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Technische Universit¨At Dresden TECHNISCHE UNIVERSITATÈ DRESDEN FakultatÈ Informatik TUD/ FI98/ 09 - October1998 Daniel Kirsten Technische Berichte Grundlagen der Programmierung Technical Reports Institut fÈur Softwaretechnik I A Connection between the Star Problem ISSN 1430-211X and the Finite Power Property in Trace Monoids Technische UniversitatÈ Dresden FakultatÈ Informatik D-01062 Dresden Germany URL: http://www.inf.tu-dresden.de/ A Connection b etween the Star Problem and the Finite Power Prop erty in Trace Monoids Daniel Kirsten Department of Computer Science Dresden University of Technology D Dresden Germany DanielKirsteninftudresdende httpwwwinftudresdendedk Octob er Abstract This pap er deals with a connection b etween two decision problems for recognizable trace languages the star problem and the nite p ower prop erty problem Due to a theorem by Richomme from we know that b oth problems are decidable in trace monoids which do not contain a C submonoid It is not known whether the star problem or the nite p ower prop erty are decidable in the C or in trace monoids containing a C In this pap er we show a new connection b etween these problems Assume a trace monoid IM I which is isomorphic to the Cartesian Pro duct of two disjoint trace monoids IM I and IM I Assume further a recognizable language L in IM I such that 1 1 2 2 every trace in L contains at least one letter in and at least one letter in Then the 1 2 main theorem of this pap er asserts that L is recognizable i L has the nite p ower prop erty This work has b een supp orted by the p ostgraduate program Sp ecication of discrete pro cesses and systems of pro cesses by op erational mo dels and logics of the German Research Community Deutsche Forschungsgemeinsch aft Contents Intro duction Formal Overview Monoids Languages and Traces Recognizable Sets Some Decision Problems for Trace Languages Main Results Semigroups and Ideals Basic Denitions and Notions Finite Semigroups without Prop er Ideals Finite Semigroups with Prop er Ideals A Classication Pro duct Automata Denitions Connections to Subsemigroups and Ideals Pro of of Theorem Q do es not have prop er left ideals Q fullls assertion C Q fullls assertion D Q fullls assertion B Completion of the Pro of Conclusions and Future Steps Acknowledgments Intro duction Free partially commutative monoids also called trace monoids were intro duced by Cartier and Foata in In Mazurkiewicz prop osed trace monoids as a p otential mo del for concurrent pro cesses which marks the b eginning of a systematic study of trace monoids by mathematicians and theoretical computer scientists see eg the recent surveys A part of the research in trace theory deals with examinations of wellknown classic results for free monoids in the framework of traces One main stream in trace theory is the study of recognizable trace languages which can b e considered as an extension of the well studied concept of regular languages in free monoids A ma jor step in this research is Ochmanskys PhD thesis from Some of the results concerning regular languages in free monoids can b e generalized to recognizable languages in trace monoids However there is one ma jor dierence The iteration of a recognizable trace language do es not necessarily yield a recognizable language This fact raises the so called star problem Given a recognizable language L is L recognizable In general it is not known whether the star problem is decidable The nite p ower prop erty problem for short FPPP is related to the star problem Given a recognizable language L has L the nite p ower prop erty for short FPP ie is there a natural n numb er n such that L L L L The strategy to achieve partial solutions of these problems by examining connections b etween them turned out to b e fruitful The main result after a stream of publications is a theorem stated by Richomme in saying that b oth the star problem and the FPP are decidable in trace monoids which do not contain a particular submonoid called C It is not known whether the star problem and the FPP are decidable in trace monoids with a C submonoid Although the FPP for nite trace languages is obviously decidable decidability of the star problem for nite trace languages is only known for just a few sp ecial cases In this pap er we establish a new connection b etween the star problem and the FPP We deal with trace monoids which are isomorphic to Cartesian Pro ducts of two trace monoids We show that for a certain class of recognizable languages in Cartesian Pro ducts the iteration of a language is recognizable i the language has the FPP The pap er is organized as follows In Section we show some concepts from algebra and formal language theory In the rst subsection we get familiar with some basic notions from algebra formal language theory and trace theory In the next subsection we deal with recog nizable sets and rational sets Then we discuss the two main problems for recognizable trace languages their connections and solutions as far as known After this discussion I explain the main results of this pap er In Section and we deal with concepts which we require to prove the main results In Section we get familiar with ideal theory which will b e a crucial to ol In Section we deal with pro duct automata After these preliminary to ols we prove the main theorems in Section In several subsec tions we deal with sp ecial cases In the last subsection we summarize these sp ecial cases Formal Overview Monoids Languages and Traces I briey intro duce the basic notions from algebra and trace theory Unless I do not state precise sources I consider the concepts and notions as wellknown By IN we denote the set of natural numb ers including zero ie IN f g We say a set K is a subset of a set L and we denote this by K L i every element of K FORMAL OVERVIEW b elongs to L We say K is a proper subset of L and we denote this by K L i every element of K b elongs to L and additionally there is an element in L which do es not b elong to K A semigroup S is an algebraic structure consisting of a set S and a binary op eration S S such that is asso ciative ie for every p q r S we have p q r p q r We call S S S S S S the underlying set and we call the operation or product of S Usually we drop the symb ol S S and denote the pro duct by juxtap osition We call a semigroup S nite i S is a nite set S We use the letter S to denote b oth the semigroup and its underlying set We call a semigroup IM a monoid i IM contains an element such that for every IM IM m IM we have m m m We call the neutral element We drop the IM IM IM IM IM index at as long as no confusion arises IM In the rest of this subsection we assume some semigroup S and some monoid IM For every natural numb er n we dene the nfold pro duct as follows For every p in S n n p yields p itself and further for n p denotes p p For every monoid IM we extend this denition by claiming that for every p IM we have p IM We extend the pro duct to subsets of S If K and L are two subsets of S the set KL contains all elements k l for some k in K and l in L We extend the nfold pro duct to sets We dene n n K K and for n we set K K K For every subset K of IM we dene K f g IM For every subset K of S we dene the nonempty iteration K as the union of the sets K K K For every subset K of IM we additionally dene the iteration of K by K and dene it by K K f g IM n n For every subset K of S and for every n we abbreviate the union K K by K n n For every subset K of IM and for every n we abbreviate K K by K S Assume two semigroups S and S Their Cartesian Pro duct is the semigroup denoted by 0 S and dened in the following way The underlying set is the Cartesian Pro duct of the underlying p 1 sets of S and S We dene the op eration comp onentwise ie for every pair of elements 0 p 1 p p p 1 2 2 The pro ducts p p and p p are the pro ducts in S and S their pro duct yields and 0 0 0 p p p 1 2 2 resp ectively The Cartesian Pro duct of two monoids yields a monoid Again assume two semigroups S and S We call a function h S S a homomorphism 0 hl hk l We call a i h preserves the pro duct ie for every k l S we have hk S S homomorphism h b etween two monoids IM and IM a monoid homomorphism i h preserves the 0 There are homomorphisms b etween two monoids neutral element ie we have h IM IM which do not preserve the neutral element We extend the notion of homomorphisms to sets If K is a subset of S we dene hK as the set of all q S such that for some p K we have hp q We denote the inverse of h by h We dene it on subsets of S If L is a subset of S h L yields the set of all m S such that hm L We call h a surjective homomorphism from S to S i hS S We call h an isomorphism i for every p S the set h fpg is a singleton Then we can regard h as a homomorphism from S to S We call the semigroups isomorphic i an isomorphism b etween S and S exists A sp ecial kind of homomorphisms are the pro jections
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