Language Recognition by Reversible Partitioned Cellular

Language Recognition by Reversible

Partitioned Cellular Automata

Kenichi Morita

Hiroshima University

20th International Workshop on Cellular Automata and

Discrete Complex Systems (AUTOMATA 2014),

Himeji, Japan, 7–9 July 2014

Outline of the Talk

• Objective: To clarify how the reversibility con-

straint affects the language accepting capability

of 1-D partitioned cellular automata (PCAs).

· · · A B C

· · ·� � � ��

Reversible system

Outline of the Talk

• Objective: To clarify how the reversibility con-

straint affects the language accepting capability

of 1-D partitioned cellular automata (PCAs).

• Result: The capability of PCAs does not de-

crease by the constraint. We have

L(RPCA) =L(PCA) =L(DLBA) =L(RDLBA).

LBA: linear-bounded automaton

D: deterministic

R: reversible

Contents

1. Partitioned cellular automaton acceptor

2. Reversible linear-bounded automaton

3. Simulating RDLBA by RPCA

Appendices

1. Partitioned cellular automaton acceptor

Bounded CA Acceptor

• Smith (1972) proposed a bounded CA acceptor.

• An input word a1a2 · · · an is given as an initial

configuration.

• It uses only n cells (the space is bounded).

• If the cell at the position 1 enters an accepting

state, we say the CA accepts the input.0 1 2 · · · · n n + 1

t = 0 # a1 a2 · · · · an #

t = 1 # x11 x1

2 · · · · x1n #

t = t1 # xt11 xt1

2 · · · · xt1n #

Past Results on Bounded CA Acceptors

• Smith (1972) showed an irreversible CA acceptor

is equivalent to a DLBA, if the computing time

is not bounded:

L(CA) = L(DLBA)

DLBA: deterministic linear-bounded automaton

• Reversible CA acceptors were studied by Kutrib

and Malcher (2008), and basic properties of real-

time ones were derived.

Partitioned Cellular Automaton (PCA)

• We use partitioned cellular automata, since it

makes easy to design reversible CAs.

• In a 3-neighbor PCA, each cell is divided into 3

parts, and a local function f is applied as below.

L L LC C CR R R

i − 1 i i + 1

︸︷︷︸

︷︸︸︷�f

• The global function is injective (hence the CA is

reversible), iff the local function is injective.8

Bounded PCA Acceptor

• An input word a1a2 · · · an is given as an initial

configuration.

• It uses n + 2 cells.

• If the left part of the first cell enters an accepting

state, we say the PCA accepts the input.0 1 2 · · n n + 1

t = 0 rs a1 a2 · · an

t = 1 l11 c11 r1

1a2 · · an

t = t1 rt10 lt11 ct1

1 rt11 lt12 ct1

2 rt12 · · · · · · lt1n ct1

n rt1n lt1n+1

Simple Example of PCA Acceptor (Irreversible)

• A PCA that accepts all well-formed parentheses.

(1) Input: ( ( ) )

0 s ( ( ) )

1 ( s ( ) )

2 ( ( s ) )

3 ( ( ) − s )

4 ( − − ) − s

5 ( − ) − − Y

6 ( ) − − Y −

7 − − Y − −

8 − Y − − −

9 Y − − − −

(2) Input: ( ( ) ( )

0 s ( ( ) ( )

1 ( s ( ) ( )

2 ( ( s ) ( )

3 ( ( ) − s ( )

4 ( − − ( s )

5 ( − − ( ) − s

6 ( − − − − Y

7 ( − − − Y −

8 ( − − Y − −

9 ( − Y − − −

10 ( Y − − − −

11 N ( − − − −

(3) Input: ( ) ( ) )

0 s ( ) ( ) )

1 ( s ) ( ) )

2 ( ) − s ( ) )

3 − − ( s ) )

4 − − ( ) − s )

5 − − − − ) − s

6 − − − ) − − Y

7 − − ) − − Y −

8 − ) − − Y − −

9 ) − − Y − − −

10 ) − Y − − − −

11 N − ) − − − −

2. Reversible linear-bounded automaton

Linear-Bounded Automaton (LBA)

• LBA is an automaton whose storage space is

bounded by the input length.

• Lange, McKenzie and Tapp (2000) showed:

L(RDLBA) = L(DLBA).

• But, their method is complex, and it is diffi-

cult to get a concrete description of a converted

RDLBA.

• We give a simpler method of converting DLBA

to RDLBA based on the method by Morita (2013).

2-Track Linear-Bounded Automaton (LBA)

• In this model, the input is kept unchanged, since

it is needed when converting DLBA into RDLBA.

Input track (read-only)� a1 a2 · · · · · · an �

q Finite-state control

Storage track (read/write)� b1 b2 · · bm # · · # �

• Its movement is defined by a set of rules:[p, [a, [b, b′]], q]: read-write rule[p, d, q]: shift rule (d ∈ {−,0,+})

• It is equivalent to a “standard” 1-track LBA.

Reversible LBA

• A “backward deterministic” LBA.

� a1 a2 · · · · · · an �

� · · · s t u′ · · · �

� a1 a2 · · · · · · an �

� · · · s′ t u · · · �

� a1 a2 · · · · · · an �

� · · · s′ t u′ · · · �

Definition of Reversible LBA

Definition 6. An LBA M = (Q,Σ,Γ, δ, �, �,#, q0, A,

N) is called a reversible LBA (RLBA) iff the follow-

ing condition holds.

∀ r1 = [p, x, q] ∈ δ, ∀ r2 = [p′, x′, q′] ∈ δ :(r1 �= r2 ∧ q = q′) ⇒ (x �∈ {−,0,+} ∧ x′ �∈ {−,0,+} ∧∀ [a, [b, c]], [a′[b′, c′]] ∈ Σ×Γ2 ∪ {[�, [�, �]], [�, [�, �]]}(x = [a, [b, c]] ∧ x′ = [a′, [b′, c′]] ⇒ [a, c] �= [a′, c′]))

For any two rules r1 and r2 in δ, if the next states q

and q′ are the same, then they are both read-write

rules and the pairs of the input symbols and the

written storage symbols [a,c] and [a′,c′] are different.

Converting DLBAs into RDLBAs

Lemma 2. Let M = (Q,Σ,Γ, δ, �, �,#, q0, A, N) be

an arbitrary DLBA. We can construct an RDLBA

M† = (Q†,Σ,Γ, δ†, �, �,#, q0, {q̂b0}, {qb0}) such that

the following holds, and thus L(M†) = L(M).

w ∈ L(M) ⇒� w �

� # # · · · · # �

� w �

� # # · · · · # �|−−M†+

w �∈ L(M) ⇒� w �

� # # · · · · # �

� w �

� # # · · · · # �|−−M†+

Proof Outline of Lemma 2

• RDLBA M† traverses a computation graph of M

using additional states.qa

q5 q0 q6 q7 q8

q9 q10 q11 q12

� �

� � ��

� � � �

q1 q2 q3

q6 q7 q8 q9

q0 q10

� �

A case M halts A case M loops

Note: Each node represents a configuration of M . But,

here, only a state of the finite-state control is written.19

Traversing a Computation Graph Reversibly

• M† has the following states for each q of M .

q is for the forward simulation.

qb is for the backward simulation.

qi1 qi2 qi3

� ��

qi1 qi2 qi3

� � �

Reversible simulation of irreversible transitions of M .

The Case M Halts in an Accepting State

q5 q0 q6 q7 q8

q9 q10 q11 q12

� �

� � ��

� � � �

�qb10

�q10

�qb4

�qb7

�qb11

�q11

�qb12

�q12

�q̂ba

�q̂b1

�q̂b3

�q̂b5

�q̂b9

�q̂9

�q̂5

�q̂b0

If M enters an accepting state qa, then M† keeps

the fact by the states of the form q̂.21

The Case M Loops

q1 q2 q3

q6 q7 q8 q9

q0 q10

� �

� qb7

� qb8

� qb9

�qb10

�q10

�qb2

�qb1

�qb6

�qb0

M† also finally goes back to M ’s initial configuration

in the rejecting state qb0. This is because RDMFA

always halts, if the initial configuration has no pre-

decessor (Lemma 1 of the paper in the proceedings).22

Example of an Irreversible DLBA

The following irreversible DLBA Mp accepts all well-

formed parentheses.

Mp = (Q, { (, ) }, {#, x}, δ, �, �,#, q0, {qa}, {qr})Q = {q0, q0,1, q0,2, q0,3, q0,4, q1, q2, q3, q4, q5, q6, qa, qr}δ = { [q0, [�, [�, �]], q0,1], [q0,1,+, q0,2], [q0,2, [ (, [#,#]], q0,1],

[q0,2, [ ), [#,#]], q0,1], [q0,2, [�, [�, �]], q0,3], [q0,3,−, q0,4],[q0,4, [ (, [#,#]], q0,3], [q0,4, [ ), [#,#]], q0,3], [q0,4, [�, [�, �]], q1],[q1,+, q2], [q2, [ (, [#,#]], q1], [q2, [ (, [x, x]], q1],[q2, [ ), [x, x]], q1], [q2, [ ), [#, x]], q3], [q2, [�, [�, �]], q5],[q3,−, q4], [q4, [ (, [x, x]], q3], [q4, [ ), [x, x]], q3],[q4, [ (, [#, x]], q1], [q4, [�, [�, �]], qr], [q5,−, q6],[q6, [ (, [x, x]], q5], [q6, [ ), [x, x]], q5], [q6, [ (, [#,#]], qr],[q6, [�, [�, �]], qa] }

(See Appendix A1.)

Example of an RDLBA Converted from Mp

M †p = (Q†, { (, ) }, {#, x}, δ†, �, �,#, q0, {q̂b

0}, {qb0})

Q† = {q, q̂, qb, q̂b | q ∈ Q}δ† = δ1 ∪ · · · ∪ δ4 ∪ δ̂1 ∪ · · · ∪ δ̂4 ∪ δA ∪ δN

δ1 = { [qb0,1, [�, [�, �]], qb

0], [q0, [�, [�, �]], q0,1], [qb0,1, [ (, [#,#]], qb

0,2], [q0,2, [ (, [#,#]], q0,1],[qb

0,1, [ ), [#,#]], qb0,2], [q0,2, [ ), [#,#]], q0,1], [qb

0,3, [�, [�, �]], qb0,2], [q0,2, [�, [�, �]], q0,3],

[qb0,3, [ (, [#,#]], qb

0,4], [q0,4, [ (, [#,#]], q0,3], [qb0,3, [ ), [#,#]], qb

0,4], [q0,4, [ ), [#,#]], q0,3],[qb

1, [�, [�, �]], qb0,4], [q0,4, [�, [�, �]], q1], [qb

1, [ (, [#,#]], qb2], [q2, [ (, [#,#]], q1],

[qb1, [ ), [x, x]], qb

2], [q2, [ ), [x, x]], q1], [qb5, [�, [�, �]], qb

2], [q2, [�, [�, �]], q5],[qb

3, [ (, [x, x]], qb4], [q4, [ (, [x, x]], q3], [qb

3, [ ), [x,#]], qb2], [q2, [ ), [#, x]], qb

4],[q4, [ ), [x, x]], q3], [qb

1, [ (, [x, x]], qb2], [q2, [ (, [x,#]], qb

4], [q4, [ (, [#, x]], q1],[qb

r , [�, [�, �]], qb4], [q4, [�, [�, �]], qr], [qb

5, [ (, [x, x]], qb6], [q6, [ (, [x, x]], q5],

[qb5, [ ), [x, x]], qb

6], [q6, [ ), [x, x]], q5], [qbr , [ (, [#,#]], qb

6], [q6, [ (, [#,#]], qr],[qb

a , [�, [�, �]], qb6], [q6, [�, [�, �]], qa] }

δ2 = { [qb0,2,−, qb

0,1], [q0,1,+, q0,2], [qb0,4,+, qb

0,3], [q0,3,−, q0,4], [qb2,−, qb

1], [q1,+, q2],[qb

4,+, qb3], [q3,−, q4], [qb

6,+, qb5], [q5,−, q6] }

δ3 = { [qb0,1, [ (, [x, x]], q0,1], [qb

0,1, [ ), [x, x]], q0,1], [qb0,1, [�, [�, �]], q0,1], [qb

0,3, [�, [�, �]], q0,3],[qb

0,3, [ (, [x, x]], q0,3], [qb0,3, [ ), [x, x]], q0,3], [qb

1, [ ), [#,#]], q1], [qb1, [�, [�, �]], q1],

[qb3, [�, [�, �]], q3], [qb

3, [ (, [#,#]], q3], [qb3, [ ), [#,#]], q3], [qb

3, [�, [�, �]], q3],[qb

5, [�, [�, �]], q5], [qb5, [ (, [#,#]], q5], [qb

5, [ ), [#,#]], q5] }δ4 = { [q0,2, [�, [�, �]], qb

0,2], [q0,2, [ (, [x, x]], qb0,2], [q0,2, [ ), [x, x]], qb

0,2], [q0,4, [ (, [x, x]], qb0,4],

[q0,4, [ ), [x, x]], qb0,4], [q0,4, [�, [�, �]], qb

0,4], [q2, [�, [�, �]], qb2], [q4, [ ), [#,#]], qb

4],[q4, [�, [�, �]], qb

4], [q6, [ ), [#,#]], qb6], [q6, [�, [�, �]], qb

6] }δ̂i = { [p̂, x, q̂] | [p, x, q] ∈ δi } (i = 1, . . . ,4)

δA = { [qa,0, q̂ba], [q̂a,0, qb

a] }δN = { [qr,0, qb

r ], [q̂r,0, q̂br ] }

(See Appendix A2.)24

3. Simulating RDLBA by RPCA

Simulating RDLBA by RPCA

Lemma 3. For any DLBA M , we can construct an

RPCA PM such that L(PM) = L(M).

Proof outline:

• The RPCA PM simulates the RDLBA M† con-

verted from M .

• Simulation method is basically the same as the

one in [Morita, Harao, 1989], where a reversible Tur-

ing machine (RDTM) is simulated by an RPCA.

Simulation Method of RDTM by RPCA

[Morita, Harao, 1989]

t a b c

t + 1 a b′ c

2t a bp c

2t + 1 a b′ q c

2t + 2 a b′cq

Remarks on Simulating RDLBA

in Bounded RPCA

• Several steps of the RDLBA M† is simulated by

one step of the RPCA PM , since the movements

of the boundary cells are limited.

• In a bounded RPCA, any computation process

necessarily loops. Thus, it comes back to the

initial configuration.

Example of RPCA Converted from RDLBA

An RPCA that simulates the RDLBA M†p, and thus

accepts all well-formed parentheses (See Appendix A3):

PMp = ((L, C, R), f, (#,#,#), q0,2, { (, ) }, {q̂b0,1})

• The number of states of a cell, and that of rules

of are both 14157 (|L| = 11, |C| = 117 and |R| = 11).

• However, the number of rules that are actually

used to simulate M†p is only 203 as listed in the

next page.

Local Rules of the RPCA PMp[[#, #, #], [#, #, #]], [[#, [ (, #], #], [#, [ (, #], #]], [[#, [ (, x], #], [#, [ (, x], #]], [[#, [ ), #], #], [#, [ ), #], #]],

[[#, [ ), x], #], [#, [ ), x], #]], [[qb0,1, [ (, #], #], [#, [ (, #, qb

0,2], #]], [[#, [ (, #], q0,2], [#, [ (, #, q0,1], #]], [[qb0,1, [ ), #], #], [#, [ ), #, qb

0,2], #]],

[[#, [ ), #], q0,2], [#, [ ), #, q0,1], #]], [[#, #, qb0,3], [q

b0,1, #, #]], [[#, #, q0,2], [q0,4, #, #]], [[#, [ (, #], qb

0,3], [#, [ (, #, qb0,4], #]],

[[q0,4, [ (, #], #], [#, [ (, #, q0,3], #]], [[#, [ ), #], qb0,3], [#, [ ), #, qb

0,4], #]], [[q0,4, [ ), #], #], [#, [ ), #, q0,3], #]], [[qb1 , #, #], [#, #, qb

0,3]],

[[q0,4, #, #], [#, #, q2]], [[qb1 , [ (, #], #], [#, [ (, #, qb

2 ], #]], [[#, [ (, #], q2], [#, [ (, #, q1], #]], [[qb1 , [ ), x], #], [#, [ ), x, qb

2 ], #]],

[[#, [ ), x], q2], [#, [ ), x, q1], #]], [[#, #, qb5 ], [qb

1 , #, #]], [[#, #, q2], [q6, #, #]], [[#, [ (, x], qb3 ], [#, [ (, x, qb

4 ], #]],

[[q4, [ (, x], #], [#, [ (, x, q3], #]], [[#, [ ), x], qb3 ], [#, [ ), #, qb

2 ], #]], [[#, [ ), #], q2], [#, [ ), x, qb4 ], #]], [[q4, [ ), x], #], [#, [ ), x, q3], #]],

[[qb1 , [ (, x], #], [#, [ (, x, qb

2 ], #]], [[#, [ (, x], q2], [#, [ (, #, qb4 ], #]], [[q4, [ (, #], #], [#, [ (, x, q1], #]], [[q4, #, #], [#, #, qb

[[#, [ (, x], qb5 ], [#, [ (, x, qb

6 ], #]], [[q6, [ (, x], #], [#, [ (, x, q5], #]], [[#, [ ), x], qb5 ], [#, [ ), x, qb

6 ], #]], [[q6, [ ), x], #], [#, [ ), x, q5], #]],

[[#, [ (, #, qbr ], #], [#, [ (, #, qb

6 ], #]], [[q6, [ (, #], #], [#, [ (, #, qr], #]], [[q6, #, #], [#, #, q̂b5 ]], [[q̂b

0,1, [ (, #], #], [#, [ (, #, q̂b0,2], #]],

[[#, [ (, #], q̂0,2], [#, [ (, #, q̂0,1], #]], [[q̂b0,1, [ ), #], #], [#, [ ), #, q̂b

0,2], #]], [[#, [ ), #], q̂0,2], [#, [ ), #, q̂0,1], #]], [[#, #, q̂b0,3], [q̂

b0,1, #, #]],

[[#, #, q̂0,2], [q̂0,4, #, #]], [[#, [ (, #], q̂b0,3], [#, [ (, #, q̂b

0,4], #]], [[q̂0,4, [ (, #], #], [#, [ (, #, q̂0,3], #]], [[#, [ ), #], q̂b0,3], [#, [ ), #, q̂b

0,4], #]],

[[q̂0,4, [ ), #], #], [#, [ ), #, q̂0,3], #]], [[q̂b1 , #, #], [#, #, q̂b

0,3]], [[q̂0,4, #, #], [#, #, q̂2]], [[q̂b1 , [ (, #], #], [#, [ (, #, q̂b

2 ], #]],

[[#, [ (, #], q̂2], [#, [ (, #, q̂1], #]], [[q̂b1 , [ ), x], #], [#, [ ), x, q̂b

2 ], #]], [[#, [ ), x], q̂2], [#, [ ), x, q̂1], #]], [[#, #, q̂b5 ], [q̂b

1 , #, #]],

[[#, #, q̂2], [q̂6, #, #]], [[#, [ (, x], q̂b3 ], [#, [ (, x, q̂b

4 ], #]], [[q̂4, [ (, x], #], [#, [ (, x, q̂3], #]], [[#, [ ), x], q̂b3 ], [#, [ ), #, q̂b

2 ], #]],

[[#, [ ), #], q̂2], [#, [ ), x, q̂b4 ], #]], [[q̂4, [ ), x], #], [#, [ ), x, q̂3], #]], [[q̂b

1 , [ (, x], #], [#, [ (, x, q̂b2 ], #]], [[#, [ (, x], q̂2], [#, [ (, #, q̂b

4 ], #]],

[[q̂4, [ (, #], #], [#, [ (, x, q̂1], #]], [[q̂4, #, #], [#, #, q̂b3 ]], [[#, [ (, x], q̂b

5 ], [#, [ (, x, q̂b6 ], #]], [[q̂6, [ (, x], #], [#, [ (, x, q̂5], #]],

[[#, [ ), x], q̂b5 ], [#, [ ), x, q̂b

6 ], #]], [[q̂6, [ ), x], #], [#, [ ), x, q̂5], #]], [[#, [ (, #, q̂br ], #], [#, [ (, #, q̂b

6 ], #]], [[q̂6, [ (, #], #], [#, [ (, #, q̂r], #]],

[[q̂6, #, #], [#, #, qb5 ]], [[#, [ (, #, qb

0,2], #], [qb0,1, [ (, #], #]], [[#, [ (, x, qb

0,2], #], [qb0,1, [ (, x], #]], [[#, [ ), #, qb

0,2], #], [qb0,1, [ ), #], #]],

[[#, [ ), x, qb0,2], #], [qb

0,1, [ ), x], #]], [[#, [ (, #, q0,1], #], [#, [ (, #], q0,2]], [[#, [ (, x, q0,1], #], [#, [ (, x], q0,2]], [[#, [ ), #, q0,1], #], [#, [ ), #], q0,2]],

[[#, [ ), x, q0,1], #], [#, [ ), x], q0,2]], [[#, [ (, #, qb0,4], #], [#, [ (, #], qb

0,3]], [[#, [ (, x, qb0,4], #], [#, [ (, x], qb

0,3]], [[#, [ ), #, qb0,4], #], [#, [ ), #], qb

0,3]],

[[#, [ ), x, qb0,4], #], [#, [ ), x], qb

0,3]], [[#, [ (, #, q0,3], #], [q0,4, [ (, #], #]], [[#, [ (, x, q0,3], #], [q0,4, [ (, x], #]], [[#, [ ), #, q0,3], #], [q0,4, [ ), #], #]],

[[#, [ ), x, q0,3], #], [q0,4, [ ), x], #]], [[#, [ (, #, qb2 ], #], [qb

1 , [ (, #], #]], [[#, [ (, x, qb2 ], #], [qb

1 , [ (, x], #]], [[#, [ ), #, qb2 ], #], [qb

1 , [ ), #], #]],

[[#, [ ), x, qb2 ], #], [qb

1 , [ ), x], #]], [[#, [ (, #, q1], #], [#, [ (, #], q2]], [[#, [ (, x, q1], #], [#, [ (, x], q2]], [[#, [ ), #, q1], #], [#, [ ), #], q2]],

[[#, [ ), x, q1], #], [#, [ ), x], q2]], [[#, [ (, #, qb4 ], #], [#, [ (, #], qb

3 ]], [[#, [ (, x, qb4 ], #], [#, [ (, x], qb

3 ]], [[#, [ ), #, qb4 ], #], [#, [ ), #], qb

[[#, [ ), x, qb4 ], #], [#, [ ), x], qb

3 ]], [[#, [ (, #, q3], #], [q4, [ (, #], #]], [[#, [ (, x, q3], #], [q4, [ (, x], #]], [[#, [ ), #, q3], #], [q4, [ ), #], #]],

[[#, [ ), x, q3], #], [q4, [ ), x], #]], [[#, [ (, #, qb6 ], #], [#, [ (, #], qb

5 ]], [[#, [ (, x, qb6 ], #], [#, [ (, x], qb

5 ]], [[#, [ ), #, qb6 ], #], [#, [ ), #], qb

[[#, [ ), x, qb6 ], #], [#, [ ), x], qb

5 ]], [[#, [ (, #, q5], #], [q6, [ (, #], #]], [[#, [ (, x, q5], #], [q6, [ (, x], #]], [[#, [ ), #, q5], #], [q6, [ ), #], #]],

[[#, [ ), x, q5], #], [q6, [ ), x], #]], [[#, [ (, #, q̂b0,2], #], [q̂b

0,1, [ (, #], #]], [[#, [ (, x, q̂b0,2], #], [q̂b

0,1, [ (, x], #]], [[#, [ ), #, q̂b0,2], #], [q̂b

0,1, [ ), #], #]],

[[#, [ ), x, q̂b0,2], #], [q̂b

0,1, [ ), x], #]], [[#, [ (, #, q̂0,1], #], [#, [ (, #], q̂0,2]], [[#, [ (, x, q̂0,1], #], [#, [ (, x], q̂0,2]], [[#, [ ), #, q̂0,1], #], [#, [ ), #], q̂0,2]],

[[#, [ ), x, q̂0,1], #], [#, [ ), x], q̂0,2]], [[#, [ (, #, q̂b0,4], #], [#, [ (, #], q̂b

0,3]], [[#, [ (, x, q̂b0,4], #], [#, [ (, x], q̂b

0,3]], [[#, [ ), #, q̂b0,4], #], [#, [ ), #], q̂b

0,3]],

[[#, [ ), x, q̂b0,4], #], [#, [ ), x], q̂b

0,3]], [[#, [ (, #, q̂0,3], #], [q̂0,4, [ (, #], #]], [[#, [ (, x, q̂0,3], #], [q̂0,4, [ (, x], #]], [[#, [ ), #, q̂0,3], #], [q̂0,4, [ ), #], #]],

[[#, [ ), x, q̂0,3], #], [q̂0,4, [ ), x], #]], [[#, [ (, #, q̂b2 ], #], [q̂b

1 , [ (, #], #]], [[#, [ (, x, q̂b2 ], #], [q̂b

1 , [ (, x], #]], [[#, [ ), #, q̂b2 ], #], [q̂b

1 , [ ), #], #]],

[[#, [ ), x, q̂b2 ], #], [q̂b

1 , [ ), x], #]], [[#, [ (, #, q̂1], #], [#, [ (, #], q̂2]], [[#, [ (, x, q̂1], #], [#, [ (, x], q̂2]], [[#, [ ), #, q̂1], #], [#, [ ), #], q̂2]],

[[#, [ ), x, q̂1], #], [#, [ ), x], q̂2]], [[#, [ (, #, q̂b4 ], #], [#, [ (, #], q̂b

3 ]], [[#, [ (, x, q̂b4 ], #], [#, [ (, x], q̂b

3 ]], [[#, [ ), #, q̂b4 ], #], [#, [ ), #], q̂b

[[#, [ ), x, q̂b4 ], #], [#, [ ), x], q̂b

3 ]], [[#, [ (, #, q̂3], #], [q̂4, [ (, #], #]], [[#, [ (, x, q̂3], #], [q̂4, [ (, x], #]], [[#, [ ), #, q̂3], #], [q̂4, [ ), #], #]],

[[#, [ ), x, q̂3], #], [q̂4, [ ), x], #]], [[#, [ (, #, q̂b6 ], #], [#, [ (, #], q̂b

5 ]], [[#, [ (, x, q̂b6 ], #], [#, [ (, x], q̂b

5 ]], [[#, [ ), #, q̂b6 ], #], [#, [ ), #], q̂b

[[#, [ ), x, q̂b6 ], #], [#, [ ), x], q̂b

5 ]], [[#, [ (, #, q̂5], #], [q̂6, [ (, #], #]], [[#, [ (, x, q̂5], #], [q̂6, [ (, x], #]], [[#, [ ), #, q̂5], #], [q̂6, [ ), #], #]],

[[#, [ ), x, q̂5], #], [q̂6, [ ), x], #]], [[qb0,1, [ (, x], #], [#, [ (, x, q0,1], #]], [[qb

0,1, [ ), x], #], [#, [ ), x, q0,1], #]], [[#, [ (, x], qb0,3], [#, [ (, x, q0,3], #]],

[[#, [ ), x], qb0,3], [#, [ ), x, q0,3], #]], [[qb

1 , [ ), #], #], [#, [ ), #, q1], #]], [[#, [ (, #], qb3 ], [#, [ (, #, q3], #]], [[#, [ ), #], qb

3 ], [#, [ ), #, q3], #]],

[[#, #, qb3 ], [q4, #, #]], [[#, [ (, #], qb

5 ], [#, [ (, #, q5], #]], [[#, [ ), #], qb5 ], [#, [ ), #, q5], #]], [[q̂b

0,1, [ (, x], #], [#, [ (, x, q̂0,1], #]],

[[q̂b0,1, [ ), x], #], [#, [ ), x, q̂0,1], #]], [[#, [ (, x], q̂b

0,3], [#, [ (, x, q̂0,3], #]], [[#, [ ), x], q̂b0,3], [#, [ ), x, q̂0,3], #]], [[q̂b

1 , [ ), #], #], [#, [ ), #, q̂1], #]],

[[#, [ (, #], q̂b3 ], [#, [ (, #, q̂3], #]], [[#, [ ), #], q̂b

3 ], [#, [ ), #, q̂3], #]], [[#, #, q̂b3 ], [q̂4, #, #]], [[#, [ (, #], q̂b

5 ], [#, [ (, #, q̂5], #]],

[[#, [ ), #], q̂b5 ], [#, [ ), #, q̂5], #]], [[#, [ (, x], q0,2], [#, [ (, x, qb

0,2], #]], [[#, [ ), x], q0,2], [#, [ ), x, qb0,2], #]], [[q0,4, [ (, x], #], [#, [ (, x, qb

0,4], #]],

[[q0,4, [ ), x], #], [#, [ ), x, qb0,4], #]], [[q4, [ ), #], #], [#, [ ), #, qb

4 ], #]], [[q6, [ ), #], #], [#, [ ), #, qb6 ], #]], [[#, [ (, x], q̂0,2], [#, [ (, x, q̂b

0,2], #]],

[[#, [ ), x], q̂0,2], [#, [ ), x, q̂b0,2], #]], [[q̂0,4, [ (, x], #], [#, [ (, x, q̂b

0,4], #]], [[q̂0,4, [ ), x], #], [#, [ ), x, q̂b0,4], #]], [[q̂4, [ ), #], #], [#, [ ), #, q̂b

4 ], #]],

[[q̂6, [ ), #], #], [#, [ ), #, q̂b6 ], #]], [[#, [ (, #, qa], #], [#, [ (, #, q̂b

a], #]], [[#, [ (, x, qa], #], [#, [ (, x, q̂ba], #]], [[#, [ ), #, qa], #], [#, [ ), #, q̂b

a], #]],

[[#, [ ), x, qa], #], [#, [ ), x, q̂ba], #]], [[#, [ (, #, q̂a], #], [#, [ (, #, qb

a], #]], [[#, [ (, x, q̂a], #], [#, [ (, x, qba], #]], [[#, [ ), #, q̂a], #], [#, [ ), #, qb

a], #]],

[[#, [ ), x, q̂a], #], [#, [ ), x, qba], #]], [[#, [ (, #, qr], #], [#, [ (, #, qb

r ], #]], [[#, [ (, x, qr], #], [#, [ (, x, qbr ], #]], [[#, [ ), #, qr], #], [#, [ ), #, qb

r ], #]],

[[#, [ ), x, qr], #], [#, [ ), x, qbr ], #]], [[#, [ (, #, q̂r], #], [#, [ (, #, q̂b

r ], #]], [[#, [ (, x, q̂r], #], [#, [ (, x, q̂br ], #]], [[#, [ ), #, q̂r], #], [#, [ ), #, q̂b

r ], #]],

[[#, [ ), x, q̂r], #], [#, [ ), x, q̂br ], #]], [[q̂b

0,1, #, #], [#, #, q̂0,2]], [[qb0,1, #, #], [#, #, q0,2]]

Computing Process of the RPCA PMp

(1) Input: ( ( ) ( ) )

t = 0 q0,2 ( ( ) ( ) )

688 q̂b0,1

( ( ) ( ) )

1378 q0,2 ( ( ) ( ) )

Computing Process of the RPCA PMp

(2) Input: ( ( ) ( )

t = 0 q0,2 ( ( ) ( )

220 qb0,1

( ( ) ( )

221 q0,2 ( ( ) ( )

Concluding Remarks

• Theorem 1:

L(RPCA) =L(PCA) =L(DLBA) =L(RDLBA).

• Reversible partitioned iterative array (RPIA) can

also be defined as another model of acceptor (see

Appendix A4).

• Future study: To characterize the capability of

RPCAs for the case computing time is limited.

References

1 Kutrib, M.: Cellular automata and language theory. In: Encyclopediaof Complexity and System Science (ed. B. Meyers), Springer-Verlag,Berlin. pp. 800–823 (2009)

2 Kutrib, M., Malcher, A.: Fast reversible language recognition usingcellular automata. Inform. Comput. 206, 1142–1151 (2008)

3 Lange, K.J., McKenzie, P., Tapp, A.: Reversible space equals deter-ministic space. J. Comput. Syst. Sci. 60, 354–367 (2000)

4 Morita, K.: Simulating reversible Turing machines and cyclic tag sys-tems by one-dimensional reversible cellular automata. Theoret. Com-put. Sci. 412, 3856–3865 (2011)

5 Morita, K.: Two-way reversible multi-head finite automata. Funda-menta Informaticae 110(1-4), 241–254 (2011)

6 Morita, K.: A deterministic two-way multi-head finite automaton canbe converted into a reversible one with the same number of heads.Proc. RC 2012, LNCS 7581 pp. 29–43 (2013)

7 Morita, K., Harao, M.: Computation universality of one-dimensionalreversible (injective) cellular automata. Trans. IEICE Japan E72,758–762 (1989)

8 Smith III, A.: Real-time language recognition by one-dimensional cel-lular automata. J. Comput. Syst. Sci. 6, 233–253 (1972)

Appendices

Computing processes of machines that accept

all well-formed parentheses:

A1. DLBA Mp

A2. RDLBA M†p

A3. RPCA PMp

A4. RPIA

A1. DLBA Mp

DLBA Mp that accepts all well-formed parentheses

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

t = 10

� ( ( ) ( ) ) �

� # # # # # # �

t = 11

� ( ( ) ( ) ) �

� # # # # # # �

t = 12

� ( ( ) ( ) ) �

� # # # # # # �

t = 13

� ( ( ) ( ) ) �

� # # # # # # �

t = 14

� ( ( ) ( ) ) �

� # # # # # # �

t = 15

� ( ( ) ( ) ) �

� # # # # # # �

t = 16

� ( ( ) ( ) ) �

� # # # # # # �

t = 17

� ( ( ) ( ) ) �

� # # # # # # �

t = 18

� ( ( ) ( ) ) �

� # # # # # # �

t = 19

� ( ( ) ( ) ) �

� # # # # # # �

t = 20

� ( ( ) ( ) ) �

� # # # # # # �

t = 21

� ( ( ) ( ) ) �

� # # # # # # �

t = 22

� ( ( ) ( ) ) �

� # # # # # # �

t = 23

� ( ( ) ( ) ) �

� # # # # # # �

t = 24

� ( ( ) ( ) ) �

� # # # # # # �

t = 25

� ( ( ) ( ) ) �

� # # # # # # �

t = 26

� ( ( ) ( ) ) �

� # # # # # # �

t = 27

� ( ( ) ( ) ) �

� # # # # # # �

t = 28

� ( ( ) ( ) ) �

� # # # # # # �

t = 29

� ( ( ) ( ) ) �

� # # # # # # �

t = 30

� ( ( ) ( ) ) �

� # # # # # # �

t = 31

� ( ( ) ( ) ) �

� # # # # # # �

t = 32

� ( ( ) ( ) ) �

� # # # # # # �

t = 33

� ( ( ) ( ) ) �

� # # # # # # �

t = 34

� ( ( ) ( ) ) �

� # # # # # # �

t = 35

� ( ( ) ( ) ) �

� # # x # # # �

t = 36

� ( ( ) ( ) ) �

� # # x # # # �

t = 37

� ( ( ) ( ) ) �

� # x x # # # �

t = 38

� ( ( ) ( ) ) �

� # x x # # # �

t = 39

� ( ( ) ( ) ) �

� # x x # # # �

t = 40

� ( ( ) ( ) ) �

� # x x # # # �

t = 41

� ( ( ) ( ) ) �

� # x x # # # �

t = 42

� ( ( ) ( ) ) �

� # x x # # # �

t = 43

� ( ( ) ( ) ) �

� # x x # x # �

t = 44

� ( ( ) ( ) ) �

� # x x # x # �

t = 45

� ( ( ) ( ) ) �

� # x x x x # �

t = 46

� ( ( ) ( ) ) �

� # x x x x # �

t = 47

� ( ( ) ( ) ) �

� # x x x x # �

t = 48

� ( ( ) ( ) ) �

� # x x x x # �

t = 49

� ( ( ) ( ) ) �

� # x x x x x �

t = 50

� ( ( ) ( ) ) �

� # x x x x x �

t = 51

� ( ( ) ( ) ) �

� # x x x x x �

t = 52

� ( ( ) ( ) ) �

� # x x x x x �

t = 53

� ( ( ) ( ) ) �

� # x x x x x �

t = 54

� ( ( ) ( ) ) �

� # x x x x x �

t = 55

� ( ( ) ( ) ) �

� # x x x x x �

t = 56

� ( ( ) ( ) ) �

� # x x x x x �

t = 57

� ( ( ) ( ) ) �

� # x x x x x �

t = 58

� ( ( ) ( ) ) �

� # x x x x x �

t = 59

� ( ( ) ( ) ) �

� x x x x x x �

t = 60

� ( ( ) ( ) ) �

� x x x x x x �

t = 61

� ( ( ) ( ) ) �

� x x x x x x �

t = 62

� ( ( ) ( ) ) �

� x x x x x x �

t = 63

� ( ( ) ( ) ) �

� x x x x x x �

t = 64

� ( ( ) ( ) ) �

� x x x x x x �

t = 65

� ( ( ) ( ) ) �

� x x x x x x �

t = 66

� ( ( ) ( ) ) �

� x x x x x x �

t = 67

� ( ( ) ( ) ) �

� x x x x x x �

t = 68

� ( ( ) ( ) ) �

� x x x x x x �

t = 69

� ( ( ) ( ) ) �

� x x x x x x �

t = 70

� ( ( ) ( ) ) �

� x x x x x x �

t = 71

� ( ( ) ( ) ) �

� x x x x x x �

t = 72

� ( ( ) ( ) ) �

� x x x x x x �

t = 73

� ( ( ) ( ) ) �

� x x x x x x �

t = 74

� ( ( ) ( ) ) �

� x x x x x x �

t = 75

� ( ( ) ( ) ) �

� x x x x x x �

t = 76

� ( ( ) ( ) ) �

� x x x x x x �

t = 77

� ( ( ) ( ) ) �

� x x x x x x �

t = 78

� ( ( ) ( ) ) �

� x x x x x x �

t = 79

� ( ( ) ( ) ) �

� x x x x x x �

t = 80

� ( ( ) ( ) ) �

� x x x x x x �

t = 81

� ( ( ) ( ) ) �

� x x x x x x �

t = 82

� ( ( ) ( ) ) �

� x x x x x x �

t = 83

� ( ( ) ( ) ) �

� x x x x x x �

t = 84

� ( ( ) ( ) ) �

� x x x x x x �

t = 85

� ( ( ) ( ) ) �

� x x x x x x �

The input is accepted

A2. RDLBA M†p

RDLBA M†p that accepts all well-formed parentheses

� ( ( ) ( ) ) �

� # # # # # # �

Direction of simulation: qi: forward, qbi : backward

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

� ( ( ) ( ) ) �

� # # # # # # �

t = 10

� ( ( ) ( ) ) �

� # # # # # # �

t = 11

� ( ( ) ( ) ) �

� # # # # # # �

t = 12

� ( ( ) ( ) ) �

� # # # # # # �

t = 13

� ( ( ) ( ) ) �

� # # # # # # �

t = 14

� ( ( ) ( ) ) �

� # # # # # # �

t = 15

� ( ( ) ( ) ) �

� # # # # # # �

t = 16

� ( ( ) ( ) ) �

� # # # # # # �

t = 17

� ( ( ) ( ) ) �

� # # # # # # �

t = 18

� ( ( ) ( ) ) �

� # # # # # # �

t = 19

� ( ( ) ( ) ) �

� # # # # # # �

t = 20

� ( ( ) ( ) ) �

� # # # # # # �

t = 21

� ( ( ) ( ) ) �

� # # # # # # �

t = 22

� ( ( ) ( ) ) �

� # # # # # # �

t = 23

� ( ( ) ( ) ) �

� # # # # # # �

t = 24

� ( ( ) ( ) ) �

� # # # # # # �

t = 25

� ( ( ) ( ) ) �

� # # # # # # �

t = 26

� ( ( ) ( ) ) �

� # # # # # # �

t = 27

� ( ( ) ( ) ) �

� # # # # # # �

t = 28

� ( ( ) ( ) ) �

� # # # # # # �

t = 29

� ( ( ) ( ) ) �

� # # # # # # �

t = 30

� ( ( ) ( ) ) �

� # # # # # # �

t = 31

� ( ( ) ( ) ) �

� # # # # # # �

t = 32

� ( ( ) ( ) ) �

� # # # # # # �

t = 33

� ( ( ) ( ) ) �

� # # # # # # �

t = 34

� ( ( ) ( ) ) �

� # # # # # # �

t = 35

� ( ( ) ( ) ) �

� # # x # # # �

t = 36

� ( ( ) ( ) ) �

� # # x # # # �

t = 37

� ( ( ) ( ) ) �

� # # x # # # �

t = 38

� ( ( ) ( ) ) �

� # # x # # # �

t = 39

� ( ( ) ( ) ) �

� # # x # # # �

t = 40

� ( ( ) ( ) ) �

� # # x # # # �

t = 41

� ( ( ) ( ) ) �

� # x x # # # �

t = 42

� ( ( ) ( ) ) �

� # x x # # # �

t = 43

� ( ( ) ( ) ) �

� # x x # # # �

t = 44

� ( ( ) ( ) ) �

� # x x # # # �

t = 45

� ( ( ) ( ) ) �

� # x x # # # �

t = 46

� ( ( ) ( ) ) �

� # x x # # # �

t = 47

� ( ( ) ( ) ) �

� # x x # x # �

t = 48

� ( ( ) ( ) ) �

� # x x # x # �

t = 49

� ( ( ) ( ) ) �

� # x x # x # �

t = 50

� ( ( ) ( ) ) �

� # x x # x # �

t = 51

� ( ( ) ( ) ) �

� # x x # x # �

t = 52

� ( ( ) ( ) ) �

� # x x # x # �

t = 53

� ( ( ) ( ) ) �

� # x x x x # �

t = 54

� ( ( ) ( ) ) �

� # x x x x # �

t = 55

� ( ( ) ( ) ) �

� # x x x x # �

t = 56

� ( ( ) ( ) ) �

� # x x x x # �

t = 57

� ( ( ) ( ) ) �

� # x x x x x �

t = 58

� ( ( ) ( ) ) �

� # x x x x x �

t = 59

� ( ( ) ( ) ) �

� # x x x x x �

t = 60

� ( ( ) ( ) ) �

� # x x x x x �

t = 61

� ( ( ) ( ) ) �

� # x x x x x �

t = 62

� ( ( ) ( ) ) �

� # x x x x x �

t = 63

� ( ( ) ( ) ) �

� # x x x x x �

t = 64

� ( ( ) ( ) ) �

� # x x x x x �

t = 65

� ( ( ) ( ) ) �

� # x x x x x �

t = 66

� ( ( ) ( ) ) �

� # x x x x x �

t = 67

� ( ( ) ( ) ) �

� # x x x x x �

t = 68

� ( ( ) ( ) ) �

� # x x x x x �

t = 69

� ( ( ) ( ) ) �

� # x x x x x �

t = 70

� ( ( ) ( ) ) �

� # x x x x x �

t = 71

� ( ( ) ( ) ) �

� x x x x x x �

t = 72

� ( ( ) ( ) ) �

� x x x x x x �

t = 73

� ( ( ) ( ) ) �

� x # x x x x �

t = 74

� ( ( ) ( ) ) �

� x # x x x x �

t = 75

� ( ( ) ( ) ) �

� x # # x x x �

t = 76

� ( ( ) ( ) ) �

� x # # x x x �

t = 77

� ( ( ) ( ) ) �

� x # # x x x �

t = 78

� ( ( ) ( ) ) �

� x # # x x x �

t = 79

� ( ( ) ( ) ) �

� x # # x x x �

t = 80

� ( ( ) ( ) ) �

� x # # x x x �

t = 81

� ( ( ) ( ) ) �

� x # # x x x �

t = 82

� ( ( ) ( ) ) �

� x # # x x x �

t = 83

� ( ( ) ( ) ) �

� x # # x x x �

t = 84

� ( ( ) ( ) ) �

� x # # x x x �

t = 85

� ( ( ) ( ) ) �

� x # # x x x �

t = 86

� ( ( ) ( ) ) �

� x # # x x x �

t = 87

� ( ( ) ( ) ) �

� # # # x x x �

t = 88

� ( ( ) ( ) ) �

� # # # x x x �

t = 89

� ( ( ) ( ) ) �

� # # # x x x �

t = 90

� ( ( ) ( ) ) �

� # # # x x x �

t = 91

� ( ( ) ( ) ) �

� x # # x x x �

t = 92

� ( ( ) ( ) ) �

� x # # x x x �

t = 93

� ( ( ) ( ) ) �

� x # # x x x �

t = 94

� ( ( ) ( ) ) �

� x # # x x x �

t = 95

� ( ( ) ( ) ) �

� x # x x x x �

t = 96

� ( ( ) ( ) ) �

� x # x x x x �

t = 97

� ( ( ) ( ) ) �

� x # x x x x �

t = 98

� ( ( ) ( ) ) �

� x # x x x x �

t = 99

� ( ( ) ( ) ) �

� x # x x # x �

t = 100

� ( ( ) ( ) ) �

� x # x x # x �

t = 101

� ( ( ) ( ) ) �

� x # x x # x �

t = 102

� ( ( ) ( ) ) �

� x # x x # x �

t = 103

� ( ( ) ( ) ) �

� x # x x # x �

t = 104

� ( ( ) ( ) ) �

� x # x x # x �

t = 105

� ( ( ) ( ) ) �

� x # x x # x �

t = 106

� ( ( ) ( ) ) �

� x # x x # x �

t = 107

� ( ( ) ( ) ) �

� x # x x # x �

t = 108

� ( ( ) ( ) ) �

� x # x x # x �

t = 109

� ( ( ) ( ) ) �

� x # x x # x �

t = 110

� ( ( ) ( ) ) �

� x # x x # x �

t = 111

� ( ( ) ( ) ) �

� x # x x # x �

t = 112

� ( ( ) ( ) ) �

� x # x x # x �

t = 113

� ( ( ) ( ) ) �

� x # x x # x �

t = 114

� ( ( ) ( ) ) �

� x # x x # x �

t = 115

� ( ( ) ( ) ) �

� # # x x # x �

t = 116

� ( ( ) ( ) ) �

� # # x x # x �

t = 117

� ( ( ) ( ) ) �

� # # x x # x �

t = 118

� ( ( ) ( ) ) �

� # # x x # x �

t = 119

� ( ( ) ( ) ) �

� x # x x # x �

t = 120

� ( ( ) ( ) ) �

� x # x x # x �

t = 121

� ( ( ) ( ) ) �

� x # x x # x �

t = 122

� ( ( ) ( ) ) �

� x # x x # x �

t = 123

� ( ( ) ( ) ) �

� x # x x # x �

t = 124

� ( ( ) ( ) ) �

� x # x x # x �

t = 125

� ( ( ) ( ) ) �

� x # x # # x �

t = 126

� ( ( ) ( ) ) �

� x # x # # x �

t = 127

� ( ( ) ( ) ) �

� x # x # # x �

t = 128

� ( ( ) ( ) ) �

� x # x # # x �

t = 129

� ( ( ) ( ) ) �

� x # x x # x �

t = 130

� ( ( ) ( ) ) �

� x # x x # x �

t = 131

� ( ( ) ( ) ) �

� x # x x x x �

t = 132

� ( ( ) ( ) ) �

� x # x x x x �

t = 133

� ( ( ) ( ) ) �

� x # x x x # �

t = 134

� ( ( ) ( ) ) �

� x # x x x # �

t = 135

� ( ( ) ( ) ) �

� x # x x x # �

t = 136

� ( ( ) ( ) ) �

� x # x x x # �

t = 137

� ( ( ) ( ) ) �

� x # x x x # �

t = 138

� ( ( ) ( ) ) �

� x # x x x # �

t = 139

� ( ( ) ( ) ) �

� x # x x x # �

t = 140

� ( ( ) ( ) ) �

� x # x x x # �

t = 141

� ( ( ) ( ) ) �

� x # x x x # �

t = 142

� ( ( ) ( ) ) �

� x # x x x # �

t = 143

� ( ( ) ( ) ) �

� x # x x x # �

t = 144

� ( ( ) ( ) ) �

� x # x x x # �

t = 145

� ( ( ) ( ) ) �

� x # x x x # �

t = 146

� ( ( ) ( ) ) �

� x # x x x # �

t = 147

� ( ( ) ( ) ) �

� x # x x x # �

t = 148

� ( ( ) ( ) ) �

� x # x x x # �

t = 149

� ( ( ) ( ) ) �

� x # x x x # �

t = 150

� ( ( ) ( ) ) �

� x # x x x # �

t = 151

� ( ( ) ( ) ) �

� # # x x x # �

t = 152

� ( ( ) ( ) ) �

� # # x x x # �

t = 153

� ( ( ) ( ) ) �

� # # x x x # �

t = 154

� ( ( ) ( ) ) �

� # # x x x # �

t = 155

� ( ( ) ( ) ) �

� x # x x x # �

t = 156

� ( ( ) ( ) ) �

� x # x x x # �

t = 157

� ( ( ) ( ) ) �

� x # x x x # �

t = 158

� ( ( ) ( ) ) �

� x # x x x # �

t = 159

� ( ( ) ( ) ) �

� x # x x x # �

t = 160

� ( ( ) ( ) ) �

� x # x x x # �

t = 161

� ( ( ) ( ) ) �

� x # x # x # �

t = 162

� ( ( ) ( ) ) �

� x # x # x # �

t = 163

� ( ( ) ( ) ) �

� x # x # # # �

t = 164

� ( ( ) ( ) ) �

� x # x # # # �

t = 165

� ( ( ) ( ) ) �

� x # x # # # �

t = 166

� ( ( ) ( ) ) �

� x # x # # # �

t = 167

� ( ( ) ( ) ) �

� x # x # # # �

t = 168

� ( ( ) ( ) ) �

� x # x # # # �

t = 169

� ( ( ) ( ) ) �

� x # x # # # �

t = 170

� ( ( ) ( ) ) �

� x # x # # # �

t = 171

� ( ( ) ( ) ) �

� x # x # # # �

t = 172

� ( ( ) ( ) ) �

� x # x # # # �

t = 173

� ( ( ) ( ) ) �

� x # x # # # �

t = 174

� ( ( ) ( ) ) �

� x # x # # # �

t = 175

� ( ( ) ( ) ) �

� x # x # # # �

t = 176

� ( ( ) ( ) ) �

� x # x # # # �

t = 177

� ( ( ) ( ) ) �

� x # x # # # �

t = 178

� ( ( ) ( ) ) �

� x # x # # # �

t = 179

� ( ( ) ( ) ) �

� # # x # # # �

t = 180

� ( ( ) ( ) ) �

� # # x # # # �

t = 181

� ( ( ) ( ) ) �

� # # x # # # �

t = 182

� ( ( ) ( ) ) �

� # # x # # # �

t = 183

� ( ( ) ( ) ) �

� x # x # # # �

t = 184

� ( ( ) ( ) ) �

� x # x # # # �

t = 185

� ( ( ) ( ) ) �

� x # x # # # �

t = 186

� ( ( ) ( ) ) �

� x # x # # # �

t = 187

� ( ( ) ( ) ) �

� x # x # # # �

t = 188

� ( ( ) ( ) ) �

� x # x # # # �

t = 189

� ( ( ) ( ) ) �

� x # x # # # �

t = 190

� ( ( ) ( ) ) �

� x # x # # # �

t = 191

� ( ( ) ( ) ) �

� x # x # x # �

t = 192

� ( ( ) ( ) ) �

� x # x # x # �

t = 193

� ( ( ) ( ) ) �

� x # x # x # �

t = 194

� ( ( ) ( ) ) �

� x # x # x # �

t = 195

� ( ( ) ( ) ) �

� x # x # x # �

t = 196

� ( ( ) ( ) ) �

� x # x # x # �

t = 197

� ( ( ) ( ) ) �

� x # x x x # �

t = 198

� ( ( ) ( ) ) �

� x # x x x # �

t = 199

� ( ( ) ( ) ) �

� x # x x x # �

t = 200

� ( ( ) ( ) ) �

� x # x x x # �

t = 201

� ( ( ) ( ) ) �

� x # x x x x �

t = 202

� ( ( ) ( ) ) �

� x # x x x x �

t = 203

� ( ( ) ( ) ) �

� x # x x x x �

t = 204

� ( ( ) ( ) ) �

� x # x x x x �

t = 205

� ( ( ) ( ) ) �

� x # x x x x �

t = 206

� ( ( ) ( ) ) �

� x # x x x x �

t = 207

� ( ( ) ( ) ) �

� x # x x x x �

t = 208

� ( ( ) ( ) ) �

� x # x x x x �

t = 209

� ( ( ) ( ) ) �

� x # x x x x �

t = 210

� ( ( ) ( ) ) �

� x # x x x x �

t = 211

� ( ( ) ( ) ) �

� x # x x x x �

t = 212

� ( ( ) ( ) ) �

� x # x x x x �

t = 213

� ( ( ) ( ) ) �

� x x x x x x �

t = 214

� ( ( ) ( ) ) �

� x x x x x x �

t = 215

� ( ( ) ( ) ) �

� x x x x x x �

t = 216

� ( ( ) ( ) ) �

� x x x x x x �

t = 217

� ( ( ) ( ) ) �

� x x x # x x �

t = 218

� ( ( ) ( ) ) �

� x x x # x x �

t = 219

� ( ( ) ( ) ) �

� x x x # # x �

t = 220

� ( ( ) ( ) ) �

� x x x # # x �

t = 221

� ( ( ) ( ) ) �

� x x x # # x �

t = 222

� ( ( ) ( ) ) �

� x x x # # x �

t = 223

� ( ( ) ( ) ) �

� x x x # # x �

t = 224

� ( ( ) ( ) ) �

� x x x # # x �

t = 225

� ( ( ) ( ) ) �

� x x x # # x �

t = 226

� ( ( ) ( ) ) �

� x x x # # x �

t = 227

� ( ( ) ( ) ) �

� x x x # # x �

t = 228

� ( ( ) ( ) ) �

� x x x # # x �

t = 229

� ( ( ) ( ) ) �

� x x x # # x �

t = 230

� ( ( ) ( ) ) �

� x x x # # x �

t = 231

� ( ( ) ( ) ) �

� x x x # # x �

t = 232

� ( ( ) ( ) ) �

� x x x # # x �

t = 233

� ( ( ) ( ) ) �

� x x x # # x �

t = 234

� ( ( ) ( ) ) �

� x x x # # x �

t = 235

� ( ( ) ( ) ) �

� # x x # # x �

t = 236

� ( ( ) ( ) ) �

� # x x # # x �

t = 237

� ( ( ) ( ) ) �

� # x x # # x �

t = 238

� ( ( ) ( ) ) �

� # x x # # x �

t = 239

� ( ( ) ( ) ) �

� # x # # # x �

t = 240

� ( ( ) ( ) ) �

� # x # # # x �

t = 241

� ( ( ) ( ) ) �

� # x # # # x �

t = 242

� ( ( ) ( ) ) �

� # x # # # x �

t = 243

� ( ( ) ( ) ) �

� # x # # # x �

t = 244

� ( ( ) ( ) ) �

� # x # # # x �

t = 245

� ( ( ) ( ) ) �

� # x # # # x �

t = 246

� ( ( ) ( ) ) �

� # x # # # x �

t = 247

� ( ( ) ( ) ) �

� # x # # # x �

t = 248

� ( ( ) ( ) ) �

� # x # # # x �

t = 249

� ( ( ) ( ) ) �

� # x # # # x �

t = 250

� ( ( ) ( ) ) �

� # x # # # x �

t = 251

� ( ( ) ( ) ) �

� # x # # # x �

t = 252

� ( ( ) ( ) ) �

� # x # # # x �

t = 253

� ( ( ) ( ) ) �

� # x # # # x �

t = 254

� ( ( ) ( ) ) �

� # x # # # x �

t = 255

� ( ( ) ( ) ) �

� # x # # # x �

t = 256

� ( ( ) ( ) ) �

� # x # # # x �

t = 257

� ( ( ) ( ) ) �

� # # # # # x �

t = 258

� ( ( ) ( ) ) �

� # # # # # x �

t = 259

� ( ( ) ( ) ) �

� # # # # # x �

t = 260

� ( ( ) ( ) ) �

� # # # # # x �

t = 261

� ( ( ) ( ) ) �

� # x # # # x �

t = 262

� ( ( ) ( ) ) �

� # x # # # x �

t = 263

� ( ( ) ( ) ) �

� # x x # # x �

t = 264

� ( ( ) ( ) ) �

� # x x # # x �

t = 265

� ( ( ) ( ) ) �

� # x x # # x �

t = 266

� ( ( ) ( ) ) �

� # x x # # x �

t = 267

� ( ( ) ( ) ) �

� # x x # # x �

t = 268

� ( ( ) ( ) ) �

� # x x # # x �

t = 269

� ( ( ) ( ) ) �

� # x x # # x �

t = 270

� ( ( ) ( ) ) �

� # x x # # x �

t = 271

� ( ( ) ( ) ) �

� x x x # # x �

t = 272

� ( ( ) ( ) ) �

� x x x # # x �

t = 273

� ( ( ) ( ) ) �

� x # x # # x �

t = 274

� ( ( ) ( ) ) �

� x # x # # x �

t = 275

� ( ( ) ( ) ) �

� x # # # # x �

t = 276

� ( ( ) ( ) ) �

� x # # # # x �

t = 277

� ( ( ) ( ) ) �

� x # # # # x �

t = 278

� ( ( ) ( ) ) �

� x # # # # x �

t = 279

� ( ( ) ( ) ) �

� x # # # # x �

t = 280

� ( ( ) ( ) ) �

� x # # # # x �

t = 281

� ( ( ) ( ) ) �

� x # # # # x �

t = 282

� ( ( ) ( ) ) �

� x # # # # x �

t = 283

� ( ( ) ( ) ) �

� x # # # # x �

t = 284

� ( ( ) ( ) ) �

� x # # # # x �

t = 285

� ( ( ) ( ) ) �

� x # # # # x �

t = 286

� ( ( ) ( ) ) �

� x # # # # x �

t = 287

� ( ( ) ( ) ) �

� # # # # # x �

t = 288

� ( ( ) ( ) ) �

� # # # # # x �

t = 289

� ( ( ) ( ) ) �

� # # # # # x �

t = 290

� ( ( ) ( ) ) �

� # # # # # x �

t = 291

� ( ( ) ( ) ) �

� x # # # # x �

t = 292

� ( ( ) ( ) ) �

� x # # # # x �

t = 293

� ( ( ) ( ) ) �

� x # # # # x �

t = 294

� ( ( ) ( ) ) �

� x # # # # x �

t = 295

� ( ( ) ( ) ) �

� x # x # # x �

t = 296

� ( ( ) ( ) ) �

� x # x # # x �

t = 297

� ( ( ) ( ) ) �

� x # x # # x �

t = 298

� ( ( ) ( ) ) �

� x # x # # x �

t = 299

� ( ( ) ( ) ) �

� x # x # # x �

t = 300

� ( ( ) ( ) ) �

� x # x # # x �

t = 301

� ( ( ) ( ) ) �

� x x x # # x �

t = 302

� ( ( ) ( ) ) �

� x x x # # x �

t = 303

� ( ( ) ( ) ) �

� x x x # # x �

t = 304

� ( ( ) ( ) ) �

� x x x # # x �

t = 305

� ( ( ) ( ) ) �

� x x x # # x �

t = 306

� ( ( ) ( ) ) �

� x x x # # x �

t = 307

� ( ( ) ( ) ) �

� x x x # x x �

t = 308

� ( ( ) ( ) ) �

� x x x # x x �

t = 309

� ( ( ) ( ) ) �

� x x x # x # �

t = 310

� ( ( ) ( ) ) �

� x x x # x # �

t = 311

� ( ( ) ( ) ) �

� x x x # x # �

t = 312

� ( ( ) ( ) ) �

� x x x # x # �

t = 313

� ( ( ) ( ) ) �

� x x x # x # �

t = 314

� ( ( ) ( ) ) �

� x x x # x # �

t = 315

� ( ( ) ( ) ) �

� x x x # x # �

t = 316

� ( ( ) ( ) ) �

� x x x # x # �

t = 317

� ( ( ) ( ) ) �

� x x x # x # �

t = 318

� ( ( ) ( ) ) �

� x x x # x # �

t = 319

� ( ( ) ( ) ) �

� x x x # x # �

t = 320

� ( ( ) ( ) ) �

� x x x # x # �

t = 321

� ( ( ) ( ) ) �

� x x x # x # �

t = 322

� ( ( ) ( ) ) �

� x x x # x # �

t = 323

� ( ( ) ( ) ) �

� x x x # x # �

t = 324

� ( ( ) ( ) ) �

� x x x # x # �

t = 325

� ( ( ) ( ) ) �

� x x x # x # �

t = 326

� ( ( ) ( ) ) �

� x x x # x # �

t = 327

� ( ( ) ( ) ) �

� # x x # x # �

t = 328

� ( ( ) ( ) ) �

� # x x # x # �

t = 329

� ( ( ) ( ) ) �

� # x x # x # �

t = 330

� ( ( ) ( ) ) �

� # x x # x # �

t = 331

� ( ( ) ( ) ) �

� # x # # x # �

t = 332

� ( ( ) ( ) ) �

� # x # # x # �

t = 333

� ( ( ) ( ) ) �

� # x # # x # �

t = 334

� ( ( ) ( ) ) �

� # x # # x # �

t = 335

� ( ( ) ( ) ) �

� # x # # x # �

t = 336

� ( ( ) ( ) ) �

� # x # # x # �

t = 337

� ( ( ) ( ) ) �

� # x # # x # �

t = 338

� ( ( ) ( ) ) �

� # x # # x # �

t = 339

� ( ( ) ( ) ) �

� # x # # x # �

t = 340

� ( ( ) ( ) ) �

� # x # # x # �

t = 341

� ( ( ) ( ) ) �

� # x # # x # �

t = 342

� ( ( ) ( ) ) �

� # x # # x # �

t = 343

� ( ( ) ( ) ) �

� # x # # x # �

t = 344

� ( ( ) ( ) ) �

� # x # # x # �

t = 345

� ( ( ) ( ) ) �

� # x # # x # �

t = 346

� ( ( ) ( ) ) �

� # x # # x # �

t = 347

� ( ( ) ( ) ) �

� # x # # x # �

t = 348

� ( ( ) ( ) ) �

� # x # # x # �

t = 349

� ( ( ) ( ) ) �

� # # # # x # �

t = 350

� ( ( ) ( ) ) �

� # # # # x # �

t = 351

� ( ( ) ( ) ) �

� # # # # x # �

t = 352

� ( ( ) ( ) ) �

� # # # # x # �

t = 353

� ( ( ) ( ) ) �

� # x # # x # �

t = 354

� ( ( ) ( ) ) �

� # x # # x # �

t = 355

� ( ( ) ( ) ) �

� # x x # x # �

t = 356

� ( ( ) ( ) ) �

� # x x # x # �

t = 357

� ( ( ) ( ) ) �

� # x x # x # �

t = 358

� ( ( ) ( ) ) �

� # x x # x # �

t = 359

� ( ( ) ( ) ) �

� # x x # x # �

t = 360

� ( ( ) ( ) ) �

� # x x # x # �

t = 361

� ( ( ) ( ) ) �

� # x x # x # �

t = 362

� ( ( ) ( ) ) �

� # x x # x # �

t = 363

� ( ( ) ( ) ) �

� x x x # x # �

t = 364

� ( ( ) ( ) ) �

� x x x # x # �

t = 365

� ( ( ) ( ) ) �

� x # x # x # �

t = 366

� ( ( ) ( ) ) �

� x # x # x # �

t = 367

� ( ( ) ( ) ) �

� x # # # x # �

t = 368

� ( ( ) ( ) ) �

� x # # # x # �

t = 369

� ( ( ) ( ) ) �

� x # # # x # �

t = 370

� ( ( ) ( ) ) �

� x # # # x # �

t = 371

� ( ( ) ( ) ) �

� x # # # x # �

t = 372

� ( ( ) ( ) ) �

� x # # # x # �

t = 373

� ( ( ) ( ) ) �

� x # # # x # �

t = 374

� ( ( ) ( ) ) �

� x # # # x # �

t = 375

� ( ( ) ( ) ) �

� x # # # x # �

t = 376

� ( ( ) ( ) ) �

� x # # # x # �

t = 377

� ( ( ) ( ) ) �

� x # # # x # �

t = 378

� ( ( ) ( ) ) �

� x # # # x # �

t = 379

� ( ( ) ( ) ) �

� # # # # x # �

t = 380

� ( ( ) ( ) ) �

� # # # # x # �

t = 381

� ( ( ) ( ) ) �

� # # # # x # �

t = 382

� ( ( ) ( ) ) �

� # # # # x # �

t = 383

� ( ( ) ( ) ) �

� x # # # x # �

t = 384

� ( ( ) ( ) ) �

� x # # # x # �

t = 385

� ( ( ) ( ) ) �

� x # # # x # �

t = 386

� ( ( ) ( ) ) �

� x # # # x # �

t = 387

� ( ( ) ( ) ) �

� x # x # x # �

t = 388

� ( ( ) ( ) ) �

� x # x # x # �

t = 389

� ( ( ) ( ) ) �

� x # x # x # �

t = 390

� ( ( ) ( ) ) �

� x # x # x # �

t = 391

� ( ( ) ( ) ) �

� x # x # x # �

t = 392

� ( ( ) ( ) ) �

� x # x # x # �

t = 393

� ( ( ) ( ) ) �

� x x x # x # �

t = 394

� ( ( ) ( ) ) �

� x x x # x # �

t = 395

� ( ( ) ( ) ) �

� x x x # x # �

t = 396

� ( ( ) ( ) ) �

� x x x # x # �

t = 397

� ( ( ) ( ) ) �

� x x x # x # �

t = 398

� ( ( ) ( ) ) �

� x x x # x # �

t = 399

� ( ( ) ( ) ) �

� x x x # x # �

t = 400

� ( ( ) ( ) ) �

� x x x # x # �

t = 401

� ( ( ) ( ) ) �

� x x x # x x �

t = 402

� ( ( ) ( ) ) �

� x x x # x x �

t = 403

� ( ( ) ( ) ) �

� x x x # x x �

t = 404

� ( ( ) ( ) ) �

� x x x # x x �

t = 405

� ( ( ) ( ) ) �

� x x x # x x �

t = 406

� ( ( ) ( ) ) �

� x x x # x x �

t = 407

� ( ( ) ( ) ) �

� x x x # x x �

t = 408

� ( ( ) ( ) ) �

� x x x # x x �

t = 409

� ( ( ) ( ) ) �

� x x x x x x �

t = 410

� ( ( ) ( ) ) �

� x x x x x x �

t = 411

� ( ( ) ( ) ) �

� x x x x x x �

t = 412

� ( ( ) ( ) ) �

� x x x x x x �

t = 413

� ( ( ) ( ) ) �

� x x x x x x �

t = 414

� ( ( ) ( ) ) �

� x x x x x x �

t = 415

� ( ( ) ( ) ) �

� x x x x x x �

t = 416

� ( ( ) ( ) ) �

� x x x x x x �

t = 417

� ( ( ) ( ) ) �

� x x x x x x �

t = 418

� ( ( ) ( ) ) �

� x x x x x x �

t = 419

� ( ( ) ( ) ) �

� x x x x x x �

t = 420

� ( ( ) ( ) ) �

� x x x x x x �

t = 421

� ( ( ) ( ) ) �

� x x x x x x �

t = 422

� ( ( ) ( ) ) �

� x x x x x x �

t = 423

� ( ( ) ( ) ) �

� x x x x x x �

t = 424

� ( ( ) ( ) ) �

� x x x x x x �

t = 425

� ( ( ) ( ) ) �

� x x x x x x �

t = 426

� ( ( ) ( ) ) �

� x x x x x x �

t = 427

� ( ( ) ( ) ) �

� x x x x x x �

t = 428

� ( ( ) ( ) ) �

� x x x x x x �

t = 429

� ( ( ) ( ) ) �

� x x x x x x �

The simulated DLBA accepts the input

t = 430

� ( ( ) ( ) ) �

� x x x x x x �

t = 431

� ( ( ) ( ) ) �

� x x x x x x �

t = 432

� ( ( ) ( ) ) �

� x x x x x x �

t = 433

� ( ( ) ( ) ) �

� x x x x x x �

t = 434

� ( ( ) ( ) ) �

� x x x x x x �

t = 435

� ( ( ) ( ) ) �

� x x x x x x �

t = 436

� ( ( ) ( ) ) �

� x x x x x x �

t = 437

� ( ( ) ( ) ) �

� x x x x x x �

t = 438

� ( ( ) ( ) ) �

� x x x x x x �

t = 439

� ( ( ) ( ) ) �

� x x x x x x �

t = 440

� ( ( ) ( ) ) �

� x x x x x x �

t = 441

� ( ( ) ( ) ) �

� x x x x x x �

t = 442

� ( ( ) ( ) ) �

� x x x x x x �

t = 443

� ( ( ) ( ) ) �

� x x x x x x �

t = 444

� ( ( ) ( ) ) �

� x x x x x x �

t = 445

� ( ( ) ( ) ) �

� x x x x x x �

t = 446

� ( ( ) ( ) ) �

� x x x x x x �

t = 447

� ( ( ) ( ) ) �

� x x x x x x �

t = 448

� ( ( ) ( ) ) �

� x x x x x x �

t = 449

� ( ( ) ( ) ) �

� x x x x x x �

t = 450

� ( ( ) ( ) ) �

� x x x x x x �

t = 451

� ( ( ) ( ) ) �

� x x x x x x �

t = 452

� ( ( ) ( ) ) �

� x x x x x x �

t = 453

� ( ( ) ( ) ) �

� x x x x x x �

t = 454

� ( ( ) ( ) ) �

� x x x x x x �

t = 455

� ( ( ) ( ) ) �

� x x x x x x �

t = 456

� ( ( ) ( ) ) �

� x x x x x x �

t = 457

� ( ( ) ( ) ) �

� x x x x x x �

t = 458

� ( ( ) ( ) ) �

� x x x x x x �

t = 459

� ( ( ) ( ) ) �

q̂b0,4

� x x x x x x �

t = 460

� ( ( ) ( ) ) �

q̂b0,3

� x x x x x x �

t = 461

� ( ( ) ( ) ) �

q̂0,3

� x x x x x x �

t = 462

� ( ( ) ( ) ) �

q̂0,4

� x x x x x x �

t = 463

� ( ( ) ( ) ) �

� x x x x x x �

t = 464

� ( ( ) ( ) ) �

� x x x x x x �

t = 465

� ( ( ) ( ) ) �

� # x x x x x �

t = 466

� ( ( ) ( ) ) �

� # x x x x x �

t = 467

� ( ( ) ( ) ) �

� # x x x x x �

t = 468

� ( ( ) ( ) ) �

� # x x x x x �

t = 469

� ( ( ) ( ) ) �

� # x # x x x �

t = 470

� ( ( ) ( ) ) �

� # x # x x x �

t = 471

� ( ( ) ( ) ) �

� # x # x x x �

t = 472

� ( ( ) ( ) ) �

� # x # x x x �

t = 473

� ( ( ) ( ) ) �

� # x # x x x �

t = 474

� ( ( ) ( ) ) �

� # x # x x x �

t = 475

� ( ( ) ( ) ) �

q̂b0,4

� # x # x x x �

t = 476

� ( ( ) ( ) ) �

q̂b0,3

� # x # x x x �

t = 477

� ( ( ) ( ) ) �

q̂b0,4

� # x # x x x �

t = 478

� ( ( ) ( ) ) �

q̂b0,3

� # x # x x x �

t = 479

� ( ( ) ( ) ) �

q̂0,3

� # x # x x x �

t = 480

� ( ( ) ( ) ) �

q̂0,4

� # x # x x x �

t = 481

� ( ( ) ( ) ) �

q̂0,3

� # x # x x x �

t = 482

� ( ( ) ( ) ) �

q̂0,4

� # x # x x x �

t = 483

� ( ( ) ( ) ) �

� # x # x x x �

t = 484

� ( ( ) ( ) ) �

� # x # x x x �

t = 485

� ( ( ) ( ) ) �

� # x # x x x �

t = 486

� ( ( ) ( ) ) �

� # x # x x x �

t = 487

� ( ( ) ( ) ) �

� # # # x x x �

t = 488

� ( ( ) ( ) ) �

� # # # x x x �

t = 489

� ( ( ) ( ) ) �

� # # # x x x �

t = 490

� ( ( ) ( ) ) �

� # # # x x x �

t = 491

� ( ( ) ( ) ) �

� # x # x x x �

t = 492

� ( ( ) ( ) ) �

� # x # x x x �

t = 493

� ( ( ) ( ) ) �

� # x x x x x �

t = 494

� ( ( ) ( ) ) �

� # x x x x x �

t = 495

� ( ( ) ( ) ) �

� # x x x x x �

t = 496

� ( ( ) ( ) ) �

� # x x x x x �

t = 497

� ( ( ) ( ) ) �

� # x x x # x �

t = 498

� ( ( ) ( ) ) �

� # x x x # x �

t = 499

� ( ( ) ( ) ) �

� # x x x # x �

t = 500

� ( ( ) ( ) ) �

� # x x x # x �

t = 501

� ( ( ) ( ) ) �

� # x x x # x �

t = 502

� ( ( ) ( ) ) �

� # x x x # x �

t = 503

� ( ( ) ( ) ) �

� # x x x # x �

t = 504

� ( ( ) ( ) ) �

� # x x x # x �

t = 505

� ( ( ) ( ) ) �

� # x x x # x �

t = 506

� ( ( ) ( ) ) �

� # x x x # x �

t = 507

� ( ( ) ( ) ) �

q̂b0,4

� # x x x # x �

t = 508

� ( ( ) ( ) ) �

q̂b0,3

� # x x x # x �

t = 509

� ( ( ) ( ) ) �

q̂b0,4

� # x x x # x �

t = 510

� ( ( ) ( ) ) �

q̂b0,3

� # x x x # x �

t = 511

� ( ( ) ( ) ) �

q̂0,3

� # x x x # x �

t = 512

� ( ( ) ( ) ) �

q̂0,4

� # x x x # x �

t = 513

� ( ( ) ( ) ) �

q̂0,3

� # x x x # x �

t = 514

� ( ( ) ( ) ) �

q̂0,4

� # x x x # x �

t = 515

� ( ( ) ( ) ) �

� # x x x # x �

t = 516

� ( ( ) ( ) ) �

� # x x x # x �

t = 517

� ( ( ) ( ) ) �

� # x x x # x �

t = 518

� ( ( ) ( ) ) �

� # x x x # x �

t = 519

� ( ( ) ( ) ) �

� # # x x # x �

t = 520

� ( ( ) ( ) ) �

� # # x x # x �

t = 521

� ( ( ) ( ) ) �

� # # # x # x �

t = 522

� ( ( ) ( ) ) �

� # # # x # x �

t = 523

� ( ( ) ( ) ) �

� # # # x # x �

t = 524

� ( ( ) ( ) ) �

� # # # x # x �

t = 525

� ( ( ) ( ) ) �

� # # # x # x �

t = 526

� ( ( ) ( ) ) �

� # # # x # x �

t = 527

� ( ( ) ( ) ) �

q̂b0,4

� # # # x # x �

t = 528

� ( ( ) ( ) ) �

q̂b0,3

� # # # x # x �

t = 529

� ( ( ) ( ) ) �

q̂b0,4

� # # # x # x �

t = 530

� ( ( ) ( ) ) �

q̂b0,3

� # # # x # x �

t = 531

� ( ( ) ( ) ) �

q̂b0,4

� # # # x # x �

t = 532

� ( ( ) ( ) ) �

q̂b0,3

� # # # x # x �

t = 533

� ( ( ) ( ) ) �

q̂b0,4

� # # # x # x �

t = 534

� ( ( ) ( ) ) �

q̂b0,3

� # # # x # x �

t = 535

� ( ( ) ( ) ) �

q̂0,3

� # # # x # x �

t = 536

� ( ( ) ( ) ) �

q̂0,4

� # # # x # x �

t = 537

� ( ( ) ( ) ) �

q̂0,3

� # # # x # x �

t = 538

� ( ( ) ( ) ) �

q̂0,4

� # # # x # x �

t = 539

� ( ( ) ( ) ) �

q̂0,3

� # # # x # x �

t = 540

� ( ( ) ( ) ) �

q̂0,4

� # # # x # x �

t = 541

� ( ( ) ( ) ) �

q̂0,3

� # # # x # x �

t = 542

� ( ( ) ( ) ) �

q̂0,4

� # # # x # x �

t = 543

� ( ( ) ( ) ) �

� # # # x # x �

t = 544

� ( ( ) ( ) ) �

� # # # x # x �

t = 545

� ( ( ) ( ) ) �

� # # # x # x �

t = 546

� ( ( ) ( ) ) �

� # # # x # x �

t = 547

� ( ( ) ( ) ) �

� # # # x # x �

t = 548

� ( ( ) ( ) ) �

� # # # x # x �

t = 549

� ( ( ) ( ) ) �

� # # x x # x �

t = 550

� ( ( ) ( ) ) �

� # # x x # x �

t = 551

� ( ( ) ( ) ) �

� # # x x # x �

t = 552

� ( ( ) ( ) ) �

� # # x x # x �

t = 553

� ( ( ) ( ) ) �

� # # x x # x �

t = 554

� ( ( ) ( ) ) �

� # # x x # x �

t = 555

� ( ( ) ( ) ) �

� # # x x # x �

t = 556

� ( ( ) ( ) ) �

� # # x x # x �

t = 557

� ( ( ) ( ) ) �

� # # x x # x �

t = 558

� ( ( ) ( ) ) �

� # # x x # x �

t = 559

� ( ( ) ( ) ) �

� # x x x # x �

t = 560

� ( ( ) ( ) ) �

� # x x x # x �

t = 561

� ( ( ) ( ) ) �

� # x x x # x �

t = 562

� ( ( ) ( ) ) �

� # x x x # x �

t = 563

� ( ( ) ( ) ) �

� # x x # # x �

t = 564

� ( ( ) ( ) ) �

� # x x # # x �

t = 565

� ( ( ) ( ) ) �

� # x x # # x �

t = 566

� ( ( ) ( ) ) �

� # x x # # x �

t = 567

� ( ( ) ( ) ) �

� # x x x # x �

t = 568

� ( ( ) ( ) ) �

� # x x x # x �

t = 569

� ( ( ) ( ) ) �

� # x x x x x �

t = 570

� ( ( ) ( ) ) �

� # x x x x x �

t = 571

� ( ( ) ( ) ) �

� # x x x x # �

t = 572

� ( ( ) ( ) ) �

� # x x x x # �

t = 573

� ( ( ) ( ) ) �

� # x x x x # �

t = 574

� ( ( ) ( ) ) �

� # x x x x # �

t = 575

� ( ( ) ( ) ) �

� # x x x x # �

t = 576

� ( ( ) ( ) ) �

� # x x x x # �

t = 577

� ( ( ) ( ) ) �

� # x x x x # �

t = 578

� ( ( ) ( ) ) �

� # x x x x # �

t = 579

� ( ( ) ( ) ) �

� # x x x x # �

t = 580

� ( ( ) ( ) ) �

� # x x x x # �

t = 581

� ( ( ) ( ) ) �

� # x x x x # �

t = 582

� ( ( ) ( ) ) �

� # x x x x # �

t = 583

� ( ( ) ( ) ) �

q̂b0,4

� # x x x x # �

t = 584

� ( ( ) ( ) ) �

q̂b0,3

� # x x x x # �

t = 585

� ( ( ) ( ) ) �

q̂b0,4

� # x x x x # �

t = 586

� ( ( ) ( ) ) �

q̂b0,3

� # x x x x # �

t = 587

� ( ( ) ( ) ) �

q̂0,3

� # x x x x # �

t = 588

� ( ( ) ( ) ) �

q̂0,4

� # x x x x # �

t = 589

� ( ( ) ( ) ) �

q̂0,3

� # x x x x # �

t = 590

� ( ( ) ( ) ) �

q̂0,4

� # x x x x # �

t = 591

� ( ( ) ( ) ) �

� # x x x x # �

t = 592

� ( ( ) ( ) ) �

� # x x x x # �

t = 593

� ( ( ) ( ) ) �

� # x x x x # �

t = 594

� ( ( ) ( ) ) �

� # x x x x # �

t = 595

� ( ( ) ( ) ) �

� # # x x x # �

t = 596

� ( ( ) ( ) ) �

� # # x x x # �

t = 597

� ( ( ) ( ) ) �

� # # # x x # �

t = 598

� ( ( ) ( ) ) �

� # # # x x # �

t = 599

� ( ( ) ( ) ) �

� # # # x x # �

t = 600

� ( ( ) ( ) ) �

� # # # x x # �

t = 601

� ( ( ) ( ) ) �

� # # # x x # �

t = 602

� ( ( ) ( ) ) �

� # # # x x # �

t = 603

� ( ( ) ( ) ) �

q̂b0,4

� # # # x x # �

t = 604

� ( ( ) ( ) ) �

q̂b0,3

� # # # x x # �

t = 605

� ( ( ) ( ) ) �

q̂b0,4

� # # # x x # �

t = 606

� ( ( ) ( ) ) �

q̂b0,3

� # # # x x # �

t = 607

� ( ( ) ( ) ) �

q̂b0,4

� # # # x x # �

t = 608

� ( ( ) ( ) ) �

q̂b0,3

� # # # x x # �

t = 609

� ( ( ) ( ) ) �

q̂b0,4

� # # # x x # �

t = 610

� ( ( ) ( ) ) �

q̂b0,3

� # # # x x # �

t = 611

� ( ( ) ( ) ) �

q̂0,3

� # # # x x # �

t = 612

� ( ( ) ( ) ) �

q̂0,4

� # # # x x # �

t = 613

� ( ( ) ( ) ) �

q̂0,3

� # # # x x # �

t = 614

� ( ( ) ( ) ) �

q̂0,4

� # # # x x # �

t = 615

� ( ( ) ( ) ) �

q̂0,3

� # # # x x # �

t = 616

� ( ( ) ( ) ) �

q̂0,4

� # # # x x # �

t = 617

� ( ( ) ( ) ) �

q̂0,3

� # # # x x # �

t = 618

� ( ( ) ( ) ) �

q̂0,4

� # # # x x # �

t = 619

� ( ( ) ( ) ) �

� # # # x x # �

t = 620

� ( ( ) ( ) ) �

� # # # x x # �

t = 621

� ( ( ) ( ) ) �

� # # # x x # �

t = 622

� ( ( ) ( ) ) �

� # # # x x # �

t = 623

� ( ( ) ( ) ) �

� # # # x x # �

t = 624

� ( ( ) ( ) ) �

� # # # x x # �

t = 625

� ( ( ) ( ) ) �

� # # x x x # �

t = 626

� ( ( ) ( ) ) �

� # # x x x # �

t = 627

� ( ( ) ( ) ) �

� # # x x x # �

t = 628

� ( ( ) ( ) ) �

� # # x x x # �

t = 629

� ( ( ) ( ) ) �

� # # x x # # �

t = 630

� ( ( ) ( ) ) �

� # # x x # # �

t = 631

� ( ( ) ( ) ) �

� # # x x # # �

t = 632

� ( ( ) ( ) ) �

� # # x x # # �

t = 633

� ( ( ) ( ) ) �

� # # x x # # �

t = 634

� ( ( ) ( ) ) �

� # # x x # # �

t = 635

� ( ( ) ( ) ) �

� # # x x # # �

t = 636

� ( ( ) ( ) ) �

� # # x x # # �

t = 637

� ( ( ) ( ) ) �

� # # x x # # �

t = 638

� ( ( ) ( ) ) �

� # # x x # # �

t = 639

� ( ( ) ( ) ) �

q̂b0,4

� # # x x # # �

t = 640

� ( ( ) ( ) ) �

q̂b0,3

� # # x x # # �

t = 641

� ( ( ) ( ) ) �

q̂b0,4

� # # x x # # �

t = 642

� ( ( ) ( ) ) �

q̂b0,3

� # # x x # # �

t = 643

� ( ( ) ( ) ) �

q̂b0,4

� # # x x # # �

t = 644

� ( ( ) ( ) ) �

q̂b0,3

� # # x x # # �

t = 645

� ( ( ) ( ) ) �

q̂0,3

� # # x x # # �

t = 646

� ( ( ) ( ) ) �

q̂0,4

� # # x x # # �

t = 647

� ( ( ) ( ) ) �

q̂0,3

� # # x x # # �

t = 648

� ( ( ) ( ) ) �

q̂0,4

� # # x x # # �

t = 649

� ( ( ) ( ) ) �

q̂0,3

� # # x x # # �

t = 650

� ( ( ) ( ) ) �

q̂0,4

� # # x x # # �

t = 651

� ( ( ) ( ) ) �

� # # x x # # �

t = 652

� ( ( ) ( ) ) �

� # # x x # # �

t = 653

� ( ( ) ( ) ) �

� # # x x # # �

t = 654

� ( ( ) ( ) ) �

� # # x x # # �

t = 655

� ( ( ) ( ) ) �

� # # x x # # �

t = 656

� ( ( ) ( ) ) �

� # # x x # # �

t = 657

� ( ( ) ( ) ) �

� # # x x # # �

t = 658

� ( ( ) ( ) ) �

� # # x x # # �

t = 659

� ( ( ) ( ) ) �

� # # x # # # �

t = 660

� ( ( ) ( ) ) �

� # # x # # # �

t = 661

� ( ( ) ( ) ) �

� # # x # # # �

t = 662

� ( ( ) ( ) ) �

� # # x # # # �

t = 663

� ( ( ) ( ) ) �

� # # x x # # �

t = 664

� ( ( ) ( ) ) �

� # # x x # # �

t = 665

� ( ( ) ( ) ) �

� # # x x x # �

t = 666

� ( ( ) ( ) ) �

� # # x x x # �

t = 667

� ( ( ) ( ) ) �

� # # x x x # �

t = 668

� ( ( ) ( ) ) �

� # # x x x # �

t = 669

� ( ( ) ( ) ) �

� # # x x x # �

t = 670

� ( ( ) ( ) ) �

� # # x x x # �

t = 671

� ( ( ) ( ) ) �

� # # x x x # �

t = 672

� ( ( ) ( ) ) �

� # # x x x # �

t = 673

� ( ( ) ( ) ) �

� # # x x x # �

t = 674

� ( ( ) ( ) ) �

� # # x x x # �

t = 675

� ( ( ) ( ) ) �

� # x x x x # �

t = 676

� ( ( ) ( ) ) �

� # x x x x # �

t = 677

� ( ( ) ( ) ) �

� # x x x x # �

t = 678

� ( ( ) ( ) ) �

� # x x x x # �

t = 679

� ( ( ) ( ) ) �

� # x x # x # �

t = 680

� ( ( ) ( ) ) �

� # x x # x # �

t = 681

� ( ( ) ( ) ) �

� # x x # # # �

t = 682

� ( ( ) ( ) ) �

� # x x # # # �

t = 683

� ( ( ) ( ) ) �

� # x x # # # �

t = 684

� ( ( ) ( ) ) �

� # x x # # # �

t = 685

� ( ( ) ( ) ) �

� # x x # # # �

t = 686

� ( ( ) ( ) ) �

� # x x # # # �

t = 687

� ( ( ) ( ) ) �

� # x x # # # �

t = 688

� ( ( ) ( ) ) �

� # x x # # # �

t = 689

� ( ( ) ( ) ) �

� # x x # # # �

t = 690

� ( ( ) ( ) ) �

� # x x # # # �

t = 691

� ( ( ) ( ) ) �

q̂b0,4

� # x x # # # �

t = 692

� ( ( ) ( ) ) �

q̂b0,3

� # x x # # # �

t = 693

� ( ( ) ( ) ) �

q̂b0,4

� # x x # # # �

t = 694

� ( ( ) ( ) ) �

q̂b0,3

� # x x # # # �

t = 695

� ( ( ) ( ) ) �

q̂0,3

� # x x # # # �

t = 696

� ( ( ) ( ) ) �

q̂0,4

� # x x # # # �

t = 697

� ( ( ) ( ) ) �

q̂0,3

� # x x # # # �

t = 698

� ( ( ) ( ) ) �

q̂0,4

� # x x # # # �

t = 699

� ( ( ) ( ) ) �

� # x x # # # �

t = 700

� ( ( ) ( ) ) �

� # x x # # # �

t = 701

� ( ( ) ( ) ) �

� # x x # # # �

t = 702

� ( ( ) ( ) ) �

� # x x # # # �

t = 703

� ( ( ) ( ) ) �

� # # x # # # �

t = 704

� ( ( ) ( ) ) �

� # # x # # # �

t = 705

� ( ( ) ( ) ) �

� # # # # # # �

t = 706

� ( ( ) ( ) ) �

� # # # # # # �

t = 707

� ( ( ) ( ) ) �

� # # # # # # �

t = 708

� ( ( ) ( ) ) �

� # # # # # # �

t = 709

� ( ( ) ( ) ) �

� # # # # # # �

t = 710

� ( ( ) ( ) ) �

� # # # # # # �

t = 711

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 712

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 713

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 714

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 715

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 716

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 717

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 718

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 719

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 720

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 721

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 722

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 723

� ( ( ) ( ) ) �

q̂b0,4

� # # # # # # �

t = 724

� ( ( ) ( ) ) �

q̂b0,3

� # # # # # # �

t = 725

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 726

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 727

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 728

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 729

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 730

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 731

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 732

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 733

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 734

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 735

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 736

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 737

� ( ( ) ( ) ) �

q̂b0,2

� # # # # # # �

t = 738

� ( ( ) ( ) ) �

q̂b0,1

� # # # # # # �

t = 739

� ( ( ) ( ) ) �

� # # # # # # �

A3. RPCA PMp

RPCA that accepts all well-formed parentheses

q0,2 ( ( ) ( ) )

( ) ( ) )

( q0,2 ( ) ( ) )

( (q0,1

) ( ) )

( ( q0,2 ) ( ) )

( ( )q0,1

( ( ) q0,2 ( ) )

( ( ) (q0,1

( ( ) ( q0,2 ) )

( ( ) ( )q0,1

t = 10

( ( ) ( ) q0,2 )

t = 11

( ( ) ( ) )q0,1

t = 12

( ( ) ( ) ) q0,2

t = 13

( ( ) ( ) ) q0,4

t = 14

( ( ) ( ) )q0,3

t = 15

( ( ) ( ) q0,4 )

t = 16

( ( ) ( )q0,3

t = 17

( ( ) ( q0,4 ) )

t = 18

( ( ) (q0,3

t = 19

( ( ) q0,4 ( ) )

t = 20

( ( )q0,3

t = 21

( ( q0,4 ) ( ) )

t = 22

( (q0,3

) ( ) )

t = 23

( q0,4 ( ) ( ) )

t = 24

( ) ( ) )

t = 25

q0,4 ( ( ) ( ) )

t = 26

q2 ( ( ) ( ) )

t = 27

( ) ( ) )

t = 28

( q2 ( ) ( ) )

t = 29

) ( ) )

t = 30

( ( q2 ) ( ) )

t = 31

( ()xqb4

t = 32

( ( )x qb

3( ) )

t = 33

( ( )x

t = 34

( ( )x

q4 ( ) )

t = 35

( ()xq3

t = 36

( ( q4)x

t = 37

t = 38

t = 39

t = 40

q2 ( ) )

t = 41

t = 42

( q2 ) )

t = 43

()xqb4

t = 44

( )x qb

t = 45

t = 46

t = 47

t = 48

( q4)x

t = 49

t = 50

t = 51

t = 52

t = 53

t = 54

t = 55

t = 56

t = 57

t = 58

t = 59

t = 60

t = 61

t = 62

t = 63

t = 64

t = 65

( q4(x

t = 66

t = 67

t = 68

t = 69

t = 70

t = 71

t = 72

t = 73

1( ) (

t = 74

( ) (x

t = 75

( ) (x

t = 76

( ) (x

t = 77

( ) (x

t = 78

q0,4(x

( ) (x

t = 79

( ) (x

t = 80

( ) (x

t = 81

( ) (x

t = 82

t = 83

( q4 ( ) (x

t = 84

( ) (x

t = 85

q2 ( ) (x

t = 86

t = 87

( q2 ) (x

t = 88

()xqb4

t = 89

( )x qb

t = 90

t = 91

t = 92

t = 93

t = 94

t = 95

( )x qb

t = 96

()xqb2

t = 97

t = 98

t = 99

t = 100

t = 101

t = 102

t = 103

t = 104

q0,4(x

t = 105

t = 106

t = 107

t = 108

t = 109

( q4 ( )x

t = 110

t = 111

q2 ( )x

t = 112

t = 113

( q2)x

t = 114

t = 115

t = 116

t = 117

t = 118

t = 119

( q4 ) )x

t = 120

t = 121

q2 ) )x

t = 122

t = 123

t = 124

t = 125

t = 126

t = 127

t = 128

t = 129

( )x qb

t = 130

()xqb2

t = 131

t = 132

t = 133

t = 134

t = 135

t = 136

t = 137

t = 138

q0,4(x

t = 139

t = 140

t = 141

t = 142

t = 143

( q4 ( )x

t = 144

t = 145

q2 ( )x

t = 146

t = 147

( q2)x

t = 148

t = 149

t = 150

t = 151

t = 152

t = 153

t = 154

t = 155

( )x qb

1( ) )

t = 156

()xqb2

t = 157

t = 158

t = 159

x( ) )

t = 160

t = 161

t = 162

t = 163

t = 164

q0,4(x

t = 165

t = 166

t = 167

t = 168

t = 169

( q4 ( )x

t = 170

t = 171

q2 ( )x

t = 172

t = 173

( q2)x

t = 174

t = 175

q2 ( ) )

t = 176

t = 177

( q2 ) )

t = 178

()xqb4

t = 179

( )x qb

t = 180

t = 181

t = 182

t = 183

( q4)x

t = 184

t = 185

t = 186

t = 187

t = 188

t = 189

t = 190

t = 191

t = 192

t = 193

t = 194

t = 195

t = 196

t = 197

t = 198

( q4)x

t = 199

t = 200

t = 201

t = 202

t = 203

t = 204

t = 205

t = 206

t = 207

t = 208

1( ) )

t = 209

( ) )x

t = 210

( ) )x

t = 211

( ) )x

t = 212

( ) )x

t = 213

( ) )x

t = 214

( ) )x

t = 215

( ) )x

t = 216

( ) )x

t = 217

q0,4(x

( ) )x

t = 218

( ) )x

t = 219

( ) )x

t = 220

( ) )x

t = 221

((xqb4

( ) )x

t = 222

( (x qb

( ) )x

t = 223

( ) )x

t = 224

( (x qb

1) ( ) )

t = 225

((xqb2

) ( ) )x

t = 226

) ( ) )x

t = 227

) ( ) )x

t = 228

) ( ) )x

t = 229

) ( ) )x

t = 230

) ( ) )x

t = 231

( qb0,3

) ( ) )x

t = 232

) ( ) )x

t = 233

( q0,4(x

) ( ) )x

t = 234

) ( ) )x

t = 235

q0,4 ( (x

) ( ) )x

t = 236

q2 ( (x

) ( ) )x

t = 237

) ( ) )x

t = 238

( q2(x

) ( ) )x

t = 239

) ( ) )x

t = 240

( ( qb3

) ( ) )x

t = 241

( ( )q3

( ) )x

t = 242

( ( q4 ) ( ) )x

t = 243

) ( ) )x

t = 244

q2 ) ( ) )x

t = 245

( ) )x

t = 246

3( ) )

t = 247

t = 248

q4 ( ) )x

t = 249

( ) )x

t = 250

( ) )x

t = 251

( ) )x

t = 252

( q4(x

( ) )x

t = 253

( ) )x

t = 254

( ) )x

t = 255

( ) )x

t = 256

( ) )x

t = 257

( ) )x

t = 258

) ( ) )x

t = 259

) ( ) )x

t = 260

1( ) ( ) )

t = 261

( ) ( ) )x

t = 262

( ) ( ) )x

t = 263

( ) ( ) )x

t = 264

( ) ( ) )x

t = 265

q0,4(x

( ) ( ) )x

t = 266

( ) ( ) )x

t = 267

( ) ( ) )x

t = 268

( ) ( ) )x

t = 269

) ( ) )x

t = 270

( q4 ( ) ( ) )x

t = 271

( ) ( ) )x

t = 272

q2 ( ) ( ) )x

t = 273

) ( ) )x

t = 274

( q2 ) ( ) )x

t = 275

()xqb4

( ) )x

t = 276

( )x qb

3( ) )

t = 277

t = 278

q4 ( ) )x

t = 279

( ) )x

t = 280

( q4)x

( ) )x

t = 281

( ) )x

t = 282

( ) )x

t = 283

( ) )x

t = 284

q2 ( ) )x

t = 285

t = 286

( q2 ) )x

t = 287

()xqb4

t = 288

( )x qb

t = 289

t = 290

( )x qb

t = 291

()xqb2

t = 292

t = 293

t = 294

t = 295

t = 296

t = 297

t = 298

t = 299

t = 300

t = 301

t = 302

t = 303

q0,4(x

t = 304

t = 305

t = 306

t = 307

((xqb4

t = 308

( (x qb

t = 309

t = 310

( (x qb

1) ( )

t = 311

((xqb2

) ( )x

t = 312

) ( )x

t = 313

) ( )x

t = 314

) ( )x

t = 315

) ( )x

t = 316

) ( )x

t = 317

( qb0,3

) ( )x

t = 318

) ( )x

t = 319

( q0,4(x

) ( )x

t = 320

) ( )x

t = 321

q0,4 ( (x

) ( )x

t = 322

q2 ( (x

) ( )x

t = 323

) ( )x

t = 324

( q2(x

) ( )x

t = 325

) ( )x

t = 326

( ( qb3

) ( )x

t = 327

( ( )q3

t = 328

( ( q4 ) ( )x

t = 329

) ( )x

t = 330

q2 ) ( )x

t = 331

t = 332

t = 333

t = 334

q4 ( )x

t = 335

t = 336

t = 337

t = 338

( q4(x

t = 339

t = 340

t = 341

t = 342

t = 343

t = 344

) ( )x

t = 345

) ( )x

t = 346

1( ) ( )

t = 347

( ) ( )x

t = 348

( ) ( )x

t = 349

( ) ( )x

t = 350

( ) ( )x

t = 351

q0,4(x

( ) ( )x

t = 352

( ) ( )x

t = 353

( ) ( )x

t = 354

( ) ( )x

t = 355

) ( )x

t = 356

( q4 ( ) ( )x

t = 357

( ) ( )x

t = 358

q2 ( ) ( )x

t = 359

) ( )x

t = 360

( q2 ) ( )x

t = 361

()xqb4

t = 362

( )x qb

t = 363

t = 364

q4 ( )x

t = 365

t = 366

( q4)x

t = 367

t = 368

t = 369

t = 370

q2 ( )x

t = 371

t = 372

( q2)x

t = 373

t = 374

t = 375

t = 376

t = 377

t = 378

t = 379

t = 380

t = 381

( q4)x

t = 382

t = 383

t = 384

t = 385

t = 386

t = 387

t = 388

t = 389

t = 390

t = 391

t = 392

t = 393

t = 394

t = 395

t = 396

t = 397

t = 398

t = 399

t = 400

t = 401

t = 402

(xq̂b6

t = 403

(x q̂b

t = 404

(xq̂b6

t = 405

(x q̂b

t = 406

)xq̂b6

t = 407

)x q̂b

t = 408

(xq̂b6

t = 409

(x q̂b

t = 410

)xq̂b6

t = 411

)x q̂b

t = 412

)xq̂b6

t = 413

)x q̂b

t = 414

)x q̂b

t = 415

)xq̂b2

t = 416

)x q̂b

t = 417

)xq̂b2

t = 418

(x q̂b

t = 419

(xq̂b2

t = 420

)x q̂b

t = 421

)xq̂b2

t = 422

(x q̂b

t = 423

(xq̂b2

t = 424

(x q̂b

t = 425

(xq̂b2

t = 426

t = 427

q̂b0,3

t = 428

q̂0,3

t = 429

q̂0,4(x

t = 430

q̂2(x

t = 431

t = 432

( q̂b3

t = 433

((xq̂b4

t = 434

( (x q̂b

t = 435

t = 436

( (x q̂b

t = 437

((xq̂b2

t = 438

( q̂b1

t = 439

t = 440

t = 441

q̂b0,3

t = 442

q̂b0,4

t = 443

( q̂b0,3

t = 444

q̂0,3

t = 445

( q̂0,4(x

t = 446

q̂0,3

t = 447

q̂0,4 ( (x

t = 448

q̂2 ( (x

t = 449

t = 450

( q̂2(x

t = 451

t = 452

( ( q̂b3

t = 453

t = 454

( ( q̂4 ) (x

t = 455

((xq̂1

t = 456

( (x q̂2 ) (

t = 457

)xq̂b4

t = 458

)x q̂b

t = 459

(xq̂b4

t = 460

(x q̂b

t = 461

t = 462

(x q̂b

t = 463

(xq̂b2

t = 464

)x q̂b

t = 465

)xq̂b2

t = 466

( (x q̂b

t = 467

((xq̂b2

t = 468

( q̂b1

t = 469

t = 470

t = 471

q̂b0,3

t = 472

q̂b0,4

t = 473

( q̂b0,3

t = 474

q̂0,3

t = 475

( q̂0,4(x

t = 476

q̂0,3

t = 477

q̂0,4 ( (x

t = 478

q̂2 ( (x

t = 479

t = 480

( q̂2(x

t = 481

t = 482

( ( q̂b3

t = 483

t = 484

( ( q̂b1

t = 485

t = 486

( q̂b1

( ) (x

t = 487

( ) (x

t = 488

( ( ) (x

t = 489

q̂b0,3

( ( ) (x

t = 490

q̂b0,4

( ) (x

t = 491

( q̂b0,3

( ) (x

t = 492

q̂b0,4

t = 493

( ( q̂b0,3

t = 494

q̂b0,4

t = 495

( ( ) q̂b0,3

t = 496

( ( )(x

q̂0,3

t = 497

( ( ) q̂0,4(x

t = 498

q̂0,3

t = 499

( ( q̂0,4 ) (x

t = 500

q̂0,3

t = 501

( q̂0,4 ( ) (x

t = 502

q̂0,3

( ) (x

t = 503

q̂0,4 ( ( ) (x

t = 504

q̂2 ( ( ) (x

t = 505

( ) (x

t = 506

( q̂2 ( ) (x

t = 507

t = 508

( ( q̂2 ) (x

t = 509

( ()xq̂b4

t = 510

( ( )x q̂b

t = 511

( ( )x

(xq̂b4

t = 512

( ( )x

(x q̂b

t = 513

( ( )x

t = 514

( ( )x

(x q̂4 ) )

t = 515

( ( )x

(xq̂3

t = 516

( ( )x q̂4

t = 517

( ()xq̂3

t = 518

( ( q̂4)x

t = 519

((xq̂1

t = 520

( (x q̂2

t = 521

)xq̂1

t = 522

)x q̂2

t = 523

t = 524

( q̂b3

t = 525

t = 526

( q̂4 ) )x

t = 527

(xq̂1

t = 528

(x q̂2 ) )

t = 529

)xq̂b4

t = 530

)x q̂b

t = 531

t = 532

)x q̂b

t = 533

)xq̂b2

t = 534

(x q̂b

t = 535

(xq̂b2

t = 536

)x q̂b

t = 537

)xq̂b2

t = 538

( (x q̂b

t = 539

((xq̂b2

t = 540

( q̂b1

t = 541

t = 542

t = 543

q̂b0,3

t = 544

q̂b0,4

t = 545

( q̂b0,3

t = 546

q̂0,3

t = 547

( q̂0,4(x

t = 548

q̂0,3

t = 549

q̂0,4 ( (x

t = 550

q̂2 ( (x

t = 551

t = 552

( q̂2(x

t = 553

t = 554

( ( q̂b3

t = 555

t = 556

( ( q̂b1

t = 557

t = 558

( q̂b1

( ) (x

t = 559

( ) (x

t = 560

( ( ) (x

t = 561

q̂b0,3

( ( ) (x

t = 562

q̂b0,4

( ) (x

t = 563

( q̂b0,3

( ) (x

t = 564

q̂b0,4

t = 565

( ( q̂b0,3

t = 566

q̂b0,4

t = 567

( ( ) q̂b0,3

t = 568

( ( )(x

q̂0,3

t = 569

( ( ) q̂0,4(x

t = 570

q̂0,3

t = 571

( ( q̂0,4 ) (x

t = 572

q̂0,3

t = 573

( q̂0,4 ( ) (x

t = 574

q̂0,3

( ) (x

t = 575

q̂0,4 ( ( ) (x

t = 576

q̂2 ( ( ) (x

t = 577

( ) (x

t = 578

( q̂2 ( ) (x

t = 579

t = 580

( ( q̂2 ) (x

t = 581

( ()xq̂b4

t = 582

( ( )x q̂b

t = 583

( ( )x

(xq̂b4

t = 584

( ( )x

(x q̂b

t = 585

( ( )x

t = 586

( ( )x

(x q̂b

t = 587

( ( )x

(xq̂b2

t = 588

( ( )x q̂b

t = 589

( ()xq̂b2

t = 590

( ( q̂b1

t = 591

t = 592

( q̂b1

t = 593

t = 594

( ( )x

t = 595

q̂b0,3

( ( )x

t = 596

q̂b0,4

t = 597

( q̂b0,3

t = 598

q̂b0,4

t = 599

( ( q̂b0,3

t = 600

q̂0,3

t = 601

( ( q̂0,4)x

t = 602

q̂0,3

t = 603

( q̂0,4 ( )x

t = 604

q̂0,3

t = 605

q̂0,4 ( ( )x

t = 606

q̂2 ( ( )x

t = 607

t = 608

( q̂2 ( )x

t = 609

t = 610

( ( q̂2)x

t = 611

( ()xq̂1

t = 612

( ( )x q̂2

t = 613

( ( )x

t = 614

( ( )x

( q̂b3

t = 615

( ( )x

t = 616

( ( )x

( q̂4 ) )

t = 617

( ( )x

(xq̂1

t = 618

( ( )x

(x q̂2 ) )

t = 619

( ( )x

)xq̂b4

t = 620

( ( )x

)x q̂b

t = 621

( ( )x

t = 622

( ( )x

)x q̂4 )

t = 623

( ( )x

)xq̂3

t = 624

( ( )x

(x q̂4

t = 625

( ( )x

(xq̂3

t = 626

( ( )x q̂4

t = 627

( ()xq̂3

t = 628

( ( q̂4)x

t = 629

((xq̂1

t = 630

( (x q̂2

t = 631

)xq̂1

t = 632

)x q̂2

t = 633

t = 634

( q̂b3

t = 635

t = 636

( q̂b1

t = 637

t = 638

)x q̂b

1( ) )

t = 639

)xq̂b2

t = 640

( (x q̂b

t = 641

((xq̂b2

t = 642

( q̂b1

t = 643

t = 644

t = 645

q̂b0,3

t = 646

q̂b0,4

t = 647

( q̂b0,3

t = 648

q̂0,3

t = 649

( q̂0,4(x

t = 650

q̂0,3

t = 651

q̂0,4 ( (x

t = 652

q̂2 ( (x

t = 653

t = 654

( q̂2(x

t = 655

t = 656

( ( q̂b3

t = 657

t = 658

( ( q̂b1

) ( ) )

t = 659

) ( ) )

t = 660

( q̂b1

( ) ( ) )

t = 661

( ) ( ) )

t = 662

( ( ) ( ) )

t = 663

q̂b0,3

( ( ) ( ) )

t = 664

q̂b0,4

( ) ( ) )

t = 665

( q̂b0,3

( ) ( ) )

t = 666

q̂b0,4

) ( ) )

t = 667

( ( q̂b0,3

) ( ) )

t = 668

q̂b0,4

t = 669

( ( ) q̂b0,3

t = 670

( ( )(

q̂b0,4

t = 671

( ( ) ( q̂b0,3

t = 672

( ( ) ()

q̂b0,4

t = 673

( ( ) ( ) q̂b0,3

t = 674

( ( ) ( ))

q̂b0,4

t = 675

( ( ) ( ) ) q̂b0,3

t = 676

( ( ) ( ) ) q̂b0,1

t = 677

( ( ) ( ))

q̂b0,2

t = 678

( ( ) ( ) q̂b0,1

t = 679

( ( ) ()

q̂b0,2

t = 680

( ( ) ( q̂b0,1

t = 681

( ( )(

q̂b0,2

t = 682

( ( ) q̂b0,1

t = 683

q̂b0,2

t = 684

( ( q̂b0,1

) ( ) )

t = 685

q̂b0,2

) ( ) )

t = 686

( q̂b0,1

( ) ( ) )

t = 687

q̂b0,2

( ) ( ) )

t = 688

q̂b0,1

( ( ) ( ) )

t = 689

q̂0,2 ( ( ) ( ) )

t = 690

q̂0,1

( ) ( ) )

t = 691

( q̂0,2 ( ) ( ) )

t = 692

q̂0,1

) ( ) )

t = 693

( ( q̂0,2 ) ( ) )

t = 694

q̂0,1

t = 695

( ( ) q̂0,2 ( ) )

t = 696

( ( ) (

q̂0,1

t = 697

( ( ) ( q̂0,2 ) )

t = 698

( ( ) ( )

q̂0,1

t = 699

( ( ) ( ) q̂0,2 )

t = 700

( ( ) ( ) )

q̂0,1

t = 701

( ( ) ( ) ) q̂0,2

t = 702

( ( ) ( ) ) q̂0,4

t = 703

( ( ) ( ) )

q̂0,3

t = 704

( ( ) ( ) q̂0,4 )

t = 705

( ( ) ( )

q̂0,3

t = 706

( ( ) ( q̂0,4 ) )

t = 707

( ( ) (

q̂0,3

t = 708

( ( ) q̂0,4 ( ) )

t = 709

q̂0,3

t = 710

( ( q̂0,4 ) ( ) )

t = 711

q̂0,3

) ( ) )

t = 712

( q̂0,4 ( ) ( ) )

t = 713

q̂0,3

( ) ( ) )

t = 714

q̂0,4 ( ( ) ( ) )

t = 715

q̂2 ( ( ) ( ) )

t = 716

( ) ( ) )

t = 717

( q̂2 ( ) ( ) )

t = 718

) ( ) )

t = 719

( ( q̂2 ) ( ) )

t = 720

( ()xq̂b4

t = 721

( ( )x q̂b

3( ) )

t = 722

( ( )x

t = 723

( ( )x q̂4 ( ) )

t = 724

( ()xq̂3

t = 725

( ( q̂4)x

t = 726

((xq̂1

t = 727

( (x q̂2

t = 728

)xq̂1

t = 729

)x q̂2 ( ) )

t = 730

t = 731

( q̂2 ) )

t = 732

()xq̂b4

t = 733

( )x q̂b

t = 734

t = 735

( )x q̂4 )

t = 736

()xq̂3

t = 737

( q̂4)x

t = 738

(xq̂1

t = 739

(x q̂2

t = 740

)xq̂1

t = 741

)x q̂2 )

t = 742

)xq̂b4

t = 743

)x q̂b

t = 744

)x q̂4

t = 745

)xq̂3

t = 746

)x q̂4

t = 747

)xq̂3

t = 748

(x q̂4

t = 749

(xq̂3

t = 750

)x q̂4

t = 751

)xq̂3

t = 752

( (x q̂4

t = 753

((xq̂3

t = 754

( q̂4(x

t = 755

(xq̂1

t = 756

(x q̂2

t = 757

t = 758

( q̂b3

t = 759

t = 760

( q̂b1

t = 761

t = 762

(x q̂b

1( ) (

t = 763

(xq̂b2

( ) (x

t = 764

( ) (x

t = 765

q̂b0,3

( ) (x

t = 766

q̂0,3

( ) (x

t = 767

q̂0,4(x

( ) (x

t = 768

q̂2(x

( ) (x

t = 769

( ) (x

t = 770

( q̂b3

( ) (x

t = 771

t = 772

( q̂4 ( ) (x

t = 773

(xq̂1

( ) (x

t = 774

(x q̂2 ( ) (

t = 775

t = 776

( q̂2 ) (x

t = 777

()xq̂b4

t = 778

( )x q̂b

t = 779

(xq̂b4

t = 780

(x q̂b

t = 781

t = 782

(x q̂b

t = 783

(xq̂b2

t = 784

( )x q̂b

t = 785

()xq̂b2

t = 786

( q̂b1

t = 787

t = 788

(x q̂b

t = 789

(xq̂b2

t = 790

t = 791

q̂b0,3

t = 792

q̂0,3

t = 793

q̂0,4(x

t = 794

q̂2(x

t = 795

t = 796

( q̂b3

t = 797

t = 798

( q̂4 ( )x

t = 799

(xq̂1

t = 800

(x q̂2 ( )

t = 801

t = 802

( q̂2)x

t = 803

()xq̂1

t = 804

( )x q̂2

t = 805

t = 806

( q̂b3

t = 807

t = 808

( q̂4 ) )x

t = 809

(xq̂1

t = 810

(x q̂2 ) )

t = 811

)xq̂b4

t = 812

)x q̂b

t = 813

t = 814

)x q̂b

t = 815

)xq̂b2

t = 816

(x q̂b

t = 817

(xq̂b2

t = 818

( )x q̂b

t = 819

()xq̂b2

t = 820

( q̂b1

t = 821

t = 822

(x q̂b

t = 823

(xq̂b2

t = 824

t = 825

q̂b0,3

t = 826

q̂0,3

t = 827

q̂0,4(x

t = 828

q̂2(x

t = 829

t = 830

( q̂b3

t = 831

t = 832

( q̂4 ( )x

t = 833

(xq̂1

t = 834

(x q̂2 ( )

t = 835

t = 836

( q̂2)x

t = 837

()xq̂1

t = 838

( )x q̂2

t = 839

t = 840

( q̂b3

t = 841

t = 842

( q̂b1

t = 843

t = 844

( )x q̂b

1( ) )

t = 845

()xq̂b2

t = 846

( q̂b1

t = 847

t = 848

(x q̂b

x( ) )

t = 849

(xq̂b2

t = 850

t = 851

q̂b0,3

t = 852

q̂0,3

t = 853

q̂0,4(x

t = 854

q̂2(x

t = 855

t = 856

( q̂b3

t = 857

t = 858

( q̂4 ( )x

t = 859

(xq̂1

t = 860

(x q̂2 ( )

x( ) )

t = 861

t = 862

( q̂2)x

t = 863

()xq̂1

t = 864

( )x q̂2 ( ) )

t = 865

t = 866

( q̂2 ) )

t = 867

()xq̂b4

t = 868

( )x q̂b

t = 869

t = 870

( )x q̂4 )

t = 871

()xq̂3

t = 872

( q̂4)x

t = 873

(xq̂1

t = 874

(x q̂2

t = 875

)xq̂1

t = 876

)x q̂2 )

t = 877

)xq̂b4

t = 878

)x q̂b

t = 879

)x q̂4

t = 880

)xq̂3

t = 881

)x q̂4

t = 882

)xq̂3

t = 883

(x q̂4

t = 884

(xq̂3

t = 885

( )x q̂4

t = 886

()xq̂3

t = 887

( q̂4)x

t = 888

(xq̂1

t = 889

(x q̂2

t = 890

)xq̂1

t = 891

)x q̂2

t = 892

t = 893

( q̂b3

t = 894

t = 895

( q̂b1

t = 896

t = 897

)x q̂b

1( ) )

t = 898

)xq̂b2

( ) )x

t = 899

(x q̂b

( ) )x

t = 900

(xq̂b2

( ) )x

t = 901

(x q̂b

( ) )x

t = 902

(xq̂b2

( ) )x

t = 903

( ) )x

t = 904

q̂b0,3

( ) )x

t = 905

q̂0,3

( ) )x

t = 906

q̂0,4(x

( ) )x

t = 907

q̂2(x

( ) )x

t = 908

( ) )x

t = 909

( q̂b3

( ) )x

t = 910

((xq̂b4

( ) )x

t = 911

( (x q̂b

( ) )x

t = 912

( ) )x

t = 913

( (x q̂b

1) ( ) )

t = 914

((xq̂b2

) ( ) )x

t = 915

( q̂b1

) ( ) )x

t = 916

) ( ) )x

t = 917

) ( ) )x

t = 918

q̂b0,3

) ( ) )x

t = 919

q̂b0,4

) ( ) )x

t = 920

( q̂b0,3

) ( ) )x

t = 921

q̂0,3

) ( ) )x

t = 922

( q̂0,4(x

) ( ) )x

t = 923

q̂0,3

) ( ) )x

t = 924

q̂0,4 ( (x

) ( ) )x

t = 925

q̂2 ( (x

) ( ) )x

t = 926

) ( ) )x

t = 927

( q̂2(x

) ( ) )x

t = 928

) ( ) )x

t = 929

( ( q̂b3

) ( ) )x

t = 930

( ) )x

t = 931

( ( q̂4 ) ( ) )x

t = 932

((xq̂1

) ( ) )x

t = 933

( (x q̂2 ) ( ) )

t = 934

)xq̂b4

( ) )x

t = 935

)x q̂b

3( ) )

t = 936

t = 937

)x q̂4 ( ) )

t = 938

)xq̂3

( ) )x

t = 939

( (x q̂4

( ) )x

t = 940

((xq̂3

( ) )x

t = 941

( q̂4(x

( ) )x

t = 942

(xq̂1

( ) )x

t = 943

(x q̂2

( ) )x

t = 944

( ) )x

t = 945

( q̂b3

( ) )x

t = 946

( ) )x

t = 947

( q̂b1

) ( ) )x

t = 948

) ( ) )x

t = 949

(x q̂b

1( ) ( ) )

t = 950

(xq̂b2

( ) ( ) )x

t = 951

( ) ( ) )x

t = 952

q̂b0,3

( ) ( ) )x

t = 953

q̂0,3

( ) ( ) )x

t = 954

q̂0,4(x

( ) ( ) )x

t = 955

q̂2(x

( ) ( ) )x

t = 956

( ) ( ) )x

t = 957

( q̂b3

( ) ( ) )x

t = 958

) ( ) )x

t = 959

( q̂4 ( ) ( ) )x

t = 960

(xq̂1

( ) ( ) )x

t = 961

(x q̂2 ( ) ( ) )

t = 962

) ( ) )x

t = 963

( q̂2 ) ( ) )x

t = 964

()xq̂b4

( ) )x

t = 965

( )x q̂b

3( ) )

t = 966

t = 967

( )x q̂4 ( ) )

t = 968

()xq̂3

( ) )x

t = 969

( q̂4)x

( ) )x

t = 970

(xq̂1

( ) )x

t = 971

(x q̂2

( ) )x

t = 972

)xq̂1

( ) )x

t = 973

)x q̂2 ( ) )

t = 974

t = 975

( q̂2 ) )x

t = 976

()xq̂b4

t = 977

( )x q̂b

t = 978

t = 979

( )x q̂b

t = 980

()xq̂b2

t = 981

( q̂b1

t = 982

t = 983

)x q̂b

t = 984

)xq̂b2

t = 985

(x q̂b

t = 986

(xq̂b2

t = 987

(x q̂b

t = 988

(xq̂b2

t = 989

t = 990

q̂b0,3

t = 991

q̂0,3

t = 992

q̂0,4(x

t = 993

q̂2(x

t = 994

t = 995

( q̂b3

t = 996

((xq̂b4

t = 997

( (x q̂b

t = 998

t = 999

( (x q̂b

1) ( )

t = 1000

((xq̂b2

) ( )x

t = 1001

( q̂b1

) ( )x

t = 1002

) ( )x

t = 1003

) ( )x

t = 1004

q̂b0,3

) ( )x

t = 1005

q̂b0,4

) ( )x

t = 1006

( q̂b0,3

) ( )x

t = 1007

q̂0,3

) ( )x

t = 1008

( q̂0,4(x

) ( )x

t = 1009

q̂0,3

) ( )x

t = 1010

q̂0,4 ( (x

) ( )x

t = 1011

q̂2 ( (x

) ( )x

t = 1012

) ( )x

t = 1013

( q̂2(x

) ( )x

t = 1014

) ( )x

t = 1015

( ( q̂b3

) ( )x

t = 1016

t = 1017

( ( q̂4 ) ( )x

t = 1018

((xq̂1

) ( )x

t = 1019

( (x q̂2 ) ( )

t = 1020

)xq̂b4

t = 1021

)x q̂b

t = 1022

t = 1023

)x q̂4 ( )

t = 1024

)xq̂3

t = 1025

( (x q̂4

t = 1026

((xq̂3

t = 1027

( q̂4(x

t = 1028

(xq̂1

t = 1029

(x q̂2

t = 1030

t = 1031

( q̂b3

t = 1032

t = 1033

( q̂b1

) ( )x

t = 1034

) ( )x

t = 1035

(x q̂b

1( ) ( )

t = 1036

(xq̂b2

( ) ( )x

t = 1037

( ) ( )x

t = 1038

q̂b0,3

( ) ( )x

t = 1039

q̂0,3

( ) ( )x

t = 1040

q̂0,4(x

( ) ( )x

t = 1041

q̂2(x

( ) ( )x

t = 1042

( ) ( )x

t = 1043

( q̂b3

( ) ( )x

t = 1044

) ( )x

t = 1045

( q̂4 ( ) ( )x

t = 1046

(xq̂1

( ) ( )x

t = 1047

(x q̂2 ( ) ( )

t = 1048

) ( )x

t = 1049

( q̂2 ) ( )x

t = 1050

()xq̂b4

t = 1051

( )x q̂b

t = 1052

t = 1053

( )x q̂4 ( )

t = 1054

()xq̂3

t = 1055

( q̂4)x

t = 1056

(xq̂1

t = 1057

(x q̂2

t = 1058

)xq̂1

t = 1059

)x q̂2 ( )

t = 1060

t = 1061

( q̂2)x

t = 1062

()xq̂1

t = 1063

( )x q̂2 )

t = 1064

)xq̂b4

t = 1065

)x q̂b

t = 1066

)x q̂4

t = 1067

)xq̂3

t = 1068

( )x q̂4

t = 1069

()xq̂3

t = 1070

( q̂4)x

t = 1071

(xq̂1

t = 1072

(x q̂2

t = 1073

)xq̂1

t = 1074

)x q̂2

t = 1075

)xq̂1

t = 1076

)x q̂2

t = 1077

)x q̂6

t = 1078

)xq̂5

t = 1079

)x q̂6

t = 1080

)xq̂5

t = 1081

(x q̂6

t = 1082

(xq̂5

t = 1083

)x q̂6

t = 1084

)xq̂5

t = 1085

(x q̂6

t = 1086

(xq̂5

t = 1087

(x q̂6

t = 1088

(xq̂5

t = 1089

q̂6(x

t = 1090

t = 1091

t = 1092

t = 1093

t = 1094

t = 1095

t = 1096

t = 1097

t = 1098

t = 1099

t = 1100

t = 1101

t = 1102

t = 1103

t = 1104

t = 1105

t = 1106

t = 1107

t = 1108

t = 1109

t = 1110

t = 1111

t = 1112

t = 1113

t = 1114

t = 1115

t = 1116

t = 1117

t = 1118

q0,4(x

t = 1119

t = 1120

t = 1121

t = 1122

((xqb4

t = 1123

( (x qb

t = 1124

t = 1125

( (x qb

t = 1126

((xqb2

t = 1127

t = 1128

t = 1129

t = 1130

t = 1131

t = 1132

( qb0,3

t = 1133

t = 1134

( q0,4(x

t = 1135

t = 1136

q0,4 ( (x

t = 1137

q2 ( (x

t = 1138

t = 1139

( q2(x

t = 1140

t = 1141

( ( qb3

t = 1142

( ( )q3

t = 1143

( ( q4 ) (x

t = 1144

t = 1145

q2 ) (x

t = 1146

t = 1147

t = 1148

t = 1149

t = 1150

t = 1151

t = 1152

t = 1153

t = 1154

t = 1155

( (x qb

t = 1156

((xqb2

t = 1157

t = 1158

t = 1159

t = 1160

t = 1161

t = 1162

( qb0,3

t = 1163

t = 1164

( q0,4(x

t = 1165

t = 1166

q0,4 ( (x

t = 1167

q2 ( (x

t = 1168

t = 1169

( q2(x

t = 1170

t = 1171

( ( qb3

t = 1172

t = 1173

( ( qb1

t = 1174

t = 1175

( ) (x

t = 1176

( ) (x

t = 1177

( ( ) (x

t = 1178

( ( ) (x

t = 1179

( ) (x

t = 1180

( qb0,3

( ) (x

t = 1181

t = 1182

( ( qb0,3

t = 1183

t = 1184

( ( ) qb0,3

t = 1185

( ( )(x

t = 1186

( ( ) q0,4(x

t = 1187

( ( )q0,3

t = 1188

( ( q0,4 ) (x

t = 1189

( (q0,3

t = 1190

( q0,4 ( ) (x

t = 1191

( ) (x

t = 1192

q0,4 ( ( ) (x

t = 1193

q2 ( ( ) (x

t = 1194

( ) (x

t = 1195

( q2 ( ) (x

t = 1196

t = 1197

( ( q2 ) (x

t = 1198

( ()xqb4

t = 1199

( ( )x qb

t = 1200

( ( )x

t = 1201

( ( )x

t = 1202

( ( )x

t = 1203

( ( )x

q4 ) )x

t = 1204

( ( )x

t = 1205

( ( )x

t = 1206

( ()xq3

t = 1207

( ( q4)x

t = 1208

t = 1209

t = 1210

t = 1211

t = 1212

t = 1213

t = 1214

t = 1215

( q4 ) )x

t = 1216

t = 1217

q2 ) )x

t = 1218

t = 1219

t = 1220

t = 1221

t = 1222

t = 1223

t = 1224

t = 1225

t = 1226

t = 1227

( (x qb

t = 1228

((xqb2

t = 1229

t = 1230

t = 1231

t = 1232

t = 1233

t = 1234

( qb0,3

t = 1235

t = 1236

( q0,4(x

t = 1237

t = 1238

q0,4 ( (x

t = 1239

q2 ( (x

t = 1240

t = 1241

( q2(x

t = 1242

t = 1243

( ( qb3

t = 1244

t = 1245

( ( qb1

t = 1246

t = 1247

( ) (x

t = 1248

( ) (x

t = 1249

( ( ) (x

t = 1250

( ( ) (x

t = 1251

( ) (x

t = 1252

( qb0,3

( ) (x

t = 1253

t = 1254

( ( qb0,3

t = 1255

t = 1256

( ( ) qb0,3

t = 1257

( ( )(x

t = 1258

( ( ) q0,4(x

t = 1259

( ( )q0,3

t = 1260

( ( q0,4 ) (x

t = 1261

( (q0,3

t = 1262

( q0,4 ( ) (x

t = 1263

( ) (x

t = 1264

q0,4 ( ( ) (x

t = 1265

q2 ( ( ) (x

t = 1266

( ) (x

t = 1267

( q2 ( ) (x

t = 1268

t = 1269

( ( q2 ) (x

t = 1270

( ()xqb4

t = 1271

( ( )x qb

t = 1272

( ( )x

t = 1273

( ( )x

t = 1274

( ( )x

t = 1275

( ( )x

t = 1276

( ( )x

t = 1277

( ( )x qb

t = 1278

( ()xqb2

t = 1279

( ( qb1

t = 1280

t = 1281

t = 1282

t = 1283

( ( )x

t = 1284

( ( )x

t = 1285

t = 1286

( qb0,3

t = 1287

t = 1288

( ( qb0,3

t = 1289

t = 1290

( ( q0,4)x

t = 1291

( (q0,3

t = 1292

( q0,4 ( )x

t = 1293

t = 1294

q0,4 ( ( )x

t = 1295

q2 ( ( )x

t = 1296

t = 1297

( q2 ( )x

t = 1298

t = 1299

( ( q2)x

t = 1300

( ()xq1

t = 1301

( ( )x

t = 1302

( ( )x

t = 1303

( ( )x

t = 1304

( ( )x

t = 1305

( ( )x

( q4 ) )

t = 1306

( ( )x

t = 1307

( ( )x

q2 ) )

t = 1308

( ( )x

t = 1309

( ( )x

t = 1310

( ( )x

t = 1311

( ( )x

t = 1312

( ( )x

t = 1313

( ( )x

t = 1314

( ( )x

t = 1315

( ( )x

t = 1316

( ()xq3

t = 1317

( ( q4)x

t = 1318

t = 1319

t = 1320

t = 1321

t = 1322

t = 1323

t = 1324

t = 1325

t = 1326

t = 1327

1( ) )

t = 1328

t = 1329

( (x qb

t = 1330

((xqb2

t = 1331

t = 1332

t = 1333

t = 1334

t = 1335

t = 1336

( qb0,3

t = 1337

t = 1338

( q0,4(x

t = 1339

t = 1340

q0,4 ( (x

t = 1341

q2 ( (x

t = 1342

t = 1343

( q2(x

t = 1344

t = 1345

( ( qb3

t = 1346

t = 1347

( ( qb1

) ( ) )

t = 1348

) ( ) )

t = 1349

( ) ( ) )

t = 1350

( ) ( ) )

t = 1351

( ( ) ( ) )

t = 1352

( ( ) ( ) )

t = 1353

( ) ( ) )

t = 1354

( qb0,3

( ) ( ) )

t = 1355

) ( ) )

t = 1356

( ( qb0,3

) ( ) )

t = 1357

t = 1358

( ( ) qb0,3

t = 1359

( ( )(

t = 1360

( ( ) ( qb0,3

t = 1361

( ( ) ()

t = 1362

( ( ) ( ) qb0,3

t = 1363

( ( ) ( ))

t = 1364

( ( ) ( ) ) qb0,3

t = 1365

( ( ) ( ) ) qb0,1

t = 1366

( ( ) ( ))

t = 1367

( ( ) ( ) qb0,1

t = 1368

( ( ) ()

t = 1369

( ( ) ( qb0,1

t = 1370

( ( )(

t = 1371

( ( ) qb0,1

t = 1372

t = 1373

( ( qb0,1

) ( ) )

t = 1374

) ( ) )

t = 1375

( qb0,1

( ) ( ) )

t = 1376

( ) ( ) )

t = 1377

( ( ) ( ) )

t = 1378

q0,2 ( ( ) ( ) )

It comes back to the initial configuration

A4. RPIA

Iterative Array Acceptor

• An input word a1a2 · · · an is given serially.

• It uses only n cells (i.e., the space is bounded).

• If the cell at the position 1 enters an accepting

state, we say the CA accepts the input.1 2 · · · · n n + 1

t = 0 q q · · · · q #

a1 a2 · · · an �

t = 1 x11

q · · · · q #

a2 · · · an �...

t = t1 xt11 xt1

2 · · · · xt1n #

Partitioned Iterative Array (PIA)

• An input word a1a2 · · · an is given serially.

• It uses n + 2 cells.

• If the left part of the 1-st cell enters an accepting

state, we say the PCA accepts the input.0 1 2 · · n n + 1

t = 0 B

� a1 a2 · · · an �

t = 1 B r10

a1 a2 · · · an �

t = 2 B r20 l21 c2

a2 · · · an � ...

t = t1 B rt10 lt11 ct1

1 rt11 lt12 ct1

2 rt12 · · · · · · lt1n ct1

n rt1n lt1n+1

Reversible partitioned iterative array that accepts all well-formed parentheses

� ) ) ( ) ( ( � B

� ) ) ( ) ( ( B ∗

� ) ) ( ) ( B ( ∗

� ) ) ( ) B ( ( ∗

� ) ) ( B ) ( ( ∗

� ) ) B ( ( ) ( ∗

� ) B ) ( ( ( ) ∗

� B ) ( ) ( ( ) ∗

B � ( ) ( ) ) ( ∗

B ( � ( ) ) ) ( ∗

t = 10

B ( ( � ) ) ( ) ∗

t = 11

B ( ( ) � ( ) ) ∗

t = 12

B ( ( ) ( � ) ) ∗

t = 13

B ( ( ) ( ) � ) ∗

t = 14

B ( ( ) ( ) ) � ∗

t = 15

B ( ( ) ( ) ) ∗ B

t = 16

B ( ( ) ( ) ∗ ) B

t = 17

B ( ( ) ( ∗ ) ) B

t = 18

B ( ( ) ∗ ( ) ) B

t = 19

B ( ( ∗ ) ( ) ) B

t = 20

B ( ∗ ( ) ( ) ) B

t = 21

B ∗ ( ( ) ( ) ) B

t = 22

B q0,2 ( ( ) ( ) ) B

It starts to simulate RPCA

t = 23

q0,1( ) ( ) ) B

t = 24

B ( q0,2 ( ) ( ) ) B

t = 25

B ( (q0,1

) ( ) ) B

t = 26

B ( ( q0,2 ) ( ) ) B

t = 27

B ( ( )q0,1

( ) ) B

t = 28

B ( ( ) q0,2 ( ) ) B

t = 29

B ( ( ) (q0,1

t = 30

B ( ( ) ( q0,2 ) ) B

t = 31

B ( ( ) ( )q0,1

t = 32

B ( ( ) ( ) q0,2 ) B

t = 33

B ( ( ) ( ) )q0,1

t = 34

B ( ( ) ( ) ) q0,2 B

t = 35

B ( ( ) ( ) ) q0,4 B

t = 36

B ( ( ) ( ) )q0,3

t = 37

B ( ( ) ( ) q0,4 ) B

t = 38

B ( ( ) ( )q0,3

t = 39

B ( ( ) ( q0,4 ) ) B

t = 40

B ( ( ) (q0,3

t = 41

B ( ( ) q0,4 ( ) ) B

t = 42

B ( ( )q0,3

( ) ) B

t = 43

B ( ( q0,4 ) ( ) ) B

t = 44

B ( (q0,3

) ( ) ) B

t = 45

B ( q0,4 ( ) ( ) ) B

t = 46

q0,3( ) ( ) ) B

t = 47

B q0,4 ( ( ) ( ) ) B

t = 48

B q2 ( ( ) ( ) ) B

t = 49

( ) ( ) ) B

t = 50

B ( q2 ( ) ( ) ) B

t = 51

B ( (q1

) ( ) ) B

t = 52

B ( ( q2 ) ( ) ) B

t = 53

B ( ()xqb4

( ) ) B

t = 54

B ( ( )x qb

3( ) ) B

t = 55

B ( ( )x

t = 56

B ( ( )x

q4 ( ) ) B

t = 57

B ( ()xq3

( ) ) B

t = 58

B ( ( q4)x

( ) ) B

t = 59

B ((xq1

( ) ) B

t = 60

B ( (x

( ) ) B

t = 61

B ( (x

( ) ) B

t = 62

B ( (x

q2 ( ) ) B

t = 63

B ( (x

t = 64

B ( (x

( q2 ) ) B

t = 65

B ( (x

()xqb4

t = 66

B ( (x

( )x qb

t = 67

B ( (x

t = 68

B ( (x

q4 ) B

t = 69

B ( (x

t = 70

B ( (x

( q4)x

t = 71

B ( (x

t = 72

B ( (x

t = 73

B ( (x

t = 74

B ( (x

q2 ) B

t = 75

B ( (x

t = 76

B ( (x

t = 77

B ( (x

t = 78

B ( (x

t = 79

B ( (x

q4)x B

t = 80

B ( (x

t = 81

B ( (x

t = 82

B ( (x

t = 83

B ( (x

t = 84

B ( (x

t = 85

B ( (x

t = 86

B ((xq3

t = 87

B ( q4(x

t = 88

t = 89

t = 90

t = 91

t = 92

t = 93

t = 94

t = 95

B (x qb

1( ) (

t = 96

B(xqb2

( ) (x

t = 97

( ) (x

t = 98

B qb0,3

( ) (x

t = 99

( ) (x

t = 100

B q0,4(x

( ) (x

t = 101

B q2(x

( ) (x

t = 102

( ) (x

t = 103

B ( qb3

( ) (x

t = 104

B ( (q3

t = 105

B ( q4 ( ) (x

t = 106

( ) (x

t = 107

q2 ( ) (x

t = 108

t = 109

( q2 ) (x

t = 110

()xqb4

t = 111

( )x qb

t = 112

t = 113

t = 114

t = 115

t = 116

) )x B

t = 117

( )x qb

) )x B

t = 118

()xqb2

) )x B

t = 119

) )x B

t = 120

) )x B

t = 121

B (x qb

) )x B

t = 122

B(xqb2

) )x B

t = 123

) )x B

t = 124

B qb0,3

) )x B

t = 125

) )x B

t = 126

B q0,4(x

) )x B

t = 127

B q2(x

) )x B

t = 128

) )x B

t = 129

B ( qb3

) )x B

t = 130

B ( (q3

) )x B

t = 131

B ( q4 ( )x

) )x B

t = 132

) )x B

t = 133

q2 ( )x

) )x B

t = 134

) )x B

t = 135

( q2)x

) )x B

t = 136

) )x B

t = 137

) )x B

t = 138

) )x B

t = 139

) )x B

t = 140

t = 141

( q4 ) )x B

t = 142

) )x B

t = 143

q2 ) )x B

t = 144

t = 145

t = 146

t = 147

t = 148

t = 149

t = 150

t = 151

( )x qb

t = 152

()xqb2

t = 153

t = 154

t = 155

B (x qb

t = 156

B(xqb2

t = 157

t = 158

B qb0,3

t = 159

t = 160

B q0,4(x

t = 161

B q2(x

t = 162

t = 163

B ( qb3

t = 164

B ( (q3

t = 165

B ( q4 ( )x

t = 166

t = 167

q2 ( )x

t = 168

t = 169

( q2)x

t = 170

t = 171

t = 172

t = 173

t = 174

t = 175

t = 176

t = 177

( )x qb

1( ) ) B

t = 178

()xqb2

( ) ) B

t = 179

( ) ) B

t = 180

( ) ) B

t = 181

B (x qb

x( ) ) B

t = 182

B(xqb2

( ) ) B

t = 183

( ) ) B

t = 184

B qb0,3

( ) ) B

t = 185

( ) ) B

t = 186

B q0,4(x

( ) ) B

t = 187

B q2(x

( ) ) B

t = 188

( ) ) B

t = 189

B ( qb3

( ) ) B

t = 190

B ( (q3

( ) ) B

t = 191

B ( q4 ( )x

( ) ) B

t = 192

( ) ) B

t = 193

q2 ( )x

( ) ) B

t = 194

( ) ) B

t = 195

( q2)x

( ) ) B

t = 196

( ) ) B

t = 197

q2 ( ) ) B

t = 198

t = 199

( q2 ) ) B

t = 200

()xqb4

t = 201

( )x qb

t = 202

t = 203

q4 ) B

t = 204

t = 205

( q4)x

t = 206

t = 207

t = 208

t = 209

q2 ) B

t = 210

t = 211

t = 212

t = 213

t = 214

q4)x B

t = 215

t = 216

t = 217

t = 218

t = 219

t = 220

( q4)x

t = 221

t = 222

t = 223

t = 224

t = 225

t = 226

t = 227

t = 228

) )x B

t = 229

) )x B

t = 230

1( ) )

t = 231

( ) )x B

t = 232

( ) )x B

t = 233

( ) )x B

t = 234

B (x qb

( ) )x B

t = 235

B(xqb2

( ) )x B

t = 236

( ) )x B

t = 237

B qb0,3

( ) )x B

t = 238

( ) )x B

t = 239

B q0,4(x

( ) )x B

t = 240

B q2(x

( ) )x B

t = 241

( ) )x B

t = 242

B ( qb3

( ) )x B

t = 243

B ((xqb4

( ) )x B

t = 244

B ( (x qb

( ) )x B

t = 245

B ( (x

( ) )x B

t = 246

B ( (x qb

1) ( ) )

t = 247

B ((xqb2

) ( ) )x B

t = 248

B ( qb1

) ( ) )x B

t = 249

) ( ) )x B

t = 250

) ( ) )x B

t = 251

B qb0,3

) ( ) )x B

t = 252

) ( ) )x B

t = 253

B ( qb0,3

) ( ) )x B

t = 254

) ( ) )x B

t = 255

B ( q0,4(x

) ( ) )x B

t = 256

) ( ) )x B

t = 257

B q0,4 ( (x

) ( ) )x B

t = 258

B q2 ( (x

) ( ) )x B

t = 259

) ( ) )x B

t = 260

B ( q2(x

) ( ) )x B

t = 261

) ( ) )x B

t = 262

B ( ( qb3

) ( ) )x B

t = 263

B ( ( )q3

( ) )x B

t = 264

B ( ( q4 ) ( ) )x B

t = 265

B ((xq1

) ( ) )x B

t = 266

B ( (x

q2 ) ( ) )x B

t = 267

B ( (x

( ) )x B

t = 268

B ( (x

3( ) )

t = 269

B ( (x

) )x B

t = 270

B ( (x

q4 ( ) )x B

t = 271

B ( (x

( ) )x B

t = 272

B ( (x

( ) )x B

t = 273

B ((xq3

( ) )x B

t = 274

B ( q4(x

( ) )x B

t = 275

( ) )x B

t = 276

( ) )x B

t = 277

( ) )x B

t = 278

( ) )x B

t = 279

( ) )x B

t = 280

) ( ) )x B

t = 281

) ( ) )x B

t = 282

B (x qb

1( ) ( ) )

t = 283

B(xqb2

( ) ( ) )x B

t = 284

( ) ( ) )x B

t = 285

B qb0,3

( ) ( ) )x B

t = 286

( ) ( ) )x B

t = 287

B q0,4(x

( ) ( ) )x B

t = 288

B q2(x

( ) ( ) )x B

t = 289

( ) ( ) )x B

t = 290

B ( qb3

( ) ( ) )x B

t = 291

B ( (q3

) ( ) )x B

t = 292

B ( q4 ( ) ( ) )x B

t = 293

( ) ( ) )x B

t = 294

q2 ( ) ( ) )x B

t = 295

) ( ) )x B

t = 296

( q2 ) ( ) )x B

t = 297

()xqb4

( ) )x B

t = 298

( )x qb

3( ) )

t = 299

) )x B

t = 300

q4 ( ) )x B

t = 301

( ) )x B

t = 302

( q4)x

( ) )x B

t = 303

( ) )x B

t = 304

( ) )x B

t = 305

( ) )x B

t = 306

q2 ( ) )x B

t = 307

) )x B

t = 308

( q2 ) )x B

t = 309

()xqb4

t = 310

( )x qb

t = 311

t = 312

( )x qb

t = 313

()xqb2

t = 314

t = 315

t = 316

t = 317

t = 318

t = 319

t = 320

B (x qb

t = 321

B(xqb2

t = 322

t = 323

B qb0,3

t = 324

t = 325

B q0,4(x

t = 326

B q2(x

t = 327

t = 328

B ( qb3

t = 329

B ((xqb4

t = 330

B ( (x qb

t = 331

B ( (x

t = 332

B ( (x qb

1) ( )

t = 333

B ((xqb2

) ( )x

t = 334

B ( qb1

) ( )x

t = 335

) ( )x

t = 336

) ( )x

t = 337

B qb0,3

) ( )x

t = 338

) ( )x

t = 339

B ( qb0,3

) ( )x

t = 340

) ( )x

t = 341

B ( q0,4(x

) ( )x

t = 342

) ( )x

t = 343

B q0,4 ( (x

) ( )x

t = 344

B q2 ( (x

) ( )x

t = 345

) ( )x

t = 346

B ( q2(x

) ( )x

t = 347

) ( )x

t = 348

B ( ( qb3

) ( )x

t = 349

B ( ( )q3

t = 350

B ( ( q4 ) ( )x

t = 351

B ((xq1

) ( )x

t = 352

B ( (x

q2 ) ( )x

t = 353

B ( (x

t = 354

B ( (x

t = 355

B ( (x

t = 356

B ( (x

q4 ( )x

t = 357

B ( (x

t = 358

B ( (x

t = 359

B ((xq3

t = 360

B ( q4(x

t = 361

t = 362

t = 363

t = 364

t = 365

t = 366

) ( )x

t = 367

) ( )x

t = 368

B (x qb

1( ) ( )

t = 369

B(xqb2

( ) ( )x

t = 370

( ) ( )x

t = 371

B qb0,3

( ) ( )x

t = 372

( ) ( )x

t = 373

B q0,4(x

( ) ( )x

t = 374

B q2(x

( ) ( )x

t = 375

( ) ( )x

t = 376

B ( qb3

( ) ( )x

t = 377

B ( (q3

) ( )x

t = 378

B ( q4 ( ) ( )x

t = 379

( ) ( )x

t = 380

q2 ( ) ( )x

t = 381

) ( )x

t = 382

( q2 ) ( )x

t = 383

()xqb4

t = 384

( )x qb

t = 385

t = 386

q4 ( )x

t = 387

t = 388

( q4)x

t = 389

t = 390

t = 391

t = 392

q2 ( )x

t = 393

t = 394

( q2)x

t = 395

t = 396

q2 ) B

t = 397

t = 398

t = 399

t = 400

t = 401

q4)x B

t = 402

t = 403

( q4)x

t = 404

t = 405

t = 406

t = 407

q2)x B

t = 408

t = 409

t = 410

t = 411

t = 412

q6)x B

t = 413

t = 414

t = 415

t = 416

t = 417

t = 418

t = 419

t = 420

t = 421

t = 422

B q6(x

t = 423

B q̂b5

t = 424

B(xq̂b6

t = 425

B (x q̂b

t = 426

(xq̂b6

t = 427

(x q̂b

t = 428

)xq̂b6

t = 429

)x q̂b

t = 430

(xq̂b6

t = 431

(x q̂b

t = 432

)xq̂b6

t = 433

)x q̂b

t = 434

)xq̂b6

t = 435

)x q̂b

t = 436

)x q̂b

t = 437

)xq̂b2

t = 438

)x q̂b

t = 439

)xq̂b2

t = 440

(x q̂b

t = 441

(xq̂b2

t = 442

)x q̂b

t = 443

)xq̂b2

t = 444

(x q̂b

t = 445

(xq̂b2

t = 446

B (x q̂b

t = 447

B(xq̂b2

t = 448

B q̂b1

t = 449

B q̂b0,3

t = 450

q̂0,3

t = 451

B q̂0,4(x

t = 452

B q̂2(x

t = 453

t = 454

B ( q̂b3

t = 455

B ((xq̂b4

t = 456

B ( (x q̂b

t = 457

B ( (x

t = 458

B ( (x q̂b

t = 459

B ((xq̂b2

t = 460

B ( q̂b1

t = 461

t = 462

B q̂b1

t = 463

B q̂b0,3

t = 464

q̂b0,4

t = 465

B ( q̂b0,3

t = 466

q̂0,3

t = 467

B ( q̂0,4(x

t = 468

q̂0,3

t = 469

B q̂0,4 ( (x

t = 470

B q̂2 ( (x

t = 471

t = 472

B ( q̂2(x

t = 473

t = 474

B ( ( q̂b3

t = 475

B ( ( )

t = 476

B ( ( q̂4 ) (x

t = 477

B ((xq̂1

t = 478

B ( (x q̂2 ) (

t = 479

B ( (x

)xq̂b4

t = 480

B ( (x

)x q̂b

t = 481

B ( (x

(xq̂b4

t = 482

B ( (x

(x q̂b

t = 483

B ( (x

t = 484

B ( (x

(x q̂b

t = 485

B ( (x

(xq̂b2

) )x B

t = 486

B ( (x

)x q̂b

) )x B

t = 487

B ( (x

)xq̂b2

) )x B

t = 488

B ( (x q̂b

) )x B

t = 489

B ((xq̂b2

) )x B

t = 490

B ( q̂b1

) )x B

t = 491

) )x B

t = 492

B q̂b1

) )x B

t = 493

B q̂b0,3

) )x B

t = 494

q̂b0,4

) )x B

t = 495

B ( q̂b0,3

) )x B

t = 496

q̂0,3

) )x B

t = 497

B ( q̂0,4(x

) )x B

t = 498

q̂0,3

) )x B

t = 499

B q̂0,4 ( (x

) )x B

t = 500

B q̂2 ( (x

) )x B

t = 501

) )x B

t = 502

B ( q̂2(x

) )x B

t = 503

) )x B

t = 504

B ( ( q̂b3

) )x B

t = 505

B ( ()

) )x B

t = 506

B ( ( q̂b1

) )x B

t = 507

) )x B

t = 508

B ( q̂b1

( ) (x

) )x B

t = 509

( ) (x

) )x B

t = 510

B q̂b1

( ( ) (x

) )x B

t = 511

B q̂b0,3

( ( ) (x

) )x B

t = 512

q̂b0,4

( ) (x

) )x B

t = 513

B ( q̂b0,3

( ) (x

) )x B

t = 514

q̂b0,4

) )x B

t = 515

B ( ( q̂b0,3

) )x B

t = 516

B ( ()

q̂b0,4

) )x B

t = 517

B ( ( ) q̂b0,3

) )x B

t = 518

B ( ( )(x

q̂0,3

) )x B

t = 519

B ( ( ) q̂0,4(x

) )x B

t = 520

B ( ( )

q̂0,3

) )x B

t = 521

B ( ( q̂0,4 ) (x

) )x B

t = 522

q̂0,3

) )x B

t = 523

B ( q̂0,4 ( ) (x

) )x B

t = 524

q̂0,3

( ) (x

) )x B

t = 525

B q̂0,4 ( ( ) (x

) )x B

t = 526

B q̂2 ( ( ) (x

) )x B

t = 527

( ) (x

) )x B

t = 528

B ( q̂2 ( ) (x

) )x B

t = 529

) )x B

t = 530

B ( ( q̂2 ) (x

) )x B

t = 531

B ( ()xq̂b4

) )x B

t = 532

B ( ( )x q̂b

) )x B

t = 533

B ( ( )x

(xq̂b4

) )x B

t = 534

B ( ( )x

(x q̂b

t = 535

B ( ( )x

t = 536

B ( ( )x

(x q̂4 ) )

t = 537

B ( ( )x

(xq̂3

) )x B

t = 538

B ( ( )x q̂4

) )x B

t = 539

B ( ()xq̂3

) )x B

t = 540

B ( ( q̂4)x

) )x B

t = 541

B ((xq̂1

) )x B

t = 542

B ( (x q̂2

) )x B

t = 543

B ( (x

)xq̂1

) )x B

t = 544

B ( (x

)x q̂2

) )x B

t = 545

B ( (x

) )x B

t = 546

B ( (x

( q̂b3

) )x B

t = 547

B ( (x

t = 548

B ( (x

( q̂4 ) )x B

t = 549

B ( (x

(xq̂1

) )x B

t = 550

B ( (x

(x q̂2 ) )

t = 551

B ( (x

)xq̂b4

t = 552

B ( (x

)x q̂b

t = 553

B ( (x

t = 554

B ( (x

)x q̂b

t = 555

B ( (x

)xq̂b2

t = 556

B ( (x

(x q̂b

t = 557

B ( (x

(xq̂b2

t = 558

B ( (x

)x q̂b

t = 559

B ( (x

)xq̂b2

t = 560

B ( (x q̂b

t = 561

B ((xq̂b2

t = 562

B ( q̂b1

t = 563

t = 564

B q̂b1

t = 565

B q̂b0,3

t = 566

q̂b0,4

t = 567

B ( q̂b0,3

t = 568

q̂0,3

t = 569

B ( q̂0,4(x

t = 570

q̂0,3

t = 571

B q̂0,4 ( (x

t = 572

B q̂2 ( (x

t = 573

t = 574

B ( q̂2(x

t = 575

t = 576

B ( ( q̂b3

t = 577

B ( ()

t = 578

B ( ( q̂b1

t = 579

t = 580

B ( q̂b1

( ) (x

t = 581

( ) (x

t = 582

B q̂b1

( ( ) (x

t = 583

B q̂b0,3

( ( ) (x

t = 584

q̂b0,4

( ) (x

t = 585

B ( q̂b0,3

( ) (x

t = 586

q̂b0,4

t = 587

B ( ( q̂b0,3

t = 588

B ( ()

q̂b0,4

t = 589

B ( ( ) q̂b0,3

t = 590

B ( ( )(x

q̂0,3

t = 591

B ( ( ) q̂0,4(x

t = 592

B ( ( )

q̂0,3

t = 593

B ( ( q̂0,4 ) (x

t = 594

q̂0,3

t = 595

B ( q̂0,4 ( ) (x

t = 596

q̂0,3

( ) (x

t = 597

B q̂0,4 ( ( ) (x

t = 598

B q̂2 ( ( ) (x

t = 599

( ) (x

t = 600

B ( q̂2 ( ) (x

t = 601

t = 602

B ( ( q̂2 ) (x

t = 603

B ( ()xq̂b4

t = 604

B ( ( )x q̂b

t = 605

B ( ( )x

(xq̂b4

t = 606

B ( ( )x

(x q̂b

t = 607

B ( ( )x

t = 608

B ( ( )x

(x q̂b

1) ) B

t = 609

B ( ( )x

(xq̂b2

t = 610

B ( ( )x q̂b

t = 611

B ( ()xq̂b2

t = 612

B ( ( q̂b1

t = 613

t = 614

B ( q̂b1

t = 615

t = 616

B q̂b1

( ( )x

t = 617

B q̂b0,3

( ( )x

t = 618

q̂b0,4

t = 619

B ( q̂b0,3

t = 620

q̂b0,4

t = 621

B ( ( q̂b0,3

t = 622

B ( ()x

q̂0,3

t = 623

B ( ( q̂0,4)x

t = 624

q̂0,3

t = 625

B ( q̂0,4 ( )x

t = 626

q̂0,3

t = 627

B q̂0,4 ( ( )x

t = 628

B q̂2 ( ( )x

t = 629

t = 630

B ( q̂2 ( )x

t = 631

t = 632

B ( ( q̂2)x

t = 633

B ( ()xq̂1

t = 634

B ( ( )x q̂2

t = 635

B ( ( )x

t = 636

B ( ( )x

( q̂b3

t = 637

B ( ( )x

t = 638

B ( ( )x

( q̂4 ) ) B

t = 639

B ( ( )x

(xq̂1

t = 640

B ( ( )x

(x q̂2 ) ) B

t = 641

B ( ( )x

)xq̂b4

t = 642

B ( ( )x

)x q̂b

t = 643

B ( ( )x

t = 644

B ( ( )x

)x q̂4 ) B

t = 645

B ( ( )x

)xq̂3

t = 646

B ( ( )x

(x q̂4

t = 647

B ( ( )x

(xq̂3

t = 648

B ( ( )x q̂4

t = 649

B ( ()xq̂3

t = 650

B ( ( q̂4)x

t = 651

B ((xq̂1

t = 652

B ( (x q̂2

t = 653

B ( (x

)xq̂1

t = 654

B ( (x

)x q̂2

t = 655

B ( (x

t = 656

B ( (x

( q̂b3

t = 657

B ( (x

t = 658

B ( (x

( q̂b1

t = 659

B ( (x

t = 660

B ( (x

)x q̂b

1( ) ) B

t = 661

B ( (x

)xq̂b2

( ) ) B

t = 662

B ( (x q̂b

( ) ) B

t = 663

B ((xq̂b2

( ) ) B

t = 664

B ( q̂b1

( ) ) B

t = 665

( ) ) B

t = 666

B q̂b1

( ) ) B

t = 667

B q̂b0,3

( ) ) B

t = 668

q̂b0,4

( ) ) B

t = 669

B ( q̂b0,3

( ) ) B

t = 670

q̂0,3

( ) ) B

t = 671

B ( q̂0,4(x

( ) ) B

t = 672

q̂0,3

( ) ) B

t = 673

B q̂0,4 ( (x

( ) ) B

t = 674

B q̂2 ( (x

( ) ) B

t = 675

( ) ) B

t = 676

B ( q̂2(x

( ) ) B

t = 677

( ) ) B

t = 678

B ( ( q̂b3

( ) ) B

t = 679

B ( ()

( ) ) B

t = 680

B ( ( q̂b1

) ( ) ) B

t = 681

) ( ) ) B

t = 682

B ( q̂b1

( ) ( ) ) B

t = 683

( ) ( ) ) B

t = 684

B q̂b1

( ( ) ( ) ) B

t = 685

B q̂b0,3

( ( ) ( ) ) B

t = 686

q̂b0,4

( ) ( ) ) B

t = 687

B ( q̂b0,3

( ) ( ) ) B

t = 688

q̂b0,4

) ( ) ) B

t = 689

B ( ( q̂b0,3

) ( ) ) B

t = 690

B ( ()

q̂b0,4

( ) ) B

t = 691

B ( ( ) q̂b0,3

( ) ) B

t = 692

B ( ( )(

q̂b0,4

t = 693

B ( ( ) ( q̂b0,3

t = 694

B ( ( ) ()

q̂b0,4

t = 695

B ( ( ) ( ) q̂b0,3

t = 696

B ( ( ) ( ))

q̂b0,4

t = 697

B ( ( ) ( ) ) q̂b0,3 B

t = 698

B ( ( ) ( ) ) q̂b0,1 B

t = 699

B ( ( ) ( ))

q̂b0,2

t = 700

B ( ( ) ( ) q̂b0,1

t = 701

B ( ( ) ()

q̂b0,2

t = 702

B ( ( ) ( q̂b0,1

t = 703

B ( ( )(

q̂b0,2

t = 704

B ( ( ) q̂b0,1

( ) ) B

t = 705

B ( ()

q̂b0,2

( ) ) B

t = 706

B ( ( q̂b0,1

) ( ) ) B

t = 707

q̂b0,2

) ( ) ) B

t = 708

B ( q̂b0,1

( ) ( ) ) B

t = 709

q̂b0,2

( ) ( ) ) B

t = 710

B q̂b0,1

( ( ) ( ) ) B

The simulated RPCA accepts the input

t = 711

B + ( ( ) ( ) ) B

t = 712

B ( + ( ) ( ) ) B

t = 713

B ( ( + ) ( ) ) B

t = 714

B ( ( ) + ( ) ) B

t = 715

B ( ( ) ( + ) ) B

t = 716

B ( ( ) ( ) + ) B

t = 717

B ( ( ) ( ) ) + B

t = 718

B ( ( ) ( ) ) � +

t = 719

B ( ( ) ( ) � ) +

t = 720

B ( ( ) ( � ) ) +

t = 721

B ( ( ) � ( ) ) +

t = 722

B ( ( � ) ) ( ) +

t = 723

B ( � ( ) ) ) ( +

t = 724

B � ( ) ( ) ) ( +

t = 725

� B ) ( ) ( ( ) +

t = 726

� ) B ) ( ( ( ) +

t = 727

� ) ) B ( ( ) ( +

t = 728

� ) ) ( B ) ( ( +

t = 729

� ) ) ( ) B ( ( +

t = 730

� ) ) ( ) ( B ( +

t = 731

� ) ) ( ) ( ( B +

t = 732

� ) ) ( ) ( ( Y B

t = 733

� ) ) ( ) ( ( Y B

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