A CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA

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1 A CLASS OF RECURSIVE SETS Florentn Smarandache Unversty of New Mexco 200 College Road Gallup, NM 87301, USA E-mal: In ths artcle one bulds a class of recursve sets, one establshes propertes of these sets and one proposes applcatons Ths artcle wdens some results of [1] 1) Defntons, propertes One calls recursve sets the sets of elements whch are bult n a recursve manner: let T be a set of elements and f for between 1 and s, of operatons n, such that f :T n T Let s buld by recurrence the set M ncluded n T and such that: (Def 1) 1 o ) certan elements a 1 of T, belong to M 2 o ) f ) belong to M, then f ) belong to M for all { 1,2,,s} 3 o ) each element of M s obtaned by applyng a number fnte of tmes the rules 1 o or 2 o We wll prove several propretes of these sets M, whch wll result from the manner n whch they were defned The set M s the representatve of a class of recursve sets because n the rules 1 o and 2 o, by partcularzng the elements a 1 respectvely f 1,, f s one obtans dfferent sets Remark 1 : To obtan an element of M, t s necessary to apply ntally the rule 1 (Def 2) The elements of M are called elements M -recursve (Def 3) One calls order of an element a of M the smallest natural p 1whch has the proprety that a s obtaned by applyng p tmes the rule 1 o or 2 o One notes M p the set whch contans all the elements of order p of M It s obvous that M 1 = { a 1 } s M 2 = U U f ) \ M 1 =1 n ) M 1 One wthdraws M 1 because t s possble that to M 1, and thus does not belong to M 2 One proves that for k 1 one has: f ( a,, a ) = a whch belongs j j1 jn j 1

2 s k M k +1 = U U f ) \ U M h =1 ( ) ) k h=1 where each { () =,,α n )/ α k j M q j j { 1,2,,n }; 1 q j k and at least an element a jo M k,1 j o n } The sets M p, p *, form a partton of the set M Theorem 1: M = U, where * = { 1,2,3, } M p Proof: From the rule 1 o t results that M 1 M One supposes that ths proprety s true for values whch are less than p It results that M p M, because M p s obtaned by applyng the rule 2 o to the elements of Thus U M p accordance wth the rule 3 o p 1 M =1 U M Recprocally, one has the ncluson n the contrary sense n Theorem 2: The set M s the smallest set, whch has the propertes 1 o and 2 o Proof: Let R be the smallest set havng propertes 1 o and 2 o One wll prove that ths set s unque Let s suppose that there exsts another set R' havng propertes 1 o and 2 o, whch s the smallest Because R s the smallest set havng these propretes, and because R' has these propertes also, t results that R R' ; of an analogue manner, we have R' R : therefore R = R' It s evdent that M ' R One supposes that M R for 1 < p Then (rule 3 o ), and takng n consderaton the fact that each element of M p s obtaned by applyng rule 2 o to certan elements of M, 1 < p, t results that M p R Therefore * U M p R ( p ), thus M R And because R s unque, M = R p Remark 2 The theorem 2 replaces the rule 3 o of the recursve defnton of the set M by: M s the smallest set that satsfes propretes 1 o and 2 o Theorem 3: M s the ntersecton of all the sets of T whch satsfy condtons 1 o and 2 o Proof: 2

3 I = I A A T 12 Let T 12 be the famly of all sets of T satsfyng the condtons 1 o and 2 o We note I has the propertes 1 o and 2 o because: 1) For all 1,2,,n { }, a I, because a A for all A of T 12 2) If α 1,,α n I, t results that α 1,,α n belong to A that s A of T 12 Therefore, { 1,2,,s}, f ) A whch s A of T 12, therefore f ) I for all from { 1,2,,s} From theorem 2 t results that M I Because M satsfes the condtons 1 o and 2 o, t results that M T 12, from whch I M Therefore M = I (Def 4) A set A I s called closed for the operaton f 0 f and only f for all α 0 1,,α 0 n of A, one has f 0 (α 0 1,,α 0 n 0 ) belong to A 0 (Def 5) A set A T s called M -recursvely closed f and only f: 1) { a 1 } A 2) A s closed n respect to operatons f 1,, f s Wth these defntons, the precedent theorems become: Theorem 2 : The set M s the smallest M - recursvely closed set Theorem 3 : M s the ntersecton of all M - recursvely closed sets (Def 6) The system of elements α 1,,α m, m 1 and α T for { 1,2,,m}, consttute a M -recursve descrpton for the element α, f α m = α and that each α ( { 1,2,,m}) satsfes at least one of the propretes: 1) α { a 1 } 2) α s obtaned startng wth the elements whch precede t n the system by applyng the functons f j, 1 j s defned by property 2 o of (Def 1) (Def 7) The number m of ths system s called the length of the M -recursve descrpton for the element α Remark 3: If the element α admts a M -recursve descrpton, then t admts an nfnty of such descrptons Indeed, f α 1,,α m s a M -recursve descrpton of α then a1,, a1, α1,, αm h tmes values from s also a M -recursve descrpton for α, h beng able to take all 3

4 Theorem 4: The set M s dentcal wth the set of all elements of T whch admt a M -recursve descrpton Proof: Let D be the set of all elements, whch admt a M -recursve descrpton We wll prove by recurrence that M p D for all p of * For p = 1 we have: M 1 = { a 1 }, and the a j, 1 j n, havng as M - recursve descrpton: < a j > Thus M 1 D Let s suppose that the property s true for the values smaller than p M p s obtaned by applyng the rule 2 o to the elements of p 1 U M ; α M p mples that = 1 α f ( α,, α ) and α j M hj for h j < p and j 1 n 1 j n But a j, 1 j n, admts M -recursve descrptons accordng to the hypothess of recurrence, let s have β,, j1 β js Then β j 11,,β 1s1,β 21,,β 2s2,,β n 1,,β n s n,α consttute a M -recursve descrpton for the element α Therefore f α belongs to D, then M p D whch s M = U M p D Recprocally, let x belong to D It admts a M -recursve descrpton b 1,,b t wth b t = x It results by recurrence by the length of the M -recursve descrpton of the element x, that x M For t = 1 we have b 1, b 1 = x and b 1 { a 1 } M One supposes that all elements y of D whch admt a M -recursve descrpton of a length nferor to t belong to M Let x D be descrbed by a system of length t : b 1,,b t, b t = x Then x { a 1 } M, where x s obtaned by applyng the rule 2 o to the elements whch precede t n the system: b 1,,b t 1 But these elements admt the M - recursve descrptons of length whch s smaller that t : b 1, b 1,b 2,, b 1,,b t 1 Accordng to the hypothess of the recurrence, b 1,,b t 1 belong to M Therefore b t belongs also to M It results that M D Theorem 5: Let b 1,,b q be elements of T, whch are obtaned from the elements a 1 by applyng a fnte number of tmes the operatons f 1, f 2,, or f s Then M can be defned recursvely n the followng mode: 1) Certan elements a 1,b 1,,b q of T belong to M 2) M s closed for the applcatons f, wth { 1,2,,s} 3) Each element of M s obtaned by applyng a fnte number of tmes the rules (1) or (2) whch precede Proof: evdent Because b 1,,b q belong to T, and are obtaned startng wth the elements a 1 of M by applyng a fnte number of tmes the operatons f, t results that b 1,,b q belong to M 4

5 Theorem 6: Let s have g j, 1 j r, of the operatons n j, where g j :T n j T such that M to be closed n rapport to these operatons Then M can be recursvely defned n the followng manner: 1) Certan elements a 1 de T belong to M 2) M s closed for the operatons f, { 1,2,,s} and g j, j { 1,2,,r} 3) Each element of M s obtaned by applyng a fnte number of tmes the precedent rules Proof s smple: Because M s closed for the operatons g j (wth j { 1,2,,r}), one has, that for any α j1,,α jnj from M, g j (α j1,,α jnj ) M for all j { 1,2,,r} From the theorems 5 and 6 t results: Theorem 7: The set M can be recursvely defned n the followng manner: 1) Certan elements a 1,b 1,,b q of T belong to M 2) M s closed for the operatons f ( { 1,2,,s}) and for the operatons g j ( j { 1,2,,r}) prevously defned 3) Each element of M s defned by applyng a fnte number of tmes the prevous 2 rules (Def 8) The operaton f conserves the property P ff for any elements α 1 havng the property P, f ) has the property P Theorem 8 : If a 1 have the property P, and f the functons f 1,, f s preserve ths property, then all elements of M have the property P Poof: M = U M p The elements of M 1 have the property P Let s suppose that the elements of M for < p have the property P Then the elements of M p also have ths property because M p s obtaned by applyng the operatons f 1, f 2,, f s to the elements of: U M, elements whch have the property P =1 Therefore, for any p of, the elements of M p have the property P Thus all elements of M have t Corollary 1 : Let s have the property P : x can be represented n the form F(x) If a 1 can be represented n the form F(a 1 ),, respectvely F(a n ), and f f,, 1 f s mantans the property P, then all elements α of M can be represented n the form F(α) Remark One can fnd more other equvalent def of M 2) APPLICATIONS, EXAMPLES 5

6 In applcatons, certan general notons lke: M - recursve element, M -recursve descrpton, M - recursve closed set wll be replaced by the attrbutes whch characterze the set M For example n the theory of recursve functons, one fnds notons lke: recursve prmtve functons, prmtve recursve descrpton, prmtvely recursve closed sets In ths case M has been replaced by the attrbute prmtve whch characterzes ths class of functons, but t can be replaced by the attrbutes general, partal By partcularzng the rules 1 o and 2 o of the def 1, one obtans several nterestng sets: Example 1: (see [2], pp , problem 797) Example 2: The set of terms of a sequence defned by a recurrng relaton consttutes a recursve set Let s consder the sequence: a n+ k = f (a n,a n+1 + k 1 ) for all n of *, wth a 0 = a, 1 k One wll recursvely construct the set A = { a m } m * and one wll defne n the same tme the poston of an element n the set A : 1 ) a 0 1,,a 0 k belong to A, and each a 0 (1 k ) occupes the poston n the set A ; 2 ) f a n,a n+1 + k 1 belong to A, and each a j for n j n + k 1 occupes the poston j n the set A, then f (a n,a n+1 + k 1 ) belongs to A and occupes the poston n + k n the set A 3 ) each element of B s obtaned by applyng a fnte number of tmes the rules 1 o or 2 o Example 3: Let G = e,a 1,a 2,,a p { } be a cyclc group generated by the element a Then ( G, ) can be recursvely defned n the followng manner: 1 or 2 1 ) a belongs to G 2 ) f b and c belong to G then b cbelongs to G 3 ) each element of G s obtaned by applyng a fnte number of tmes the rules Example 4: Each fnte set ML = { x 1, x 2,, x n } can be recursvely defned (wth ML T ): 1 ) The elements x 1, x 2,, x n of T belong to ML 2 ) If a belongs to ML, then f (a) belongs to ML, where f :T T such that f (x) = x ; 3 ) Each element of ML s obtaned by applyng a fnte number of tmes the rules 1 or 2 Example 5: Let L be a vectoral space on the commutatve corps K and { x 1,, x m } be a base of L Then L, can be recursvely defned n the followng manner: 1 ) x 1,, x m belong to L ; 2 ) f x, y belong to L and f a belongs to K, then x y y belong to L and a x belongs to L ; 3 ) each element of L s recursvely obtaned by applyng a fnte number of tmes the rules 1 or 2 6

7 (The operators and are respectvely the nternal and external operators of the vectoral space L ) Example 6: Let X be an A -module, and M X (M ), wth M = The sub-module generated by M s: M = x X / x= a x + + a x, a A, x M, 1,, n { 1 1 n n { }} {} x I can be recursvely defned n the followng way: 1 ) for all of { 1,2,,n}, { 1, 2,, n} x M ; 2 ) f x and y belong to M and a belongs to A, then x + y belongs to M, and ax also; 3 ) each element of M s obtaned by applyng a fnte number of tmes the rules 1 or 2 In accordance to the paragraph 1 of ths artcle, M s the smallest sub-set of X that verfes the condtons 1 and 2, that s M s the smallest sub-module of X that ncludes M M s also the ntersecton of all the subsets of X that verfy the condtons 1 and 2, that s M s the ntersecton of all sub-modules of X that contan M One also drectly refnes some classc results from algebra One can also talk about sub-groups or deal generated by a set: one also obtans some mportant applcatons n algebra Example 7: One also obtans lke an applcaton the theory of formal languages, because, lke t was mentoned, each regular language (lnear at rght) s a regular set and recprocally But a regular set on an alphabet Σ= { a 1 } can be recursvely defned n the followng way: 1 ), {}, ε {},, a 1 { a n } belong to R 2 ) f P and Q belong to R, then P Q, PQ, and P belong to R, wth P Q= { x/ x P or x Q} ; PQ = { xy / x P and y Q}, and P = U P n wth and, by conventon, P0 = { ε} n P = P P P n tmes 3 ) Nothng else belongs to R other that those whch are obtaned by usng 1 or 2 From whch many propertes of ths class of languages wth applcatons to the programmng languages wll result REFERENCES: [1] C P Popovc, L Lvovsch, H Georgescu N Ţăndăreanu, Curs de bazele nformatc (funcţ booleene ş crcute combnaţonale), Tpografa Unverstăţ dn Bucurest, 1976 [2] F Smarandache, Problèmes avec et sans problèmes!, Sompress, Fès (Maroc), 1983 n=0 7