Lindenmayer systems as a model of computations

11 Pages·0.14 MB·English

by  Yuri Ozhigov

LINDENMAYER SYSTEMS AS A MODEL OF COMPUTATIONS
Y. Ozhigov
8 Abstract.
9 LSisaparticulartypeofcomputational processessimulatinglivingtissue. They
9 use an unlimited branching process arising from the simultaneous substitutions of
1 some words instead of letters insome initial word. This combines the properties of
cellularautomata andgrammars. Itisprovedthat
n
1) The set of languages, computed in a polynomial time on such LS that all
a
replacingwordsarenotempty,isexactlyNP-languages.
J
2)Thesetoflanguages,computedinapolynomialtimeonarbitraryLS,contains
7
thepolynomialhierarchy.
1
3) The set of languages, computed in a polynomial time on a nondeterministic

versionofLS,strictlycontains thesetoflanguages, computed inapolynomialtime
1 onTuringMachineswithaspacecomplexity na,whereaispositiveinteger.
v
In particular, the last two results mean that Lindenmayer systems may be even
2
morepowerfultoolofcomputations thannondeterministicTuringMachine.
0
0
1
0
8 1.Introduction and definitions
9
/
s The simulation of nondeterminism by a physical device remains attractive but
a unattainabledream. Eventhemostpowerfulphysicalcomputationaldevice-quan-
g
tumcomputerisnotabletoachievethisobjectiveinthecomputationswithoracles
-
p as follows from the result of C.Bennett, E.Bernstein, G.Brassard and U.Vazirani
m [1]. This is mostlikely to be true also for absolute (without oracles)computations.
o Thereforelivingthings present,probablythe best realmodelofnondeterminismin
c nature.
:
v The simple mathematical model of the structures with two fundamental prop-
i erties of a Life: reproductions and determined stable interactions with the nearest
X
neighborhood, was proposed by A.Lindenmayer ([4],[5][6] ). One-dimensional Lin-
r
a denmayer systems considered in this paper is a crude model of a reproduction
because we ignore the spatial arrangement of tissue, otherwise this model is well
suited to describe the behavior of living cells. The spatial arrangement of tissue
can be often ignored for example, for the filamentous organisms.
Lindenmayersystem also bears a resemblance to an imaginarynondeterministic
computerwith the supernaturalabilities. These abilities isthe subjectfor study in
this work. We proceed with an exact definition.
Let ω = {a ,a ,...,a } be our basic alphabet, ω∗ denotes the set of all words
0 1 n
from ω and Σ = {a Aa | A ∈ ω∗} will be our basic set of words. If x ∈ ω∗, then
0 0
|x| means the length of the word x. Let A be a function of the form:
A: ω3 −→ω∗,
TypesetbyAMS-TEX
1
2 Y. OZHIGOV
where A(a ,a ,a ) = a for all i,j = 0,1,...,n. Then A is called LS. If here
i 0 j 0
∀x∈ImA |x|>0 then A is called L-system with nonempty results - LSN.
Finally, if here ∀x ∈ ImA |x| = 1, then A is an ordinary cellular automaton
(CA) (look at [10] ).
Evolutions generated by LS ( L-languages ) and their algorithmic properties
were subject for study mainly since 1971. As a typical example of mathematical
problem here we refer to an algorithmically undecidable problem of coincidence
for languages, determined by two LS ( look at the work of A.Lindenmayer [6]).
W.J.Savitchin [8]gavesomeconditions whichshouldbe imposedonthe nontrivial
set of strings to ensure that it is L-language; this conditions use stack machines.
J. van Leeuwen in [3] obtained upper bounds for space required to recognize some
extensions of L-languages ( by natural operations such as sum and intersection )
deterministicallyandnondeterministically. P.Vitanyiin[9]treatedcontextsensitive
table L-languages. Other interesting examples may be found in [7].
In the present paper LS will be considered as a particular type of computa-
tions using unlimited branching processes which arise if the length of some word
A(a a a )ismorethan1. Suchaprocessrequiresexponentionalspacebutreduces
i j k
the time for solutions of NP-complete problems. The advantage of such compu-
tational processes are their simplicity and power. The simplicity of rules for LS
allows to find them by evolutionary programming as in the work [2] of J.R.Koza.
Inthis paper westudy a polynomialcomplexityofcomputations onLS. LSmay
be considered as a sort of cellular automata with reproductions which resembles a
nondeterminism. It is founded that the set of languagescomputed in a polynomial
time on deterministic as well as on nondeterministic version of LSN is coincident
?
with NP. Thus the complexity status of LSN is clear with a precision of P = NP-
problem. The following peculiarities of LS must be mentioned explicitly.
The first is that the possibility of an empty results A(a ,a ,a ) means the an-
i j k
nihilation of some domains in a computational space and so it allows to transmit
an information faster comparatively with LSN. In the section 4 it is proved that
the setoflanguagescomputedonLSina polynomialtime containsthe polynomial
hierarchy.
The second is that this presumably makes LS inconvenient for analysis and its
complexity status is still obscure.
Let P be the set of languages, which can be computed on Turing Machines in
α
a polynomialtime with a space complexity nα, α=1,2,.... The sole lowerbound
ofcomplexity resulthavingto do with LSis thatthe setoflanguagescomputedon
the nondeterministic version of LS in polynomial time strictly includes P .
α
Remark. All inclusions established in this paper can be relativized. This property
may be traced by a reader in all constructions.
If A is LS and
(1) B =a a a ...a a
i0 i1 i2 ip ip+1
is an arbitrary word from Σ, a = a = a , the new word A(B), defined by
i0 ip+1 0
A(B)=a A A ...A a where ∀j =1,...p A =A(a ,a ,a ), is called the
0 1 2 p 0 j ij−1 ij ij+1
result of A-action on B.
Let W be the underlined occurrence in the word B:
a a ...a a a ...a a ...a a .
0 i1 is−1 is is+1 it it+1 ip 0
LINDENMAYER SYSTEMS AS A MODEL OF COMPUTATIONS 3
The base of W is the word a a ...a . Then the underlined occurrence W in
is is+1 it 1
the word A(B):
a A ...A A A ...A A ...A a
0 i1 is−1 is is+1 it it+1 ip 0
is called the descendant of W in transformation B −→A(B). If A 6= a then an
j ij
occurrence of a in B will be considered to be affected in this transformation.
ij
A sequence
B ,B ,...,B
0 1 N
is called an evolution of A if ∀i = 1,...,N B = A(B ). If B = B , the
i i−1 0
descendant of W in this evolution is defined by induction on N. This evolution is
calledcompleteifthelettera occursonlyinthewordB andnotinB ,...,B .
n N 1 N−1
WesaythatasetM ofsomewordsinalphabetω ={a ,...,a }iscomputed
1 1 n−1
on LS A if for any B ∈ω∗ the following conditions are equivalent:
1
1) there exists a complete evolution B ,B ,...,B , where a Ba =B .
0 1 N 0 0 0
2) B ∈M.
In this case for B ∈M we denote the number N from 1) by τ (B). The set M,
A
computed on A, is denoted by M .
A
The time complexity of LS A is defined by the following
t (n)= max τ (B).
A A
|B|≤n, B∈M
Let PLS ( PLSN ) denote the class of languages,computed on LS ( on LSN ) in
a polynomial time.
Now let us define all corresponding notions for a nondeterministic versionof LS
( LSN ). To denote these notions we add the letter ”N” : NLS, NLSN, etc.
For an arbitrary set C let 2C denote the set of all subsets of C. Thus, NLS is a
function of the form
A: ω3 −→2ω∗.
If ∀¯b∈ω3 ∀C ∈A(¯b) |C|>0, then A is called NLSN. If a word B from Σ has
the form (1) then the result of A-action on B is the set
A(B)={a A A ...A a | ∀j =1,...,p A ∈A(a ,a ,a )}.
0 1 2 p 0 j ij−1 ij ij+1
AsequenceB ,...B iscalledanevolutionofAif∀i=1,...,N B ∈A(B ).
0 N i i−1
A complete evolution is defined as above.
IfB ∈ω∗ letτ (B)betheminimallengthofcompleteevolutionsbeginningwith
A
B =a Ba if such an evolution exists, and τ (B)=∞ in the opposite case.
0 0 0 A
NLS A admits a word B iff τ (B) < ∞. Thereafter, the time complexity of A
A
can be defined by
t (n)=max{τ (B) | |B|≤n, τ (B)<∞}.
A A A
M denotes here the set of all words admitted by A , we say that this set is
A
computed by A.
So,NPLS(N)denotes the classofallsets,computedonNLS(N) ina polynomial
time. The first two results to be established in this paper are the following.
Theorem 1. PLSN=NPLSN=NP
Theorem 2. ∀α=1,2,... NPLS6=P
α
4 Y. OZHIGOV
2. Deterministic version of LSN
Proof of Theorem 1
The deduction of inclusion
(2) NP⊆PLSN
willbeourinitialconcern. Notethattheestablishmentofthe belongingc∈PLSN,
whereC isanarbitraryNP-completesetwouldbeample. Forexample,letSATbe
the set ofallsatisfiable formulas ofa propositionallogic. The inclusion (2) reduces
to the establishment of the following belonging:
SAT∈PLSN.
Let¯l={&,∨,¬,(,),0,1}bethebasicalphabetforapropositionallogic. Letany
Booleanvariableα be codedby the word011...10. ThenanyBooleanformulaφ
i
i times
canbeencodedin¯l andletpφqbe itscode. |Le{tzs¯}={f,t}beanauxiliaryalphabet
for Boolean values: t - true and f - false.
Let a special alphabet ω consist of the following parts:
1
1) the alphabet ¯l for the propositional logic;
2) the alphabet s¯for the Boolean values;
3) auxiliary letters α,β,b,e,h,∗,+,%,ν;
4) alphabets of doubles: L ,L ,L ,L ,L for the letters from
1 2 3 4 5
L=¯l∪{α,h,b,e,ν}∪s¯(ifx∈Lwe’lldenoteitsdoublesbyx′ ∈L ,x˜∈L ,x0 ∈
1 2
L ,x+ ∈L ,x¯∈L respectively ).
3 4 5
5)the alphabetL ×L whose elementsaredenotedby d1 ,where d ,d ∈L .
5 5 d2 1 2 5
(cid:0) (cid:1)
Lemma 1. There exists LSN A with the alphabet ω , such that for any proposi-
1 1
tional formula φ ( which has been encoded as it is pointed out above ) there exists
the following evolution of A :
1
B =a pφqa ,B ,...,B ,...
0 0 0 1 N
where:
1) N =4|pφq|2+7|pφq|+15, B has the form
N
(3) a eA eA e...eA ea ,
0 1 2 q 0
q = 2|pφq|, ∀i = 1,2,...,q A = S %pφq, S ∈ s¯∗, |S | = |pφq|, and for all i,j if
i i i i
1≤i<j ≤q then S 6=S .
i j
2) ∀j =2,3,...,N, A,B if
(4) B =A%B
j
then ∃A′,B′ : A = A′S′, B = pφqB′, where S′ ∈ s¯∗, |S′| = |pφq|, and the
underlined occurrencein the word B : A′S′%pφqB′ is not affected in the evolution
j
B ,B ,....
j j+1
3) ∀j >N B =B .
j N
LINDENMAYER SYSTEMS AS A MODEL OF COMPUTATIONS 5
Proof of Lemma 1
ItisnotdifficulttounderstandhowsuchanautomatonA canbeconstructed. It
1
mustreplicatethewordsoftheformSbkαφforvariousvaluesofS ∈s¯∗ sequentially,
where for all time |Sbk|=|pφq|, |S|=1,2,..., b is the special letter, k ∈N. And
what is essential for the present purposes, this can be done in a polynomial time
for the input data pφq by the definition of LSN.
Any LS A with an alphabet ω may be determined by the list of all records of
the form
(a,b,c)−→A(a,b,c); a,b,c∈ω,A(a,b,c)6=b,
whichmaybethoughtofasactivecommands. Inwhatfollowsanylistofcommands
is taken to be supplemented by all plausible (passive) commands of the form
(a,b,c)−→b
which are not written explicitly.
The list of commands, determining A consists of two groups:
1
Group 1.
input: a pφqa ,
0 0
output: a b...bαpφqea
0 0
|pφq|
time: ≤7|p|φ{qz|}+5.
x,y take all values from ¯l, z takes all values from ¯l and its doubles from L , ? -
2
from ω .
1
(x,a ,?)−→ea , (?,x′,y˜)−→x˜,
0 0
(?,a ,x)−→a +, (?,x′,e)−→x0,
0 0
(?,+,z)−→β+, (?,x0,?)−→x,
(?,a0,β)−→a0e, (?,x′,y0)−→x0,
(+,x,?)−→x′, (?,+,x0)−→α,
(x′,y,?)−→y˜, (?,β,α)−→b,
(?,x˜,?)−→x′, (?,β,b)−→b.
Group 2.
input: a ebqαpφqea ,
0 0
output: (3)
time: ≤4|pφq|2+10,
w,x,y,z,u take all values from L−{e}.
6 Y. OZHIGOV
y¯ y¯
(x, ,?)−→ ,
(cid:18)z¯(cid:19) (cid:18)x¯(cid:19)
(v,b,?)−→ν˜ (v ∈ω −{b,β}),
1
x¯ y¯ y¯
(?,x,y˜)−→x˜, ( , ,?)−→ , d,g ∈L∪{h},
(cid:18)d¯(cid:19) (cid:18)g¯(cid:19) (cid:18)d¯(cid:19)
(r,˜b,?)−→ν˜, (r ∈s¯∪{e,∗,a }),
0 y¯
(c,x¯, )−→x,
(ρ,x˜,?)−→x¯, ρ∈{∗,e}, (cid:18)h¯(cid:19)
(x¯,y,?)−→y¯, x6=ν, c∈{e,∗,a },
0
y¯ x¯
(x¯,y¯,e)−→ (x∈/ s¯or y 6=b), (c,ν¯, )−→f, c∈{e,∗,a },
(cid:18)y¯(cid:19) (cid:18)h¯(cid:19) 0
(x¯,e,?)−→∗∗, y¯
(x, ,?)−→y, x∈/ s¯or y 6=ν,
x¯ (cid:18)z¯(cid:19)
( ,∗,?)−→∗y, (y 6=ν),
(cid:18)y¯(cid:19) ν¯
(r, ,?)−→f, r ∈s¯,
x¯ (cid:18)x¯(cid:19)
( ,∗,?)−→∗t,
(cid:18)ν¯(cid:19) (x,∗,?)−→e,
y¯ x¯ ¯b
(w¯,x¯,(cid:18)z¯(cid:19))−→(cid:18)h¯(cid:19), (r¯,¯b,e)−→(cid:18)ν¯(cid:19),
(x,α,?)−→%.
Informal comments.
Let an evolution of A begin with the word a pφqa .
1 0 0
1) An occurrence of word xb where x ∈/ {b,β} originally arises in the output of
Group 1: a pφqa .
0 0
2)ThefirstactivecommandfromGroup2operatingintheevolutioninquestion
is (e,b,?)−→ν˜ , so the input of Group 2 has the form a pφqa .
0 0
Let φ contain Boolean variables α ,α ,...,α .
1 2 q
Lemma 2. There exists CA A such that 1) For any active command of A :
2 2
(x,y,z) −→ w y,w ∈/ ω and if t ∈ {x,z},t ∈ ω , then t = %. 2) For any input
1 1
data of the form (4) from the statement of Lemma 1 the following conditions are
equivalent:
A). There exists a complete evolution of A of the form
2
(5) Bj =Bj′,Bj′+1,...,BN′ ′,
B). B contains an occurrence of the word S%φC, where the end of the length q
j
ofthewordS constitutesthelistofvalues forBoolean variables whichmakes φtrue.
3) Any complete evolution of A of the form (5) has the length N′ =O(|pφq|2).
2
Proof of Lemma 2
A realizes a conventional procedure for the search of true value for φ. Such
2
a procedure is well known and it requires a quadratic time on Turing Machines.
Lemma 2 is proved.
LINDENMAYER SYSTEMS AS A MODEL OF COMPUTATIONS 7
Returningtothe proofofinequality(2),note thatif thelistofactivecommands
forLSNA isobtainedfromsuchlists forA andA bya simple joinofthem, then
1 2
A recognizes the satisfability of Boolean formulae in a quadratic time that implies
inequality (2).
It is evident that PLSN ⊆ NPLSN. So, the establishment of the following
inclusion
(6) NPLSN⊆NP
would suffice to accomplish the proof of Theorem 1. We shall take up this subject
now.
Given NLSN A computing the set M in a polynomial time, one need only
A
to obtain a nondeterministic cellular automaton A∗, which computes M in a
A
polynomial time.
Let
(7) B ,B ,...,B
1 2 N
be anevolutionofNLSNAandletW be someselectedandunderlinedoccurrence
i
in B : B =B′A′a A′′B′′ for each i=1,2,...,N, so that
i i i i ji i i
1) for all i=1,...,N
if B′ 6=Λ, then |A′|=N +1,
i i
if B′′ 6=Λ, then |A′′|=N +1, where Λ denotes the empty word,
i i
2) for all i = 1,...,N −1 W is contained in the descendant of W in the
i+1 i
transformation B −→B ,
i i+1
3) if U is separated occurrence of a in W then U is contained in the de-
i ji i i+1
scendant of U in the transformation B −→B .
i i i+1
In this case the sequence of occurrences
(8) W ,W ,...W
1 2 N
is called a local sequence for the evolution (7), and the sequence
(9) a ,a ,...a
ji j2 jN
is called the central sequence for evolution (7) corresponding to the local sequence
(8).
The following property of NLSN follows immediately from the definitions.
Property of localization. Let the sequence (9) be the central sequence for evolu-
tion (7) corresponding to the local sequence (8). Then for any i ∈ {1,...,N} and
for any words C′,C′′ the sequence a ,a ,...,a will be a central sequence for
i i ji ji+1 jN
evolution B′,B′ ,...,B′ , where B′ =C′A a A′′C′′.
i i+1 N i i i ji i i
This property makes it possible to simulate NLSN A on a nondeterministic
cellular automaton A∗. The work of A∗ consists of sequential cycles, each of them
has an input I and an output I with separated occurrences.
i i+1
Let I = a A′a A′′a be the input of some step number i with the separated
i 0 i ji i 0
occurrence, and a B′AB′′a ∈ A(I ) where A is the descendent of a . Let A =
0 0 i ji
A a A be some representation of A.
1 ji+1 2
8 Y. OZHIGOV
Let t (n)=O(np), p∈N, M =|I |, where I is an input of A.
A 1 1
Let A′ be the maximal from the ends of B′A, whose lengths do not exceed
i+1
t (M)+1.
A
Let A′′ be the maximal from the beginnings of A B′′, whose lengths do not
i+1 2
exceed t (M)+1. Then
A
I =a A′ a A′′ a .
i+1 0 i+1 ji+1 i+1 0
The automaton A∗ described computes the set M due to Property of localiza-
A
tion. It is readily seen that A∗ requires the time O(n2p). Theorem 1 is proved.
Note that Theorem 1 gives PLS⊆EXPTIME in particular.
3. Nondeterministic version of LSN
Theorem 2: Sketch of the proof.
A conventionalprocedure of diagonalizationwill be run by NLS. Namely, if pBq
denotes a code of CA B, we shall construct such NLS A with the time complexity
t (n)=O(n)thatforanyalphabetω′andforanyα=1,2,... thereexistc ,c >0
A 1 2
so that for any CA B in alphabet ω′ the following two conditions are equivalent.
1). There exists a complete evolution of B of the form
B′ =a pBqa ,B′,...,B′
0 0 0 1 M
where m = |pBq|,M ≤ exp(c m), ∀i = 1,...,M |B′ | ≤ mα and B contains
1 M M
a .
n−1
2). There exists a complete evolution of A of the form
B =a pBqa ,B ,...,B , h≤c mα.
0 0 0 1 h 2
I drop all details.
4. LS and the polynomial hierarchy
There is an intimate connection between LS and the well-known polynomial
hierarchy (PH).
PH is determined by the sequences of classes:
ΣP ⊆ΣP ⊆...,
0 1
ΠP ⊆ΠP ⊆...,
0 1
defined by the following induction.
Basis. ΣP =ΠP =P- isthe classofallpredicates,computedina polynomial
0 0
time on Turing Machines.
Step. 1. Let ΣP be the class of all predicates A(x ,...,x ) such that
n 1 s
A(x ,...,x )⇐⇒∃x : |x |≤p(|x |,...|x |) B(x ,...,x ,x )
1 s s+1 s+1 1 s 1 s s+1
for some polynomial p and some B ∈ΠP .
n−1
2. Let ΠP be the class of all predicates A(x ,...,x ) such that
n 1 s
A(x ,...,x )⇐⇒∀x : |x |≤p(|x |,...|x |) B(x ,...,x ,x )
1 s s+1 s+1 1 s 1 s s+1
for some polynomial p and some B ∈ΣP .
n−1
LINDENMAYER SYSTEMS AS A MODEL OF COMPUTATIONS 9
∞ ∞
Then PH= ΣP = ΠP .
n n
nS=0 nS=0
Adding the sign of coma to alphabet ω we naturally extend the class PLS of
predicates to predicates A(x ,...,x ) on ω∗×···×ω∗ for any n=1,2,....
1 n
n times
Theorem 3. PH⊆PLS. | {z }
Proof
It is sufficient to provethat for all n=0,1,... the following two inclusions take
place:
(10) ΣP ⊆PLS
n
(11) ΠP ⊆PLS.
n
This will be proved by a simultaneous induction on n.
Basis Follows from Theorem 1.
Step
1) Let us prove the inclusion (10).
GivenLSB computingapredicateB(x ,...,x ,x inapolynomialtimep by
1 s s+1 1
theinductivehypothesis,theLSAmustbeconstructedcomputinginapolynomial
time the predicate
A(x ,...,x )⇐⇒∀x : |x |≤p(|x |,...|x |).
1 s s+1 s+1 1 s
We can suppose that p and p are increasing positive functions.
1
It will be readily seen how to define such LS A with the help of new auxiliary
letters in the following sequential steps.
Step 1 Input :
x ,x ,...,x
1 2 s
Output
(12) αx˜ ,...,x˜ ,x˜ %x˜ ,...,x˜ ,x˜ %...%x˜ ,...,x˜ ,x˜ α,
1 s s+1 1 s s+2 1 s s+M
whereM ≤np(|x1|,...,|xs|)+1+1,xs+1,...,xM areallthedifferentwordsofthelength
≤p(|x |,...,|x |), x =y y ...y , all y are letters, x˜ =Ty Ty T ...Ty T,
1 s i i1 i2 isi ij i i1 i2 isi
T =β×···×β, p =p (|x |,...,|x |,p(|x |,...,|x |)). T will be a counter recog-
2 1 1 s 1 s
p1
nizin|g the{zfinish} of the work of B on the list x1,...xs+j; j =1,...,M.
Appropriate rules for A can be written as in section 2.
Step 2 Input : (12)
Output:
(13) αR %R %...%R α,
1 2 s+M
whereR istheresultoftheworkofB onx ,...,x , j =1,...,M. Thecounters
i 1 s+j
are contracted on 1 in any substitution of A in Step 2.
10 Y. OZHIGOV
Step 3
Input : (13)
Output contains the letter γ iff A(x ,...x ) is right.
1 s
Let a be the final letter for B if success. Rules for the Step 3 have the form:
n
(x,y,z)−→L , if %,a ,α∈/ {x,y,z}, or y =z =a ,
n n
(x,y,z)−→a , if only one from x,y,z is a ,
n n
(%,a ,%)−→L ,
n
(a ,%,a )−→L ,
n n
(a ,α,α)−→γ,
0
(α,%,a )−→L ,
n
(a ,%,α)−→L .
n
Inclusion (10) is proved.
Let’s prove (11).
The difference from the previous case is only in Step 3. Rules have the form:
(x,y,z)−→L , if a ∈/ {x,y,z}, or y =z =a ,
n n
in other cases (x,y,z)−→a , if a ∈{x,y,z},
n n
(α,a ,α)−→γ.
n
Theorem 3 is proved.
5. Resume
Thus, writing out sequentially the standard classes P, NP, HP and all the com-
plexity classes defined above, we obtain the following chain
? ? ? ?
P⊆NP=PLSN=NPLSN⊆PH⊆PLS⊆NPLS6=P .
α
? ?
The following two problems: P ⊆ NP and NP ⊆ PH are the famous open prob-
lems in the theory of complexity. Therefore, in view of Theorem 3, Lindenmayer
system (with the possibility of empty results) may be even more powerful tool of
computations than nondeterministic Turing Machine.
Note that the nondeterminism of LS is conceptually classical. Harnessing of
quantum mechanical effects for increasing the power of LS faces problems. The
pointisthatreproductionsofquantumstatesisforbiddenbyuncertaintyprinciple.
7. References
1. C.H.Bennett, E.Bernstein, G.Brassard,, U.Vazirani, Strengths and Weakness of Quantum
Computing,SIAMJournalofcomputing. (1997).
2. J.R.Koza,DiscoveryofrewriterulesinLindenmayersystemsandstatetransitionrulesincel-
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