Examples from Elements of Theory of Computation. Abstract. Introduction

Transcription

1 Examles from Elements of Theory of Comutation Mostafa Ghandehari Samee Ullah Khan Deartment of Comuter Science and Engineering University of Texas at Arlington, TX-7609, USA Tel: +(87)7-5688, Fax: +(87)7-784 Abstract Study of formal languages is a central toic in theoretical comuter science and engineering Results from number theory are used to give examles of regular and non-regular languages In articular Goldbach s conjecture gives examles of two non-regular languages whose concatenation is regular Introduction In Fall 00, the authors were involved in teaching Theoretical Concets, a Comuter Science (CS) foundation course at junior level Theoretical Concets or Theoretical Comuter Science (TCS) forms the very basics of the resent day CS undergraduate and graduate studies and ractical research The curriculum is designed to broaden students'ersectives on the role of CS and Mathematics in the modern world, while equiing them with both quantitative and comuter literacy skills Mathematics lays an imortant role in the understanding of how comuters work, and how they oerate It is to say without any doubt that the basic understanding of the mathematics of comuters is necessary for any comuter scientist The study of TCS involves grasing the concets of abstract machines and abstract mathematics The students, before they study these concets have already got basic hands on exerience with ractical machines, which makes the study of the theoretical concets a bit more difficult The authors of this aer use ure mathematics to exlain concets in TCS Some of the results in number theory were used to argue and show roerties of regular and non-regular languages The results obtained are related to some famous mathematical roblems For examle we use Goldbach s conjecture to discuss the concatenation of two non-regular languages, and use Fermat s last theorem to argue that a articular language can be roved non-regular We introduce some reasoning that can be used while teaching TCS to the undergraduates We assume that the reader is well versed with basic definitions and concets of formal languages (both regular and non-regular), and elementary number theory The readers are encouraged to read Elements of the Theory of Comutation for an overview of languages and related concets Proceedings of the 00 ASEE Gulf-Southwest Annual Conference Coyright 00, American Society for Engineering Education

2 in TCS, and From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Develoment for understanding of some number theory results that will be used later on We divide the aer as follows We will first introduce the necessary materials for background information in the reliminary section, followed by a classroom session where we introduce the new concets and others Finally we conclude with some interesting observations during the session Preliminaries In this section we will resent the reader with some roerties associated with regular and nonregular languages We will also introduce the uming lemma and the Goldbach s conjecture which will be used for the formal roofs of various concets Closure Proerty of Regular Languages are two regular languages then the resulting language say L, is closed under the following oerations: Union are two regular languages, then their union is regular Intersection are two regular languages, then their intersection is regular Deterministic Finite Automata (DFA) M and M such that L = L( M ) = L( M ), where M = Q,, δ, q, ), M = P,, δ,, ) Construct M,' ( 0 F ' 0 0 F ( 0 F = ( Q, δ,'( q, ), )', where Q ' = ( Q P) and δ '(( q i, j ), a) = ( qk, l ) if w L( M )' w L L Concatenation are two regular languages, then the concatenation L L is regular Comlement is a regular language, then the comlement L is regular DFA M such that L = L(M ) Construct a DFA M ' such that the final states in M are non-final states in Kleene Star is a regular language, then r is regular denoting M ' and the non-final states in M are final states in M ' L, where L is regular r is the concatenation of all zero or more r from L Proceedings of the 00 ASEE Gulf-Southwest Annual Conference Coyright 00, American Society for Engineering Education

3 Puming Lemma is an infinite regular language, then there exists a constant m > 0 such that any w L with length w m can be decomosed into three arts as w = xyz with xy m ; y and xy i z L i 0 The technique can be used to rove a language L is not regular The roof is by contradiction, ie suose L is regular, then it would satisfy the uming lemma Let w in L be a long enough string ( w m) Now all we have to do is to show that w can be written as xyz such that xy m, y and xy i z L i 0 We find secific value of i such that xy i z L, thus contradicting the uming lemma For examle if we want to show L = { a q }, where and q are rimes is not regular, then we assume that it is regular By using the uming lemma there exists n, such that if w = n and w L, then w = xyz, y e, xy i z L i 0 If i = 0, xz = q, for rimes and q, then q + i y i 0 (roduct of two rimes) Let i = q, then: q + ( q ) y = q ( q)(+ y ) = q q then = or = q If =, then q = q If = q, then q =, and q = q Thus, y + = contradicts the fact that y Classroom Snashot We know something about the closure roerties about regular languages How about nonregular languages? Can we aly the same techniques? Yes, we can but not all of the roerties Let us start with the comlement of a language If we have a non-regular language L, is L nonregular? The answer is yes and the roof is as follows: Suose L is regular then L = L should be regular, but it contradicts to the fact that L is nonregular For examle if L = { a = rime} is a non-regular language, then the comlement c of L, L = { a c = comosite} would also be a non-regular Well how about an examle of concatenation? The answer is yes, and we can roof it by using the Goldbach s conjecture But since the Goldbach s conjecture has not been roven for sufficiently large numbers, we can use a related result to the Goldbach s conjecture by Chen 4,5 The result says: Every "large" even number may be written as n = + m where is a rime and m is the roduct of two rimes So if we have two non-regular languages L such Proceedings of the 00 ASEE Gulf-Southwest Annual Conference Coyright 00, American Society for Engineering Education

4 q that L = { a = rime} = { a }, where and q are rime, then the concatenation of these two languages can be written as L L = ( a ) { a, e} Seaking of rimes, Vinogradov 6,7 has a much interesting result in number theory In 90 s he roved that any odd number can be reresented as a sum of three rime numbers Let us see if we can use his result Let us ick a language L such that L = { a = rime} and concatenate it with itself three times ( LLL = L ) Interestingly we get the final language as a regular language, ie Proceedings of the 00 ASEE Gulf-Southwest Annual Conference Coyright 00, American Society for Engineering Education L = k + { a k 0} = a( a We can give further examles using Fermat s last theorem 8 Fermat s last theorem states: n n n x + y = z has no non-zero integer solution for x, y and z when n For a history of Fermat s last theorem leading to the solution by Andrew Wiles in 995, see Fermat s Enigma 9 n As an examle we rove that the language L = { a 4 n 0} formal roof is as follows: is a non-regular language The n Assume that L = { a 4 n 0} is regular, then according to the uming lemma there exists j, such that if w L, where w = xyz, then w n, xy j and y = l, where y e Therefore we can write: xy k z L, k 0 k k xy z = xz + y, if n = 4 4 n + kl = m, let k = l n + l = m Fermat s last theorem imlies that there is no solution to the given roblem However, the uming lemma imlies that there exists a solution Thus, we obtain a contradiction Let us see if we can give examles of union and intersection Let L be two non regular languages, then L L is regular c A simle examle is the union of L = { a = rime} = { a c = comosite}, which conforms to a language L = L L = Similarly the intersection is also regular Using the same examle L = } a L { e Conclusions These examles show that mathematics is useful for the study of TCS, and results from ure mathematics can be alied to teach and further exlain concets in TCS As a final note, similar examles can also be derived for context-free languages References Lewis, HR, Paadimitriou, CH, 998, Elements of the Theory of Comutation, nd ed, Prentice-Hall, Inc Scharlau, W, 985, From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Develoment, Sringer Chen, JR, Wang, TZ, 989, On the Goldbach Problem, Acta Math Sinica, )

5 4 Chen, JR, 97, On the Reresentation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, Sci Sinica 6, Chen, JR, 978, On the Reresentation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, II, Sci Sinica, Vinogradov, I M, 97, Reresentation of an Odd Number as a Sum of Three Primes Comtes rendus (Doklady) de l'académie des Sciences de l'urss 5, Vinogradov, I, 97, Some Theorems Concerning the Theory of Primes Recueil Math, Mordell, LJ, 956, Fermat's Last Theorem, New York, Chelsea 9 Sing, S, 997, Fermat s Enigma, Walker & Co New York, NY MOSTAFA GHANDEHARI Mostafa Ghandehari received PhD in Mathematics at the University of California at Davis in 98 He attended the institute for Retraining in Comuter Science at Clarkson University during He has taught a variety of Mathematics and Comuter Science courses in the Northern California rior to coming to University of Texas at Arlington He is a lecturer in Mathematics, Comuter Science and Engineering and Civil Engineering deartments at University of Texas at Arlington SAMEE ULLAH KHAN Samee Ullah Khan is a PhD student at the Comuter Science and Engineering deartment at University of Texas at Arlington His research interests include: Combinatorics, Combinatorial Game Theory, and Adative Nash Bargaining Solutions Proceedings of the 00 ASEE Gulf-Southwest Annual Conference Coyright 00, American Society for Engineering Education