Universality of RTMs. Theorem [Bennett, 1973]. For any one-tape (irreversible) TM T, there is a garbage-less 3-tape reversible TM which simulates the former.
Universality Issues in Reversible Computing Systems and Cellular Automata Kenichi Morita Hiroshima University
Contents 1. 2. 3. 4.
Introduction Reversible Turing machines (RTMs) Reversible logic elements and circuits Reversible cellular automata (RCAs)
Even very simple reversible systems have universal computing ability!
1. Introduction
Reversible Computing • Roughly speaking, it is a “backward deterministic” computing; i.e., every computational configuration has at most one predecessor.
···
-
-
-
-
···
↑ Computational configuration
• Though its definition is rather simple, it reflects physical reversibility well.
Reversible Computing • Roughly speaking, it is a “backward deterministic” computing; i.e., every computational configuration has at most one predecessor. No such configuration exists
···
-
j -
-
-
···
↑ Computational configuration
• Though its definition is rather simple, it reflects physical reversibility well.
Several Models of Reversible Computing • • • • •
Reversible Turing machines (RTMs) Reversible logic elements and circuits Reversible cellular automata (RCAs) Reversible counter machines (RCMs) Others —————————————————— • These models are closely related each other. • Reversible computers work in a very different fashion from classical computers!
2. Reversible Turing Machines
Reversible Turing Machines (RTMs) A “backward deterministic” TM.
· · · · s · · · · · 6
q
Definition of a TM T = (Q, S, q0, qf , s0, δ) Q: S: q0 : qf : s0 : δ:
a finite set of states. a finite set of tape symbols. an initial state q0 ∈ Q. a final state qf ∈ Q. a blank symbol s0 ∈ S. a move relation given by a set of quintuples [p, s, s, d, q] ∈ Q× S × S × {−, 0, +} × Q.
Definition of an RTM A TM T = (Q, S, q0, qf , s0, δ) is called reversible iff the following condition holds for any pair of distinct quintuples [p1, s1, s1, d1, q1] and [p2, s2, s2, d2, q2]. If q1 = q2, then s1 = s2 ∧ d1 = d2 (If the next states are the same, then the written symbols must be different and the shift directions must be the same.)
Universality of RTMs Theorem [Bennett, 1973] For any one-tape (irreversible) TM T , there is a garbage-less 3-tape reversible TM which simulates the former.
A Small Universal RTM (URTM) A URTM is an RTM that can compute any recursive function. Theorem The following URTMs exist: 17-state 5-symbol URTM [Morita and Yamaguchi, 2007] 15-state 6-symbol URTM [Morita, 2008]
A Small Universal RTM (URTM) A URTM is an RTM that can compute any recursive function. Theorem The following URTMs exist: 17-state 5-symbol URTM [Morita and Yamaguchi, 2007] 15-state 6-symbol URTM [Morita, 2008]
These URTMs can simulate any cyclic tag system [Cook, 2004], which is proved to be universal.
Cyclic Tag System (CTAG) [Cook, 2004] C = (k, {Y, N }, (halt, p1, · · · , pk−1)) • k: the length of a cycle (positive integer). • {Y, N }: the alphabet used in a CTAG. • (p1, · · · , pk−1) ∈ ({Y, N }∗)k−1 : production rules. An instantaneous description (ID) is a pair (v, i), where v ∈ {Y, N }∗ and i ∈ {0,· · · ,k−1}. For any (v, i), (w, j) ∈ {Y, N }∗ × {0, · · · , k − 1}, (Y v, i) ⇒ (w, j) iff
[m = 0] ∧ [j = i + 1 mod k] ∧[w = vpi], (N v, i) ⇒ (w, j) iff [j = i + 1 mod k] ∧ [w = v].
A Simple Example of a CTAG System C1 = (3, {Y, N }, (halt, Y N , Y Y )) If an initial word N Y Y is given, the computing on C1 proceeds as follows: (N Y Y , ⇒( Y Y , ⇒( Y Y N , ⇒( Y N Y Y ,
0 1 2 0
) ) ) )
The quintuple set of the URTM(17,5) q0 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16
b $ − q2 halt ∗ − q3 b + q12 Y + q5 b − q6 Y − q3 N + q8 b − q9 N − q3
∗ − q14 b + q16 N + q0
Y $ − q1 Y − q1 Y − q2 b + q4 Y + q4
N b − q13 N − q1 N − q2 b + q7 N + q4
∗
$
∗ + q0 ∗ − q2 b + q10 ∗ + q4
b − q1 null $ + q4
Y − q6 Y + q7
N − q6 N + q7
∗ − q6 ∗ + q7
$ − q6 $ + q7
Y − q9 Y + q10 Y + q11 Y + q12 Y − q13 Y − q14 Y + q15 Y + q16
N − q9 N + q10 N + q11 N + q12 N − q13 N − q14 N + q15 N + q16
∗ − q9 ∗ + q10 ∗ + q11 ∗ + q12 ∗ − q13 b + q15 ∗ + q15 ∗ + q16
$ − q9 $ + q11 Y + q0 $ − q3 $ − q13 $ + q15 $ − q14
Simulating the CTAG C1 by the URTM(17,5) t=0 The rules of the CTAG C1
A given string
b Y Y * N Y * b $ N Y Y b b b b b 6
q0 t=6 b Y Y * N Y b * $ b Y Y b b b b b 6
q15 t = 59 b Y Y b N Y * * $ N $ Y Y N b b b 6
q11 t = 142 b Y Y * N Y * b $ N Y $ Y N Y Y b 6
q11 The final string t = 148 b Y Y * N Y * b b N Y Y $ N Y Y b 6
q1
Small UTMs and URTMs Symbols 6
•UTM(2,18)[Rogozhin,1996]
•UTM(3,9)[Kudlek, Rogozhin, 2002] •UTM(4,6)[Rogozhin,1996] •URTM(15,6)[Morita, 2008] •UTM(5,5)[Rogozhin,1996] •URTM(17,5)[Morita, Yamaguchi, 2007] •UTM(6,4)[Neary, Woods,2007] •UTM(9,3)[Neary, Woods,2007] •UTM(18,2)[Neary, Woods,2007] -
States
3. Reversible Logic Elements
Reversible Logic Element A logic element whose function is described by a one-to-one mapping. (1) Reversible logic elements without memory (i.e., reversible logic gates): • Toffoli gate [Toffoli, 1980] • Fredkin gate [Fredkin and Toffoli, 1982] • etc. (2) Reversible logic elements with memory: • Rotary element (RE) [Morita, 2001] • etc.
Rotary element (RE) A 2-state 4-input-line 4-output-line element.
H-state n n
? 6
V-state n n
? 6
e w - e w -
w w-
? 6
s
s
e - e
? 6
s s
(Remark) We assume signal “1” is given at most one input line.
Operations of an RE • Parallel case: t=0 t=1 ? 6
? 6
-
-
-
--
? 6
? 6
• Orthogonal case: t=0
t=1
? 6
? 6
-
-
? 6
6-
? ? 6
Logical Universality of a Rotary Element A Fredkin gate can be composed of REs and delay elements. 10
?
p
-
q
5
c
1
1
?
?
-
-
6 ?
-
? -
3
6 ?
-
3
-
6
?
6 ?
-y
-c
3
5
-x
3
(Remark) But, this is not a good method to use REs.
Any Reversible Turing Machine Can Be Composed Only of REs [Morita, 2001] ?
-
?
-
?
6
6
6
6 -
6 6
6
6
6
6
6 6
6
-
?
?
?
-
6 ?
?
6 -
6 ?
-
-
-
?
-
6
6
-
6
-
?
-
?
?
-
?
-
-
?
6
? 6
6
-
?
6
6
-
?
6
?
? 6
6
-
?
6
6 -
?
6 -
-
6 6
? 6
-
-
-
?
?
-
6 ?
6
-
6
-
-
-
-
6
-
6
?
6
?
?
?
-
-
-
-
6
-
• • •
-
6
?
6
6 ?
-
6
6
?
-
6
-
?
6 ?
?
-
6
-
6 ? -
?
-
-
?
6
-
6
6
?
-
-
6
6
?
6
6 -
6
6 6 6
6
?
6
?
6
-
?
-
-
? -
6
?
6
6
-
-
-
6
6 ?
-
-
?
6 -
6
-
6
6
?
6
?
-
6
-
?
-
6
q0
6
?
6
?
-
? -
6 ?
•-
-
6
Accept
Begin
6
-
Reject
?
0 1 0 · ·
? 6
? 6
-
A Simple Example of an RTM Tparity Tparity = (Q, {0, 1}, q0, qacc, 0, δ) Q = {q0, q1, q2, qacc, qrej} δ = {[ [ [ [ [
q0, 0, 1, R, q1 ], q1, 0, 1, N, qacc ], q1, 1, 0, R, q2 ], q2, 0, 1, N, qrej ], q2, 1, 0, R, q1 ] }.
A Simple Example of an RTM Tparity
t=0 0 1 1 0 6
q0
A Simple Example of an RTM Tparity
t=1 1 1 1 0 6
q1
A Simple Example of an RTM Tparity
t=2 1 0 1 0 6
q2
A Simple Example of an RTM Tparity
t=3 1 0 0 0 6
q1
A Simple Example of an RTM Tparity
t=3 1 0 0 1 6
qacc
Reject Accept
t=0
Begin
head 0
1
1
0
Reject Accept
t = 1402
Begin
1
head 1
1
0
Reject Accept
t = 2816
Begin
1
0
head 1
0
Reject Accept
t = 5000
Begin
1
0
0
head 0
Reject Accept
t = 6875
Begin
1
0
0
head 1
Billiard Ball Model (BBM) – A reversible physical model of computing – [Fredkin and Toffoli, 1982]
c
cx
R
R
cx
R
x
R R
c
Realization of an RE by BBM [Morita, 2008] n
n
w
s0
w I
n1
V
I
s1 I
I
-
n0
R
w1
I
R
R
H
I 6
w0
?
R
e
I
?
R
I
R I -
6
e0
e1
6 ?
s
s
R
R
R
R
I
e
Parallel Case t=0
t=1
? 6
? 6 6
-
-
? 6
? 6
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Movements of Balls (State: V , Input: s) n
n
6
w w
e V -
?
s
s
e
Orthogonal Case t=0
t=1
? 6
? 6
-
-
? 6
-? 6
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
n
w
6
s0
e
R
w
-
s1 I
I I
H 6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0
e
R
w
-
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
Movements of Balls (State: H, Input: s) n
w
n
6
s0 R
w
e V -
s1 I
I I
6
R
?
s
s
e
3. Reversible Cellular Automata
Reversible Cellular Automata (RCAs) • It is a CA whose global function is one-to-one. • A kind of spatio-temporal model of a physically reversible space. • In spite of the strong restriction of reversibility, they have rich ability of computing. – Computation-universality – Self-reproduction – Synchronization – etc.
Partitioned Cellular Automata • 1D Partitioned CA (PCA) L C R L C R L C R t
f
?
t+1 i−1 i+1 i A local function f of a 1D PCA. • We can design RCAs easily using PCAs.
Universal Reversible CAs — 1D Case — • On infinite configurations: 24-state RPCA • On finite configurations: 98-state RPCA
[Morita, 2008] [Morita, 2007]
cf. 1D Universal Irreversible CAs: • On infinite configurations: 2-state CA (ECA of rule 110) [Cook, 2004] • On finite configurations: 7-state CA (a modified model) [Lindgren et al., 1990]
Universal Reversible CAs — 2D Case — • On infinite configurations: – 2-state Margolus-neighbor RCA [Margolus, 1984] – 16-state RPCAs [Morita and Ueno, 1992] – 8-state triangular RPCA [Imai and Morita, 1998]
An 8-State Triangular RPCA T1 [Imai and Morita, 1998] • It has an extremely simple local function: →
•
→
•
•
•
→
••
•
•
→
•• •
•
A Fredkin Gate in a Triangular 8-State RPCA T1
Universal Reversible CAs — 2D Case — • On infinite configurations: – 2-state Margolus-neighbor RCA [Margolus, 1984] – 16-state RPCAs [Morita and Ueno, 1992] – 8-state triangular RPCA [Imai and Morita, 1998] • On finite configurations: – 81-state RPCA [Morita and Ogiro, 2001]
A 34-State Universal RPCA P3 P3 = (Z2, {0, 1, 2}4, g3, (0, 0, 0, 0)) ◦
•
•
•
•
◦ •
→
→
◦
◦
(a) •
• •
(b)
• →
•
• →
• • •
(d)
◦ •
(e)
→
(c)
•
◦
• →
◦ • •
→
◦ ◦ • ◦
◦ • •
• ◦
◦ →
◦ ◦ •
(f)
(g)
(h)
◦
•
◦
•
→
• ◦ ◦
(i)
◦ • ◦
(j)
• →
• ◦ • •
(k)
→
• ◦ •
(l)
◦ ◦ • • ◦ w
◦
→
z y
◦
w x → z yx
(m)
The rule scheme (m) represents 33 rules not specified by (a)–(l) (w, x, y, z ∈ { blank, ◦, • } = {0, 1, 2}).
Reversible Counter Machine in P3 Space aaaa aaaa aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaa aaaa
aaaq aqqqa a a aqqaq aqaa a a aa aa aa aq qqaa a a aqaq qa aaaa aaaaq q aaaa aaaaaa aq qa aa q q qq qqqq qq aa aa aaaa aaqaa qq qaaaa aaaa aa aa
aaaa aaaa
aaaa aaaa
aaaa aaaa aaaa aaaaa a aa aa aaaa aaaa a a aaaa aaaa aaaa aaaa
aaaa aaaa
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Movie of an RCM(2) in P3
Self-Reproduction of a Worm in 2D RCA [Morita and Imai, 1996]
Self-Reproduction of a Loop in 3D RCA [Imai, Hori and Morita, 2002]
Concluding Remarks • We saw even very simple reversible systems have computation-universality. • Computation can be carried out in a very different way from that of conventional computers. • We expect that further studies on them will give new insights for future computing.
Thank you for your attention!
