1st cubes are different and ( = ( ).This contradicts Proposition 2.2, so we can conclude that V ∉ £ R [4-HSPA].In the case of four-dimension, we can show that V ∉ £ R [4-HSPA] by using the same technique.This completes the proof of Theorem 2.1.□

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We next show the recognizability of four-dimensional connected tapes by real-time 4-HSPA's by using the same technique of Ref. [3].Let x in {0,1} (4) .A maximal subset P of N 4 satisfying the following conditions is called a 1component of x.
Let Tc be the set of all the four-dimensional connected tapes.Then, we have Theorem 2.2.Tc ∉ £ R [4-HSPA].

Conclusion
The technique of AR (augmented reality) or VR(virtual reality) progresses like the Pokemon GO and the Virtual Cinema in the world.The virtual technique will spread steadily among our societies in future.Thus, we think that the study of four-dimensional automata is very meaningful as a computational model of fourdimensional pattern processing.In this paper, we proposed a homogeneous systolic pyramid automaton with four-dimensional layers, and investigated a relationship between the accepting powers of homogeneous systolic pyramid automaton with fourdimensional layers(4-HSPA) and one-way fourdimensional cellular automata (1-4CA) in real time, and showed that real-time 4-HSPA's are less powerful than real time 1-4CA's.
It will be interesting to investigate about an alternating version or synchronized alternating version of homogeneous systolic pyramid automaton with fourdimensional layers.Moreover, we think that there are many interesting open problems for digital geometry.Among them, the problem of connectedness, especially topological component is one of the most interesting topics [17].
Finally, we would like to hope that some unsolved problems concerning this paper will be explicated in the near future.