Non-Deterministic Communication Complexity of Regular Languages

Non-Determini stic Comm unica tion Complexi t y of Regu lar Langua ges Anil Ada Scho ol of Computer S c ienc e McGil l University, Montr´ eal Jan uary , 2008 A thesis submitted to the F aculty of Gr aduate St udies and Researc h in partial fulfillm ent of the requir emen ts of the degr ee of Mast er o f Sci ence. Copyrigh t c  Anil A d a 2 007. Abstract The notion of comm unication complexit y w as in tro duce d b y Y ao in his sem- inal pap er [Y ao79]. In [BFS86], Babai F rankl and Simon dev elop ed a ric h structure of comm unication complexit y classes to understand the relation- ships b etw een v arious mo dels of comm unication comple xit y . This made it apparen t that comm unication complexity w as a self-con tained mini-w orld within complexit y theory . In this thesis, w e study the place of regular lan- guages within this mini-w o rld. In particular, w e are intere sted in the non- deterministic comm unication complex it y of regular la ng uages. W e sho w that a regular language has either O (1 ) or Ω(log n ) non-determi- nistic complexit y . W e obtain sev eral linear low er b ound results whic h co ver a wide range o f regular languag es ha ving linear non-deterministic complexit y . These lo w er b o und results also imply a result in semigroup theory: w e obtain sufficien t conditions for not b e ing in the p ositiv e v ariet y P ol ( C om ). T o obtain our results, w e use algebraic tec hniques. In t he study o f regular languages, the algebraic p oin t of view pioneered b y Eilen b erg ([Eil74]) has led to man y intere sting r esults. Vie wing a semigroup as a computationa l device that recognizes la ng ua ges has prov en to b e prolific from b oth semigroup theory and f ormal languages p ers p ectiv es. In this thesis, we prov ide further instances of suc h m utualism. i R ´ esum ´ e La not io n de complexit ´ e de commun ication a d’ab ord ´ et ´ e introduite par Y ao [Y ao79]. Les tra v aux fondateurs de Babai et al. [BFS86] ont d ´ ev oil´ e une ric he structures de classes de complexit ´ e de comm unication qui p ermetten t de mieux comprendre la puissance de div ers mo dles de complexit ´ e de commu- nication. Ces r´ esultats on t fait de la complexit ´ e de comm unication une sorte de maquette p etite c helle du m onde de la complexit. Dans ce m´ emoire, nous ´ etudions la place des langages r´ eguliers dans cette maquette. Plus pr ´ ecis ´ emen t, nous c herc herons d ´ eterminer la complexit ´ e de comm unication non-d ´ eterministe de ces langag es. Nous montrons qu’un la ngage r´ egulier a une complexit ´ e de comm uni- cation soit O (log n ), soit Ω(log n ). Nous ´ etablissons de plus des b ornes inf ´ erieures lin´ eaires sur la complexit ´ e non-d ´ eterministe d’une v aste classe de langages. Celles-ci fournissen t ´ egalemen t des conditions suffisan tes p our qu’un langage donn´ e n’appartienne pas la v a ri ´ et ´ e p ositive P ol ( C om ). Nos r´ esultats se basent sur des tec hniques alg´ ebriques. Dans l’ ´ etude des langages r´ eguliers, le p o int de vue alg´ ebrique, d ´ ev elopp ´ e initialement par Eilen b e rg [Eil74] s’est r´ ev ´ el ´ e comme un outil central. En effet, on p eut v oir un semigroup e fini comme une machine capable de reconna ˆ ıtre des langag es et cette p ers p ectiv e a p ermis des a v anc ´ ees tan t en th´ eorie des semigroup es qu’en th ´ eorie de s langages formels. Da ns ce m ´ emoire, nous ´ etablissons de nouv eaux exemples de ce mutualisme. ii Ac k no wledgmen ts First, I w ould like to express my deep est grat it ude to Prof. Denis Th ´ erien. I thank him fo r trusting me and accepting me as his studen t. His en th usiasm for complexit y theory and mathematics was highly con ta g ious and is one of the reasons I am in this field. I also thank him for supp orting me financially . I am indebted to m y co- sup ervisor Prof. Pas cal T esson for many things. I thank him for in tro ducing me to the sub ject of this thesis. I am gra t eful for the extremely useful discussions we had, which taugh t me a lot of things. I also thank him for his constructiv e commen ts on the earlier drafts of the thesis. I am v ery fortunate to hav e met Ark adev Ch attopadhy a y , L´ aszl´ o Egri, Na vin Go y al and Mark Mercer in the complexit y theory group. Thanks to m y office mate L´ aszl´ o for discussions ab out mathematics and man y other topics. I thank Ark adev for discussing the thesis pro blem with me and for sharing his k een insigh t on v arious things. Thanks to Na vin a nd Mark for generously sharing their kno wledge. I ha ve learned a lot from all o f them. I thank the academ ic and administrativ e staff of the computer scien ce departmen t. I ha v e met man y w onderful p eople ov er the y ears and I f eel privileged t o b e a part of this family . I w ould also lik e to thank all m y friends in Mon t r ´ eal for making lif e fun for me here. iii Finally , biggest thanks go to m y parents. Their lo ve and support nev er w av ered and this has made ev erything p ossible. iv Con ten ts 1 In tro duction 1 1.1 Computational Complexit y Theory . . . . . . . . . . . . . . . 1 1.2 Comm unicatio n Complexit y . . . . . . . . . . . . . . . . . . . 3 1.3 Algebraic Auto mata Theory . . . . . . . . . . . . . . . . . . . 4 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Comm unication Complexit y 9 2.1 Deterministic Mo del . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Lo w er Bound T ec hniques . . . . . . . . . . . . . . . . . 12 2.2 Non-Deterministic Mo del . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 P ow er of Non-Determinism . . . . . . . . . . . . . . . . 23 2.2.3 Lo w er Bound T ec hniques . . . . . . . . . . . . . . . . . 24 2.3 Other Mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Comm unicatio n Complexit y Class es . . . . . . . . . . . . . . . 28 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Algebraic Aut omata Theory 32 3.1 Monoids - Automata - R egular Languag es . . . . . . . . . . . 33 3.1.1 Monoids: A Computatio nal Mo del . . . . . . . . . . . 33 v 3.1.2 The Syn tactic Monoid . . . . . . . . . . . . . . . . . . 36 3.1.3 V arieties . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Ordered Monoids . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Recognition b y Ordered Monoids . . . . . . . . . . . . 45 3.2.2 The Syn tactic Ordered Monoid . . . . . . . . . . . . . 46 3.2.3 V arieties . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Comm unication Complexit y of Regular Languages 51 4.1 Algebraic Appro a c h to Comm unication Complexit y . . . . . . 52 4.2 Complexit y Bounds for R egular Languages and Monoids . . . 56 5 Conclusion 69 A F acts Ab out L 5 75 Bibliograph y 80 vi Chapter 1 In tro duction 1.1 Computation al C omplexit y The o ry The theory of computation is one of the fundamen tal branc hes of computer science that is concerned with the computability and complexit y of pro b- lems in differen t computational mo dels . Computabilit y theory fo cus es on the question of whether a problem can b e solv ed in a certain computational mo del. On t he other hand, complexit y theory seeks to determine ho w m uc h resource is sufficien t and necessary f o r a computable problem to b e solv ed in a computational mo del. Simply put, a c omputing devic e is a machine that p erforms calculations automatically: it can b e as complicated as a p ersonal computer and as simple as an a utomatic do or. In theoretical computer science, a c omputational mo d e l is a pure mathematical definition whic h mo dels a real-world computing de- vice. This abstraction is necessary in order to rigoro usly study computation, its pow er and limitations. The most studied computational mo del (whic h ph ysically corresp onds to the eve ryda y computers w e use) is the T uring Ma- c hine. The most studied resources are time and space (memory) measured with resp ect to t he input size. Based on these resources, differen t complexit y 1 1.1. Computational Complexit y Theory 2 classes can b e defined. F or instance, P and N P ar e classes of problems that can b e solv ed in p olynomial (in the input size) time using a deterministic a nd a non-deterministic T uring Mac hine resp ectiv ely . Whether t hese class es are equal or not is without a doubt one of the biggest op e n questions in computer science a nd mathematics. There are v arious interes ting computational mo de ls including (but not limited to) T uring Machines , finite automata, con text-free grammars, b o olean circuits and branc hing programs. Their countles s applications span com- puter science. F or instance, when designing a new pro g ramming lang uage one w ould find g r a mmars useful. F inite automata and r egula r languages ha v e applications in string searc hing and pattern matching. When trying to come up with an efficien t algorithm, the theory of NP-completeness can be insigh tf ul. Many cryptogra phic proto cols rely on theoretical principles . All these applications aside, the mat hematical elegance and aesthetic inherent in t heory o f computation is enough to attract man y minds a round the w orld. And p erhaps the main reason that computer science is called a “science” is b ecause of the study o f theoretical foundations of computer science. Despite t he in tense efforts of man y researc hers, our understanding and kno wledge o f computational complexity is quite limited. Similar to the P v ersus N P question, there are many ot her core questions (in different com- putational models) that b eg to be answ ered. The fo cus of r esearc h in com- plexit y theory is t w ofold. G iv en a certain problem, a computational mo del and a resource: • What is the maxim um amoun t o f resource w e need to solv e the pro blem in the computational mo del? • What is the minim um amoun t of resource w e need to solv e the problem in the computational mo del? The ultimate goal is to find match ing upp er and lo we r b ounds. The first 3 CHAPTER 1. In tro duction question can b e answ ered b y depicting a metho d 1 of solving the problem and analyzing the a mo unt of resource this metho d consum es. Almost alw a ys, the more c hallenging question is the second one. Pro ving results of the form “Problem p requires at least x resource.” requires us to a rgue against all p ossible metho ds that solv e the problem. In most computatio na l mo dels, this is in trinsically hard. Y et it should b e also noted that complexit y theory is a relatively new field and therefore can b e considered as an a menable discipline of mathematics. 1.2 Comm uni cation Complexity In this thesis, w e will b e studying a computational mo del whic h em ulates distributed computing: comm unication proto cols. In this mo del, there are usually t w o computers tha t are trying to collab oratively ev aluate the v alue of a g iv en function. The difficulty is tha t the input is distributed among the t w o computers in a predetermined adv ersarial w a y so that neither computer can ev aluate the v alue of the function b y itself. Therefore, in order to deter- mine the v alue of the function, these computers need to comm unicate ov er a net w ork. The comm unication will b e carried o ut according to a proto col that has b een agreed up on b eforehand. The resource w e are in terested in is the n umber of bits that is comm unicated i.e. w e w ould like to determine the c ommunic ation c om plexity of a given function. As an example, consider tw o files that reside in tw o computers. Supp ose w e w a nted to kno w if these tw o files were copies of eac h other. How man y bits w o uld the computers need to comm unicate in order to conclude that the files are the same or not? What is the b est pro t o col for the computers to accomplish this task? Note that the scenario here is quite differen t from information the o ry . In 1 In the T uring Machine computationa l mo del, the metho d is ca lled an algorithm . 1.3. Algebraic Automata Theory 4 information theory , the goal is to robustly transmit a predetermined mes- sage thro ug h a no isy c hannel and there is no function to b e computed. In the comm unication complexit y setting, the c hannel of communication is not noisy . What is sen t through the c hannel is determined b y the proto col and it usually changes according to the inputs o f the computers and the comm u- nication history . There are v arious models for comm unication complexit y . The first de- fined was the 2 - pla y er deterministic mo de l. Since then, non- deterministic, randomized, multi-part y , distributional, sim ultaneous and man y more mo d- els hav e b een defined and analyzed. Although the mathematical theory of comm unication complexit y was first in tro duced in light of its applications to par a llel computers ([Y ao79]), it has been sho wn to ha v e many more applications wh ere the need for com- m unication is not explicit. These applications include time/space low er b ounds for VLSI chips ([KN97]), time/space t r a deoffs for T uring Mac hines ([BNS92]), data structures ([KN97]), b o olean circuit low er b ounds ([Gro92], [HG91],[Nis93],[RM97]), pseudorandomness ([BNS92]), separation of pro of systems ([BPS07 ]) and lo w er bounds on the size of p olytop es represen ting N P -complete pro blems ([Y an91]). 1.3 Algebraic Auto mata Theory One of the fundamen t a l (and simplest) computational mo dels is the finite automaton and it is usually t he first model in theory of computation t ha t computer science students are in tro duced to. The w ord “finite” r efers to the memory of the mac hine and finite a uto mata are mo de ls for computers with an extremely limited amoun t of memory (fo r example a n automatic do or). Ev en though it is a quite limited mo del, its w ell-known a pplications include text pro ce ssing, compilers and hardw are design. 5 CHAPTER 1. In tro duction In a n utshell, finite automata are abstract machin es suc h that giv en a w o r d ov er some alphab et as a n input, it either accepts or rejects the w o rd after pro cessing eac h letter of the w ord sequen tially . The set of a ll w ords that a finite auto maton accepts is called the language cor r espo nding to the finite automaton and w e say t ha t the language is r e c o gnize d by this auto maton. A language recognized b y some finite automat on is called a r e gular l a nguage . Algebra has alw a ys b ee n a n imp ort a n t to ol in the study of computational complexit y . In the study of regular languages, semigroup theory 2 has b een the do minan t to ol. It should b e men tioned that semigroups hav e shed new ligh t not only on regular languages but o n computatio nal t heory in general. On top of this, it is also tr ue that computational theory has led to adv ances in the study of semigroup theory ([TT04]). The link b et w een semigroups and regular langua g es has b een established b y viewing a semigroup as a computatio nal machine that accepts/rejects w o r ds o v er some a lphab et. In this context, it is not difficult to pro v e that the f a mily of languages that finite semigroups recognize is exactly the regular languages. In fact, the connection b et w een finite automata a nd semigroups is m uc h more profound. There are sev eral reasons wh y this p oin t of view is b eneficial. First of all, the semigroup approa ch to regular langua ges allo ws one to use to ols fr o m semigroup theory while inv estigating the prop erties o f these languag es. Eilen b erg sho w ed that there is a one to one corresp ondence b et w een certain robust and natural classes of languages and semigroups. This has organized and heigh tened our understanding of regular languages. F urthermore, in certain computational mo dels, the complexit y of a regular language can b e parametrized by the complexit y o f the corresp onding semi- group a nd so this provides us alternate a v enu es to analyze the complexit y of regular languages. Often the com binato rial descriptions of regular la nguages suffice to obtain upp er b ounds o n their complexit y . The alg ebraic p oint o f 2 A s e mig roup is a set eq uipped with a bina ry a sso ciative o p eration. 1.4. Outline 6 view pro v es to b e useful when proving hardness results. Comm unication complexit y is one o f the computational mo dels where this is the case. 1.4 Outline In this t hesis, w e study the non-deterministic comm unication complexit y of regular languages. The ultimate goa l is to find functions f 1 ( n ) , f 2 ( n ) , ..., f k ( n ) suc h that each regular language has Θ( f i ( n )) non-deterministic comm unica- tion complexit y for some i ∈ { 1 , 2 , ..., k } . F urthermore, we w ould like a c har- acterization of the languages with Θ( f i ( n )) complexit y for all i ∈ { 1 , 2 , ..., k } . In [TT03 ], this goal w as reac hed for the fo llo wing comm unication mo d- els: deterministic, sim ultaneous, probabilistic, sim ultaneous probabilistic and Mo d p -coun ting. Obtaining a similar result fo r the non-deterministic mo del requires a refinemen t of the t ec hniques used in [TT03 ]. The study of the non-deterministic comm unication complexit y of regular languages from an algebraic p oin t of view is imp ortant for sev eral reasons. W e can summarize it b y stating that it increases our understanding of regular languages a nd non- deterministic commun ication complexit y . F rom regular languages p ersp ectiv e, our results yield suffic ien t algebraic conditions for not b eing in a certain class of languages. T his is a n in teresting result within algebraic automata theory . F urthermore, giv en t he fact that comm unication complex it y has many ties with other computat io nal mo dels, understanding the commu nication complexit y of regular languages helps us understand the p ow er of regular langua g es in differen t computational mo dels and where they stand within the complexit y theory fra me. F rom a comm unication complexit y p ersp ectiv e, there are se v eral in ter- esting consequences. In [TT03], it w as sho wn that in the regular languages setting, Θ(lo g log n ) probabilistic communic ation complexit y coincides with Θ(log n ) sim ultaneous communic ation complexit y . Results ab out the non- 7 CHAPTER 1. In tro duction deterministic comm unication complexit y leads to further suc h corresp on- dences whic h allow s us the compare different comm unication mo dels within the r egula r languages fra mework. Eve n though regular languages are “sim- ple” with resp ect to T uring Mac hines for example, they pr ovide a non-trivial case-study of non-deterministic communication complexit y since there are b oth “hard” and “easy” regular languag es with r esp ect to this mo del. There- fore, a complete c hara cterization of regular languages in this mo del is lik ely to force one to dev elop new lo w er b ound tec hniques and study functions (for example promise functions) ot her than the commonplace ones whic h ha ve b een intens iv ely studied. Through the notion of p r o gr ams over monoids ([Bar86]) , a connection b e- t w een algebraic automata theory and circuit complexit y has b een formed. F or example algebraic c haracterizations of some of the most studied circuit classes AC 0 , AC C 0 and N C 1 ha v e b een obt a ined ([BT87]) . The connection b et w een comm unication complex it y and circuit complexit y is w ell kno wn. Curren tly , tec hniques fro m comm unication complexit y pro vide one o f the most p o w erful to ols for pro ving circuit lo w er b ounds ([Gro92],[HG91],[Nis93],[RM97]). Al- gebraic ch aracterization o f regular la nguages with resp ect to communic ation complexit y completes a full circle and further strengthens our understanding of the three fields. Circuits Comm unication Complexit y Algebraic Automat a Theory 1.4. Outline 8 The breakdo wn of the thesis is as f ollo ws. In Chapter 2, w e giv e an in- tro duction t o comm unication complexity and presen t the fundamen tal tec h- niques in this field. Chapter 3 is dev oted t o the basics of algebraic auto mata theory . The main purp ose o f these tw o c hapters is t o deliv er t he bac kground material needed for Chapter 4. In Chapter 4, w e presen t the r esults obtained ab out the non-deterministic communic ation complexit y of regular languages. Finally we conclude in Chapter 5. Chapter 2 Comm unication Complexi t y In this chapter, w e presen t the notio n of commu nication complexit y as in- tro duced b y Y ao in [Y ao79]. W e start in Section 2.1 with the deterministic mo del in whic h we lo ok at the f undamen ta l concepts. In Section 2.2, we mo ve to the non-deterministic model which is the mo del o f interes t for this w ork. In Section 2.3, we briefly men tion other p opular communic ation mo dels. In Section 2.4, w e introduce the notion o f a r eduction whic h plays a key role in o ur arg umen ts in Chapter 4. W e also define communic ation complex it y classes and see a b eautiful analogy b e t w een t hese classes and T uring Mac hine classes. Finally we summarize this c ha pt er in Section 2.5 . W e refer the reader to the muc h celebrated b o ok by Kushilevitz and Nisan [KN97] fo r an in depth surv ey of the sub ject. One can also find and excellen t in tro duction in the lecture notes b y Ra n Raz [Raz04]. W e mostly use the notation used in [KN97]. 9 2.1. Deterministic Mo del 10 2.1 Determinis tic Mo del 2.1.1 Definition In the t w o- part y comm unication complexit y mo del, we ha ve t wo pla y ers (usu- ally referred to as Alice and Bob) and a function f : X × Y → Z . Alice is giv en x ∈ X and Bob is giv en y ∈ Y . Both know the function f and their goal is to c ol lab or ative l y compute f ( x, y ) i.e. they b oth w an t to know the v alue f ( x, y ). In order to do this they ha v e to commun icate (fo r most func- tions) since neither of them see the whole input. W e are only in terested in the n um b e r of bits that they need to comm unicate to compute f ( x, y ). Th us the complexity o f their individual computatio ns are irrelev ant and we assume that b o th Alice and Bob hav e unlimited computational p o w er. The communic ation of Alice and Bob is carried out according to a proto col P that b oth play ers ha v e a greed up o n b eforehand. The proto c ol P specifies in eac h step the v alue o f the next bit communic ated as a function of t he input of the play er who sends it and the sequence of previously comm unicated bits b y the t w o pla ye rs. The prot o col also determines who sends the next bit as a function of the bits comm unicated th us fa r. More formally , a proto col is a 5- tuple of functions ( c A , c B , n, f A , f B ) suc h that: • At eac h step of the comm unication, c A tak es as input the commun ica- tion history thus far and the input for Alice and returns the bit that Alice will comm unicate (similarly for c B and Bo b). • n tak es as input the comm unication history t hus far and decides whether the communic ation is o v er or not. If not, it decides who sp eaks next. • After the commun ication is o v er, f A tak es as input t he comm unication history a nd the input for Alic e and returns one bit (similarly for f B 11 CHAPTER 2. Comm unication Complexit y and Bob). This bit is the o utput of the proto c ol and the v alues of f A and f B should b e the same. Unless stated otherwise, the functions w e consider in this c hapter are of the f orm f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } . Since the output of the function is just one bit, w e can assume tha t the last bit comm unicated is t his v alue. Let P ( x, y ) b e the output of the protocol P , i.e. the last bit commu- nicated. Then we sa y that P is a proto col for f if for all x, y ∈ { 0 , 1 } n , P ( x, y ) = f ( x, y ) . The cost of P is defined as cost ( P ) := max ( x,y ) ∈ X × Y n um b er of bits communic ated for ( x, y ) . The deterministic c o mmunic ation c omplex ity of a function f , denoted as D ( f ), is defined a s D ( f ) := min P proto col for f cost ( P ) Example 2.1. Define the EQUALITY function as E Q ( x, y ) := ( 1 if x = y , 0 otherwise. An ob vious upp er b ound for D ( E Q ) is n + 1 since one of the play ers, sa y Alice, can just send all her bits to Bob and Bob can simply compare the string he has with the string Alice has sen t. If the strings are equal he can send 1 t o Alice and otherwise he can send 0. In f act, this proto col gives an upp er b ound for an y b o olean function. Once o ne of the pla y ers knows a ll the input, s/he can compute the v alue of the function and send this v alue t o the other play er. The num b er of bits comm unicated is n + 1. In t uit ively , one exp ects that D ( E Q ) = n + 1 , i.e. n + 1 is a lso a low er b ound for D ( E Q ). Although this in tuition is correct, ho w can one rigorously pro v e this low er b ound? 2.1. Deterministic Mo del 12 000 001 010 01 0 100 101 110 111 000 1 0 1 1 0 0 0 1 001 0 0 0 0 1 0 1 0 010 0 0 0 0 0 0 1 1 011 1 0 0 1 1 0 0 1 100 1 0 0 0 0 0 1 0 101 0 1 1 0 0 0 0 1 110 0 0 0 1 0 1 1 1 111 0 0 1 1 1 0 0 1 Figure 2.1: Example of an input matrix and a 0-mono c hromatic rectangle. 2.1.2 Lo w er Bound T ec hniques As in an y other computational mo del, pro ving tigh t lo we r b ounds for the complexit y of explicit functions in the comm unication mo del is usually a non-trivial task. Nev ertheless, there are a n um b er of effectiv e tec hniques one can use to accomplish this. No w w e explore three of these metho ds: the disjoin t co v er metho d, the rectangle size metho d and the fo oling set metho d. F or a function f , define the input matrix by A f xy = f ( x, y ) where the rows are indexed b y x ∈ X and the columns are indexed b y y ∈ Y . W e sa y that R is a r e ctangle if R = S × T for some S ⊆ X and T ⊆ Y . This is equiv alent to say ing that ( x 1 , y 1 ) ∈ R a nd ( x 2 , y 2 ) ∈ R together imply ( x 1 , y 2 ) ∈ R . W e sa y tha t R is mono chr om atic with resp ect to f if for some z ∈ Z w e hav e A f xy = z for all ( x, y ) ∈ R (see Figur e 2.1.2). Giv en f , let P b e a proto col f or f . F or simplicit y let us assume tha t the pla y ers send bits alternately . Also without loss of generality w e can assume Alice ( who has the input x ) sends the first bit. Th us a t step 1, the proto c ol 13 CHAPTER 2. Comm unication Complexit y partitions X × Y = ( X 0 × Y ) ∪ ( X 1 × Y ) such that ∀ x ∈ X 0 , Alice sends 0 , ∀ x ∈ X 1 , Alice sends 1 . A t the second step it is Bob’s turn to send a bit so the pro t o col partitions b oth X 0 × Y = ( X 0 × Y 00 ) ∪ ( X 0 × Y 01 ) and X 1 × Y = ( X 1 × Y 10 ) ∪ ( X 1 × Y 11 ). Here, if the first commu nicated bit w as a 0, then ∀ y ∈ Y 00 , Bob sends 0 , ∀ y ∈ Y 01 , Bob sends 1 , and if the fir st comm unicated bit w as a 1, then ∀ y ∈ Y 10 , Bob sends 0 , ∀ y ∈ Y 11 , Bob sends 1 . In general, if it is Alice’s turn to sp e ak and the bits comm unicated th us far are b 1 , b 2 , ...b k , then Alice partitions X b 1 ,...,b k − 1 × Y b 1 ,...,b k in to X b 1 ,...,b k , 0 × Y b 1 ,...,b k and X b 1 ,...,b k , 1 × Y b 1 ,...,b k . A proto col partitioning tree nicely illustrates what happ ens (see Figure 2.2). Observ e that eac h no de in the proto col partit io ning tree is a rectangle and t w o no des in tersect if and only if one is the ancestor of the other. In particular, the lea v es of t he tree are disjoin t rectangles. The same bits are comm unicated fo r all the inputs in a leaf so P ( x, y ) is the same for all these inputs, i.e. the lea v es are mono c hromatic. The height of t he tree is equal t o cost ( P ). Th us w e ha v e pro v ed the follo wing lemma whic h is a k ey combina- torial prop ert y of a proto c ol. Lemma 2.1. A pr oto c ol P for f with cost ( P ) = c p artitions the inp ut matrix A f into at most 2 c mono chr om atic r e ctangles. 2.1. Deterministic Mo del 14 Alice Bob Alice X × Y X 0 × Y 0 X 0 × Y 00 0 X 000 × Y 00 · · · 0 X 001 × Y 00 · · · 1 X 0 × Y 01 1 X 010 × Y 01 · · · 0 X 011 × Y 01 · · · 1 X 1 × Y 1 X 1 × Y 10 0 X 100 × Y 10 · · · 0 X 101 × Y 10 · · · 1 X 1 × Y 11 1 X 110 × Y 11 · · · 0 X 111 × Y 11 · · · 1 Figure 2.2: Proto col pa r t it ioning tree. 15 CHAPTER 2. Comm unication Complexit y A mono chr omatic disjo i n t c ove r is a pa rtition of a matr ix in to disjoin t mono c hromatic rectangles. W e denote b y C D ( f ), the minim um n um b er of rectangles in an y mono c hromatic disjoint co ve r of A f . With this definition and the previous lemma at hand, we can presen t the first low er b ound tech- nique. Corollary 2.2 (D isjoin t Co v er Method) . F or a func tion f we have D ( f ) ≥ log 2 C D ( f ) . With this to o l it is no w easy t o sho w a linear lo w er b ound for D ( E Q ). Observ e that the input matrix fo r EQUALITY is a 2 n b y 2 n iden tity ma- trix. No 1-mono chromatic rectangle can con t a in more than one 1. Thus an y mono c hromatic disjoin t co v er has 2 n 1-mono chromatic rectangles and at least one 0-mono c hromatic rectangle. So D ( E Q ) ≥ ⌈ log 2 (2 n + 1) ⌉ = n + 1 as predicted. Although ev ery proto col for a function induces a mono c hromatic disjoin t co v er of the input matrix, simple examples show that the con v erse is not true. So if some of the mono chromatic disjoin t cov ers do not corresp ond to an y proto col, how go o d can the disjoint cov er metho d b e? The next theorem states that the gap is not v ery larg e. Theorem 2.3. F or a function f , we have D ( f ) = O ( log 2 2 C D ( f )) . Pr o of. F or an y function f , w e presen t a proto col for it with complexit y O (log 2 2 C D ( f )) . The proto c ol consists o f at most lo g 2 C D ( f ) rounds a nd in eac h round at most log 2 C D ( f ) + O (1) bits are communicated. The basic idea is as follo ws: Alice and Bob agree up o n an optimal disjoin t mono c hromatic co ve r b efo re- hand. They try to figure out whether ( x, y ) lies in a 0- mono c hromatic rect- angle or a 1-mono c hromatic rectangle. The proto col proceeds in rounds. If f ( x, y ) = 1 then in eac h round they successfully eliminate a t least half of the 2.1. Deterministic Mo del 16 0-mono chromatic r ectangles. A t the end, all 0-mono c hromatic r ectangles are eliminated and they conclude f ( x, y ) = 1. If on the other hand f ( x, y ) = 0, then in one of the rounds they a re not able to eliminate a t least half of the 0-mono chromatic rectangles. A t this p oint they conclude f ( x, y ) = 0. Before giving the details of a r o und, we mak e t w o crucial observ ations. The first observ ation implies the second one. The correctness o f the proto col fol- lo ws fr o m the second observ ation. Observ ation 1 : Supp ose R 0 = S 0 × T 0 is a 0-mono c hromatic rectangle and R 1 = S 1 × T 1 is a 1-mono chromatic rectangle. Then either R 0 and R 1 are disjoin t in rows or they are disjoint in columns, i.e. either S 0 and S 1 are disjoin t or T 0 and T 1 are disjoin t. Observ ation 2 : Let C b e a n y collection of 0-mono c hromatic rec tangles and R 1 an y 1-mono c hromatic rectangle. Then either - R 1 in tersects with at mo st half of the rectangles in C in ro ws or - R 1 in tersects with at mo st half of the rectangles in C in columns. Otherwise there is at least one rectangle R 0 in C suc h that R 0 and R 1 in tersect b oth in rows and columns. This contradicts the first o bserv ation. No w w e can describ e how a round is carried out. Initially C con ta ins all the 0-mono chromatic rectangles. A. If C = ∅ then Alice comm unicates to Bob that f ( x, y ) = 1 and the pro- to col ends. Otherwise, Alice tries to find a 1-mo no c hromatic rectangle R 1 = S 1 × T 1 suc h that x ∈ S 1 and R 1 in tersects with at most half of the rectangles in C in ro ws. If suc h a rectangle exists, then Alice sends it s name (log 2 C D ( f ) bits) to Bob and they b o t h up date C so it con tains all the rectangles that intersec t with R 1 in ro ws (the other rectangles cannot con ta in ( x, y )). At this p oint the r ound is o v er since they succes sfully eliminated at least half of the rectangles in C . If Alice is unable find such a rectangle t hen she comm unicates this to Bob. 17 CHAPTER 2. Comm unication Complexit y B. At this p oint w e kno w Alice could not find a 1 -mono chromatic rect- angle to end the round so Bob tries to end the round by finding a 1-mono chromatic rectangle R 1 = S 1 × T 1 suc h that y ∈ T 1 and R 1 in tersects with at most half of the rectangles in C in columns. If he finds suc h a rectangle, he comm unicates its name to Alice and they b oth up date C so it con tains all the rectangles that interse ct with R 1 in columns. After this p oin t the round is ov er. If he cannot find suc h a rectangle this means bo th Alice a nd Bob fa iled and therefore he com- m unicates to Alice that f ( x, y ) = 0 b ecause by t he second observ ation, he kno ws that there is no 1 -mono chromatic rectangle containing ( x, y ). In most cases it is hard to exactly determine C D ( f ). So the natural next step is to find lo wer b ounds on C D ( f ) whic h in turn giv es lo we r b o unds on D ( f ). (This is actually what we did for the EQUALITY function.) An obvious wa y of b o unding (from b elow) the n um b er of mono c hromatic rectangles needed in a mono c hromatic disjoint co v er is to b ound (fro m ab ov e) the size of eve ry mono c hromatic rectangle. In other w ords, if ev ery mono chro- matic rectangle in the input matrix has size less than or equal to s , then we need at least 2 2 n /s mono c hromatic rectangles in a mono c hromatic disjoint co v er of the matrix. Here ‘size’ r efers to the n um b er o f pairs ( x, y ) in the rectangle and we can interpret this as a measure µ . The ab ov e actually generalizes to any kind of measure. Prop osition 2.4 (Rectangle Size Method) . L et µ b e a me asur e define d on the sp ac e X × Y . If al l mono chr omatic r e ctangles R (with r esp e ct to f ) ar e such that µ ( R ) ≤ s , then D ( f ) ≥ log 2 ( µ ( X × Y ) /s ) . In p articular, if µ is a pr ob ability and every mono chr omatic r e ctangle R sat- isfies µ ( R ) ≤ ǫ , then D ( f ) ≥ log 2 1 /ǫ . 2.1. Deterministic Mo del 18 Example 2.2. Let us see an application of the rectangle size metho d by pro ving a linear lo w er b o und fo r the commun ication complexity of the DIS- JOINTNESS function. W e define DISJOINTNESS as D I S J ( x, y ) := ( 1 if x ∩ y = ∅ , 0 otherwise. where x and y are view ed a s subsets o f [ n ] ( x i = 1 if x con tains the elemen t i ∈ [ n ]). W e claim that an y 1-mono c hromatic rectangle R = S × T has size at most 2 n . It is easy to show that the n um b e r o f ( x, y )’s suc h that x ∩ y = ∅ is P n j =0  n j  2 n − j = 3 n . No w if fo r all x and y that in tersect w e set µ ( x, y ) = 0 and f o r all x and y that are disjoin t w e set µ ( x, y ) = 1 then µ ( X × Y ) = 3 n and the ab ov e claim together with Prop osition 2.4 imply D ( DI S J ) = Ω( n ). Pr o of of claim: Suppose | S | = k and | ∪ x ∈ S x | = c . Then clearly k ≤ 2 c . Also | T | ≤ 2 n − c since ev ery set in T m ust be disjoint fr om ev ery set in S . Th us the size of the rectangle is | S | · | T | ≤ k 2 n − c ≤ 2 c 2 n − c = 2 n . The last low er b ound t echniq ue w e lo ok at in this section is the w ell- kno wn fo oling se t techniq ue. It is a direct conse quence o f the disjoin t cov er metho d and in f act it is a sp ecial case of Prop osition 2.4 . First w e mak e the formal definition of a fo oling set. Definition 2.5. A set F ⊆ X × Y is a fo oling set for f if the following conditions are satisfied. 1. F or all ( x, y ) ∈ F , f ( x, y ) = z for some z ∈ Z . 2. F or all distinc t ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ F either f ( x 1 , y 2 ) 6 = z or f ( x 2 , y 1 ) 6 = z . By the definition of a fo oling set, no t w o elemen ts in F can b e in the same mono c hromatic rectangle. Therefore there mus t b e at least | F | man y mono c hromatic rectangles in any mono chromatic disjoin t co v er o f the input matrix. So by Corollary 2.2 we get the following fact. 19 CHAPTER 2. Comm unication Complexit y Lemma 2.6 (F o o ling Set Metho d) . If F is a fo olin g set for f then D ( f ) ≥ log 2 | F | . T o see that the f o oling set metho d is indeed a sp ecial case of Prop osition 2.4, for a fo oling set F , let µ ( x, y ) = c > 0 for ev ery ( x, y ) ∈ F and for ev ery ( x, y ) / ∈ F set µ ( x, y ) = 0. T hen an y mono chromatic rectangle R satisfi es µ ( R ) ≤ c and therefore D ( f ) ≥ log 2 ( µ ( X × Y ) /c ) = log 2 ( c | F | /c ) = log 2 | F | . Example 2.3. Define the LESS-THAN function as LT ( x, y ) := ( 1 if x ≤ y , 0 otherwise. where x and y are view ed as binary n um b ers . W e can show that LT has linear deterministic comm unication complexit y by the fo oling set tec hnique. Let F = { ( x, x ) : x ∈ { 0 , 1 } n } . It is easy to see that F is a fo oling set. Clearly | F | = 2 n and this prov es our claim. In fact F is also a f o oling set for the EQUALITY function. F rom our discussion in this section, it is clear that we can exploit the nice com bina t orial structure o f proto cols to prov e tigh t lo w er b ounds f or explicit functions. In the next sec tion, w e see that most of the tec hniques s een in this section can b e applied to the non- deterministic mo del as w ell. 2.2 Non-Dete rministic Mo del 2.2.1 Definition The definition o f the non-deterministic communic ation mo del is analogous to the non-deterministic mo del in the T uring Mac hine w orld. There are sev eral 2.2. Non-D eterministic Mo del 20 w ays of defining non-determinism, all of whic h are equiv alen t. Here w e will presen t the one that b est suits our needs. In t uit ively , non-determinism can b e view ed as a certificate v erification pro cess 1 : A third pla yer (referred to as Go d) giv es a pro of (bit string) that f ( x, y ) = z to b oth Alice and Bob. If indeed f ( x, y ) = z , then Alice and Bo b m ust b e able to con vince thems elv es that this is the case b y communic ating with eac h o ther. If o n the other hand f ( x, y ) 6 = z , then the ve rification pro- cess should fail and Alice and Bob should b e able to conclude that the pro of w a s wrong. W e consider the bits sen t by G o d as a par t of the commun icated bits. More formally , in the non-deterministic setting, Alice and Bo b comm uni- cate a ccording to a non-deterministic pro to col P z . This pro to col differs f r o m the de terministic one as follo ws. P z tak es three inputs, x, y and s , where x and y are perceiv ed as the inputs for Alice and Bo b resp ectiv ely , and s is some bit string whic h w e think of as the “pro of string”. P z sp ecifies in eac h step the v alue of the next bit communic ated as a f unction of the input of the pla y er who sends it, the sequence o f previously comm unicated bits as w ell as s . It also determines who will send the next bit as a function of the comm unicated bits thus far. So it differs from a deterministic pro to col b ecause what a play er sends also dep ends on the string s . W e will denote the output of the proto col b y P z ( x, y , s ). W e say that P z is a non-deterministic pr o to col f or f if f or all ( x, y ) suc h that f ( x, y ) = z , there exists a string s suc h that P z ( x, y , s ) = z , and for all ( x, y ) such that f ( x, y ) 6 = z we hav e P z ( x, y , s ) 6 = z fo r an y s . The t he cost of P z is defined as cost ( P z ) := m ax ( x,y ): f ( x,y )= z min s : P z ( x,y ,s )= z | s | + no of comm unicated bits for ( x, y , s ). 1 Equiv alently one can view it as a communication game in whic h the play ers are allow ed to take non-deter ministic s teps. 21 CHAPTER 2. Comm unication Complexit y 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 Figure 2.3: An example of a mono c hromatic co v er o f the 1-inputs. W e define the non-deterministic c omm unic ation c om plexity of f as N 1 ( f ) := min P 1 non-deterministic protocol for f cost ( P 1 ) The co-non- deterministic communic ation complexit y of f is defined similarly and is denoted b y N 0 ( f ). Example 2.4. Let us sho w N 0 ( D I S J ) = O ( lo g 2 n ) by exhibiting a pro of and a v erification proto col. Go d can pro v e D I S J ( x, y ) = 0 by telling Alice and Bo b the index i in whic h x and y intersec t. This pro of is O (log 2 n ) bits long a nd Alice and Bob can convince themselv es that the pro of is correct b y exc hanging the bits x i and y i . If D I S J ( x, y ) = 1, then g iv en a ny index as a pro of, Alice and Bob can detect that the pro of is wrong. (Any other kind of pro of is considered as a wrong pro of.) Unlik e the deterministic comm unicatio n complexit y , w e can get an exact c ha r acterization o f non-deterministic communic ation complexit y in terms of mono c hromatic rectangles. W e denote b y C z ( f ) the minim um num b er of z -mo no c hromatic rectangles in any mono chromatic co v er of the z -inputs of f (observ e that here w e dropp e d the word “disjoint” since w e allow the rectangles to in tersect). This quan tit y exactly determines N z ( f ). Prop osition 2.7. log 2 C z ( f ) ≤ N z ( f ) ≤ lo g 2 C z ( f ) + 2 . Pr o of. 2.2. Non-D eterministic Mo del 22 • log 2 C z ( f ) ≤ N z ( f ) W e hav e seen in the deterministic case that a certain commun ication pattern corresp onds to a certain mono c hromatic rectangle. This situa- tion is not muc h differen t in t he non- deterministic mo del. In t his case, what Alice and Bob send in each step also dep ends on the pro of bits. So for ev ery fixed pro o f string, there corresp onds a proto col partitioning tree as in Figure 2.2. No w observ e that for this particular pro o f, ev ery comm unication pat- tern that con vinces Alice and Bob leads to a z -mono chromatic rectan- gle. (Other comm unicatio n patterns ma y not lead to a mono c hromatic rectangle since the pro of w e fixed may not b e a pro of for all ( x, y ) with f ( x, y ) = z .) So eac h con vincing comm unication pattern (including the pro of ) corresp onds to a z -mono c hromatic rectangle. Since fo r ev- ery ( x, y ) suc h that f ( x, y ) = z there mus t b e a pro of that conv inces Alice and Bob, a ll the conv incing comm unication pa t terns together cor- resp ond to a co v ering of the z -inputs. Here the rectangles are a llow ed to in tersect since for some ( x, y ) with f ( x, y ) = z , there might b e more than one pro of that leads Alice and Bo b to b e con vinced. There are at most 2 N z ( f ) comm unication patterns and therefore C z ( f ) ≤ 2 N z ( f ) . • N z ( f ) ≤ log 2 C z ( f ) + 2 Fix any o ptimal mono c hromatic co ver of the z -inputs. If Go d sends Alice and Bob the name of a mono c hromatic rectangle R = S × T that ( x, y ) lies in, then Alice can c hec k that x ∈ S a nd if so, she can send 1 to Bob. Bob can similarly c hec k if y ∈ T a nd send 1 to Alice if this is the case. 23 CHAPTER 2. Comm unication Complexit y 2.2.2 P o w er of Non-Determinism A natural question that arises in this context is: how m uc h p ow er do es non-determinism give ? Non-determinism in the finite automaton computa- tional mo del do es not give extra p ow er with resp ect to the class of languag es recognized. In the T uring Machine mo del, it is not know n whether non- determinism prov ides significan tly more p ow er. In t he comm unication com- plexit y mo del w e can answ er t his question and pro v e that non-determinism is strictly more p ow erful. First w e observ e that the g ap b e t w een determinism and no n- determinism cannot b e more than exp onen tial. Prop osition 2.8. F or a n y z ∈ { 0 , 1 } , D ( f ) ≤ C z ( f ) + 1 . Pr o of. Alice and Bob agree on an optimal cov er of the z -inputs. Alice com- m unicates to Bob the z -mono c hromatic rectangles that x lies in (this requires C z ( f ) bits of comm unicatio n). Bob, with this information, can determine if there is a z -mono c hromatic rectangle that ( x, y ) lies in and send the answ er to Alice. The ab ov e is actually tigh t. F or example the EQUALITY function sat- isfies D ( E Q ) = n + 1 a nd N 0 ( E Q ) ≤ log 2 n + 2 (similar proto col to the one in Example 2.4). Can it b e the case that b oth N 0 ( f ) a nd N 1 ( f ) a r e exp onen tially smaller than D ( f )? Th e answ er to this question is giv en by the next theorem. Theorem 2.9. F or every function f : X × Y → { 0 , 1 } , D ( f ) = O ( N 0 ( f ) N 1 ( f )) . The pro of of this theorem is the same as t he pro of of Theorem 2 .3. It w a s shown in [F ur87] that this b ound is tight. Observ e that there are t w o reasons why non-determinism is more pow erful than determinism: 2.2. Non-D eterministic Mo del 24 1. non- determinism is one sided in the sense that w e only need to co v er the z -inputs, 2. the z -monochromatic rectangles in the cov er are a llo w ed to o v erlap. F rom our discuss ion ab o v e it should b e clear that the ultimate p o w er comes from the first p o in t. In the EQUALITY example w e see that it is “ easy” to co v er the 0-inputs in the sense that w e do not need exp onen t ia lly man y 0- mono c hromatic rectangles to co v er the 0-inputs. The hardness lies in cov ering the 1-inputs. The exp onen tial g a p is a pro duc t of this fa ct. The p o w er of a co v er aga inst a disjoin t cov er is o nly quadratic as implied by Theorem 2.9. 2.2.3 Lo w er Bound T ec hniques In the deterministic mo del, w e sa w the rectangle size metho d as a lo w er b ound tec hnique. It is clear that the same approac h giv es a lo wer bo und for the non- deterministic communication complexit y as w ell. If ev ery z - mono c hromatic rectangle has size less than or equal to s and there are k z -inputs, then w e need at least k /s man y rectangles to co ver these inputs. The no n-deterministic vers ion of Prop osition 2.4 is as fo llo ws. Prop osition 2.10. L e t K ⊆ X × Y b e the set of al l z -in p uts and let µ b e a me asur e define d on the sp ac e K . If al l z -mo n o chr omatic r e ctangles R satisfy µ ( R ) ≤ s , then N z ( f ) ≥ log 2 ( µ ( K ) /s ) . It can b e sho wn that the rectangle size metho d in the non-deterministic case is almost tigh t. Supp o se w e c ho ose the b est p ossible measure µ (i.e. the one that giv es t he b es t b ound) and the maxim um size (with resp ect to µ ) o f a z -mono c hromatic rectangle is s . Then we hav e: Theorem 2.11 (see [KN97]) . N z ( f ) ≤ lo g 2 ( µ ( K ) /s ) + log 2 n + O (1) . 25 CHAPTER 2. Comm unication Complexit y There a r e examples that sho w that w e cannot do b etter than this. The f act tha t we can use the rectangle size metho d here implies that w e can a lso use the fo oling set metho d. Ho wev er, as the next prop osition sho ws, the quality of the fo oling set metho d is questionable. Prop osition 2.12 (see [KN97]) . A lm o st al l functions f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } satisfy N 1 ( f ) = Ω( n ) but the size of their lar gest fo oling s et is O ( n ) . W e finish off this section b y lo oking at the non-deterministic comm unica- tion complexit y o f the PROMI SE-DISJOINTNESS function. This function is defined the same wa y as the D ISJOINTNESS function but the input space is differen t: it is the union of the f ollo wing t w o sets A and B . A := { ( x, y ) ∈ { 0 , 1 } n × { 0 , 1 } n : x ∩ y = ∅} . B := { ( x, y ) ∈ { 0 , 1 } n × { 0 , 1 } n : | x ∩ y | = 1 } . In other w ords Alice a nd Bob are promise d that if they get an input that in tersects, then the size of the intersec tion is exactly 1. No w w e sho w that the PROMISE -DISJOINTNESS ( P D I S J ) function has linear non- deterministic complexit y . This fact is used in Chapter 4 to pro v e linear lo w er b ounds for the complexit y of certain regular languages. T o sho w the linear lo wer b ound, we use a result tha t implies a linear lo w er b ound o n the randomized comm unication complexity of the DISJOINTNESS function. Before w e can state this result, w e first need to define t w o measures on { 0 , 1 } n × { 0 , 1 } n . µ A ( x, y ) := ( 1 | A | if ( x, y ) ∈ A , 0 otherwise. µ B ( x, y ) := ( 1 | B | if ( x, y ) ∈ B , 0 otherwise. 2.3. Other Mo dels 26 Lemma 2.13 (see [Raz04]) . F or any r e ctangle R = S × T , if µ A ( R ) > 2 − n/ 100 then µ B ( R ) > 1 100 µ A ( R ) (for n lar ge enough). In pa rticular, if µ A ( R ) > 2 − n/ 100 then R con tains elem en ts from b oth A and B . Therefore to cov er the inputs in A with 1-mono c hromatic rectangles 2 , w e need exp onen tially man y rectangles. This sho ws N 1 ( P D I S J ) = Ω( n ). 2.3 Other Mo dels In this se ction, we men tion some of the most in teresting and w ell-studied comm unication complex it y mo dels. Randomized Comm unication Complexit y In the randomized setting, Alice a nd Bob b oth hav e a ccess to random bit strings t ha t are generated a ccording to some proba bility distribution. These random strings are priv at e to them and are indep endent. What Alice and Bob comm unicate dep ends on these random strings as w ell as their input and the previously communic ated bits. W e sa y that P is a proto col for f with ǫ error if the follow ing holds. ∀ ( x, y ) ∈ X × Y , Pr[ P ( x, y ) = f ( x, y )] ≥ 1 − ǫ The cost of P is defined as the maximum n umber o f bits comm unicated where the maxim um is tak en ov er all p ossib le ra ndom strings and all inputs ( x, y ). 2 In this s etting 1- mo no chromatic rectangles can co ntain any element from ( X × Y ) \ B 27 CHAPTER 2. Comm unication Complexit y The r a ndomized comm unication complexit y of f is R ( f ) := min P protocol for f wi th error 1 / 3 cost ( P ) . One can also define the one sided error randomized complexity . W e sa y that P is a proto col f or f with one sided ǫ error if the follo wing holds. ∀ ( x, y ) ∈ X × Y with f ( x, y ) = 0 , Pr[ P ( x, y ) = 0] = 1 and ∀ ( x, y ) ∈ X × Y with f ( x, y ) = 1 , Pr[ P ( x, y ) = 1] ≥ 1 − ǫ. Then the one sided randomized comm unication complexit y of f is R 1 ( f ) := min P protocol for f wi th one sided error 1 / 2 cost ( P ) . There are also v aria tions of the ra ndomized mo del in whic h Alice and Bob ha v e access to one public random string. (F or a comparison see [New91].) Distributional Comm unication Complexit y In this setting, the definition of the cost o f a proto col and the comm unica- tion complexit y of a f unction are the same as t he deterministic mo del. The difference is that w e relax the condition ∀ ( x, y ) ∈ X × Y , P ( x, y ) = f ( x, y ) to Pr µ [ P ( x, y ) = f ( x, y ) ] ≥ 1 − ǫ for a giv en probability distribution µ on the input space X × Y a nd a constan t ǫ . The distributional communic ation complexit y of a function is denoted by D µ ǫ ( f ). 2.4. Comm unication Complexit y Classes 28 Multipart y Comm unication Complexit y A natural w a y o f generalizing the tw o pla y er mo del to k -pla y ers is as follow s. k -play ers try to compute a function f : X 1 × X 2 × ... × X k → Z where pla y er i gets x i ∈ X i and comm unication is established b y broadcasting (ev ery pla y er receiv es the commu nicated bit). Observ e tha t as the n umber of play ers increases, the p ow er of the mo del decreases. Another wa y of generalizing the t w o part y mo de l to k play ers w as pro- p osed in [CFL8 3]. This mo del is referred to as “ num b er on the forehead” mo del b ecause here eac h play er i sees eve ry input but x i . This can b e view ed as eac h pla y er having their input on their f orehead and not b eing able to see it. The p o w er of this mo del increases as the n um b er of pla y ers increase s. In this setting, coming up with low er b ounds is considerably harder. How ev er, these low er b ounds imply low er b ounds in other computational mo dels suc h as circuits and bounded-width br a nc hing pro grams. This is one of the rea- sons wh y this mo del has a t t racted more in terest than the natural generaliza- tion men tioned previously . There are applications in time-space tradeoffs for T uring Mac hines ([BNS92]), length-width tra deoffs for branc hing programs ([BNS92]), circuit complexit y ([HG 91], [G ro92], [Nis93 ], [G ro98]), pro of com- plexit y ([BPS07]) and pseudorandom generators ([BNS92]), to cite only a few. 2.4 Comm uni cation Complexity Classes It is po ssible to define complexit y classes with resp ect t o communic ation com- plexit y once w e settle what it means to be “easy” or “tractable”. Comm unica- tion complexit y classes w ere intro duced in [BFS86] in whic h “ t r actable” was defined to b e pol y log ( n ) complexit y . That is, a f unction is tractable if its com- plexit y is O (log c n ) for some constan t c . F rom this foundation, one can build 29 CHAPTER 2. Comm unication Complexit y comm unication complexit y classes analogous to P , N P , coN P , B P P , RP and man y more. F or example P cc = { f : D ( f ) = pol y log ( n ) } . The corresp on- dence b etw een some of t he complexit y classes and the complexit y measures can b e summarized as follow s. Complexit y class P cc N P cc coN P cc B P P cc RP cc Complexit y measure D N 1 N 0 R R 1 The relationship b etw een these classes are m uch b etter kno wn than their T uring Machine coun terparts since pro ving lo wer b ounds for explicit func- tions is easier in the commun ication w orld. W e ha ve seen that the function NOT-EQUALITY satisfies D ( N E Q ) = n + 1 and N 1 ( N E Q ) ≤ lo g 2 n + 1. This pro v es P cc 6 = N P cc . Since N 0 ( N E Q ) = O ( n ), w e ha v e coN P cc 6 = N P cc . Theorem 2.9 show s that P cc = N P cc ∩ coN P cc . It can also b e sho wn that P cc 6 = RP cc and N P cc * B P P cc . R emark. It is a lso p ossible to define analo g s of the p olynomial hierarc h y . Reducibilit y and completenes s are fundamen tal concepts in the T uring Mac hine computational mo del so it is natural to define t he comm unication complexit y analogs. The idea of reduction is as follow s. Giv en tw o functions f and g , f r e duc es to g if Alice and Bob can priv ately con vert their inputs x and y to x ′ and y ′ suc h that f ( x, y ) = 1 if and only if g ( x ′ , y ′ ) = 1. Supp ose f reduces to g and that the inputs of length n are conv erted in to inputs of length t ( n ). Then it is clear that if the communication complexit y of g is O ( h ( n )) then the com- m unication complexit y o f f is O ( h ( t ( n ))). If the communic ation complex it y of f is Ω( h ( n )) then the comm unication complexit y of g is Ω( h ( t − 1 ( n ))). Reductions of par ticular inte rest with resp ect to t he comm unication com- plexit y classes are those with t ( n ) = 2 log c 2 n for some constan t c . The f ormal definition as giv en in [BFS8 6] is as fo llo ws. Definition 2.14. Let t = 2 log c 2 n for some constan t c . A r e ctangular r e duction 2.5. Summary 30 from a function f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } to a function g : { 0 , 1 } t × { 0 , 1 } t → { 0 , 1 } is a pair of functions a : { 0 , 1 } n → { 0 , 1 } t and b : { 0 , 1 } n → { 0 , 1 } t suc h that f ( x, y ) = 1 if and only if g ( a ( x ) , b ( y )) = 1. F rom this definition it is clear that if there is a rectangular reduction fro m f to g and g ∈ P cc then f ∈ P cc . The same is true for N P cc (and in fact for ev ery lev el of the p o lynomial hierarch y). Example 2.5. F or a fixed cons tan t q > 1, define the INNER-PRODUCT function a s follows . I P q ( x, y ) := ( 1 if P n i =1 x i y i ≡ 0 mo d q , 0 otherwise. W e exhibit a reduction from P D I S J to I P q suc h that an input of length n is conv erted into an input of length n + q . Since q is a constant, this prov es that N 1 ( I P q ) = Ω( n ). Giv en x and y , Alice and Bo b eac h app end q 1’s a t the end of their inputs to obtain x ′ and y ′ . If P D I S J ( x, y ) = 1 then clearly I P q ( x ′ , y ′ ) = 1 . If on the other hand P D I S J ( x, y ) = 0, then we kno w that x a nd y in tersect only at one p osition and therefore x ′ and y ′ will inters ect in q + 1 p ositions. This implies I P q ( x ′ , y ′ ) = 0. Ha ving established the definition of a reduction, w e can define the notion of completeness . F or a class of functions C , w e sa y f ∈ C is c om plete in C if there is a rectangular reduction fr o m ev ery function in C to f . In [BFS86], a complete f unction is found in eve ry lev el o f the p olynomial hierarc hy . 2.5 Summary In t his c hapter w e to ok a glimpse at the mini-world within complexit y the- ory: comm unication complexit y . The main f o cus in this area has b een pro v- ing tigh t low er bounds for sp ecific functions. W e sho w ed three low er b o und 31 CHAPTER 2. Comm unication Complexit y tec hniques for the deterministic mo del. These w ere the disjoin t co v er metho d, rectangle size metho d and the fo oling set metho d. W e in tr o duced the non- deterministic mo del and sa w that the non- deterministic comm unication com- plexit y of a function w as essen tially the n um b er of z -mono c hromatic rectan- gles needed to co v er the z - inputs. W e saw that the rectangle size metho d, and therefore the fo o ling set metho d w ere also applicable in this setting. W e lo ok ed at the p o w er o f non- determinism and o bserv ed that the p ossible exp o nen tia l gap b etw een the deterministic and the non- deterministic com- plexit y arose fro m the f a ct that non- determinism w a s one sided. Later we touc hed o n some other comm unication models: randomized complexit y , dis- tributional complexit y and m ultiparty complexit y . F inally w e defined some of the comm unication complexit y class es, P cc , N P cc , coN P cc , B P P cc , RP cc , b y considering pol y log ( n ) complexity as tractable. Natural definitions of reducibilit y and completeness w ere a lso in tro duced. The deterministic and the non-deterministic comm unication complexities of the functions seen in this chapter are summarized with the following table. E Q N E Q LT DI S J P D I S J I P q D Θ( n ) Θ( n ) Θ( n ) Θ( n ) Θ( n ) Θ( n ) N 1 Θ( n ) Θ(log 2 n ) Θ( n ) Θ( n ) Θ( n ) Θ( n ) Chapter 3 Algebraic Automata Theory In this chapter, we in t r o duce the reader to alg ebraic automata theory b y presen ting the fundamental concepts in this area. The heart of this theory is viewing a mo no id as a language recognizer. Therefore w e b egin this c hapter in Section 3.1 b y explaining ho w a monoid can b e view ed as a computationa l mac hine. Later w e define the syn tactic monoid of a language whic h is anal- ogous to the minimal aut o maton. Then we define v arieties and state the v ariet y theorem whic h establishes a one to one correspondence b e t w een v a - rieties of finite monoids and v ar ieties of regular languages. This con v eys the in timate relationship b etw een finite monoids and regular languages. In Sec- tion 3.2 , w e extend the theory to ordered monoids since (as w e see in Chapter 4) this provides the prop er fra mew ork to analyze the non-deterministic com- m unication complexit y o f regular languages. W e assume that the reader has basic know ledge in automata theory . F or more details on the sub jects cov ered in this c hapter, see [Pin86] and [Pin97 ] for the ordered case. 32 33 CHAPTER 3. Algebraic Automata Theory 3.1 Monoids - Automata - Regular L ang uages 3.1.1 Monoids: A Computational Mo del Before w e can presen t ho w a monoid can b e view ed as a computationa l mac hine, w e first need to fo r mally define a monoid and a morphism b et w een monoids. A semigr oup ( S , · ) is a set S together with an asso ciative binary op eration defined on this set. A monoid ( M , · ) is a semigroup that ha s an iden tity: ∃ 1 M ∈ M whic h satisfies 1 M · m = m · 1 M = m for an y m ∈ M . W e denote a monoid b y its underlying set and write m 1 m 2 instead of m 1 · m 2 when there is no ambiguit y ab out the o p eration. O bserv e that a group is just a monoid in whic h eac h elemen t has an inv erse. Giv en t w o monoids M and N , a function ϕ : M → N is a morphism if ϕ (1 M ) = 1 N and if ϕ preserv es the op eration, i.e. ϕ ( mm ′ ) = ϕ ( m ) ϕ ( m ′ ) f or an y m, m ′ ∈ M . W e assume that the monoids w e ar e dealing with are finite, with the exception of the free monoid Σ ∗ whic h consists of all w ords (including the empt y word ǫ ) ov er the alphab et Σ, with the underlying op era t ion b ein g concatenation. Observ e t hat any function ϕ : Σ → M extends uniquely t o a morphism Φ : Σ ∗ → M . One branc h of algebraic graph theory studies the connection b et w een groups a nd corresp onding Ca yley graph represen t a tions of the groups. Sim- ilarly , monoids also hav e g raph r epresen tations. Giv en a monoid M , w e can construct a lab eled m ultidigraph G = ( V , A ) as fo llo ws. Let V b e the under- lying se t of the monoid and let ( m 1 , m 2 ) ∈ A with lab el m 3 if m 1 m 3 = m 2 . See Figure 3.1 for an example. No w the corresp ondence b e t w een monoids and automata should b e clear since w e can easily view the g r a ph of M as an automaton whic h recognizes a language ov er the alphab et M . All we need to do is declare the v ertex 1 M as the initial state and agree up on a set of accepting vertice s F ⊆ M . Observ e 3.1. Monoids - Automata - Regular Languages 34 1 2 0 0 1 0 2 0 2 1 1 2 Figure 3.1: Gra ph of ( Z 3 , +). that the graph of a monoid accepts a word m 1 m 2 ...m n iff m 1 · m 2 · ... · m n ∈ F . In fact, once w e fix a function ϕ : Σ → M , the graph o f M recognizes a language ov er the alphab et Σ: replace eac h arc’s lab el by its preimage under ϕ (now an arc can ha v e more than one lab el). A w ord s 1 s 2 ...s n ∈ Σ ∗ is accepted iff ϕ ( s 1 ) · ϕ ( s 2 ) · ... · ϕ ( s n ) ∈ F . If w e allow the set of accepting states to v ary and the function ϕ : Σ → M to v ary (for fixed Σ and M ) then by viewing the monoid’s graph as an automaton, we see that a single monoid can b e used to recognize a family of languages o v er Σ. Each languag e in the fa mily correspo nds to a fixed set of accepting states and a fix ed function ϕ . This leads to the more formal definition of recognitio n b y a monoid. W e sa y that a language L ⊆ Σ ∗ is r e c o gni ze d b y a finite monoid M if there exists a morphism Φ : Σ ∗ → M and an a ccepting set F ⊆ M suc h tha t L = Φ − 1 ( F ). Similarly , w e say tha t Φ : Σ ∗ → M r e c o gn i z es L if there exists F ⊆ M suc h that L = Φ − 1 ( F ). See Figure 3.2 for an alternative w a y of viewing M as a language recognizer. Giv en an y monoid morphism Ψ : M → N , the nucle ar c o ngruenc e with resp ect to Ψ is denoted by ≡ Ψ and is defined by m ≡ Ψ m ′ if Ψ( m ) = Ψ( m ′ ). 35 CHAPTER 3. Algebraic Automata Theory ϕ s 1 s 2 s n m 1 m 2 m n Multiply in M m In F ? Y es or No Figure 3.2: Another w a y o f viewing a monoid as a machin e. W e sa y t hat a set of w ords is homo gen e ous with resp ect to L if either ev ery w o r d in the set is in L o r none of the words is in L . No w observ e that Φ : Σ ∗ → M recognizes L if and only if the nucle ar congruence classes of Φ are ho mo g eneous. This fact is used in the up c oming pro ofs. F rom the earlier discussion, w e can conclude tha t if L is recognized b y a finite mono id, then it is recognized by a finite automaton (the graph of the monoid) and therefore it is regular. In fact, the con v erse is also true. Theorem 3.1. L is r e gular if and only if a finite monoid r e c o gnizes L . Pr o of. If L is regular t hen it is recognized by a finite automaton. The def- inition o f a n automat on includes the transition f unction δ : Q × Σ → Q where Q is the set of states. This function can b e naturally extended to δ : Q × Σ ∗ → Q . In other w o rds, ev ery w ord in Σ ∗ defines a function from Q to Q . Let δ w : Q → Q , q 7→ δ ( q , w ), b e the function corresp o nding to the w o r d w . Then it is easy to see that the set T := { δ w : w ∈ Σ ∗ } is a mono id with the op eration b eing comp osition of functions. F urthermore T is finite since Q is finite. W e call T the tr ansformation m o noid of the automaton. W e claim that the transformation monoid T recognizes L . T o see this let 3.1. Monoids - Automata - Regular Languages 36 Φ : Σ ∗ → T b e the canonical mapping: w 7→ δ w . Φ is a morphism since Φ( uv ) = δ uv = δ u ◦ δ v = Φ( u ) ◦ Φ( v ) . If Φ( u ) = Φ( v ) then u ∈ L iff v ∈ L so the n uclear congr uence classes are homogeneous and th us Φ recognizes L , whic h means T recognizes L . Theorem 3 .1 constitutes the foundat io n of algebraic auto mata theory . It sho ws that finite monoids and finite automat a hav e the same computational p ow er with resp ect to the class of la nguages recognized. The pro of rev eals the strong link b etw een mono ids and automata. In fact, this link can b e seen to b e m uch stronger via the relation b etw een the com binato rial prop erties 1 of L and the alg ebraic prop erties of a monoid recognizing L . With the purp os e of exploring this relation, w e define the syn tactic monoid of a regula r language. 3.1.2 The Syn tactic Monoid F or ev ery regula r langua g e there is a minimal automaton that recognizes it. Similarly , ev ery regular lang uage has a “minimal” monoid that r ecognizes it. W e call this monoid t he syn tactic monoid and it is unique. The syntactic c ongruenc e asso ciated with a language L ⊆ Σ ∗ is denoted b y ≡ L and x ≡ L y if for all u, v ∈ Σ ∗ w e hav e uxv ∈ L iff uy v ∈ L . It is straigh tforw ard to che c k that this relatio n is indeed a congruence. The syn- tactic monoid of L is the quotient monoid Σ ∗ / ≡ L and is denoted b y M ( L ). Let [ w ] represen t the cong r uence class of w with resp ect to the syn ta ctic congruence. There is a w ell-defined op eratio n: [ w ][ u ] = [ w u ], so the canon- ical surjectiv e mapping Φ : Σ ∗ → M ( L ), w 7→ [ w ], is a morphism. Observ e that a n y congruence class of ≡ L is homogeneous (i.e. the n uclear congru- ence classes are homogeneous) so Φ r ecognizes L . W e call Φ the syntact ic morphism . 1 Regular la nguages are definable b y reg ula r expressio ns which are combinatorial de- scriptions of the language . 37 CHAPTER 3. Algebraic Automata Theory It is quite easy to v erify the following fact. Prop osition 3.2. L is r e gular if and only if its syntactic monoid is finite. W e sa y that a monoid N divides a mono id M (denoted b y N ≺ M ) if t here exists a surjectiv e mor phism f rom a submonoid 2 of M on to N . In tuitiv ely , this means that the multiplic ativ e structure of N is em b edded in M . The syn tactic monoid of L recognizes L and is the minimal monoid with this prop erty with resp ect to division. Prop osition 3.3. M ( L ) r e c o gn i z e s L a n d divide s any other monoid that also r e c o gni zes L . Pr o of. W e hav e already pro v ed that M ( L ) recognizes L so we pro v e the second statement. Let M b e any monoid that recognize s L . So there exists a morphism Ψ : Σ ∗ → M recognizing L . Let Φ b e the syn tactic morphism. T o sho w M ( L ) divides M , we find a surjectiv e morphism Υ from a submonoid N of M on to M ( L ). Before defining Υ w e first prov e the f o llo wing claim: if Ψ( a ) = Ψ( b ) then Φ( a ) = Φ( b ). Supp ose not, so there exists a, b suc h that Ψ( a ) = Ψ( b ) but Φ( a ) 6 = Φ( b ). By the definition of Φ this means that without loss of generalit y , there exists u, v suc h that uav ∈ L but ubv / ∈ L . W e ha v e Ψ( uav ) = Ψ( u )Ψ( a )Ψ( v ) = Ψ( u )Ψ( b )Ψ( v ) = Ψ( ubv ) so Ψ maps uav and ubv to the same elemen t. Since n uclear congruence classes (with resp ect to Ψ) must b e homogeneous and uav ∈ L but ubv / ∈ L , we get a contradiction. No w let N := Ψ(Σ ∗ ). So N is a submonoid of M . Define Υ : N → M ( L ), Ψ( w ) 7→ Φ( w ), i.e. Φ = Υ ◦ Ψ. 2 A submonoid is a subset that contains the identit y and is clo sed under the op eration. 3.1. Monoids - Automata - Regular Languages 38 Σ ∗ M ( L ) N Φ Ψ Υ By claim Υ is well-define d. Since Φ is su rjectiv e, Υ is surjectiv e. F ur- thermore, Υ(Ψ( u )Ψ( v )) = Υ(Ψ( uv )) = Φ( uv ) = Φ( u )Φ( v ) = Υ(Ψ( u ))Υ(Ψ( v )) and so Υ is a morphism. If M ≺ N and N ≺ M t hen M is isomorphic to N . So as claimed b efore, for ev ery regular language there is a unique (up to isomorphism) canonical monoid, the syntactic monoid, a ttac hed to it. An in teresting prop ert y of M ( L ) is that it is isomorphic to the transformation monoid of the minimal automaton recognizing L . In the next subsection, we intro duce the notio n of languag e a nd monoid v arieties. The com binatorial prop erties of a language are reflected on the algebraic prop erties o f M ( L ) and v arieties are the pro p er framework to for- malize this. 3.1.3 V arieties W e first give a brief o v erview of v arieties and ho w monoid v arieties and language v arieties relate to each other. A v ariet y of languages is a family of languages that satisfy certain condi- tions. Similarly a v ariety o f monoids is a family o f monoids satisfying certain conditions. The v ariety theorem states that there is a one to one corresp on- dence b et w een v arieties of regular la ng uages and v a r ieties of finite monoids: 39 CHAPTER 3. Algebraic Automata Theory a v ariety of monoids V corresp onds to the v ariety of regular la nguages V con- sisting of all the languages whose syn ta ctic monoid is in V . Consequen tly , w e a r e able to state results of the fo rm: “A regular lang uage belongs to the language v a riet y V if and only if its syn tactic monoid b elongs to the mo no id v ariet y V .” Man y classes of la ng uages tha t are defined combin atorially form language v arieties a nd man y classes of mono ids that a re defined algebraically form monoid v a rieties. So fr o m ab ov e w e can hop e to reach results of the form: “A regular language has the com binatoria l prop erty P if and only if its syn tactic monoid has the alg ebraic prop ert y Q .” Sc h ¨ utze n b erger was the first to establish suc h a result: A regular lan- guage is star-free 3 if and o nly if its syn tactic monoid is finite and ap erio dic 4 ([Sc h6 5]). Seve ral imp ortan t classes of regula r langua ges admit a similar al- gebraic c haracterization. This often yields decidabilit y results whic h a re not kno wn to b e obtainable b y other means. F or instance , b y a result of Mc- Naugh ton and P ap ert ([MP71 ]), w e k no w that r egula r languages de finable b y a first-order formula a re exactly the star-free languages. This implies that w e can decide if a r egula r language is first-order definable b y c hec king if its syn tactic monoid is ap erio dic and this is the only kno wn w a y of doing this. These types of algebraic ch aracterizations of regular langua ges a lso prov ide one with p o w erful algebraic to ols when a na lyzing and prov ing results ab out regular languages. V arieties of Finite Monoids A va rie ty of finite m o noids is a family of finite monoids V that satisfies the follo wing tw o conditions: 3 A language is star-free if it can b e defined by a extended reg ular expressio n without the K leene sta r op era tion. 4 A mo no id is ap erio dic if no subset of it forms a non-trivial g roup. 3.1. Monoids - Automata - Regular Languages 40 (i) V is closed under division: if M ∈ V and N ≺ M then N ∈ V , (ii) V is closed under direct pro duct: if M 1 , M 2 ∈ V then M 1 × M 2 ∈ V . Example 3.1. The followin g are some examples o f v arieties of monoids: • I is the trivial v ariet y consisting of only the trivial monoid I = { 1 } . • M is the v ariety containing all finite monoids. • Com is t he v ariety o f all comm utativ e monoids. • G is the v ariety of groups. • A is the v ariet y of ap eriodic mono ids. • J is the v ariety of monoids M that satisfy M m 1 M = M m 2 M = ⇒ m 1 = m 2 . W e call these monoids J -trivial. There is a con venie n t w a y of defining v arieties of monoids through iden- tities . The not ion of identities can b e presen ted in t w o wa ys. One in v o lv es top ological sem igroups (see [Pin97 ]), whic h w e wish to a v oid. Therefore w e use t he presen t a tion whic h w e think is more in tuitiv e. Let Σ b e a coun t a ble alphab et and u, v tw o words in Σ ∗ . W e say tha t a monoid M satisfie s the iden tit y u = v if for all morphisms ϕ : Σ ∗ → M w e ha v e ϕ ( u ) = ϕ ( v ). This means that if w e replace the letters of u and v with arbitrary (but consisten t) elemen ts of M then w e will arriv e at an equalit y in M . F or example a mono id is commutativ e if and only if it satisfies the iden tity ab = ba . It can b e sho wn that the fa mily of finite monoids consisting of the monoids that satisfy the iden tit y u = v forms a v ariet y . This v ariet y is denoted by V ( u, v ). 41 CHAPTER 3. Algebraic Automata Theory No w let ( u n , v n ) n> 0 b e a sequence of pa ir of w or ds in Σ ∗ . Define W := lim n →∞ V ( u n , v n ) = [ m> 0 \ n ≥ m V ( u n , v n ) . Observ e that M ∈ W if and only if there exists an n 0 > 0 suc h that for ev ery n > n 0 , M satisfies u n = v n . Here w e sa y that W is ultimately define d by the sequence of identities ( u n = v n ) n> 0 . Theorem 3.4 (see [Pin86]) . Every variety of monoids is ultimately defi ne d by some se quenc e of e quations. F or example the v ariety of comm utativ e monoids is ultimately defined b y the constan t sequenc e ( ab = ba ) n> 0 . A less trivial example is the v a riet y of ap erio dic monoids. It can b e shown that a finite monoid is a p erio dic if and only if for eac h m ∈ M there exists n ≥ 0 s uc h that m n = m n +1 . Consequen tly , the v ariet y of ap erio dic monoids is ultimately defined b y the sequence ( a n = a n +1 ) n> 0 . The v ariety of comm uta t ive ap erio dic monoids is ultimately defined by the sequence a 1 = a 2 , ab = ba, a 2 = a 3 , ab = ba, a 3 = a 4 , ab = ba, ... In suc h a case, for clarit y , w e say that the v ariet y is ultimately defined b y t w o sequences. An elemen t m of a mono id is called idemp otent if m · m = m . In finite monoids, idemp ot en ts pla y a k ey role. F or instance, ev ery non-empt y monoid con tains an idemp oten t. I ndeed, if w e tak e an y elemen t m of the monoid, then there exists a num b er n > 0 suc h that m n is an idempotent (in fact this is t he unique idempoten t generated b y m ). This implies that for an y finite monoid, t here is a num b er k > 0 suc h tha t for ev ery elemen t m in the monoid, we hav e that m k is an idempo ten t . W e call k an exp onent of M . Observ e that if k is an exp onent o f M then for any n ≥ k , n ! is also an exp o nen t of M . 3.1. Monoids - Automata - Regular Languages 42 W e use n ! in many sequence s o f iden tities that ultimately define v arieties of monoids. F or example, the sequence ( x n ! y x n ! = x n ! ) n> 0 ultimately defines the v ariety of lo c al ly trivial monoids . F ro m this, it should b e clear that a monoid M is lo cally trivial if and only if for ev ery idemp ot ent e ∈ M and ev ery elemen t m ∈ M w e ha v e eme = e . As a con v ention, a sequ ence of equations in v o lving n ! is written b y replacing n ! with ω . So for example we use x ω y x ω = x ω as an abbreviation for ( x n ! y x n ! = x n ! ) n> 0 . It is easy to se e that the v ariet y of groups G is ultimately defined by x ω = ǫ . Giv en a sequenc e of iden tities E , we denote by [[ E ]] the v ariety that is ultimately defined b y E . So f o r example we ha v e G = [[ x ω = ǫ ]] and the v ariet y of lo cally trivial mono ids is [[ x ω y x ω = x ω ]]. V arieties of Regular Languages Before w e can define a v a r iet y of regular languages w e need some preliminary definitions. A class of r e gular la n guages is a function C t ha t maps ev ery alphab et Σ to a set of regular languages in Σ ∗ . A set of languages in Σ ∗ that is closed under finite in tersection, finite union a nd complemen tation is called a b o ole a n algebr a . No w a variety o f r e gular languages is a class of r egula r langua g es V that satisfies the follow ing conditions: (i) F or an y alphab et Σ, V (Σ) is a b o olean algebra. (ii) V is closed under in vers e morphisms: giv en any a lphab ets Σ and Γ, f o r an y morphism Φ : Σ ∗ → Γ ∗ , if L ∈ V (Γ) then Φ − 1 ( L ) ∈ V (Σ). (iii) V is closed under left and r igh t quotien ts: for L ∈ V (Σ) a nd s ∈ Σ, w e ha v e s − 1 L := { w ∈ L | sw ∈ L } and Ls − 1 := { w ∈ L | w s ∈ L } are in V (Σ). 43 CHAPTER 3. Algebraic Automata Theory Example 3.2. The followin g a re some examples of v arieties of regular lan- guages: • The trivial v a r iety: V (Σ) = {∅ , Σ ∗ } . • The v ariet y of a ll regular languages (eac h alphab et is mapp ed to all the regular languages o v er this alphab et). • The v ariet y o f star-fr ee languages. • The v a r iet y o f piecewise testable langua g es: A la nguage is called pie c e- wise testable if there exists a k ∈ N suc h that membership of an y w ord in the language dep ends on the set of sub w o rds 5 of length at mo st k o ccurring in tha t w ord. The V ariet y Theorem F or a giv en finite monoid v ariety V , let V (Σ) b e the set o f langua ges in Σ ∗ whose syn tactic monoid b elongs to V . Alternativ ely , w e can define V as follo ws. Prop osition 3.5. L et C (Σ) b e the set of languag e s over Σ that is r e c o g n ize d by a monoid in V . Then V = C . Pr o of. V ⊆ C : If L ∈ V then M ( L ) ∈ V . M ( L ) recognizes L so L ∈ C . C ⊆ V : If L ∈ C then there exists M ∈ V recognizing L . M ( L ) ≺ M and V is closed under division so M ( L ) ∈ V . Therefore L ∈ V . No w w e can state the v ariet y theorem due to Eilen b erg ([Eil74]). Theorem 3.6 (The V ariet y Theorem) . V is a variety of languages and the mapping V 7→ V is one to one. 5 A w ord u = a 1 ...a n is a subw ord of a word x if x = x 0 a 1 x 1 a 2 ...a n x n for some words x 1 , ..., x n . 3.2. Ordered Monoids 44 In ligh t of this theorem, one can hop e to explicitly mak e suc h corresp on- dences. Tw o imp ortan t corresp ondence results are the follow ing. Theorem 3.7 ([Sc h65 ]) . The monoid varie ty A c o rr esp onds to the variety of star-fr e e languages. Equivalently, a r e gular lan g uage is star-fr e e if and only if its syntactic monoid is ap erio dic. Theorem 3.8 ([Sim75]) . The monoid varie ty J c orr esp onds to the variety of pie c ew ise testable languages. Eq uiva lently, a r e gular language is pie c ewis e testable if a nd on ly if its syntactic monoid is J -trivial. F urthermore w e can restate Theorem 3.1 as follo ws. Theorem 3.9. Th e monoid variety M c orr esp onds to the v a ri e ty of al l r e g- ular languages. Eq uiva lently, L is r e gular if and only i f its syntactic m onoid is fini te. 3.2 Ordered Monoid s In the previous section, w e hav e seen that w e can classify regular languages in terms of the monoids that recognize them. W e w ere able to obta in alge- braic c haracterizations for certain classes of langua g es: v arieties of languages. Man y intere sting com binatorially defined classes of languages form v arieties. But there are other combinatorially defined classes of langua ges that do not form a v ariety . Of pa rticular in terest are families of lang uages t ha t are not necessarily closed under complemen tation but satisfy the other prop erties of a v ariet y . W e call suc h f amilies “p ositiv e v arieties of languages”. Is it p os- sible to get a similar algebraic c haracterization for these languag es as w ell? In particular, is there a result similar to Eilen b e rg’s v ar iety theorem that p ermits us to treat p ositiv e v arieties? F ortunately the answ ers to the ab ov e questions are “ye s”. The idea is to attach an o r der on the monoids and adapt the definition of recognition 45 CHAPTER 3. Algebraic Automata Theory b y monoids to ordered mo no ids. This p oin t of view is a generalization of the unordered case and allo ws us to mak e a one to o ne corr espo ndence b e- t w een v a rieties of ordered monoids and p ositiv e v arieties of lang ua ges. This extension was in tro duced in [Pin95 ]. In t uit ively sp eaking, the syn tactic mono id has less information than the minimal automa t o n. One reason f or this is that in the minimal automa- ton the accepting states are predetermined, but in the syn tactic monoid the accepting set is not. As we see in the next subsection, t he order on the monoid restricts the w ay w e can c ho os e the accepting set and consequen tly the ordered syn tactic monoid reco vers some of the missing information. This restriction lets us analyze classes of languages that are not closed under com- plemen tatio n. In this section, we go o ver the definitions a nd the results seen th us far, and presen t the analogous or dered coun terparts. W e start with the notion of recognition by o rdered monoids. Then w e define the syn tactic o rdered monoid. La ter we lo o k at v arieties of ordered monoids, p ositive v arieties of languages and the v ariety theorem that establishe s a one to one corresp on- dence b etw een these o rdered monoid v arieties and p ositive language v arieties. 3.2.1 Recognition b y Or d ered Monoids An o r der relation on a set S is a relation that is reflexiv e, a n ti-symmetric and transitiv e and it is denoted by ≤ . W e say that ≤ is a stable or der r e lation on a monoid M if for a ll x, y , z ∈ M , x ≤ y implies z x ≤ z y and xz ≤ y z . An or der e d mono i d ( M , ≤ M ) is a monoid M to gether with a stable or der relation ≤ M that is defined on M . A morphism of or der e d monoids Φ : ( M , ≤ M ) → ( N , ≤ N ) is a morphism b et w een M and N that also pr eserv es the order relation, i.e. for a ll m, m ′ ∈ M , m ≤ M m ′ implies Φ( m ) ≤ N Φ( m ′ ). The free monoid Σ ∗ will alw a ys b e equipp ed with the equality relation. 3.2. Ordered Monoids 46 Observ e that an y morphism Φ : Σ ∗ → M is also a morphism o f ordered monoids Φ : (Σ ∗ , =) → ( M , ≤ M ) for any stable order ≤ M and vice v ersa. A subset I ⊆ M is called an or der ide al if for an y y ∈ I , x ≤ M y implies x ∈ I . Observ e that ev ery order ideal I in a finite monoid M has a generating set x 1 , ..., x k suc h that I = h x 1 , ..., x k i := { y ∈ M : ∃ x i with y ≤ M x i } . No w the concept of r ecognizabilit y is v ery similar to the unordered case. W e say that a lang ua ge L ⊆ Σ ∗ is recognized by an o rdered monoid ( M , ≤ M ) if there exists a morphism of ordered monoids Φ : (Σ ∗ , =) → ( M , ≤ M ) and an order ideal I ⊆ M suc h that L = Φ − 1 ( I ). Equiv alen tly , L is recognized b y ( M , ≤ M ) if there exists a morphism Φ : Σ ∗ → M and an order ideal I ⊆ M suc h that L = Φ − 1 ( I ). Observ e that this is a generalization of the unordered case in the sense that an y mono id is an ordered monoid with the equalit y o rder (the trivial order) and an y subset of t he monoid is an order ideal with resp ect to equalit y . Also note that in the unordered case, if L is recognized b y M , then so is the complemen t of L . In the ordered case, since w e require the a ccepting set to b e an order ideal, this statemen t is no longer true. This restriction on the accepting set allo ws the ordered monoid to k eep more information ab out the a utomaton recognizing L . In this sense, one can think of the ordered case as a refinemen t of the unordered case. 3.2.2 The Syn tactic Ordered M onoid The definition o f the syn tactic congruence with resp ect to L is as exactly as b efore: x ≡ L y if for all u, v ∈ Σ ∗ w e ha ve u xv ∈ L iff uy v ∈ L . Also the syn tactic monoid is the quotien t monoid M ( L ) = Σ ∗ / ≡ L . T o b e a ble to get a similar v ariety theorem for classes of languages not closed under complemen tation, w e need to define a stable order on M ( L ) that allo ws us to obtain an ordered counte rpart of the v ariety theorem. First, break up ≡ L : Let x  L y if for all u, v ∈ Σ ∗ , uy v ∈ L = ⇒ uxv ∈ L . 47 CHAPTER 3. Algebraic Automata Theory So x ≡ L y if and only if x  L y and y  L x . Now  L induces a well-define d stable order ≤ L on M ( L ) given by [ x ] ≤ L [ y ] if and o nly if x  L y . It is straigh tforw ard to c hec k tha t this is indeed a we ll-defined stable or der. The o rdered monoid ( M ( L ) , ≤ L ) is the syntactic or der e d monoid of L . W e sa y that an ordered monoid ( N , ≤ N ) d i v i d es an ordered monoid ( M , ≤ M ) if there exists a surjectiv e mo r phism o f o r dered monoids from a submonoid 6 of ( M , ≤ M ) onto ( N , ≤ N ). No w w e state and pro v e the analog of Prop osition 3.3. W e g iv e the pro of to demons trate that sligh t mo difications to the original pro of suffices to obtain the pro of fo r the o rdered coun terpar t . Prop osition 3.10. ( M ( L ) , ≤ L ) r e c o gni z e s L and is the minimal or der e d monoid with this pr op erty. Pr o of. Let Φ : Σ ∗ → M ( L ) b e the surjectiv e canonical mapping: w 7→ [ w ], i.e. Φ is the syn ta ctic mor phism. W e already kno w that the congruence classes are homog eneous so a ll w e need to sho w is t ha t the accepting set I is an order ideal, i.e. we need to show that if [ y ] ∈ I and [ x ] ≤ L [ y ], then [ x ] ∈ I . Since [ x ] ≤ L [ y ], we hav e x  L y and so fo r all u, v ∈ Σ ∗ , uy v ∈ L = ⇒ u xv ∈ L . In particular y ∈ L = ⇒ x ∈ L . Sin ce [ y ] ∈ I , y ∈ L and t herefore x ∈ L and so [ x ] ∈ I as required. Let ( M , ≤ M ) b e any mono id that recognizes L . So there exists a mor- phism Ψ : Σ ∗ → M and an or der ideal I ⊆ M suc h that L = Ψ − 1 ( I ). Let Φ b e defined a s ab ov e. T o sho w ( M ( L ) , ≤ L ) divides ( M , ≤ M ) w e find a sur- jectiv e morphism of ordered monoids Υ from a submonoid of ( M , ≤ M ) o n to ( M ( L ) , ≤ L ). 6 A s ubmonoid o f ( M , ≤ M ) is a submonoid of M with the order b eing the r estriction of ≤ M to the submonoid. 3.2. Ordered Monoids 48 W e let N := Ψ( M ) s o ( N , ≤ M ) is a sub monoid o f ( M , ≤ M ). Define Υ to b e the same function as the one we defined in the pro o f of Prop osition 3.3, so Υ is suc h that Υ : ( N , ≤ M ) → ( M ( L ) , ≤ L ), Ψ ( w ) 7→ Φ( w ), i.e. Φ = Υ ◦ Ψ. As sho wn b efore, Υ is a w ell-defined surjectiv e morphism. What remains to b e sho wn is that Υ is a morphism of ordered monoids. F or this, w e need to sho w Ψ( a ) ≤ M Ψ( b ) = ⇒ Υ(Ψ( a )) ≤ L Υ(Ψ( b )), i.e. Ψ( a ) ≤ M Ψ( b ) = ⇒ Φ( a ) ≤ L Φ( b ). Supp ose the ab ov e is not true. So there exists a and b suc h that Ψ( a ) ≤ M Ψ( b ) but Φ( a )  L Φ( b ). This means tha t a  L b and therefore there exists u, v ∈ Σ ∗ suc h that ubv ∈ L but uav / ∈ L . O n the o t her hand, since ≤ M is a stable order w e ha v e Ψ( uav ) = Ψ( u ) Ψ ( a )Ψ( v ) ≤ M Ψ( u )Ψ( b )Ψ( v ) = Ψ( u bv ) . ubv ∈ L implies that Ψ( ubv ) ∈ I and b y ab ov e and the f act that I is an order ideal w e m ust ha v e that Ψ( uav ) ∈ I . This is a con tra diction since uav / ∈ L . 3.2.3 V arieties The definition of an ordered monoid v ariet y is iden tical to the unor dered case. W e say that a family of ordered monoids V is a variety of or de r e d monoids if it is closed under division of ordered monoids and finite direct pro duct 7 . Similar to unordered monoid v ar ieties, v arieties of ordered monoids can b e defined using iden tities. W e say that ( M , ≤ M ) satisfies the iden t it y u ≤ v if and only if for ev ery morphism ϕ : Σ ∗ → M w e ha v e ϕ ( u ) ≤ M ϕ ( v ). Let V ( u, v ) b e the v ariety of ordered mo no ids that satisfy the iden tity u ≤ v . Then giv en a sequence of pair of words ( u n , v n ) n> 0 , W := lim V ( u n , v n ) is said to b e ultimately define d by this sequence . 7 The order in a finite dir ect product M 1 × ... × M n is giv en by ( m 1 , ..., m n ) ≤ ( m ′ 1 , ..., m ′ n ) iff m i ≤ m ′ i ∀ i ∈ [ n ]. 49 CHAPTER 3. Algebraic Automata Theory Theorem 3.11 ([PW96]) . Every variety of or der e d monoids is ultimately define d by some se quenc e of identities. No w w e define p ositiv e v ariet y of languages. A set o f languag es in Σ ∗ that is closed under finite inte rsection and finite union is called a p ositive b o ole an algebr a . So it differs from a b oolean algebra b ecause w e do not require the set to b e closed under complemen tation. A class of languages V is called a p ositive variety of l a nguages if it is a p ositiv e b o olean algebra, is closed under in v erse morphisms and is closed under left a nd right quotients. F or a giv en v ariet y o f finite ordered monoids V , let V (Σ) b e the set of languages ov er Σ whose syn ta ctic o r dered monoid b elongs to V . As b e fore, this is equiv alen t to saying that V (Σ) is the set of languages ov er Σ that are recognized by an ordered monoid in V . Theorem 3.12 (The V ariety Theorem [Pin95]) . V is a p ositive variety of languages and the mapping V 7→ V is one to one. No w w e giv e tw o explicit correspo ndences. The inte rested reader can find the pro ofs in [Pin95]. Let Γ b e a subset of the alphab et Σ. Define L (Γ) as L (Γ) := \ a ∈ Γ Σ ∗ a Σ ∗ . This is equiv alen t to sa ying that L (Γ) is the set of w o r ds that contain at least one o ccu rrence of eac h letter in Γ. A monoid is idemp oten t if ev ery elemen t in the mo no id is idemp oten t. Theorem 3.13. A language in Σ ∗ is a finite union of languages of the form L (Γ) for Γ ⊆ Σ if and only if it is r e c o gnize d by a finite c ommutative idem- p otent or der e d monoid ( M , ≤ M ) i n which the identity is the gr e atest elemen t with r esp e ct to the or der. 3.2. Ordered Monoids 50 A language L is a shuffle ide al if it satisfies the following prop erty: if a w o r d w has a sub word in L , then w is in L . Theorem 3.14. A language is a shuffle ide al if a n d only if it is r e c o gnize d by a finite or der e d monoid in wh i c h the identity is the gr e atest el e ment. W e conclude t his chapter by p oin ting out t hat our main interes t is in p os - itiv e v arieties of languages (and consequen tly in or dered monoids) b e cause regular la ng uages hav ing O ( f ) non- deterministic commu nication complexit y form a p os itiv e v ariety of languages (see next c hapter). F or the comm unica- tion mo dels studied in [TT03], regular languages ha ving O ( f ) comm unication complexit y form a v ariety of languages and so the theory of ordered monoids is not necessary . F rom no w on, w e abandon unor dered monoids and work with the more general theory of o rdered monoids. Chapter 4 Comm unication Complexi t y of Regular Languages The main goal of this chapter is to prov e upp er and low er b ounds on the non- deterministic comm unication complexit y of regular languag es. In Section 4.1 , w e formally define the comm unication complexit y of finite or dered monoids and regular languages. W e pro v e t w o theorems that establish the soundness of a n algebraic approach to the comm unication complexit y of regular lan- guages. In Section 4 .2, w e presen t a fo rm of the definition of rectangular reductions and in tro duce lo c al rectangular reductions. Then w e presen t up- p er a nd low er b ound results f o r regular languages in whic h the lo w er b o unds are established using rectangular reductions from three functions w e ha v e seen in Chapter 2. W e a lso state an in triguing conjecture that giv es an ex- act c haracterization of the non-deterministic comm unication complexit y of regular languages. 51 4.1. Algebraic Approach to Communic ation Complexit y 52 4.1 Algebraic Approac h to C omm unic ation Complexit y In Ch apter 2 , w e studied the communic ation complexit y of functions that ha v e 2 explicit inputs, each b eing an n -bit string. In order to define the comm unication complexit y of a monoid and a r egula r language, w e need to generalize the definition of communication complexit y to include functions that ha v e a single input string. Suppose a function f has one n - bit string x 1 ...x n as input and let A ∪ B b e a partition of [ n ]. Then the comm unication complexit y of f with resp ect to this partition is t he comm unication complex- it y of f when Alice receiv es the bits x i for all i ∈ A and Bob receiv es the bits x j for all j ∈ B . F or instance, in the non-deterministic mo del w e denote this b y N 1 AB ( f ). In this case, the no n-deterministic comm unication complexit y of f is defined as N 1 ( f ) := max A,B N 1 AB ( f ) where t he maxim um is tak en ov er all p ossible partitions of [ n ]. The partition that achie v es this maxim um is called a wors t c ase p artition . Note that the commun ication complexit y definitions and results seen th us far apply to functions that hav e inputs that are strings o f length n ov er a ny fixed alphab et. That is, the requiremen t of bit strings as inputs can b e relaxed. W e define the comm unication complexity o f a finite ordered monoid using the w orst-case partitioning notion. The comm unication complexit y of a pair ( M , I ) where M is a finite ordered monoid and I is an order ideal in M is the comm unication complexit y of the monoid ev aluation problem corr espo nding to M a nd I : Alic e is giv en m 1 , m 3 , ..., m 2 n − 1 and Bob is giv en m 2 , m 4 , ..., m 2 n suc h that eac h m i ∈ M . They w an t to decide if the pro duct m 1 m 2 ...m 2 n is in I . The comm unication complexit y of M is the maxim um comm unication complexit y of ( M , I ) where I ranges ov er all order ideals in M . Observ e that 53 CHAPTER 4. Comm unication Complexit y o f Regular Languag es if for example Alic e we re to rece iv e m i and m i +1 , then she could m ultiply these monoid elemen ts and treat them as one monoid eleme n t. This is why for a w orst-case partitio n, Alice and Bob should not get consecutiv e monoid elemen ts. Similarly , w e define the communic ation complexit y of a regular language L ⊆ Σ ∗ as the communic ation complexit y of the language problem corre- sp onding to L : Alice is give n a 1 , a 3 , ..., a 2 n − 1 and Bob is giv en a 2 , a 4 , ..., a 2 n suc h that eac h a i ∈ Σ ∪ { ǫ } where ǫ represen ts the empt y w ord in Σ ∗ (also referred to as the empt y letter). They w an t to determine if a 1 a 2 ...a 2 n ∈ L . The w a y the input is distributed corresp onds to the worst-case partition since w e a llow a i to b e empt y letters. As men tioned in Chapter 1 , our aim is to find functions f 1 ( n ) , ..., f k ( n ) suc h that eac h regular language has Θ( f i ( n )) non-deterministic communi- cation complexit y for some i ∈ { 1 , 2 , ..., k } . W e w ould also lik e a c harac- terization of the la nguages with Θ( f i ( n )) complexit y for all i ∈ { 1 , 2 , ..., k } . The next tw o results sho w that suc h a c haracterization can b e o btained b y lo oking a t t he algebraic prop erties of regular languages. Theorem 4.1. L et L ⊆ Σ ∗ b e a r e gular lan guage with M ( L ) = M . We have N 1 ( M ) = Θ( N 1 ( L )) . Theorem 4.2. F or any incr e asing function f : N → N , the c la ss of or der e d monoids V s uch that e ach m o noid M ∈ V satisfies N 1 ( M ) = O ( f ) forms a variety of or der e d mon o ids. These tw o theorems tog ether with the v ariet y t heorem imply that the class of languages V suc h that fo r a n y L ∈ V w e hav e N 1 ( L ) = O ( f ) forms a p o s- itiv e v ariety of languages. So a characterization in terms of p ositiv e v a rieties is p ossible. F urthermore, observ e that comm unication complexit y of monoids parametrize the comm unication complexit y of regular lang ua ges. Bo unds on monoids yield b ounds on regular languages and v ice v ersa. When pro ving 4.1. Algebraic Approach to Communic ation Complexit y 54 suc h b ounds, carefully choosing b etw een the tw o directions can considerably simplify the a na lysis. Us ually w e find that upp er b ound ar g umen ts are easier to establish with the com binatorial descriptions of a language whereas low er b ound argumen ts are easie r to establish with the algebraic desc riptions of the corresp onding syn tactic monoid. Pr o of of T he or em 4 . 1. First w e sho w that N 1 ( L ) = O ( N 1 ( M )). F or this, w e presen t a non-deterministic pro t o col for L . Supp ose Alice is giv en a 1 a 2 ...a n and Bob is given b 1 b 2 ...b n . Let Φ b e the syntactic mo r phism and let I b e t he accepting order ideal. The proto col is a s follows : Alice computes Φ( a 1 ) , ..., Φ( a n ) and Bob computes Φ( b 1 ) , ..., Φ( b n ). Using the proto col for the monoid ev aluation problem of ( M , I ) , they can decide at O ( N 1 ( M )) cost if Φ( a 1 )Φ( b 1 ) ... Φ( a n )Φ( b n ) = Φ( a 1 b 1 ...a n b n ) ∈ I . This determines if a 1 b 1 ...a n b n is in L or not. No w w e sho w that N 1 ( M ) = O ( N 1 ( L )). W e presen t a proto col for ( M , I ) where I = h i 1 , ..., i k i is some order ideal in M . Before presen ting the proto col, w e first fix some notation and definitions. Again let Φ b e the syn tactic morphism. F or eac h monoid elemen t m , fix a w o rd that is in the preimage of m under Φ, and denote it by w m . Let Y a := { ( u , v ) : uav ∈ L } . Recall that a  L b if for all u, v ∈ Σ ∗ , ubv ∈ L = ⇒ uav ∈ L . So Φ( a ) ≤ L Φ( b ) iff a  L b iff Y b ⊆ Y a . F or eac h Y a and Y b with Y b * Y a , pic k ( u, v ) such that ( u, v ) ∈ Y b but ( u, v ) / ∈ Y a . Let K b e the set of all these ( u, v ). One can t hink of K as con taining a witness for Y b * Y a for eac h suc h pair. Note that K is finite. No w pad eac h w m and eac h w ord app earing in a pair in K with the empt y letter ǫ so that eac h o f these words hav e the same length. Observ e t ha t this length is a cons tan t that do es not depend o n the length of the input that Alice a nd Bob will receiv e. 55 CHAPTER 4. Comm unication Complexit y o f Regular Languag es No w assuming that Alice and Bob hav e agreed up on the definitions made th us far, the proto col is as follows . Supp o se Alice is giv en m a 1 , m a 2 , ..., m a n and Bob is given m b 1 , m b 2 , ..., m b n . F or each i j they w a n t to determine if m a 1 m b 1 ...m a n m b n ≤ L i j . This is equiv alen t to determining if w m a 1 m b 1 ...m a n m b n  L w i j , and this is equiv alen t to w m a 1 w m b 1 ...w m a n w m b n  L w i j . If this is not the case, then Y w i j * Y w m a 1 w m b 1 ...w m a n w m b n and so there will be a witnes s of this in K , i.e. there exists ( u, v ) suc h that uw i j v ∈ L but uw m a 1 w m b 1 ...w m a n w m b n v / ∈ L . If indeed w m a 1 w m b 1 ...w m a n w m b n  L w i j then for each ( u, v ) ∈ K with u w i j v ∈ L , w e will hav e uw m a 1 w m b 1 ...w m a n w m b n v ∈ L . Using t he proto c ol for L , Alice a nd Bob can c hec k which o f the tw o cases is true. The fo llowing sho ws ho w Alice and Bob’s inputs lo ok like b efore running the proto col fo r L . Note that each blo c k has the same constant length. Alice Bob u w m a 1 ǫǫ...ǫ ... w m a n ǫǫ...ǫ v ǫǫ...ǫ ǫǫ...ǫ w m b 1 ... ǫǫ...ǫ w m b n ǫǫ...ǫ The pro of of Theorem 4.2 follo ws from the follow ing t w o lemmas. The first lemma sho ws that V is closed under finite direct pro duct. The second lemma show s that V is closed under division of monoids. Lemma 4.3. L et ( M , ≤ M ) and ( N , ≤ N ) b e or d e r e d monoids . Then N 1 ( M × N ) ≤ N 1 ( M ) + N 1 ( N ) . Pr o of. Any order ideal in M × N will b e of the form I × J where I is an order ideal in M and J is an o r der ideal in N . Therefore testing whether a pro duct of elemen ts in M × N is in an order ideal I × J or no t can b e done b y testing if t he pro duct of the first co or dina t e elemen ts is in I and t esting if the pro duct of the second co ordinate elemen ts is in J . 4.2. Complexit y Bounds for Regular Languages and Monoids 56 Lemma 4.4. L et ( M , ≤ M ) and ( N , ≤ N ) b e or der e d monoids such that N ≺ M . Then N 1 ( N ) ≤ N 1 ( M ) . Pr o of. Since N ≺ M , there is a surjectiv e morphism φ from a submonoid M ′ of M on to N . Denote b y φ − 1 ( n ) a fixed elemen t from the preimage o f n . Let I b e an order ideal in N . A proto col for ( N , I ) is as follo ws. Alice is giv en n a 1 , n a 2 , ..., n a t and Bob is giv en n b 1 , n b 2 , ..., n b t . They w ant to decide if n a 1 n b 1 ...n a t n b t ∈ I . This is equiv alen t to deciding if φ − 1 ( n a 1 ) φ − 1 ( n b 1 ) ...φ − 1 ( n a t ) φ − 1 ( n b t ) ∈ φ − 1 ( I ) . It can b e easily seen that φ − 1 ( I ) is a n or der ideal in M ′ so Alice and Bob can use the proto col fo r M ′ to decide if the ab ov e is true. Therefore w e ha v e N 1 ( N ) ≤ N 1 ( M ′ ). It is straigh tforw ard to c hec k that N 1 ( M ′ ) ≤ N 1 ( M ) and so N 1 ( N ) ≤ N 1 ( M ) as required. 4.2 Complexit y Boun d s for Regular Languages and Monoids In this section, w e presen t upp er and low er b ounds f o r the non- deterministic comm unication complexit y of certain classes of langua ges. Upp er b ounds are established b y presen ting a n appropriate proto col whereas lo wer b ound argumen ts are based on rectangular reductions from the follo wing functions: LESS-THAN, PR OMISE-DISJOINTNESS, INNER-PRODUCT. In Chapter 2, w e hav e seen that eac h of these functions require linear comm unication in the non-deterministic mo del. W e ha v e also seen the definition o f a rectangular reduction. W e g iv e here a fo rm of this definition whic h sp ecifically suits our needs in this section. Definition 4.5. Let f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , M a finite ordered monoid and I an order ideal in M . A r e c tan g ular r e duction of length t 57 CHAPTER 4. Comm unication Complexit y o f Regular Languag es from f to ( M , I ) is a sequence of 2 t functions a 1 , b 2 , a 3 , ..., a 2 t − 1 , b 2 t with a i : { 0 , 1 } n → M and b i : { 0 , 1 } n → M and suc h tha t for eve ry x, y ∈ { 0 , 1 } n w e hav e f ( x, y ) = 1 if and o nly if the pro duct a 1 ( x ) b 2 ( y ) ...b 2 t ( y ) is in I . Suc h a reduction transforms an input ( x, y ) of the function f in to a se- quence of 2 t monoid elemen ts m 1 , m 2 , ..., m 2 t where the o dd-index ed m i are obtained as a function of x only and the ev en-indexed m i are a function of y . W e write f ≤ t r ( M , I ) to indicate tha t f has a rectangular reduction of length t to ( M , I ). When t = O ( n ) w e omit the sup erscript t . It should b e clear that if f ≤ t r ( M , I ) and f ha s communication complexit y Ω( g ( n )), then ( M , I ) has comm unicatio n complex it y Ω( g ( t − 1 ( n ))). Most of the reductions w e use here are sp ecial kinds of rectangular reduc- tions. W e call these reductions lo c al r e ctangular r e d uctions . In a lo cal rectan- gular reduction, Alice conv erts each bit x i to a sequence of s monoid elemen ts m a i, 1 , m a i, 2 , ..., m a i,s b y applying a fixed function a : { 0 , 1 } → M s . Similarly Bob con v erts eac h bit y i to a sequence of s monoid elemen ts m b i, 1 , m b i, 2 , ..., m b i,s b y applying a fixed function b : { 0 , 1 } → M s . f ( x, y ) = 1 if and only if m a 1 , 1 m b 1 , 1 ...m a 1 ,s m b 1 ,s ......m a n, 1 m b n, 1 ...m a n,s m b n,s ∈ I W e often view the ab o v e pro duct as a w ord o ver M . The reduction t r ansforms an input ( x, y ) into a sequence of 2 sn monoid elemen ts. Let a ( z ) k denote the k th co ordinate of the tuple a ( z ). W e sp ecify this kind of lo cal transformation with a 2 × 2 s matrix: a (0) 1 b (0) 1 ... ... a (0) s b (0) s a (1) 1 b (1) 1 ... ... a (1) s b (1) s . It is conv enien t to see what happ ens for all p ossib le v alues o f x i and y i and the follo wing table sh o ws the w ord that cor r espo nds to these p ossibilities. 4.2. Complexit y Bounds for Regular Languages and Monoids 58 F or simplicit y let us assume s is ev en. x i y i corresp onding w ord 0 0 a (0) 1 b (0) 1 ...a (0) s b (0) s 0 1 a (0) 1 b (1) 1 a (0) 2 b (1) 2 ...a (0) s b (1) s 1 0 a (1) 1 b (0) 1 a (1) 2 b (0) 2 ...a (1) s b (1) s 1 1 a (1) 1 b (1) 1 ...a (1) s b (1) s No w w e are ready to presen t the upp er and lo w er b ound results. Lemma 4.6. If M is c o m mutative then N 1 ( M ) = O (1) . Pr o of. Let I be an order ideal in M . Supp ose Alice is giv en m a 1 , ..., m a n and Bob is giv en m b 1 , ..., m b n . They wan t to decide if m a 1 m b 1 ...m a n m b n ∈ I . Since M is comm uta tiv e, t his is equiv alen t to determining if m a 1 m a 2 ...m a n m b 1 m b 2 ...m b n ∈ I . So Alice can priv at ely compute the pro duct m a 1 ...m a n and send the result m to Bob. Observ e that this requires a constan t n um b er of bits to b e comm u- nicated since the size of M is a constant. Bob can c hec k if mm b 1 ...m b n ∈ I and send the o ut come to Alice. Lemma 4.7. If M is not c ommutative then for any or der on M we have N 1 ( M ) = Ω(log n ) . Pr o of. Since M is not commutativ e, there must b e a, b ∈ M suc h that ab 6 = ba . Therefore either ab  M ba or ba  M ab . Without loss of generality assume ba  M ab . Let I = h ab i . W e sho w that LT ≤ 2 n r ( M , I ). Alice gets x and constructs a sequence of 2 n monoid elemen ts in whic h a is in p o sition x and 1 M is in ev erywhere else. Bob gets y and constructs a sequence of 2 n monoid elemen ts in whic h b is in p osition y and 1 M is in ev erywhere else. If x ≤ y then the pro duct of the monoid elemen ts will b e ab whic h is in I . If x > y then the pro duc t will b e ba which is not in I . Denote b y C om t he p ositiv e language v ariety corresp onding to the v ariety of comm uta tiv e monoids Com . The ab o v e t w o results sho w that regular 59 CHAPTER 4. Comm unication Complexit y o f Regular Languag es languages that hav e constant non-deterministic communic ation complexit y are exactly those languages in C om . The next step is to determine if t here are regular languages outside of C om that ha v e O (log n ) non-deterministic complexit y . F or this, w e first need the definition of a p olyn omial closure. The p olynomial closur e of a set o f languages L in Σ ∗ is a f amily of lan- guages such that each of these languages are finite unions of languages of the form L 0 a 1 L 1 ...a k L k where k ≥ 0, a i ∈ Σ and L i ∈ L . If V is a v ariet y of la nguages, then w e denote b y P ol ( V ) the class of languages suc h tha t for ev ery alphab et Σ, P ol ( V )(Σ) is the p olynomial closure of V (Σ). P ol ( V ) is a p ositiv e v ariety of languages ( [PW95]) . Lemma 4.8. If L is a language of P ol ( C om ) then N 1 ( L ) = O (lo g n ) . Pr o of. Supp ose L is a union of t languages of the form L 0 a 1 L 1 ...a k L k . Alice and Bob kno w b eforehand the v alue of t and the structure o f eac h of these t languages. So a proto col for L is as follows . Assume Alice is giv en x a 1 , ..., x a n and Bob is given x b 1 , ..., x b n . Go d com- m unicates to Alice and Bob whic h of the t languages the w ord x a 1 x b 1 ...x a n x b n resides in. Th is r equires a constan t n um b er of bits to b e comm unicated since t is a constan t. Now t hat Alice and Bob kno w the L 0 a 1 L 1 ...a k L k the w ord is in, G o d communic ates the p ositions of each a i . This requires k log n bits of comm unicatio n where k is a constant. The v alidit y of the information comm unicated by Go d can b e immediately c heck ed by Alice and Bob. All they ha v e to do is c hec k if the w o rds in b et w een the a i ’s b elong to the righ t languages. Since these languages are in C om , this can b e done in constan t comm unication as prov ed in Lemma 4.6. Therefore in total w e require only O (log n ) comm unication. 4.2. Complexit y Bounds for Regular Languages and Monoids 60 F rom the ab o v e pro of, w e see that we can actually afford to comm unicate O (log n ) bits to chec k that the words betw een the a i ’s b elong to the corre- sp onding language. In other w ords, we could ha v e L i ∈ P ol ( C om ). Note that this do e s not matter since P ol ( P ol ( C om )) = Pol ( C om ). Denote b y  x L 0 a 1 L 1 ...a k L k  the n um b er of factorizations of the w ord x as x = w 0 a 1 w 1 ...a k w k with w i ∈ L i . When the a i and the L i are suc h that for any x w e ha v e  x L 0 a 1 L 1 ...a k L k  ∈ { 0 , 1 } , then we sa y that the concate- nation L 0 a 1 L 1 ...a k L k is unambig uous . W e denote by U P ol ( V ) the v ari- et y of languages that is disjo int unions o f the unam biguous concatenations L 0 a 1 L 1 ...a k L k with L i ∈ V (in some sense, there is only o ne witness for x in U P ol ( V )). Similarly w e denote b y M p P ol ( V ) the langua g e v ariet y generated b y the languages { x |  x L 0 a 1 L 1 ...a k L k  = j mo d p } for some 0 ≤ j ≤ p − 1 and L i ∈ V . Observ e that for P ol ( C om ) we ha v e  x L 0 a 1 L 1 ...a k L k  unrestricted. Denote b y U P the sub class of N P in whic h the num b er of accepting paths (or num b er of witnesses) is exactly one. W e kno w tha t U P cc = P cc ([Y an91]). F rom [TT03] w e kno w that regular langua g es ha ving O (log n ) deterministic comm unication complexit y are exactly those languages in U P ol ( C om ) and regular languages ha ving O (log n ) Mo d p coun t ing comm unicatio n complexity are exactly tho se languages in M p P ol ( C om ). F urthermore, it was shown that any regular language o ut side of U P ol ( C om ) has linear deterministic complexit y and any regula r language outside of M p P ol ( C om ) has linear Mo d p coun t ing complexit y . So with resp ect to regular languages, U P cc = P cc = U P ol ( C om ) and M od p P cc = M p P ol ( C om ). Similarly w e conjecture that with resp ect to regular languages N P cc = P ol ( C om ). Conjecture 4.9. If L ⊆ Σ ∗ is a r e gular language that is not in P ol ( C om ) , 61 CHAPTER 4. Comm unication Complexit y o f Regular Languag es then N 1 ( L ) = Ω( n ) . T hus we have N 1 ( L ) =        O (1) if and only i f L ∈ C om ; Θ(log n ) if an d only if L ∈ P ol ( C om ) but not in C o m ; Θ( n ) otherwise. As men tio ned in Chapter 2, the gap b etw een deterministic a nd no n- deterministic communic ation complexit y of a function can be exp onen tially large. How ev er, it has b een sho wn that the deterministic communic ation complexit y of a function f is b ounded ab ov e b y t he pro du ct c N 0 ( f ) N 1 ( f ) for a constan t c (Theorem 2.9), and that this b ound is optimal. The ab o v e conjecture, together with the result of [TT03 ] implies the following m uch tigh ter relatio n for regular langua ges. Corollary 4.10 (to Conjecture 4.9 ) . If L is a r e gular language then D ( L ) = max { N 1 ( L ) , N 0 ( L ) } . F or any v ariety V , we hav e that P ol ( V ) ∩ co − P ol ( V ) = U P ol ( V ) ([Pin97]). This implies that N 1 ( L ) = O ( lo g n ) and N 0 ( L ) = O ( lo g n ) iff D ( L ) = O (log n ), proving a sp ecial case of the ab ov e corollary . An imp ortan t question that arises in this context is the following. What do es it mean to b e o utside o f P ol ( C om )? In order to prov e a linear lo w er b ound for the regular languages outside of P ol ( C om ), w e need a con venie n t algebraic description for the syn tactic monoids of these languages since (ig- noring the exceptions) lo w er b ound argumen ts rely on these algebraic pro p- erties. One suc h description exists based on a result o f [PW95] that describ es the ordered mo no id v ariet y corresp onding to P ol ( C om ). Befo r e stating this description, we fix some notatio n. If M is a monoid, we write M = h G, R i to indicate that M has the presen tation h G, R i where G is the generating set and R is the set of relations. F or instance, a cy clic g roup of order n has the presen tation h{ x } , x n = 1 i 4.2. Complexit y Bounds for Regular Languages and Monoids 62 and the dihedral group of order 2 n has the presen tation h{ x, y } , x n = 1 , y 2 = 1 , xy x = y i . F or an y w ∈ G ∗ , w e denote by ev al ( w ) the elemen t of M that w corresp o nds to. Observ e that the transformation monoid corresp onding to an a utomaton has a presen tatio n in which the generating set consists of the letters of the alpha b et. The relat io ns dep end on the par t icular automaton and can b e determined b y analyzing the state transition function eac h w ord induces. Lemma 4.11. Supp ose L is not in P ol ( C om ) and M = h G, R i is the syntac- tic or der e d monoid of L w ith exp one nt ω . Then ther e exists u , v ∈ G ∗ such that (i) for any monoid M ′ ∈ C om and any morphism φ : M → M ′ , we have φ ( ev al ( u ) ) = φ ( ev al ( v )) and φ ( ev al ( u )) = φ ( ev al ( u 2 )) , (ii) ev al ( u ω v u ω )  ev al ( u ω ) . Although w e cannot y et prov e the conjecture, we can still sho w linear lo w er bo unds for certain classes o f regular languages out side of P ol ( C om ). Our first low er b ound captures regular langua ges that come v ery close to the description g iv en in the previous lemma. A word w is a shuffle o f n w ords w 1 , ..., w n if w = w 1 , 1 w 2 , 1 ...w n, 1 w 2 , 1 w 2 , 2 ...w n, 2 w 1 ,k w 2 ,k ...w n,k with k ≥ 0 and w i, 1 w i, 2 ...w i,k = w i is a partition of w i in to sub w ords fo r 1 ≤ i ≤ n . Lemma 4.12. If M = h G, R i and u, v ∈ G ∗ is such that (i) u = w 1 w 2 for w 1 , w 2 ∈ G ∗ , (ii) v is a shuffle of w 1 and w 2 , 63 CHAPTER 4. Comm unication Complexit y o f Regular Languag es (iii) ev al ( u ) is an ide mp otent, (iv) ev al ( uv u )  ev al ( u ) , then N 1 ( M ) = Ω( n ) . Observ e that the conditions of this lemma imply t he conditions of Lemma 4.11: since ev al ( u ) is idemp oten t, for a n y monoid M ′ ∈ Com and a n y morphism φ : M → M ′ , w e ha v e φ ( ev al ( u )) = φ ( ev al ( u 2 )) and since v is a sh uffle of w 1 and w 2 w e ha ve φ ( ev al ( u ) ) = φ ( ev al ( v )). Also, since ev al ( u ) is idemp oten t, ev al ( u ω ) = ev al ( u ), and in this case ev al ( uv u )  ev al ( u ) is equiv alen t to ev al ( u ω v u ω )  ev al ( u ω ). Pr o of of L emma 4.12. W e sho w tha t P D I S J ≤ r ( M , I ) where I = h ev al ( u ) i . Since v is a sh uffle of w 1 and w 2 , there exists k ≥ 0 suc h that v = w 1 , 1 w 2 , 1 w 1 , 2 w 2 , 2 ...w 1 ,k w 2 ,k . The reduction is essen tially linear and is giv en by the follo wing matrix when k = 3. The transformatio n easily generalizes to any k . w 1 ǫ ǫ ǫ ǫ w 2 , 1 ǫ w 2 , 2 ǫ w 2 , 3 w 1 , 1 w 2 , 1 w 1 , 2 w 2 , 2 w 1 , 3 w 2 , 3 ǫ ǫ ǫ ǫ . x i y i corresp onding w ord 0 0 w 1 w 2 , 1 w 2 , 2 w 2 , 3 = u 0 1 w 1 w 2 , 1 w 2 , 2 w 2 , 3 = u 1 0 w 1 , 1 w 1 , 2 w 1 , 3 w 2 , 1 w 2 , 2 w 2 , 3 = u 1 1 w 1 , 1 w 2 , 1 w 1 , 2 w 2 , 2 w 1 , 3 w 2 , 3 = v After x and y ha v e b een transformed in to w ords, Alice prep ends her w ord with u and app ends it with | u | man y ǫ ’s, where | u | denotes the length of the word u . Bob prep ends his word with | u | many ǫ ’s and app ends it with u . Let a ( x ) b e the w o rd Alice has and let b ( y ) b e the w o r d Bob 4.2. Complexit y Bounds for Regular Languages and Monoids 64 a b b a a, b Figure 4 .1: The minimal automaton recognizing the la ng uage whose syn tactic ordered monoid is B A + 2 . has after these transfor ma t io ns. Observ e that if P D I S J ( x, y ) = 0, then there exists i suc h tha t x i = y i = 1. By the transformation, this means that a ( x ) 1 b ( x ) 1 a ( x ) 2 b ( x ) 2 ...a ( x ) s b ( x ) s is of the f orm u...uv u...u and since ev al ( u ) is idemp otent, ev a l ( a ( x ) 1 b ( x ) 1 a ( x ) 2 b ( x ) 2 ...a ( x ) s b ( x ) s ) = ev al ( uv u )  ev al ( u ). On the other hand if P D I S J ( x, y ) = 1, then b y the transforma t ion, a ( x ) 1 b ( x ) 1 a ( x ) 2 b ( x ) 2 ...a ( x ) s b ( x ) s is of the fo rm u ...u and so ev al ( a ( x ) 1 b ( x ) 1 a ( x ) 2 b ( x ) 2 ...a ( x ) s b ( x ) s ) = ev al ( u ) . The abov e result giv es us a corollar y ab out the monoid B A + 2 whic h is defined to b e the syn ta ctic or dered monoid o f the regular la nguage recognized b y the automaton in Figure 4.1. The unordered syn ta ctic monoid o f the same language is denoted b y B A 2 and is kno wn a s the Brandt monoid (see [Pin97 ]). Corollary 4.13. N 1 ( B A + 2 ) = Ω( n ) . Pr o of. It is easy to v erify by lo oking at the transformation monoid of the automaton that B A + 2 = h{ a, b } , aa = bb, aab = aa, baa = aa, aaa = a, aba = 65 CHAPTER 4. Comm unication Complexit y o f Regular Languag es a, bab = b i . The only thing w e need to kno w ab out the order relation is that ev al ( aa ) is greater than an y other elemen t. This can be deriv ed from the definition of the syntactic ordered monoid (Subs ection 3.2.2) since for any w 1 and w 2 , w 1 aaw 2 is not in L . So w 1 aaw 2 ∈ L = ⇒ w 1 xw 2 ∈ L trivially holds f or an y word x . Let u = ab and v = ba . These u and v satisfy the four conditions of the previous lemma. The last condition is satisfied b e cause ev al ( uv u ) = ev al ( abbaab ) = ev al ( aa ) and ev al ( ab ) 6 = ev al ( aa ). Therefore N 1 ( B A + 2 ) = Ω( n ). Denote b y U − the syn tactic ordered monoid of the regular language ( a ∪ b ) ∗ aa ( a ∪ b ) ∗ , and denote b y U the unordered syn tactic monoid. Also let U + b e the syn tactic ordered monoid of the complemen t of ( a ∪ b ) ∗ aa ( a ∪ b ) ∗ . Observ e tha t N 1 ( U − ) = O ( lo g n ) since all w e need to do is c hec k if there are t w o consecutiv e a ’s. By an argumen t similar to the one for Corollary 4.13, one can sho w that N 1 ( U + ) = Ω( n ). Our next linear lo w er b ound result is for non- comm utativ e groups. Lemma 4.14. If M is a n on-c ommutative gr oup then N 1 ( M ) = Ω( n ) . Pr o of. Since M is non-commutativ e, there exists a, b ∈ M suc h that the comm uta tor [ a, b ] = a − 1 b − 1 ab 6 = 1. This means that [ a, b ] has order q > 1. Let m ∈ M b e suc h that there is no m ′ ∈ M with m ′ 6 = m and m ′ ≤ m . Denote by I the order ideal t ha t just con tains m . There is a reduction from I P q to ( M , I ). The reduction is esse n tially lo cal. Alice and Bob will apply the tra nsfor ma t io n giv en b y t he following matrix. 1 1 1 1 a − 1 b − 1 a b . 4.2. Complexit y Bounds for Regular Languages and Monoids 66 x i y i corresp onding w ord 0 0 1 0 1 b − 1 b = 1 1 0 a − 1 a = 1 1 1 a − 1 b − 1 ab After, Alice will a pp end m to her tra nsformed input and Bob will app end 1 to his. Observ e tha t the pro duct of the monoid elemen ts ev aluates to m if and o nly if P 1 ≤ i ≤ n x i y i ≡ 0 mo d q i.e. t he pro duct is ≤ m if and only if P 1 ≤ i ≤ n x i y i ≡ 0 mo d q . T o obtain our last linear low er b ound result, w e need the fo llo wing fact. Prop osition 4.15. A ny stabl e or der define d on a gr oup G m ust b e the trivial or der ( e quality). Pr o of. Supp ose the claim is false. So there exis ts a, b ∈ G suc h tha t a 6 = b and a ≤ b . This implies 1 ≤ a − 1 b =: g . If 1 ≤ g then g ≤ g 2 , g 2 ≤ g 3 and so on. Therefore w e hav e 1 ≤ g ≤ g 2 ≤ ... ≤ g k = 1 . This can only b e true if 1 = g , i.e. a = b . W e say that M is a T q monoid if there exists idemp oten ts e, f ∈ M suc h that ( ef ) q e = e but ( ef ) r e 6 = e when q do es not divide r . Lemma 4.16. If M is a T q monoid for q > 1 then N 1 ( M ) = Ω( n ) . Pr o of. Observ e tha t { e, ef e, ( ef ) 2 e, ..., ( ef ) q − 1 e } forms a subgroup with iden- tit y e b ecause since e is idemp otent, w e ha v e ( ef ) i e · ( ef ) j e = ( ef ) i + j e . Therefore an y order on M m ust induce an equality order o n this set. Let I = h e i . W e sho w I P q ≤ r ( M , I ) via the follow ing lo cal reduction. e ( ef ) q ( ef ) q e e f e . 67 CHAPTER 4. Comm unication Complexit y o f Regular Languag es x i y i corresp onding w ord 0 0 e ( ef ) q ( ef ) q e = e 0 1 e ( ef ) q f e = e 1 0 e ( ef ) q e = e 1 1 ef e Observ e that the pro duct of the monoid elemen ts ev aluates to ( ef ) P 1 ≤ i ≤ n x i y i ≡ 0 mo d q e, whic h is equal to e if a nd only if I P q ( x, y ) = 1. Com bining our linear low er b ound results together with Lemma 4.4, w e can conclude the follo wing. Theorem 4.17. If M is a T q monoid for q > 1 or is divid e d by one of B A + 2 , U + or a n o n-c ommutative gr o up, then N 1 ( M ) = Ω( n ) . W e underline the relev ance of the ab ov e result by stating a theorem whic h w e b orro w f rom [TT05]. Theorem 4.18. If M is s uch that D ( M ) 6 = O (log n ) then M is either a T q monoid for some q > 1 or is divid e d by one of B A 2 , U or a non-c ommutative gr oup. The three linear lo w er b ound results imply the following result, whic h giv es us three sufficien t conditions for not b eing in P ol ( C om ). Theorem 4.19. L e t L b e a r e gular language with syntactic or der e d monoid M = h G, R i . If one of the fo l lowing hold s, then L is not in P ol ( C om ) . 1. The r e exists u, v ∈ G ∗ such that u = w 1 w 2 , v is a s h uffle of w 1 and w 2 , ev al ( u ) is an i d emp otent and ev al ( uv u )  ev al ( u ) . 2. M is divide d by a non-c ommutative gr oup. 4.2. Complexit y Bounds for Regular Languages and Monoids 68 3. M is a T q monoid for q > 1 . In particular, if M ( L ) is a T q monoid or is divided b y o ne of B A + 2 , U + or a non-commutativ e gr o up, then L is not in P ol ( C om ). Chapter 5 Conclusion The fo cus of this thesis has b een the non-deterministic comm unication com- plexit y of regular languag es. Regular langua g es are, in some sense, the sim- plest languages with resp ect to the usual time/space complexity framew ork, but in the communic ation complexit y mo del, they require a no n- trivial study as there are complete regular languages for ev ery level of the comm unication complexit y p olynomial hierarc h y . This fact can b e deriv ed from the results in [Bar86] and [BT87]. In [TT03], a complete characterization of the comm uni- cation complexit y of regular la nguages w as established in the deterministic, sim ultaneous, probabilistic, sim ultaneous probabilistic and Mo d p -coun ting mo dels. In order to get a similar algebraic c haracterization fo r the non- deterministic mo del, one needs the notion o f ordered monoids, a more general theory than the one used in [TT03], to b e able to deal with classes of lan- guages tha t are not closed under complemen tation. This thesis presen ts the fundamen ta ls of comm unication complexit y , monoid theory as w ell as ordered monoid theory and obtains b ounds on the non-deterministic comm unication complexit y of regular languages. Our results constitute the first steps tow ards a complete classification for the non- deterministic comm unication complexit y o f regular la nguages. W e 69 70 kno w exactly whic h r egula r languages hav e constan t non-deterministic com- m unication complexit y . W e know that there is a considerable complexit y gap b et w een t ho se languages having constant no n- deterministic complexit y and the rest of the regular languag es since if a regula r la nguage do es not hav e constan t complexit y t han it ha s Ω(log n ) complexit y . W e also obta in three linear lo w er b ound results and the imp ortance of these results are highlighted b y Theorem 4.18. These results also pro vide us with sev eral sufficien t condi- tions f o r not b eing in the v ariety P ol ( C om ), whic h is a result v ery in teresting from an algebraic automata theory p oin t of view. This r esult also exemplifies ho w computational complexit y can b e use d to ma ke progress in semigroup theory . Our ultimate ob jectiv e is to get a complete c haracterization of the non- deterministic communic ation complexit y o f regular languag es. W e conjecture that regular languages in P ol ( C om ) are the only languages that ha ve O (log n ) complexit y and any other regular language m ust hav e Ω( n ) complexit y . The linear low er bo und argument presen ts a real c hallenge. A natural next step to take is to explicitly find a regular language t ha t is not in P ol ( C om ) for whic h our current linear lo wer b ound arguments do not apply and try to either prov e a linear low er b ound fo r this sp ecific language or show that it requires O ( n ǫ ) complexit y for a constan t ǫ < 1 ( which would dispro v e our conjecture). A linear low er b ound argumen t for this lang ua ge is lik ely to apply t o some o t her languages outside of P ol ( C om ), if not all. The regular language recognized b y the automaton in Figure 5.1 is an example of a regular language that is outside of P ol ( C om ) and for whic h w e cannot get a linear lo w er b ound nor a sublinear upp er b ound. W e call this language L 5 . In t he App endix, we prov e that L 5 is not in P ol ( C om ) and v arious other facts ab out L 5 . An inte resting prop ert y of L 5 is that it b elongs to P ol ( N il 2 ) where N il 2 denotes the v ariet y of lang uages that corresp o nd to the v ariet y of nilpo t en t 71 CHAPTER 5. Conclusion a, b a a b b a, b a b 1 2 3 4 5 Figure 5.1: An auto ma t o n recognizing a language L 5 outside of P ol ( C om ). groups of class 2. Nilp oten t groups of class 2 are usually considered as “al- most” commutativ e g r o ups. In some sense, this says that ev en though L 5 is not in P ol ( C om ), it is v ery “close” to it. W e prop ose sev eral intuitiv e reasons of wh y prov ing a linear lo w er b o und for this regular language can b e c hallenging (assuming that the linear low er b ound is indeed true). W e also suggest p o ssible a pproac hes to ov ercome the difficulties. All of these tie with the imp o rtance of the problem w e are studying from a comm unication complexit y p oint of view as w ell as from a semigroup theory p oin t of view. First of all, from Chapter 2 w e kno w that the b est low er b ound tec hnique w e hav e for non-determinism is the rectangle size metho d. Inheren t in this metho d is the requiremen t to find the b es t p ossible distribution. Needless to say , this can b e quite hard. And ev en if the b es t distribution was know n, b ounding the size of an y 1-mono chromatic rectangle can b e a non-trivial task. Putting these tw o things together, the rectangle size metho d do es not seem to considerably simplify our task of b ounding the size of the optimum co v ering of the 1 -inputs. 72 Consider the set S of all functions hav ing Ω( n ) non-deterministic com- m unication complexit y . Define an equiv alence relation on these functions : f ≡ g if there is a rectangula r reduction of length O ( n ) from f to g and from g to f . W e can turn S/ ≡ in to a pa rtially o rdered set (p oset) b y defining the order [ f ] < [ g ] if there is a rectangular reduction of length O ( n ) from f to g . It certainly w ould not b e surprising if there w ere regular languages app earing in the low er lev els of a c ha in in this p oset and this w ould suggest that obtaining a lo w er b ound for these languages can b e difficult. If the ab o v e is indeed true, then what can b e done ab o ut this? A natural step would b e to find functions tha t are at t he same lev el or b elow the regular language at hand, and try to get a reduction that w ould prov e the language has linear non-deterministic complexit y . This raises our intere st in promise functions. Let f b e a b o olean function with the domain { 0 , 1 } n × { 0 , 1 } n . A promise function P f is a function that has a domain D that is a strict subset of f ’s domain and is suc h tha t for any ( x, y ) ∈ D , P f ( x, y ) = f ( x, y ). An example of a promise function is the PROMISE -DISJOINTNESS function, P D I S J . Promise functions are intere sting b ecause through a pro mise, w e can define functions that reside in the low er lev els of a c ha in. This in r eturn can ma ke a reduction p ossible from the promise function to the regular language o f in ter- est. F or instance, P D I S J is a promise function whic h lies b elow D I S J a nd I P q (Example 2.5). Of course an imp ortant p oin t when defining a promise function is that w e need the pro mise function to hav e Ω( n ) complexit y . In some sense, through the pr o mise, we w ould lik e t o eliminate the easy in- stances a nd k eep the instances that mak e the function hard. A t first, there migh t b e no reason to b eliev e that obtaining a linear lo w er b ound f or the promise function is an y easier than obtaining a lo w er b ound for the regular language. Nev ertheless, the purp o se of this line o f attack is the following. By putting a promise on a w ell-known , w ell-studied function (t ha t make s a 73 CHAPTER 5 . Conclusion reduction p os sible), we ma y b e able to utilize (or impro v e) the v arious tech- niques and ideas dev elop ed f or the analysis of the original function to pro v e a low er b ound on the promise function. No w we define a promise function, PR OMISE-INNER-PR ODUCT ( P I P 2 ), suc h that there is a reduction fro m this function to L 5 (see App endix). P I P 2 is the same f unction as I P 2 but has a restriction on the ( x, y ) for whic h I P 2 ( x, y ) = 0. W e only allow the 0 -inputs whic h satisfy the follow ing tw o conditions: ∀ i, x i = 0 and y i = 1 = ⇒ I P 2 ( x 1 ...x i − 1 , y 1 ...y i − 1 ) = 0 ∀ i, x i = 1 and y i = 0 = ⇒ I P 2 ( x 1 ...x i − 1 , y 1 ...y i − 1 ) = 1 It remains an o p en pro blem to prov e a line ar lo w er b ound, or a sublinear upp er b ound on P I P 2 . The fa ct is that little is kno wn ab out pr o mise functions. One promise function we kno w of is P D I S J . As a consequence of the celebrated work of Razb orov ([Raz92]), whic h sho ws that the distributional comm unication com- plexit y of the DISJOINTNESS function is Ω( n ), w e kno w that N 1 ( P D I S J ) = Ω( n ) as w ell. Giv en t he description of regular languages outside of P ol ( C om ) (Lemma 4.11), P D I S J is one of t he first functions one tries to get a reduc- tion from, where the reduction is a s in the pro of of Lemma 4.12. This hop e is h urt by the fa ct that suc h a reduction do es not exist from P D I S J to L 5 (see App e ndix). W e b eliev e that more att ention should b e giv en to promise functions since the study of t hese functions is lik ely to fo r ce us to deve lop new tec hniques in comm unication complexit y and give us more insigh t in this area. F urther- more, give n the connection of comm unication complexit y with many other areas in computer science, pro mise functions a r e b ound to hav e useful appli- cations. F or instance , in a v ery recen t w ork of G´ al and Gopalan ([GG07]), 74 comm unication complexit y b ounds fo r a promise function is used to pro v e b ounds o n streaming algorithms. W e hav e lo oke d at our question f rom a comm unication complexit y p er- sp ectiv e. Now w e lo ok at it from a semigroup theory p ersp ectiv e. The ke y to making progress on our question can b e finding a more conv enien t description of what it means to b e outside of P ol ( C om ). The description that w e hav e (Lemma 4.11) a ctually applies to P ol ( V ) for any v ariety V , and it is based on a complicated result of [PW95] that mak es use of a deep com binatorial result of semigroup theory ([Sim89],[Sim90],[Sim92 ]). Since w e are only in terested in P o l ( C om ) where C om is a relatively simple v ariet y , it ma y b e p ossible to obtain a more useful description that allows us to show comm unication complexit y bounds. W e conclude that, in a n y case, the resolution of our question will probably lead to a dv ances in either comm unication complexit y or semigroup theory , if not b o t h. App endix A F acts Ab out L 5 a, b a a b b a, b a b 1 2 3 4 5 In this app endix, w e pro ve some of the facts a b out the regula r language L 5 that we mentioned in Chapter 5. W e start with the fact that L 5 is not in P ol ( C om ). F or this, w e need a result tha t describ es the ordered monoid v ariet y corresp onding to P ol ( C om ). This description in v olv es the Mal’cev pro duct and top ological issues whic h w e c ho os e to av oid for simplicit y . The in terested reader can find the neces sary information ab out these in [Pin97 ]. Here w e will s tate a r estricted v ersion of this result whic h suffices for o ur needs. Lemma A.1. L et L b e a language in P ol ( C om ) and let M = h G, R i b e the 75 76 syntactic or der e d monoid of L with ex p onent ω . Th e n for any u, v ∈ G ∗ with the pr op erty that any monoi d M ′ ∈ Co m and any mo rphism φ : M → M ′ satisfies b oth φ ( ev al ( u )) = φ ( ev al ( v )) and φ ( ev al ( u )) = φ ( ev al ( u 2 )) , we must have ev al ( u ω v u ω ) ≤ ev al ( u ω ) . Prop osition A .2. L 5 is not in P ol ( C om ) . Pr o of. Consider the transformatio n monoid of L 5 , whic h is t he syn ta ctic monoid. Let u = abab and v = bbaa . Observ e that ev al ( u ) is an idempo ten t and this u and v satisfy the condition in the lemma. W e sho w ev al ( uv u )  ev al ( u ). Observ e that ev al ( uv u ) = ev al ( v ) so we wan t to sho w ev al ( v )  ev al ( u ). If the o pp osite w as true, then b y the definition of the syn t a ctic ordered monoid (Subsection 3.2.2), we m ust hav e for an y w 1 and w 2 , w 1 uw 2 ∈ L = ⇒ w 1 v w 2 ∈ L . In particular, for w 1 = ǫ and w 2 = aa , w e would hav e uaa ∈ L = ⇒ v aa ∈ L . It is true that uaa ∈ L but v aa / ∈ L . No w w e sho w that t he PROMISE-INNE R-PRODUCT f unction that we defined in Chapter 5 reduces to L 5 . Prop osition A .3. P I P 2 ≤ L 5 Pr o of. The reduction is linear and is give n b y the following matrix. a ǫ ǫ b a b ǫ ǫ a b ǫ a b a ǫ b . x i y i corresp onding w ord state tra nsition 0 0 abab 1 → 1 , 2 → 5 , 3 → 3 , 4 → 5 , 5 → 5 0 1 abaaab 1 → 1 , 2 → 5 , 3 → 5 , 4 → 5 , 5 → 5 1 0 abbb 1 → 5 , 2 → 5 , 3 → 3 , 4 → 5 , 5 → 5 1 1 ababab 1 → 3 , 2 → 5 , 3 → 1 , 4 → 5 , 5 → 5 . After this transformation is applied, Alice a pp ends her w o rd with b a nd Bob app ends his w o r d with ǫ . No w if P I P 2 ( x, y ) = 1 then the transformed word 77 CHAPTER A. F acts Ab out L 5 m ust end up at state 5. This is b ecause, from state 1, w e either en ter state 5 and stay t here forev er, or we go to state 3 when x i = y i = 1. If we w ere in state 3 already , x i = y i = 1 ta kes us ba c k to state 1. P I P 2 ( x, y ) = 1 implies that there are an o dd num b er o f indices fo r whic h x i = y i = 1. So af t er the linear transformat ion, assuming w e do not end up in state 5, w e w ould end up in state 3 . The b app ended at t he end of the transformed w ord w ould ensure that we end up in state 5. If P I P 2 ( x, y ) = 0, then we do not w an t t o end up at state 5. Observ e that the promise ensures w e nev er en t er state 5 and since there are ev en nu m b er of indices for whic h x i = y i = 1, we m ust end up at state 1 . The b app ended at the end of t he w ord just t a k es us from state 1 to 2. Observ e that w e can restrict the 1-inputs of P I P 2 the same w a y w e re- stricted the 0- inputs and the reduction w ould trivially w ork for this case as w ell. Putting a promise on b oth the 0 - inputs and the 1- inputs ma y help analyzing the complexit y of P I P 2 . Prop osition A.4. Ther e is no lo c al r e duction fr o m P D I S J to L 5 such that the r e d uction is of the form x i y i c orr esp ond ing wor d 0 0 u ω 0 1 u ω 1 0 u ω 1 1 v . wher e u and v satisfy the c onditions of L emma 4.11. Pr o of. (Sketch) . Since u ω is an idemp oten t, it m ust induce a state transition function in whic h either 1. k → k , j → j and o t her states are sen t to 5 , or 2. k → k and other states are sen t to 5, or 78 3. ev ery state is sent to 5. Observ e that it cannot b e t he case that u ω is a partial identit y on more than t w o states. If u ω satisfies condition 2, then we cannot hav e ev a l ( u ω v u ω )  ev al ( u ω ). Suppose this is the case. Then there exists w 1 , w 2 suc h t hat w 1 u ω w 2 ∈ L and w 1 u ω v u ω w 2 / ∈ L . Since the latter is true, it mus t b e the case that w 1 tak es state 1 to k and w 2 m ust tak e k to a state other than 5. These w 1 and w 2 do not satisfy w 1 u ω w 2 ∈ L , so w e g et a contradiction. This sho ws we cannot hav e condition 2. Similarly , one can sho w that u ω cannot satisfy condition 3 , whic h leav es us with condition 1. This means u ω is either ( abab ) k or ( baba ) k for some k > 0. W e assume it is ( abab ) k . The argumen t for ( baba ) k is v ery similar. Giv en u ω = ( abab ) k , and the fa ct that w e wan t to satisfy ev al ( u ω v u ω )  ev al ( u ω ), one can sho w that the state transition f unction induced b y v m ust b e one of the fo llo wing. 1. 1 → 3 and any other state is sen t to 5. 2. 3 → 1 and any other state is sen t to 5. 3. 1 → 3 and 3 → 1 and an y ot her state is sent to 5. Supp ose v satisfies condition 1. Case 1: v = ( ab ) 2 t − 1 for t > 0. Consider the matrix represen tation of the lo cal reduction. In this mat r ix A , w e count the parit y of the a ’s in t w o wa ys and get a con tradiction. First w e count it b y lo oking at the ro ws. The first row m ust pro duc e the w or d u ω = ( abab ) k and the sec ond ro w mus t pro duce the w o r d v = ( ab ) 2 t − 1 so in total w e ha ve o dd n umber of a ’s. No w w e count the parit y of a ’s b y lo oking at A 1 , 1 A 2 , 2 A 1 , 3 A 2 , 4 ... and A 2 , 1 A 1 , 2 A 2 , 3 A 1 , 4 ... . Bo t h of these m ust pro duce the w ord ( abab ) k so in total w e m ust hav e an ev en n um b er of a ’s. Case 2: v = ( ab ) 2 t bb... for t ≥ 0. Let c b e the column where w e find the second b in the second row. Giv e v alue 1 to en tries of A whic h are a and give 79 CHAPTER A. F acts Ab out L 5 v alue -1 to entries of b . Other en tries (the ǫ ’s) get v alue 0. In terms of these v alues w e ha v e c X i =1 A 2 ,i = − 2 and c X i =1 A 1 ,i ∈ { 0 , 1 } . 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