Colonies - the simplest grammar systems
Alica I(elemenov6
Institute of Computer Science, Faculty of Philosophy and Science, Silesian University in Opava, Czech Republic
Department of Computer Science, Faculty of Pedagogl,, Catolic tJniversity in RuZomberok, Slovakia
a l i c a . k e l e m e n o v a @ f p f . s l u . c z
Abstract. Brief introduction to the colonies, the grammars systems in- teracting on common passive environment with components individually producing finite languages, is presented. An overview of the topic is com- pleted by large number of references.
I(eywords: grammar system, colony, generative power, hierarchy
1 Introduction
Colonies were introduced in [19], where basic fact on the motivation can be found. Overviews on the topic are in [20], [29]. The present paper was prepared to introduce the topic, namely to call attention to sequential and parallei models of colonies. It ends with the list of research topics inside the theory of colonies and with actual open problems. For the further information on the topic we rec- ommend www.sztaki.hu/mms/bib.html, where also abstracts of the listed papers can be found.
2 Colony - general model
A colony is a collection of (very simple) grammars operating in a common string.
By "very simple" we mean a grammar producing a fi,ni,te language.
First, we present general model of a colony. Different acting possibiiities realized by different derivation steps, due to different motivations leacl to various variants of colonies.
D e f i n i t i o n 1 . A c o l o n y C is a S-tuple C : ( V , T , f ) , w h e r e (i) V is arr, alphabet of the colony, and
(ii) T e V is a terminal alphabet of the colony,
( i i i ) f : { ( S , , F t ) t S t e V , F i e ( V - , 5 i ) . , 4 i s f i n i t e , 1 ( i < r } .
A par'r (St, Fi) i,s called l-th component of C, Si is r,ts start symbol and, Fi is the language of i-tlt component.
1 5 8 A. Kelenrenovd
I''lote 1. a) Any symbol of the strings from 4 can occur as a start symbol .9i of another component of the colony, j + i .
b) Conrponcrrts can bc detcrrrrincd in more dctaiJ., by cornplete spccificatiorrs of (regular) grammars Gr - (l{r,Ti, St, Pt) producing from ,Si words of Fi, L(Gi) : 4. See the original definition in [19]. This case is useful for study processes in the ievel of components.
c) By the definition, V:7 is also possible. This special case is discussed in [28], [ 1 0 ]
An activity of components in a colony is realized by string transformation on a cornmon tape.
Generally, denote by =5 the relation on strings representing an elementary string transformation realized by components and called a derivation step. Spe- cific types of r introduced la,ter define various variants of colonies.
As usual, =5. stays for a reflexive anci transitive closure of =5 . It rep- resents string transformations, called derivation, realized by finite sequences of elementary transformations.
L a n g u a g e L , ( C , t l s ) d e t e r m i n e d by a colony C: ( V , T , f ) a n d a n i n i t i a l string (axiom) wo e V* by =5 derivation consists of all terminal strings derived from the axiom, i.e.
L , ( C , u ; 0 ) : { r l , o + . u , u € T " } .
By COL* w€ denote the class of all languages generated by colonies with S derivation.
The subscript tr, above, can be omitted in the case when it is clear which derivation step is considered.
Basic differences among various possibilities of a behaviour of a colony, due to the number of components used in one step. The sequential model and parallel models discussed in the next sections are two examples of behaviour of colonies. The intermediate cases, colonies working in teams are discussed in [10],
l42l
3 Sequentional Colonies
In the sequential models of a colony an elementary change of strings is realized by single conponent. Sequential colonies can differ in the amount of symbols rewritten by a component in one derivation step.
Basic derivation step =:; corresponds to rewriting a single symbol. Total derivation step i; corresponds to a parallel behaviour of the chosen component.
3.1 Sequential colonies with sequentialy acting components b mode of derivation
The simplest case of rewriting in sequential colonies is the case wheu chosen component rervrites exactly one letter in a derivation step. Formally:
Colonies - the Simplest Grantmar Systents
Definition 2. For r,A e V* we defi,ne basic deriuation step:
r + y i f f r : r t s i r z , U : r 1 Z T 2 a n d z € 4 f o r s o m e i, , L < i 1 > n . In accordance r,vith general case) the language of the colony with derivation step -!+ ir clenotecl La(C,trs) and COLb is the corresponding class of all languages.
T h e o r e m 1 . C O L u - - C F
Proof. Let L e C,F. There is a context free grammar G : (1V,7, P, ,S) such t h a t I : L ( G ) . T o c o n s t r u c t a c o l o n y C : ( V , T , f ) s u c h th a t L ( G ) - L 6 ( C , r ) for some ?r) we have to determine finite Languages of components. To do it we eliminate of all rules with direct recursion fron G. Let l/ : {Ze t A e ,A/}, wlrere Z s are new nonterminals.
lVe replace rules of the type A > ()) where A occurs in a by two rules A -, aA, ZA --- A,
where a4 denotes the word obtained from a by replacing all occurrences of A by z,t.
Denote bv G resulting grammar G : (N U /V, T, P,.9). Evidently L(G) : L(G) C o n s i d e r t h e c o l o n y C : ( V , T , f ) , w h e r e V : N U , n / U ? a n d
F : { ( X , F . y ) : X e l V U l / , { . v : { a : X - - a e P } . E v i d e n t l y , C i s t h e c o l o n y a n d L ( G ) - L 6 ( C , , 5 ) .
Let L € COL6. There is a colony C : (V,T,f) and an a-xiom ?r0 such tha*"
L : Lu(C,.0). To construct an equivalent context-free grammar we have to determine the set of nonterminals .l/ to be strictly disjoin rvith the terminal set 7 . C o n s i d e r 7 : { X : X e V a T } U { S } , w h e r e X a r e n e w p a i r w i s e d i f f e r e n t nonterminals and S # V is a new start symbol.
Associate rvith a word u e V* following set of words
W : { u t r r . . . L l r r t n ' t l ? } + 1 : ' l l ) : 1 l , I T 1 . . . l ) 1 L n t l n : r l , U i € V " r I i e V O : t } .
Words u e W diffcrs from tl only in some letters from V nT, not necessary all, which are replaced by con'esponding barred letters.
C o n s t r u c t th e g r a m m a r G : ( ( V _ T ) U V , T , P , , S ) , r v h e r e P : { S , u ) o } U { S - + ? L i u e W s }
U { X - - + ? , : X e V - 7 , ( X , F * ) e f , u € F y , u e l V } U { X - - - + L L : X e V ) 7 , ( X , F x ) e f , w e F x , u e W - } .
It means that rules of the grammar are constructed from components of the colony in such a way, that they rewrite the start symbol of the component by a rvord fronr ass<-rciatcd finitc language. In case when original start symbol of some ( omponent was also a terminal symbol, corresponding barred letter is used for t h e d e r i v a t i o n . E v i d e n t l y , L ( G ) - L a ( C , w ) . I llote 2. In the previous proof we presented possibility to transform a colony to an equivalent context-free granrmar and vice versa. This makes possible to trans- form many known results on context-free grammars and languages to colonies.
Normal form theorems, resuits on e-rules and many others are exampies of 1 5 9
160 A. Kelemenovd
above mentioned properties. Descriptional complexity of colonies measured by the number of conrponents necessary to produce the language is discussed in
[1e]
The case T - V and axiom is a letter, corresponds to languages of the sentential fbrms.
3.2 Sequential colonies with parallely acting components t mode of derivation
In this section we discuss a sequential model of colony with parallelism inside the components. Parallelism at the level of a single component, denoted here by
$, rneans that in one clerivation step one component rewrites all appearances of its start symbol ^9; in an actual string to a (not necessary same) word of Fi- D e f i n i t i o n 3 . F o r a c o l o n y C : ( V , T , f ) a n d r , A e V * u ' e d e f i , n e a t e r m i n a l deriuation step
t
4 ---\ ?'
& - y i f f r : r 1 S ; r 2 S i r 3 . . . r r r S i r y n a r t r t r z . . . r r n * r € ( V - { S , } ) . ,
A : I I W I I 2 W 2 T J . . . I r n U r n T q l ! t
where wi € Fi, for each j,I < j < m a n d f o r s o m e i, l S i 1 n.
In accordance with general case, a language of a colony with derivation step is denoted Lt(C,ur6) and COLt is the corresponding class of all languages.
Erample 1. Consider a colony with - V - { A . B , a } ,
- T - { a } a n d
- r : { ( A , { B B } ) , ( 8 , { A } ) , ( 8 , { o } ) } .
A + B B + A A + B B B B J , n o o o
is an exanrple of derivation in C.
L 1 ( C . A ) : { o ' ' t i > 1 } .
We compare class COLt with class COLb as well as with t:lasses of languages generated by other known models of parallel grammars, nameiy L systems and Indian parallel grammars.
Theorem 2. lL}l COLa C COLI
Proof. LeL C be a colony with b mode of derivation. Let C be a colony equivalent to C rvith all components having different start symbols. We can construct such a colony by cumulating all components of C with the same start symbol .9 to
t
Colonies - the Simplest Grammar Systems 161 one component (,5, F) of C. Set -F of new component is the union of the original sets { of cumulated components from C.
I t h o l d s L , , ( C , w ) : L r ( a , u ) , i . e . t h e w o r d s p r o d u c e d b y a c o l o n y C w i t h b mode of derivation a,re identical with that of t mode of derivation. The derivations of the same word in these colonies can differ only in their length.
The inclusion in the theorem is proper. The language in presented in previous
e x a m p l e i s i n C O L I - C O L ; . I
Furthermore we compare colonies with t mode of determined by Indian parallel grammars.
An Indian parailel grammar is a quadruple G : CF grammars with derivation step j4:
r 4 A i f f . r - r 1 A r 2 A r z . . . r r n A r n r * r t
! : frIUIZU.XZ . . . frrnutrrn*It
r t r z . . . r r n t r e ( V - { A } ) .
a n d A " > w e P , f o r s o m e A e I ' 1 .
By L(I P) *. denote the class of all Ianguages generated by Indian parallel grammar.
I{ote 3. [13] f(IP) and CF are incomparable.
T h e o r e m 3 . L ( I P ) C C O L t
Proof . Let L be a language given by an Indian parallel grammar G : (N, T, P, S) . We can assume that none of the rules contain direct recursion, i.e. for A --+ a e P, a does not contain A.
To construct C : (V,T,f) we put V : TU-f/ and for every A -- a € P the pair ( A , {o}) € f. We have Lr(C,S) : L.
Inciusion in the theorem is proper. Example of the ianguage in COLt - L(IP)
is the Dyck language. !
Another type of parallel grammars are L-systems.
A n E T } L s y s t e m is a q u a d r u p l e G : ( V , 7 , P , , 5 ) , w h e r e V ' i s a t o t a l a l p h a b e t , T c V i s a t e r m i n a l a l p h a b e t , , S € V , P : { P t : 1 S i < n } a n d P , : { A - + ' t D :
A e V , u e V * \ .
Nloreover, Pi, | < i < n contains Derivation step 4 of I system
r
r + y i f f r : t r \ r 2 . . . ! x m , 'A : a t A z . . . ! J n " , w h e r e r r , T 2 , . . . , T r n € l / a n d r i - At € P,, for some i and f.or j,l < j < m.J V J
Theorem 4. COLI C ET}L
Proof. Stronger result, the equality COLl - ET\Ltlj was proved in [10], where ET}Lg are languages generated bv tables 4 which rewrites at most one letter
of V- to non identical strins. f
derivation with languages (l/,7, P, ,S) specified as itr
at ieast one rule for each A e V.
specifies totally parallel rewriting:
4
t 6 2 A. Kelemenova
Parallel Colonies
For a derivation in parallel model of colony is typical that several components are active in a derivation step of the colony. Parallel colonies were introduced in the principle that components of a colony, which can work on the tape musf work simulttrneously and each component rewrites at most one occurrence of its start symboi. To formaiize mentioned requirement the case when more components have the same associated nonterminal requires a speciai discussion:
* If (S,.F,), and (S,Fi) are two components of a colony C and if (at ieast) two symbols 5 appear in a current string, then both these components rnust be used, each rewriting one occurrence of 5.
* If only one 5 appears in a current string, then each component can be used.
but not both in parallel, hence in such a case we discuss trvo possibilities (i) derivation is blocked - strongly competitiue parallel way of derivation, and (ii) derivation continues and maximal number of components is used, non- deterministically chosen from all the components which can be used - weakly competiti,ue parallel way of derivation.
This corresponds to the following derivation steps:
For r, A € V" define a strongly competi,tiue parallel derivation step by r ' + s p A i f f i : r r S h r z S i r . . . r t S t * ' r k + r ,
U : l t j t Z i r T 2 Z i r . . . I k Z i x t t e + t , Z i i € F l r r I < j < k ,
i " f i , f o r a l l u * u , L I u , u I k ,
(one component is allowed to rewrite at most one occurrence of its start symbol)
l"ls, > 0 implies t -- ii for some 7, | < j < k
(if component F; can be used, then it must be used).
In accordance with general case, the language of a colony with derivation step 4 is denotecl Lrr(C,tr,s) and COLre is the corresponding class of all languages.
Erarnple 2. Let
C : ( { A , B , C , D , 0 , 1 } ' , { 0 , 1 } , F ) , w h e r e
f : { ( A , { 0 8 , 1 C } ) ,( A , { 0 C , i B } ) , ( 8 , { A , E } ) , ( C , { A , E } ) , ( 8 , { r } ) , ( 8 , { r } ) }
In the colony we can derive for example
AA 4 0B0C jS oao.4 j+ IICILB l5 orgorr g oror
Let the derivation start with
A A + 0 B I B .
Next sentential form contains either singie occurrence cf. -.1 ,-..r -8. The colony ha,s
Colonies - the Simplest Gramntar Systems 163 two components to rewrite A and two components to rewrite E. In both cases d e r i v a t i o n s t o p s '
L , o ( c , A A ) - { w w : u € { 0 , r } * } .
For r, A € V* define a weakly competi,ti,ue parallel derivation step by
L D p
r + y i f f r : r r S ; . r r z S b . . . r x S t . * r t x ; r t
l J : r t Z ; r I 2 Z i r . . . f r k Z i x f r k + L Z i i F t , L S j S k , i u # i , f o r a l l u * u , l 1 u , , u I k ,
(one component is allowed to rewrite at most one occurrence of its startsymbol)
l"ls, > 0 implies St :,S1, for some j, L < j < k, k is the maximal integer with the previous properties.
In accordance with general case, the language of a colony with derivation step S is denoted. L-r(C , tus ) and C O L-e is the corresponding class of all languages.
Erample 3. Let
C : ( { S , A , B , C , D , E , F , a , b , c } , { a , b , c } , f) , w h e r e
f : { ( s , { A B C } ) , (y , { z } ) , ( 2 , { Y } ) ,
( A , {a D , X } ) , ( 8 , { b E , X } ) , (C , {c F , X } ) , ( D , { A } ) , ( 8 , { B } ) , ( 4 { C } ) ,
( X , {u}), (X, {e}), (X, {)'})}.
A succesfull derivation in the colony ends by rewriting all occurrences of X by e. There are at most three occurrences of X in sentential forms produced by the colony but only words containing at most two X-es can be rewritten to terminal words.
Successful derivation:
S ry ABC # XUncF ry bBcC $ 6yssF ry bccC
lDD IDD
j fissx + bcc Blocked derivation
S ry ABC 3 anUncF ry aAbBcC 3 axuxcx 3 aby
L - , ( C , S ) : { a i V c k l i , j , k ) 0 , i * j o r j + k o r i l k } .
If thc start syrnbols of all components are different in a paralle] colony, then both 4 and i!$ clefine the same relation denoteci by =5. For the generative power of COL, we have
Theorem 5. a) COL, : CF b) COLp c COL"e c) COL, C COL-,
t64 A. Kelemenovd
P r o o f . a ) L e t L e CF, L: L(G) for a context-free g r a m m a r G : ( / V , T , P , S ) . Assume, that A does not appear in z for each rule A --) z € P and .A/ : { A t , 4 2 , . . . , A , r } . C o n s i d e r t h e c o l o n y
C : ( N t ) T , T , f ) , f : { ( A t , { t ; A o - - - + z e P } ) , f < i < n } .
Clearly, C conta,ins exactly the rules of G and Lr(C, S) S L(G).Also the converse inclusion is true. Take a derivation D ; $ 4* r tn G. The order of applying the rules in D can be modified in such a way to obtain another derivation, D', described by the same derivation tree as D (hence producing the same string r), but consisting of subderivations u + r.' with respect to G such that for every Ai e IV which appears in u, exactly one occurrence of it is rewritten. The derivation D' is a parallel derivation with respect to C, hence r e Lo(C, S). We have obtained the inclusion CF C COL,.
Conversely, let L e COL,. There is a colony C -- (V,T,f) and an a-xiom u such that tr : Lr(C,tr,'). To construct an equivalent context-free grammar we have to determine set of nonterminals l/ to be strictly disjoin with the terminal s e t 7 . E x a c t l y a s f o r C O L a w e c o n s t r u c t G -- ((V _ T)UV,T,P,,S). Rules of the grammar are constructed from components of the colony in such a way, that they rewrite start symbol of the component by a word from associated finite Ianguage. In the case when original start symbol of some component was also a terminal symbol, corresponding barred letter is used for the derivation.
The inclusion.Lo(C,r) e LG) is obvious, the converse inclusion can be ob- tained as previously, hence Lo(C,w) : L(G) and COLp e CF.
b) and c) Clearly, COLe e COL,e n COL-e.
For I"o * Lsp(C,AA) from the Example 4.1 we have L,p e COL,e - COLp.
F o r L * o : L - p ( C , , 5 ) f r o m t h e E x a m p l e 4 . 2 w e h a v e L . p e COL,,,p-COLp. r To specify the generative power of parallel colonies more detailly we use ma- trix grammars with appearance checking An ET\L system is a construct G - ( E , T , H , - ) , w h e r e f i s a v o c a b u l a r y , T g E , w e . E + , a n d . I / i s a f i n i t e s e t crf talrlcs, thalt is of firiite substitritions lr ; E* --' 2t-. For h e H and r € D*
we define the l-iimited irnage l't(r) as the set of all strings in ,!* which can be obtained from r by replacing for each a e E which appears in r exactly one occurrence by an element of h(a). Thus, the 1-limited generated language is
L r t ( G ) : { z e T * ; z e h n ( h u - t ( . . . ( h r ( r ) ) . . . ) ) , r , ) 0 , h i € H , f o r a l l 7 } Note l. A matrir granxrnar (with appearance checking) is a construct G : (l/,7, IuI,S,F), where l/ is a nonterminal vocabulary, ? is a terminal vocabulary, ,S € .A/ is the a symbol, /'f'1 is a finitc sct of firritc scquerrccs clf context-frec rules ar,nd ,F is a set of occurrences of rules in IuI. The derivations start from ,5 and consist o f s t e p s o f u s i n g m a t r i c e s in f u I ; w h e n using a matrix (Ar * rr,...,-Ln - frn), all the rules are used, in this ortier, and this means the symbols A; arc effectivcly rewritten by ,, if they appear in the current string, with the possibility to skip the rules A; - 11 which appeerr in F but Ai does trot appear in the current strirrg. and l-lirnited ET0I systems.
Colonies - the Simplest Grammar Systems t 6 5 We denot e by IVI ATo. the family of languages generated by matrix grammars with appearance checking (without using e-rules) and by LIET\L the family of languages generated by 1-limited ET\L systems. It is known that C F c lvI ATo" C C S, strict inclusions for s free grammars. IIET\L is strictly included in IvI ATo".
Theorem 6. ll2l COL,e e IvI AT". and COL*, c IIET}L
Open problems: What is the relation between COL-. and COLro.Is the inclusion COL"e C L,IATo" proper?
What is the influence of E in components of the colony on the generative power.
5 Further Research Topics and Open Problems
Large nunrbcr of problenrs were invcstigated fbr colonies. !!i: trriefly surnruarize them with references to the original papers.
First we mention various variants of the basic model of a colony. The choice of tcrminal alphabets influences the generative capacity. Cases T - I/ and V etT * 0 for sequential colonies \vere discussed in [10], [28].
Results on parallel colonies can be found in [12], [43].
The role of e rules in different models has to be discussed. No resuit in this direction was done and at least for some types of colonies the absence of E rules will reduce their generative power.
Descriptional complexity of colonies characterized for example by the number of components were discussed in [19] ,1421.
Topics like stability and inference were open in [19] and [a3], respectively.
There are several extensions or modifications of the basic model. Between se- quential colonies with t and b mode of derivation the colonies with k active components in one step can be treated [42].
Consequences of restriction of using components in the sense of frequency or fixcd period of forbidden activity of components werc trcated in [19], [23] ,122).
Diffcrent motivations leads to more significant modifications of the basic model.
Colonies rvith point mutations, PNI colonies [41], [40], [30], [31] have speciai type of rules allowing to add at most one symbols and moreover also position of corrr- ponents in the environment plays role in this model.
Symbiosis and parasitism were inspiration for models introduced in ff fl.
Inspirations from automata theory inflrrenced papers [1], i2].
Unreliable colonies are introduced and discussed in [14], [15], [16].
One can found also treatment to use ideas of colonies to modei low level economv [ 3 5 ]
Application of the idea in linguistics due to [2a],[3].
Coionies have constant environment. I,Iodels, rvhere environment can be changed.
by inner rttles, namely e-colonies and eco-colonies, were also introcluced. These models caIIIe rip comitining ideas concerning the internal behaviorrr of the envi- ronrnent in the eco-grammar svstems 181, [29], [38], [38] and single lerter rewriting
166 A. Kelemenovd
behaviour of the components of the colonies. [a6] , [47), [48].
In last few years one very interesting type of colonies was introduced on the base of membrane systems. So called P colonies are studied in [25], [21], [4], [5], [7],
[ e ] , [ 2 7 ] . References
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