t = 0 t = 8 (2 cycles)

Transcription

1 A omputation-universal two-dimensional 8-state triangular reversible ellular automaton Katsunobu Imai, Kenihi Morita Faulty of Engineering, Hiroshima University, Higashi-Hiroshima , Japan fimai, Abstrat A reversible ellular automaton (RCA) is a ellular automaton (CA) whose global funtion is injetive and every onguration has at most one predeessor. Margolus showed that there is a omputation-universal two-dimensional 2-state RCA. But his RCA has nonuniform neighbor, so Morita and Ueno proposed 16-state omputation-universal RCA using partitioned ellular automata (PCA). Beause PCA an be regarded as a sublass of standard CA, their models has standard neighbor. In this paper, we show that the number of states of Morita and Ueno's models an be reduible. To derease the number of states from their models with preserving isotropi and bit-preserving properties, we usedtriangular 3- neighbor, and thus 8-state RCA an be possible. This is the smallest state two-dimensional RCA under the ondition of isotropi property on the framework of PCA. We show that our model an simulate basi iruit elements suh as unit wires, delay elements, rossing wires, swith gates and inverse swith gates. And it is possible to onstrut a Fredkin gate by ombining these elements. Sine Fredkin gate is known to be a universal logi gate, our model has omputation-universality. Keywords: ellular automata, reversibility, omputation-universality, onservative logi. 1

2 1 Introdution A reversible ellular automaton (RCA) is a ellular automaton (CA) whose global funtion is injetive and every onguration has at most one predeessor. Reversibility is a very strong onstraint, but omputation-universality ofrca has been showed [2]. On two-dimensional CA, their omputation-universality an be proved by embedding universal logi elements. For example, omputation-universality of the game of life was proved by onstruting AND, OR, NOT, and fan-out gates on its ellular spae. Sequenes of glider patterns was used as signal arriers [1]. Using this approah, one an onstrut small state omputationuniversal CA. But on RCA, erasing informations are inhibited and suh irreversible logi gates an not be embedded diretly. Margolus proposed a two-dimensional omputation-universal RCA (BBMCA) [4]. He realized its universality by embedding Fredkin and Tooli's Billiard Ball Model (BBM). BBM is a omputing model in whih logial operations are performed by elasti ollisions of balls. They showed that 3-input, 3-output reversible and bit-preserving Fredkin gate (F-gate) an be embedded in their BBM, and ombining F-gates and unit delays, any logi iruits an be onstruted using BBM. Although BBMCA is simple RCA, it has non-uniform neighbor. So Morita and Ueno proposed a dierent type of 4-neighbor 16-state omputation-universal RCA [5]. They used a partitioned ellular automaton (PCA). It is regarded as a sublass of standard CA. In PCA, the injetivity of a global funtion is equivalent to the injetivity of loal funtion and it makes ease of onstruting RCA [6]. In this paper, we showthatthenumber of states of Morita and Ueno's models an be redued. To derease the number of states from their models with preserving isotropi and bit-preserving properties, we used triangular 3-neighbor, and thus 8-state RCA an be possible. This is the smallest state two-dimensional RCA under the ondition of isotropi property on the framework of PCA. 2 Computation-universal RCA and BBM 2.1 Fredkin Gate On BBMCA and 16-state RPCA models, BBM is used for showing their omputation-universality. First we make a brief desription about BBM and a Fredkin gate. A Fredkin gate (F-gate) is a basi element in the theory of Conservative Logi proposed by Fredkin and Tooli [3]. It is reversible and bit-preserving logi gate (Fig. 1). They showed that AND, OR, NOT, and fan-out gate an be onstruted by an F-gate and any iruits an be onstruted by F-gates and unit delays. Figure 1: A Fredkin gate They also introdued a swith gate (S-gate) and its inverse gate (Fig. 2). An S-gate is a 2-input, 3-output reversible and bit-preserving logi gate. An S-gate swithes the input x by 2

3 the ontrol signal. They showed that S-gates an be onstruted on their Billiard Ball Model (BBM) and using two S-gates and two inverse S-gates, it is possible to onstrut an F-gate (Fig. 3). x y x y+z x z Figure 2: An S-gate and an inverse S-gate Figure 3: A realization of a Fredkin gate by S-gates and inverse S-gates 2.2 Computation-universal RPCA Though BBMCA uses a non-uniform neighborhood, Morita and Ueno onstruted 16-State twodimensional omputation-universal RCA using the framework of partitioned ellular automata (PCA) [5]. PCA is regarded as the sublass of standard CA. Eah ell is partitioned into the equal number of parts to the neighborhood size and the information stored in eah part is sent to only one of the neighboring ells. In PCA, injetivity of global funtion is equivalent to injetivity ofloal funtion, thus a PCA is reversible if its loal funtion is injetive [6]. Their models used 4-neighbor PCA and g. 4 shows its domain and range of the loal funtion. Figure 4: Domain and range of loal funtion in 2D 4-neighbor PCA They proposed two models and g. 5 is the loal funtion of one of their models. Their RPCA has following properties. (i) bit-preserving: the number of \1" ells (i.e. blak ells) on both side of the transition rules is the same. (ii) isotropi: the loal funtion is invariant under the rotation of 90, 180, 270 degrees(90-degree isotropi). 3

4 Figure 5: The loal funtion of 2D 16-state 4-neighbor RPCA They proved its omputation-universality by onstruting S-gates, Inverse S-gates and F-gates on its ellular spae. 3 Two-dimensional triangular RPCA state triangular RPCA To derease the number of states from Morita and Ueno's model with preserving bit-preserving and isotropi properties, we used triangular 3-neighbor, and thus 8-state PCA an be possible. Fig. 6 shows domain and range of a loal funtion. This is the smallest state two-dimensional PCA under the ondition of isotropi property. Figure 6: Domain and range of loal funtion in 3-neighbor triangular PCA There are nine bit-onserving and 120-degree isotropi loal funtion by ombining four rules out of eight rules depited in g. 7. And there are ve dierent loal funtions exluding symmetri ases. Figure 7: Isotropi transition rules in 3-neighbor triangular PCA 4

5 A omputation-universal 8-state model We show that the RPCA whih loal funtion is given by g. 8 is omputation universal. Figure 8: The loal funtion of 3-neighbor triangular RPCA (a) (b) Figure 9: Stable bloks First, we onstrut signal transmission wires. On the proof of omputation-universality of the game of life, gliders were used to enode signals [1]. In the same way, BBMCA [4] and 16-state RPCA models [5] used \ball"s. They propagate on a quiesent ellular spae as signal arriers. Although on this model, it is diult to onstrut simple patterns propagating on a quiesent ellular spae, there are two simple stable bloks shown in g. 9, and ombining (b)-type bloks, it is possible to onstrut signal transmission wires (Fig. 10). Ablok as a signal arrier is shown in gray olor. The gray blok t = 0 t = 8 (2 yles) Figure 10: A data transmission wire takes 4 steps (regarded as 1 yle) to move to the next dent (the part shown as the number 4). A 4 steps (1 yle) delay element is shown in g. 11 (a) This wire has a ave and signals take 4 steps in travelling through it. This delay element an be used as synhronization element for hanging olumns/rows of transmission wires. 5

6 The transmission wire depited in g. 11 (b) hanges its row by two lines. The wire ontains one right turning and one left turning, so signals are propagating on the wire with stritly 4 step (1 yle) delays. Then if the straight wire ontains a delay element (Fig. 11 (a)), both signals an be synhronized Figure 11: A delay element When a iruit ontains feedbak loop like g. 12, the transmission wire turns right/left six times, and thus output signal phase of the feedbak wire diers from that of input signal by 2 steps(0:5 yles). But delay element of 2 steps an be onstruted by the rossing of two some logi elements Figure 12: A feedbak wire signals. Fig. 13(a) is a ;2 steps phase shifter. The stable star shaped blok above the horizonal wire has a \n", and it turns around the 6 branhes of the blok in 30 steps. When this n rosses the arriving signal along the wire, it advanes the phase of the signal 2 steps. So this ;2 steps phase shifter aepts signals every 30 steps. And ombining a delay element, +2 steps shifter an be also available (Fig. 13(b) ). This model uses onneted stable bloks for transmission wires and we need speial struture to realize rossing wires. Fig. 14 shows a module for rossing wires. Rotating \n"s swith signals from two input wires to eah output wires. This rossing element aepts signals every 30 steps We showed that it an be possible to realize signal transmission wires, delay elements, phase shifters and rossing wires. Next we show the funtion of S-gates and inverse S-gates an be available. 6

7 Figure 13: Phase shifters (a) ;2, (b) +2 Figure 14: A rossing element A ell an be regarded as a 3-input, 3-output gate. Although an F-gate is also a 3-input, 3- output gate, it annot be possible to realize under the isotropi ondition. But S-gate and inverse S-gate an be onstruted by isotropi ondition. Fig. 15 shows the input/output relations of an S-gate and an inverse S-gate. Fig. 16 shows an S-gate with synhronizing phase shifters. This element takes 40 steps (10 yles) to generate output signals and input must be given every 30 steps. Although all input/output signals in g. 16 are synhronized, it does not t the denition of the S-gate in g. 2. Beause the output terminal of signal \" is plaed in the opposite side of this element. Then we show a omplete pattern of an S-gate in g. 17. In this pattern, an output signal \" travells along a feedbak wire and a rossing element, and it has a long delay against other output signals. so the right part of this patters are used to synhronize all output signals. This element takes 140 steps to generate output signals and inputs must be given every 30 steps. Beause an S-gate and an inverse S-gate are symmetri, an inverse S-gate an be onstruted by reeting a pattern of this S-gate with simple modiations of rotating \n"s. It is shown in g. 18. Combining iruit elements mentioned above, an F-gate is onstruted by g. 3 (g. 19). It takes 704 steps (176 yles) to simulate an F-gate. 4 Conlusion In this paper, we have onstruted an 8-state triangular RPCA whih has omputation universality. It is the smallest state RPCA with bit-preserving and isotropi loal funtions. It is an 7

8 Figure 15: Constrution of S-gate and inverse S-gate x x x Figure 16: An S-gate with phase shifter open problem to nd other 8-state universal RPCA. Referenes [1] Berlekamp, E., Conway, J., Guy, R.: Winning Ways for Your Mathematial Plays, Vol. 2, Aademi Press, New York (1982). [2] Tooli, T.: Computation and onstrution universality of reversible ellular automata, J. Comput. Syst. Si., 15 (1977) 213{231. [3] Fredkin, E., Tooli, T.: Conservative logi, Int. J. Theoretial Physis, 21, 3/4(1982) 219{ 253. [4] Margolus, N.: Physis-like models of omputation, Physia, D10 (1984) 81{95. [5] Morita, K., Ueno, S: Computation-universal models of two-dimensional 16-state reversible ellular automata, IEICE Trans. Inf. & Syst., E75-D (1992) 141{147. [6] Morita, K.,Harao, M.: Computation universality of one-dimensional reversible (injetive) ellular automata, Trans. IEICE, E72 (1989) 758{762. 8

9 Figure 17: A onguration of S-gate Figure 18: A onguration of inverse S-gate 9

10 Figure 19: A onguration of F-gate 10