The Accepting Power of Finite Automata Over Groups Victor Mitrana Faculty of Mathematics University of Bucharest Str Academiei Bucharest ROMANIA Ralf STIEBE Faculty of Computer Science University of Magdeburg POBox D Magdeburg GERMANY Turku Centre for Computer Science TUCS Technical Rep ort No Novemb er ISBN ISSN Abstract Some results from are generalized for nite automata over arbitrary groups The accepting p ower is smaller when ab elian groups are considered in comparison with the nonab elian groups We prove that this is due to the commutativity Each language accepted by a nite automaton over an ab elian group is actually a unordered vector language Finally deterministic nite automata over groups are investigated TUCS Research Group Mathematical Structures of Computer Science Intro duction One of the oldest and most investigated machine in the automata theory is the nite automaton Many fundamental prop erties have b een established and many problems are still op en Unfortunately the nite automata without any external control have a very limited accepting p ower Dierent directions of research have b een con sidered for overcoming this limitation The most known extension added to a nite automata is the pushdown memory In this way a considerable increas ing of the accepting capacity has b een achieved The pushdown automata are able to recognize all contextfree languages Another simple and natural extension related somehow to the pushdown memory was considered in a series of pap ers namely to asso ciate to each conguration an element of a given group but no information regarding the asso ciated element is allowed This value is stored in a counter An input string is accepted if and only if the automaton reaches a designated nal state with its counter containing the neutral element of the group Thus new characterizations of unordered vector languages or context free languages have b een rep orted These results are in a certain sense unexp ected since in such an automaton the same choice is available regardless the content of its counter More precisely the next action is determined just by the input symb ol currently scanned and the state of the machine In this pap er we shall consider only acceptors with a oneway input tap e read from left to right and a counter able to store elements from a given group The aforementioned pap ers deal with nite automata over very well dened groups eg the additive group of integers the multiplicative group of nonnull rational numb ers or the free group The aim of this pap er is to provide some general results regardless the asso ciated group We shall prove that the addition of an ab elian group to a nite automaton is less p owerful than the addition of the multiplicative group of rational numb ers An interchange lemma p oints out the main reason of the p ower decrease of nite automata over ab elian groups Characterizations of the contextfree and recursively enumerable languages classes are set up in the case of nonab elian groups As far as the deterministic variants of nite automata over groups are concerned we shall show their considerable lack of accepting p ower Preliminaries We assume the reader familiar with the basic concepts in automata and formal language theory and in the group theory For further details we refer to and resp ectively For an alphab et we denote by the free monoid generated by under the op eration of concatenation the empty string is denoted by and + the semigroup fg is denoted by The length of x is denoted by jxj Let K M e b e a group under the op eration denoted by with the neutral element denoted by e An extended nite automaton EFA shortly over the group K is a construct A Z K q F 0 where Z q F have the same meaning as for a usual nite automaton 0 namely the set of states the input alphab et the initial state and the set of nal states resp ectively and Z fg P Z M f This sort of automaton can b e viewed as a nite automaton having a counter in which any element of M can b e stored The relation q m s a q s Z a fg m M means that the automaton A changes its current state s into q by reading the symb ol a on the input tap e and writes in the register x m where x is the old content of the register The initial value registered is e We shall use the notation q aw m j s w m r i s r q a A for all s q Z a fg m r M The reexive and transitive closure Sometimes the subscript identifying of the relation j is denoted by j A A the automaton will b e omitted when it is selfundersto o d The word x is accepted by the automaton A if and only if there is a nal state q such that q x e j q e In other words a string is 0 accepted if the automaton completely reads it and reaches a nal state when the content of the register is the neutral element of M The language accepted by an extended nite automaton over a group A as ab ove is LA fx jq x e j q e for some q F g 0 A For two groups K M e and K M e we dene the triple 1 1 1 1 2 2 2 2 K K M M e e with m m n n m n m n 1 2 1 2 1 2 1 2 1 2 1 1 1 2 2 2 It is wellknown that K K is also a group 1 2 We are going to provide some results that will b e useful in what follows The notation LRE G identies the class of regular languages Theorem For any group K LEFAK LRE G i al l nitely gen erated subgroups of K are nite Proof Let K b e a group such that any nitely generated subgroup of K is nite Let A Z K z F b e an EFA over K M e We denote 0 by X the nite subset of M X fm M j z m z a for some z z Z a fgg Let H hX i e b e the subgroup generated by X We construct the nite automaton with moves B Z hX i z e 0 F feg with z m a fz m n j z n z a for all z Z z m i m M a fg One can easily prove that z w e j 0 A z e w j z m which implies LA LB 0 B It remains to prove that for any innite group K nitely generated exists an EFA over K accepting a nonregular language Let K hX i e b e such a group with the nite set of generators X Consider the deterministic EFA 1 A fz g Y K z fz g with Y X fx j x X g and z a z a for all a Y The following facts ab out LA are obvious z m For any m hX i exist a word v Y such that z v e j A For any v Y exists a word w Y such that v w LA For any k the set z mg X fm hX i j v Y jv j k z v e j k A is nite As a consequence of these facts and of the inniteness of hX i we obtain For all k there is a word v such that v v LA for all v Y jv j k k k But one can easily prove that for any regular language L Y exists k such that for all v w L there is w Y jw j k with v w L Hence LA cannot b e regular 2 A nitely generated ab elian group is nite if all its elements are of nite order Hence for an ab elian group K LEFAK LRE G i all ele ments of K have nite order This is not necessarily true for nonab elian groups We can however prove a pumping lemma which is very similar to the pumping lemma for regular languages Lemma Let K be some group without elements of innite order For any language L LEFAK there is a constant n such that for al l x L jxj n there exist a decomposition x uv w and a natural number q iq +1 with juv j n jv j uv w L for al l i Moreover if K has the nite exponent p then q can uniformly be chosen as q p Proof Let A Z K z F b e an EFA over K We cho ose n jZ j 0 Now consider a word x with jxj n Similar to the pro of of the pumping lemma for regular languages it can b e shown that there is a decomp osition x uv w juv j n jv j such that z uv w e j z v w m j z w m m j f e z Z f F 0 1 1 2 A A A q iq +1 Now cho ose q such that m e Obviously any word uv w is accepted 2 by A 2 As a consequence of the ab ove pumping lemma we obtain Theorem For any group K LEFAK contains the language L n n fa b j n g i at least one element of K has an innite order Proof Let K M e If M contains an element m of innite order then the nitely generated subgroup hmi e is isomorphic to ZZ hence L is in LEFAK If all elements of M have nite order a simple application of the ab ove pumping lemma yields L LEFAK 2 For a group K let F K denote the family of all nitely generated sub groups of K Theorem For any group K LEFAK LEFAH HF (K) Proof Let K M e b e a group The inclusion LEFAK LEFAH HF (K) holds since LEFAK LEFAH for any subgroup H of K On the other hand let A Z K z F b e an EFA over K The 0 group H hX i e where X fm M jq m z a for some q z Z a fgg is a nitely generated subgroup of K Obviously during any computation in the counter of A app ear only elements of hX i Therefore the automaton A can b e viewed as an automaton over H More precisely A Z H z F accepts the same language as A do es This proves the 0 second inclusion and thus the theorem 2 EFA over ab elian groups Valence grammars and EFA have initially b een intro duced for the groups k Z ZZ k and Q Q fg In what follows we shall k show that the accepting capacity of EFA do es not increase if we consider arbitrary ab elian groups instead of Q Thus every language accepted by an EFA over an ab elian group is a unordered vector language The deep er reason of this fact is the following fundamental result in the group theory Theorem A nitely generated abelian group is the direct product of a nite number of cyclic groups As a consequence a nitely generated ab elian group is either nite or isomorphic to a group Z H where k is a p ositive integer and H