on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

eigenvalue intervals for a two point boundary value problem on a measure chain we study the existence of eigenvalue intervals for the second order differential equation on a measure chain x t p t f x t 0 t t1 t2 satisfying the boundary conditions x t1 x t1 0 and x t2 x t2 0 where f is a positive function and p a nonnegative function that is allowed to vanish on some subintervals of t1 t2 of the measure chain the methods involve applications of a fixed point theorem for operators on a cone in a banach space

the ehrenfeucht conjecture a algebra framework for its proof abstract we describe an algebraic framework including in particular the affine monoid of a given ring within this framework it is a simple matter to deduce ehrenfeucht s conjecture from hilbert s basis theorem

on the equivalence problem of compositions of morphisms and inverse morphisms on it is shown that the equivalence problem of h 1h on the family of context free languages is decidable where h is the family of morphisms and h 1 is the family of inverse morphisms

a class of the first order impulsive dynamic equations on time scales abstract in this paper the first order impulsive dynamic equations on time scales is considered the existence and uniqueness of solution for a class of nonlinear impulsive dynamic equations on time scales is proved

on undecidability bounds for matrix decision problems in this paper we consider several reachability problems such as vector reachability membership in matrix semigroups and reachability problems in piecewise linear maps since all of these questions are undecidable in general we work on lowering the bounds for undecidability in particular we show an elementary proof of undecidability of the reachability problem for a set of 5 two dimensional affine transformations then using a modified version of a standard technique we also prove that the vector reachability problem is undecidable for two rational matrices in dimension 11 the above result can be used to show that the system of piecewise linear functions of dimension 12 with only two intervals has an undecidable set to point reachability problem we also show that the zero in the upper right corner problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23

binary generalized post correspondence problem we give a new proof for the decidability of the binary post correspondence problem pcp originally proved in 1982 by ehrenfeucht karhumki and rozenberg our proof is complete and somewhat shorter than the original proof although we use the same basic copyright 2002 elsevier science b v all rights reserved

example of undecidable problems for 2 generator matrix semigroups abstract we show that the following two problems are undecidable given two square matrices decide whether the semigroup that they generate contains the zero matrix and whether it contains a matrix having a zero in the right upper corner

systemes entiers d equations sur un alphabet fini et conjecture d ehrenfeucht the following conjecture has been posed by a ehrenfeucht for each language l over a finite alphabet there exists a finite subset of f of l such that for each pair of morphisms of into a free monoid the equality is true for all in l if and only if it is true for all in f such a finite subset f of l is called a test set of l a system s of equations over the alphabet is called entire if there exists a morphism of into a free monoid such that s 1 we have introduced for each morphism of in a free monoid such that is finite and 1 1 1 a generalized automaton m called the shuffle automaton of which recognizes a rational language k over a finite alphabet and a pair of morphisms of into such that 1 k nivat s 1966 characterization of rational relations is a motivation for this construction and implies that all entire systems over a finite alphabet are rational culik ii and salomaa 1978 have proved that every rational language over a finite alphabet admits a test set culik ii and karhumaki 1983 have shown that if s is a system of equations over a finite alphabet and if there exists a language l over a finite alphabet and morphisms and from into such that s l then if l admits a test set f then t f is a finite equivalent subsystem of s this gives an algorithm which determines for each entire system of equations over a finite alphabet a finite subsystem this result can be improved by using the following theorem spehner 1984 each finitely generated submonoid m of a free monoid has a finite m presentation i e there exists a finite alphabet and a finite part of such that m is isomorphic to the quotient monoid of by the smallest malcev congruence containing is a malcev congruence of if the quotient monoid is embeddable in a group the algorithm for determining the shuffle automaton m and the finite equivalent subsystem of s 1 works with a maximum efficiency because m can be embedded in the minimal automaton of the language k

reachability problems in quaternion matrix and rotation semigroups we examine computational problems on quaternion matrix and rotation semigroups it is shown that in the ultimate case of quaternion matrices in which multiplication is still associative most of the decision problems for matrix semigroups are undecidable in dimension two the geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3 sphere in particular we show that the reachability of the rotation problem is undecidable on the 3 sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers

a homomorphic characterization of principal semiafls without using intersection with regular sets abstract if one is asked to consider which one of the semiafl or afl operations differs most from the remaining ones a set theoretically well grounded conclusion would be intersection with regular sets inspired by the result of culik fich and salomaa 1 we shall show that this operation can be replaced by catenation with an endmarker the basis of the paper is the result that any free principal semiafl m l can be presented in the form m l h 4 h 3 1 h 2 h 1 1 l each h 1 a homomorphism h 2 and h 3 free h 4 length preserving where is a symbol not in the alphabet of l if m l is not free we get the same representation but now h 4 erases the last letter from each word in h 3 1 h 2 1 h 1 1 l and is length preserving for other letters for the family of free regular languages we may choose l the last section contains a homomorphic characterization of regular star languages

on test sets for checking morphism equivalence on languages with fair distribution of letters abstract a test set for a language l is a finite subset t of l with the property that each pair of morphisms that agrees on t also agrees on l some results concerning test sets for languages with fair distribution of letters are presented the first result is that every d0l language with fair distribution of letters has a test set the second result shows that every language l with fair distribution has a test set relative to morphisms g h which have bounded balance on l these results are generalizations of results of culik ii and karhumaki 1983

extension of the decidability of the marked pcp to instances with unique blocks in the post correspondence problem pcp an instance h g consists of two morphisms h and g and the problem is to determine whether or not there exists a nonempty word w such that h w g w here we prove that the pcp is decidable for instances with unique blocks using the decidability of the marked pcp also we show that it is decidable whether an instance satisfying the uniqueness condition for continuations has an infinite solution these results establish a new and larger class of decidable instances of the pcp including the class of marked instances

nilpotent fusion systems abstract let p be a prime and let f be a saturated fusion system over a finite p group p a fusion system f is said to be nilpotent if f f p p we give various criteria for a saturated fusion system f to be nilpotent which generalize the analogous criteria for a finite group to be p nilpotent

optimal control for a class of strongly nonlinear impulsive equations in banach spaces abstract in this paper we consider the optimal control problem of system governed by a class of strongly nonlinear impulsive evolution equations based on the existence of strongly nonlinear impulsive evolution equations which contain nonlinear monotone operators and nonmonotone perturbations we prove the existence of optimal controls for the general lagrange problem the necessary conditions for optimality are also presented

refining the hierarchy of blind multicounter languages and twist closed trios we introduce the new families k r rbc of languages accepted in quasi realtime by one way counter automata having k blind counters of which at least r are reversal bounded it is proved that these families form a strict and linear hierarchy of semi afls within the the family blind m c1 of blind multicounter languages with generator c1 w a1 b1 w a1 w b1 this thereby combines the families blind and rbc from 13 to one strict hierarchy and generalizes and sharpens greibachs results the strict inclusions between the k counter families k r rbc are proved using linear algebra techniques we also study the language theoretic monadic operation twist 18 20 in connection with the semi afls of languages accepted by multicounter and multipushdown acceptors all restricted to reversal bounded behavior it is verified that the family k r rbc is twist closed if and only if r 0 in which case k 0 rbc m ck ck being the k fold shuffle of disjoint copies of c1 we characterize the family m pal of languages accepted in quasi realtime by nondeterministic one way reversal bounded multipushdown acceptors as the least twist closed trio mtwist pal generated by the set of palindromes pal w a b w wrev

precircular codes and periodic biinfinite words the aim of decoding or factorizing in a single way biinfinite words with a finitary language leads us to consider two distinct notions of factorization for biinfinite words for one of these notions requiring singleness of decoding periodic biinfinite words leads us to study circular codes the other notion leads us to define the new notion of precircular codes precircular codes are defined and analyzed the family of precircular codes is compared to the family of circular codes and to the family of almost circular codes the structure of two element precircular codes is determined codes and precircular codes are characterized by their way of acting upon periodic biinfinite words

nagumo type existence results for second order nonlinear dynamic bvps abstract in this paper we present a nagumo condition for time scales equations that includes as a particular case a type of the classical one for differential equations involving a certain restriction the established nagumo condition further allows us to prove the existence of at least one solution lying between a pair of lower and upper solutions of a nonlinear second order dynamic equation with nonlinear functional boundary value conditions

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

on the combinatorial problem of codes composed of two words resume nous montrons que si x est un code compose de deux mots x et y alors un mot de x suffisamment long admettant deux x interp etations disjointes est necessairement facteur d un mot de x ou de y ou de y ou d une puissance d un mot imprimitif appartenant a x y xy

equations over finite sets of words and equivalence problems in automata theory abstract several observations concerning equations over monoids of finite sets of words are made monoids considered are the monoid of finite prefix codes the monoid of all finite sets and the monoid of all finite multisets as well as some others as an application some known decidability results in automata theory are extended finally several open questions are raised

a note on p nilpotence and solvability of finite groups abstract in this note we first give some examples to show that some hypotheses of some well known results for a group g to be p nilpotent solvable and supersolvable are essential and cannot be removed second we give some generalizations of two theorems in a ballester bolinches x guo some results on p nilpotence and solubility of finite groups j algebra 228 2000 491 496

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

test sets for finite substitutions abstract ehrenfeucht s conjecture that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s has recently been verified in this paper we extend this result from endomorphisms to certain types of finite substitutions by applying konig s lemma and embeddings of the monoid into a free algebra

periodicity forcing words the dual post correspondence problem asks for a given word if there exists a non periodic morphism g and an arbitrary morphism h such that g h thus satisfies the dual pcp if and only if it belongs to a non trivial equality set words which do not satisfy the dual pcp are called periodicity forcing and are important to the study of word equations equality sets and ambiguity of morphisms in this paper a prime subset of periodicity forcing words is presented it is shown that when combined with a particular type of morphism it generates exactly the full set of periodicity forcing words furthermore it is shown that there exist examples of periodicity forcing words which contain any given factor prefix suffix finally an alternative class of mechanisms for generating periodicity forcing words is developed resulting in a class of examples which contrast those known already

unary multiple equality sets the languages of rational matrices unary multiple equality sets are equality sets of finite sets of homomorphisms which map into the free monoids over single letter alphabets these languages can equally well be defined by rational matrices unary multiple equality sets have extraordinary language theoretic properties they coincide with the commutative multiple equality sets each unary multiple equality set is precisely classified by the rank of its defining matrices and all fundamental decision questions are polynomially decidable the least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata finally crossing free unary multiple equality sets are introduced which vary from unary multiple equality sets in a similar way as the one sided dyck set varies from the two sided dyck set

binary equality sets are generated by two words we show that the equality set eq g h of two non periodic binary morphisms g h a is generated by at most two words if the rank of eq g h is two then and begin and end with different letters this in particular implies that any binary language has a test set of cardinality at most two

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

impulsive nonlocal differential equations through differential equations on time scales abstract we propose a non standard approach to impulsive differential equations in banach spaces by embedding this type of problems into differential dynamic problems on time scales we give an existence result for dynamic equations and as a consequence we obtain an existence result for impulsive differential equations

nonlinear integro differential equations and optimal control problems on time scales this paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro differential equation on time scales using the contraction mapping principle the existence and uniqueness of a weak solution of the nonlinear integro differential equation on time scales are proved discussing l t 1 strong weak lower semicontinuity of integral functional on time scales we give sufficient conditions for the existence of optimal control by the virtue of the integration by parts formula on time scales the necessary conditions of optimality are derived

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

relative undecidability in term rewriting i the termination hierarchy for a hierarchy of properties of term rewriting systems related to termination we prove relative undecidability for implications x y in the hierarchy the property x is undecidable for term rewriting systems satisfying y for most implications we obtain this result for term rewriting systems consisting of a single rewrite rule

equality sets of binary d0l sequences abstract we study d0l sequences over a binary alphabet if s s n n 0 and t t n n 0 are d0l sequences their equality set e s t is defined by e s t n 0 s n t n we show that if s and t are d0l sequences over a binary alphabet then their equality set e s t is eventually periodic

marked pcp is decidable we show that the marked version of the post correspondence problem where the words on a list are required to differ in the first letter is decidable on the other hand we prove that the pcp remains undecidable if we only require the words to differ in the first two letters thus we locate the decidability undecidability boundary between marked and 2 marked pcp

decidability of termination of grid string rewriting rules termination of string rewriting is known undecidable termination of string rewriting with only one rule is neither known decidable nor known undecidable this paper presents a decision procedure for rules u rightarrow v such that some letter b from u occurs as often or less often in v we call such rules grid rules by far most rules are grid rules grid rules cover all rules which terminate by a total division order thus total division orders are shown to be irrelevant for the termination problem of one rule string rewriting

decidable and undecidable problems about quantum automata we study the following decision problem is the language recognized by a quantum finite automaton empty or nonempty we prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds this result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and nonstrict thresholds both lead to undecidable problems

on three element codes abstract we show that three element codes have some special properties which do not hold even for four element codes firstly for each three element code a if u and v are words in pref xa pref ya with x y a x y then one of them is a prefix of the other i e among the words which can be covered in two different ways from left to right there exists a unique maximal possibly infinite element secondly each three element code has a bounded delay in at least one direction

on the equivalence problem for binary dol systems it is shown that to test whether two dol sequences in the binary case coincide it is enough to test whether four first words of these sequences are the same the result is optimal

representations of language families by homomorphic equality operations and generalized equality sets abstract in this paper the operations of homomorphic equality and inverse homomorphic equality are introduced and are studied in all detail as part of formal language theory these operations are defined from n tuples of homomorphisms by a matching of the homomorphic images they model a simple and powerful equality mechanism and incorporate the notion of generalized equality sets single homomorphisms on free monoids are natural and simple operations however combining n tuples of homomorphisms into a homomorphic equality or inverse homomorphic equality may result in very complex morphic mappings which map e g to the set of solutions of an instance of the post correspondence problem in this paper particular emphasis is laid upon strong representation theorems in the general case every recursively enumerable set can be obtained from the trivial set by applying an inverse homomorphism a inverse homomorphic equality of two nonerasing homomorphisms and an erasing homomorphism this is a uniform morphic representation of the recursively enumerable sets with special regular sets as a basis and it is minimal in its constituents hence the recursively enumerable sets are the smallest class containing all regular sets of the form and closed under homomorphic equality or under inverse homomorphic equality and homomorphism every nonempty language is a generator of the recursively enumerable sets when the full trio operations are extended to include homomorphic equality or inverse homomorphic equality in a similar manner the regular sets can be characterized in terms of homomorphic equality and inverse homomorphic equality operations with bounded balance finally in the nonerasing case the target are classes of the form h l bnp for a class of languages l h l bnp is the smallest class containing l and closed under nonerasing homomorphic equality or under inverse homomorphic equality and nonerasing homomorphism it includes the intersection closure of l every language in h l bnp can be obtained from a language in l by a nonerasing homomorphic equality or an inverse homomorphic equality of three homomorphisms and followed by a nonerasing homomorphism provided that l is closed under inverse homomorphism and endmarking this representation goes through to trio generators of l which are the generators of h l bnp if the trio operations are extended to include nonerasing homomorphic equality or inverse homomorphic equality the operations of homomorphic equality and inverse homomorphic equality are based on generalized equality sets these are equality sets of finite sets of homomorphisms which are used forwards or backwards we investigate basic language theoretic properties of these sets including the star event property commutativity and hardest sets and provide a complete classification of the generalized equality sets according to the numbers of homomorphisms that are used forwards or backwards

higher order dynamic equations on measure chains wronskians disconjugacy and interpolating families of functions this paper introduces generalized zeros and hence disconjugacy of nth order linear dynamic equations which cover simultaneously as special cases among others both differential equations and difference equations we also define markov fekete and descartes interpolating systems of functions the main result of this paper states that disconjugacy is equivalent to the existence of any of the above interpolating systems of solutions and that it is also equivalent to a certain factorization representation of the operator the results in this paper unify the corresponding theories of disconjugacy for nth order linear ordinary differential equations and for nth order linear difference equations

decision problems for semi thue systems with a few rules we show that the accessibility problem the common descendant problem the termination problem and the uniform termination problem are undecidable for 3 rules semi thue systems as a corollary we obtain the undecidability of the post correspondence problem for 7 rules

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

termination and derivational complexity of confluent one rule string rewriting systems it is not known whether the termination problem is decidable for one rule string rewriting systems though the confluence of such systems is decidable by wrathall in word equations and related topics lecture notes in computer science vol 572 1992 pp 237 246 in this paper we develop techniques to attack the termination and complexity problems of confluent one rule string rewriting systems with given such a system we associate another rewriting system over another alphabet the behaviour of the two systems is closely related and the termination problem for the new system is sometimes easier than for the original system we apply our method to systems of the special type apbq t where t is an arbitrary word over a b and give a complete characterization for termination we also give a complete analysis of the derivational complexity for the system apbq bnam

p nilpotence of finite groups and minimal subgroups in this paper it is proved that a finite group g is p nilpotent if every minimal subgroup of p op g is permutable in p and ng p is p nilpotent and when p 2 either 2 p op g p 1 p op g or p is quaternion free where p is a prime dividing the order of g and p is a sylow p subgroup of g by using this result we may get a series of corollaries for p nilpotence which contain some known results some other applications of this result are also given

necessary conditions of optimality for a class of optimal control problems on time scales a class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered by the lebesgue d integral theory and the sobolev type space h 1 on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced therefore the necessary conditions of optimality are presented some typical examples are given for demonstration