on the state complexity of operations on two way finite automata the paper investigates the effect of basic language theoretic operations on the number of states in two way deterministic finite automata 2dfas if m and n are the number of states in the 2dfas recognizing the arguments of the following operations then their result requires the following number of states at least m no m n and at most 4m n const for union at least m no m n and at most m n 1 for intersection at least mn 2 n logm and at most 2mm 12nn 1 for concatenation at least 1n2n21 and at most 2o nn 1 for kleene star square and projections between n 1 and n 2 for reversal exactly 2n for inverse homomorphisms all results are obtained by first establishing high lower bounds on the number of states in any 1dfas recognizing these languages and then using these bounds to reason about the size of any equivalent 2dfas

a very hard log space counting class abstract we consider the logarithmic space counting and optimization classes l span l and opt l which are defined analogously to their polynomial time counterparts we obtain complete functions for these three classes in terms of graphs and finite automata we show that l and opt l are both included in nc 2 but that surprisingly span l seems to be a much harder class than l and opt l we demonstrate that span l functions can be computed in polynomial time if and only if p p and all the classes of the polynomial time hierarchy are included in p this result follows from the fact that span l and p are very similar span l p and any function in p can be represented as the difference of a function in fl and a function in span l nevertheless the inclusion p span l would imply nl p np we furthermore investigate restrictions of the classes opt l and span l

on a structural property in the state complexity of projected regular languages a transition is unobservable if it is labeled by a symbol removed by a projection the present paper investigates a new structural property of incomplete deterministic finite automata a number of states incident with an unobservable transition and its effect on the state complexity of projected regular languages we show that the known upper bound can be met only by automata with one unobservable transition up to unobservable multi transitions we improve this upper bound by taking into consideration the structural property of minimal incomplete automata and prove the tightness of new upper bounds special attention is focused on the case of finite languages the paper also presents and discusses several fundamental problems which are still open

on the zeros of the riemann zeta function in the critical strip ii we describe extensive computations which show that riemann s zeta function t s has exactly 200 000 001 zeros of the form a it in the region 0 t 81 702 130 19 all these zeros are simple and lie on the line a this extends a similar result for the first 81 000 001 zeros established by brent in math comp v 33 1979 pp 1361 1372 counts of the numbers of gram blocks of various types and the failures of rosser s rule are given

succinct representation of regular languages by boolean automata ii abstract boolean automata are a generalization of finite automata in the sense that the next state the result of the transition function given a state and a letter is not just a single state deterministic automata or a set of states nondeterministic automata but a boolean function of the states boolean automata accept precisely the regular languages also they correspond in a natural way to certain language equation involving complementation as well as to sequential networks in a previous note we showed that for every n 1 there exists a boolean automaton b n with n states such that the smallest deterministic automaton for the same language has 2 2 n states in the present note we will show a precisely attainable lower bound on the succinctness of representing regular languages by boolean automata namely we will show that for every n 1 there exists a reduced automaton d n with n states such that the smallest boolean automaton accepting the same language has also n states

on the state minimization of fuzzy automata this paper investigates the minimization problem of fuzzy automata aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one the decision version of the minimization problem is as follows given a fuzzy automaton cal a and a natural number k i e a pair langle cal a k rangle is there a k state fuzzy automaton equivalent to cal a we prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton to this end we introduce the concept of systems of fuzzy polynomial equations present a procedure for finding solutions of these systems and finally reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations it is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms the term minimization there did not mean state minimization in our sense since these algorithms did not aim at a minimal fuzzy automaton but found reasonably small fuzzy automata

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

complexity of multi head finite automata origins and directions multi head finite automata were introduced and first investigated by rabin and scott in 1964 and rosenberg in 1966 since that time a vast literature on computational and descriptional complexity issues on multi head finite automata documenting the importance of these devices has been developed although multi head finite automata are a simple concept their computational behavior can be already very complex and leads to undecidable or even non semi decidable problems on these devices such as for example emptiness finiteness universality equivalence etc additionally the conversions between different types of multi head finite automata induce in most cases size bounds that cannot be bounded by any recursive function so called non recursive trade offs these strong negative results trigger the study of subclasses and alternative characterizations of multi head finite automata for a better understanding of the nature of non recursive trade offs and thus the borderline between decidable and undecidable problems in the present paper we tour a fragment of this literature

optimal state reductions of automata with partially specified behaviors nondeterministic finite automata with don t care states namely states which neither accept nor reject are considered a characterization of deterministic automata compatible with such a device is obtained furthermore an optimal state bound for the smallest compatible deterministic automata is provided it is proved that the problem of minimizing deterministic don t care automata is np complete and pspace hard in the nondeterministic case the restriction to the unary case is also considered

two way unary automata versus logarithmic space we show that if l nl the classical logarithmic space classes then each unary 2nfa a two way nondeterministic finite automaton can be converted into an equivalent 2dfa a deterministic two way automaton keeping the number of states polynomial unlike other results of this kind here the deterministic simulation is valid for inputs of all lengths not only polynomially long ones this shows a connection between the standard logarithmic space complexity and the state complexity of two way unary automata it indicates that l could be separated from nl by proving a superpolynomial gap in the number of states for the conversion from unary 2nfas to 2dfa moreover without any unproven assumptions we show that each n state unary 2nfa can be simulated by an equivalent 2ufa an unambiguous 2nfa with a polynomial number of states

denseness for norm attaining operator valued functions abstract in this note we offer a short constructive proof for hilbert spaces of lindenstrauss famous result on the denseness of norm attaining operators specifically we show given any a l h there is a sequence of rank 1 operators k n such that a k n is norm attaining for each n and k n converges in norm to zero we then apply our construction to establish denseness results for norm attaining operator valued functions

on measuring nondeterminism in regular languages abstract it is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language this paper studies situations intermediate between forbidding nondeterminism and allowing it the amount of nondeterminism used by a finite automaton is quantified so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied these intermediate situations are shown always to lie between two extremes 1 there are no savings as the amount of nondeterminism increases incrementally so that savings occur only when the amount of nondeterminism becomes unlimited 2 each increment of nondeterminism results in additional savings the number s of states decreasing approximately as s 1 i until exponential savings have been achieved after about i log s log log s increments

quantum finite automata we first outline main steps and achievements along bertoni s research path in quantum finite automata theory from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power next we choose to focus on bertoni s studies on quantum finite automata descriptional complexity in particular we expand on a statistical framework for the synthesis of succinct quantum finite automata discussing its adaptation to the case of multiperiodic events and languages we then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages finally we introduce some promise problems for multiperiodic inputs showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata we build small size monte carlo quantum finite automata for multiperiodic languages we analyze the modular architecture of such monte carlo quantum finite automata we solve promise problems on multiperiodic inputs by small quantum finite automata

automaticity ii descriptional complexity in the unary case abstract let and be finite alphabets and let be a map from to then the deterministic automaticity of a n is defined to be the size of the minimum finite state machine that correctly computes on all inputs of size in a previous paper shallit and breitbart examined the properties of this measure of descriptional complexity in the case 2 in this paper we continue the study of automaticity focusing on the case where 1 we prove that a n l n where l we also prove that a n n 2 log l n 2 log l log l n for almost all functions in the nondeterministic case we show that there exists a c such that for almost all unary languages l we have n l n cn log n for all sufficiently large n the proof is based on a new enumeration method for languages accepted by unary q state nfas if l is not a regular language then it follows from a result of karp that lim sup n a l n n 1 2 we conjecture that l 0 then this bound can be improved to 5 1 2 finally we give some lower bounds for nondeterministic automaticity for nonregular languages

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

on the computational complexity of problems related to distinguishability sets we study the computational complexity of problems related to distinguishability sets for regular languages roughly speaking the distinguishability set mathsf d l for a not necessarily regular language l consists of all words w that are a common suffix of a word xw in l and of a word yw that does not belong to l in particular we investigate the complexity of the representation problem i e deciding for given automata a and b whether b accepts the distinguishability set of l a it is shown that this problem and some of its variants are highly intractable namely pspace complete in fact determining the size of an automaton for mathsf d l a is already pspace complete on the other hand questions related to the hierarchy induced by iterated application of the mathsf d operator turn out to be much easier for instance the question whether for a given automaton a the accepted language is equal to its own distinguishability set i e whether l a mathsf d l a holds is shown to be nl complete as a byproduct of our investigations we found a compelling characterization of synchronizing automata namely that a minimal automaton a is synchronizing if and only if mathsf d l a mathsf d 2 l a

vitali s and harnack s type results for vector valued functions in this brief note we extend vitali s theorem for holomorphic functions obtained by arendt and nikolski to nets of functions of sheaves of smooth vector valued functions as a consequence we also extend a harnack s theorem for compact operator valued harmonic functions recently obtained by enflo and smithies to bounded operator valued harmonic functions avoiding the assumption that the hilbert space h where the operators are defined is separable

descriptional complexity of unambiguous input driven pushdown automata it is known that a nondeterministic input driven pushdown automaton idpda a k a visibly pushdown automaton a k a nested word automaton with n states can be transformed to an equivalent deterministic automaton with 2 n 2 states b von braunmuhl and r verbeek 1983 8 and that this size is necessary in the worst case r alur and p madhusudan 2009 4 this paper demonstrates that the same worst case 2 n 2 size blow up occurs when converting a nondeterministic idpda to an unambiguous one and an unambiguous idpda to a deterministic one in addition the methods developed in this paper are used to demonstrate that the descriptional complexity of complementation for nondeterministic idpdas is 2 n 2 and that the descriptional complexity of homomorphisms for deterministic idpdas is 2 n 2

inequalities for proper contractions and strictly dissipative operators abstract some inequalities for proper contractions and strictly dissipative operators on a hilbert space are proved these inequalities for operators are related to the classical harnack s inequalities for analytic functions with a positive real part

on the limits of the communication complexity technique for proving lower bounds on the size of minimal nfa s in contrast to the minimization of deterministic finite automata dfa s the task of constructing a minimal nondeterministic finite automaton nfa for a given nfa is pspace complete moreover there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal nfa s since one is unable to efficiently estimate the size of a minimal nfa in an efficient way one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal nfa for a given regular language here we consider the robust and most successful lower bound proof technique that is based on communication complexity in this paper it is proved that even a strong generalization of this method fails for some concrete regular languages to fail is considered here in a very strong sense there is an exponential gap between the size of a minimal nfa and the achievable lower bound for a specific sequence of regular languages the generalization of the concept of communication protocols is also strong here it is shown that cutting the input word into 2 o n 1 4 pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata it seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

a hitchhiker s guide to descriptional complexity through analytic combinatorics nowadays increasing attention is being given to the study of the descriptional complexity in the average case although the underlying theory for such a study seems intimidating one can obtain interesting results in this area without too much effort in this gentle introduction we take the reader on a journey through the basic analytical tools of that theory giving some illustrative examples using regular expressions additionally new asymptotic average case results for several e nfa constructions are presented in a unified framework it turns out that asymptotically and in the average case the complexity gap between the several constructions is significantly larger than in the worst case furthermore one of the e nfa constructions approaches the corresponding e free nfa construction asymptotically and on average

the monadic quantifier alternation hierarchy over grids and graphs the monadic second order quantifier alternation hierarchy over the class of finite graphs is shown to be strict the proof is based on automata theoretic ideas and starts from a restricted class of graph like structures namely finite two dimensional grids considering grids where the width is a function of the height we prove that the difference between the levels k 1 and k of the monadic hierarchy is witnessed by a set of grids where this function is k 1 fold exponential we then transfer the hierarchy result to the class of directed or undirected graphs using an encoding technique called strong reduction it is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first order closure of existential monadic second order logic we also verify that these graph properties even belong to the complexity class nlog which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

descriptional complexity of two way pushdown automata with restricted head reversals two way nondeterministic pushdown automata 2pda are classical nondeterministic pushdown automata pda enhanced with two way motion of the input head in this paper the subclass of 2pda accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects in particular the effect of reducing the number of pushdown reversals to a constant number is of interest it turns out that this reduction leads to an exponential blow up in case of nondeterministic devices and to a doubly exponential blow up in case of deterministic devices if the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped the resulting trade offs are no longer bounded by recursive functions and so called non recursive trade offs are shown

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

alternation in two way finite automata abstract the study of alternation in two way finite automata 2 fa s has been quite non systematic since the 1970 s various authors with a variety of motivations have studied 2 fa s with various types of alternation and a variety of names creating a fairly long list of sporadic contributions with little internal consistency this article attempts to organize the subject into a single unifying framework we start with a detailed account of all contributions to date that reveals the large variety of approaches and the lack of consistency between them we then identify and name four types of automata that these contributions have really studied over the years general two way boolean finite automata 2 b fa s monotone 2 b fa s basic 2 b fa s and monotone basic or alternating 2 b fa s 2 a fa s next we identify four different ways by which authors have described how such automata compute each offering a distinct view onto their operation the circuit view where computation is modelled by a circuit of boolean gates the formula view where computation is modelled by a boolean formula over configuration variables the run view where decisions are determined by the existence of appropriate trees of configuration goal pairs called runs and the most classic tree view where computation is modelled by a tree of configurations after carefully defining each 2 b fa type and each view we prove the following first that within each type every two of the four views are equivalent to each other in the strong sense that each of them closely mimics the other at every step of the computation second that not all types of 2 b fa s are equivalent although general 2 b fa s are as powerful as monotone 2 b fa s and basic 2 b fa s are as powerful as 2 a fa s up to polynomial differences in the number of states general 2 b fa s may need exponentially fewer states than 2 a fa s

an alternating hierarchy for finite automata we study the polynomial state complexity classes 2 s k and 2 p k that is the hierarchy of problems that can be solved with a polynomial number of states by two way alternating finite automata 2afas making at most k 1 alternations between existential and universal states starting in an existential or universal state respectively this hierarchy is infinite for k 2 3 4 both 2 s k 1 and 2 p k 1 are proper subsets of 2 s k and of 2 p k since the conversion of a one way s k or p k alternating automaton with n states into a two way automaton with a smaller number of alternations requires 2 n 4 o k states the same exponential blow up is required for converting a s k bounded 2afa into a p k bounded 2afa and vice versa that is 2 s k and 2 p k are incomparable in the case of s k bounded 2afas the exponential gap applies also for intersection while in the case of p k bounded 2afas for union the same results are established for one way alternating finite automata this solves several open problems raised in c kapoutsis size complexity of two way finite automata in proc develop lang theory in lect notes comput sci vol 5583 springer verlag 2009 pp 47 66

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

separating exponentially ambiguous finite automata from polynomially ambiguous finite automata we resolve an open problem raised by ravikumar and ibarra siam j comput 18 1989 pp 1263 1282 on the succinctness of representations relating to the types of ambiguity of finite automata we show that there exists a family of nondeterministic finite automata an over a two letter alphabet such that for any positive integer n an is exponentially ambiguous and has n states whereas the smallest equivalent deterministic finite automaton has 2n states and any smallest equivalent polynomially ambiguous finite automaton has 2n 1 states

a measure and orientation preserving homeomorphism with approximate jacobian equal 1 almost everywhere we construct an almost everywhere approximately differentiable orientation and measure preserving homeomorphism of a unit n dimensional cube onto itself whose jacobian is equal to 1 almost everywhere moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms

the multidimensional luzin theorem each measurable map of an open set u subset of r n to r n is equal almost everywhere to the gradient of a continuous almost everywhere differentiable function defined on r n that vanishes together with its gradient outside u c 2007 elsevier inc all rights reserved

on the equivalence and containment problems for unambiguous regular expressions regular grammars and finite automata the known proofs that the equivalence and containment problems for regular expressions regular grammars and nondeterministic finite automata are pspace complete sm depend upon consideration of highly unambiguous expressions grammars and automata here we prove that such dependence is inherent deterministic polynomial time algorithms are presented for the equivalence and containment problems for unambiguous regular expressions unambiguous regular grammars and unambiguous finite automata the algorithms are then extended to ambiguity bounded by a fixed k our algorithms depend upon several elementary observations on the solutions of systems of homogeneous linear difference equations with constant coefficients and their relationship with the number of derivations of strings of a given length n by a regular grammar

complexity of some problems from the theory of automata 1 1 given a finite alphabet l the regular events over x are those accepted by a finite state automaton by kleene s theorem a subset w of s is a regular event if and only if it can be constructed from the finiteword sets by boolean operations together with concatenation and operation in some sense the regular events are quite simple because they are accepted by a machine with no storage capacity nevertheless in recent years much attention has been paid to special subclasses of the class of regular events in this paper we shall be concerned with star free events events of dot depth one and piecewise testable events star free events are constructed like regular events from the finite word sets but with the restriction that the operation is not allowed events of dot depth one and piecewise testable events are star free of a very simple form and will be defined below star free events have been characterized in the work of schiitzenberger 1965 in terms of their syntactic monoid algebraic characterizations of the other two classes have been given simon 1975 knast 1983 from an algorithmic point of view these algebraic characterizations do not yield efficient procedures because computing the syntactic monoid of a regular event given e g by an automaton is obviously time consuming in this paper we investigate the complexity of three problems which we now describe in the style of garey and johnson 1979 we refer the reader to aho hopcroft and ullman 1983 garey and johnson 1979 for standard concepts of complexity theory

dot depth monadic quantifier alternation and first order closure over grids and pictures this paper presents results from two different areas the first area is monadic second order logic mso over finite structures in particular over the so called grids these are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors for this logic we study the expressive power of the alternation of existential and universal monadic second order quantifiers i e set quantifiers in matz et al information and computation lics 97 1999 to appear it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness in this paper we strengthen this result in several ways firstly we show that there are mso formulas that have a very restricted form of k 1 set quantifiers but are not equivalent to a formula with k quantifiers secondly we show that if we fix the number of such alternations the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one this was previously known only for coloured grids thirdly we investigate how an additional prefix of first order i e element quantifiers influences the expressive power of mso formulas the second area that this paper is concerned with is two dimensional formal language theory we study how the alternation of first and monadic second order quantifications on the one hand relates to the dot depth measure of two dimensional i e picture languages on the other hand that measure is the two dimensional version of the classical notion of dot depth for one dimensional starfree word languages we show that the hierarchy induced by this dot depth cuts through the hierarchy given by monadic second order quantifications in particular beyond each level of the monadic second order alternation hierarchy there is a starfree picture language

finite rank harmonic operator valued functions abstract the purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a hilbert space is expressible as a harmonic matrix valued function we show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix valued function we also prove that for any natural number n a harmonic function with values in the rank n non negative operators is expressible as a matrix valued function and we give examples showing that these decomposition theorems fail when various hypotheses are relaxed

complexity of universality and related problems for partially ordered nfas abstract partially ordered nfas ponfas are nfas where cycles occur only in the form of self loops a ponfa is universal if it accepts all words over its alphabet deciding universality is pspace complete for ponfas we show that this remains true when restricting to fixed alphabets this is nontrivial since standard encodings of symbols in e g binary can turn self loops into longer cycles a lower conp complete complexity bound is obtained if all self loops in the ponfa are deterministic we find that such restricted ponfas rponfas characterize r trivial languages and establish the complexity of deciding if the language of an nfa is r trivial the limitation to fixed alphabets is essential even in the restricted case deciding universality of rponfas with unbounded alphabets is pspace complete consequently we obtain the complexity results for inclusion and equivalence problems finally we show that the languages of rponfas are definable by deterministic one unambiguous regular expressions

state complexity of operations on two way finite automata over a unary alphabet the paper determines the number of states in two way deterministic finite automata 2dfa over a one letter alphabet sufficient and in the worst case necessary to represent the results of basic language theoretic operations on 2dfas with a certain number of states it is proved that i intersection of an m state 2dfa and an n state 2dfa requires between m n and m n 1 states ii union of an m state 2dfa and an n state 2dfa between m n and 2m n 4 states iii kleene star of an n state 2dfa g n o n 2 states where g n e 1 o 1 n l n n is the maximum value of lcm p 1 p k for p i

determination of finite automata accepting subregular languages we investigate the descriptional complexity of the nondeterministic finite automaton nfa to the deterministic finite automaton dfa conversion problem for automata accepting subregular languages such as combinational languages definite languages and variants thereof strictly locally testable languages star free languages ordered languages prefix suffix and infix closed languages and prefix suffix and infix free languages most of the bounds for the conversion problem are shown to be tight in the exact number of states that is the number is sufficient and necessary in the worst case otherwise tight bounds in order of magnitude are shown

classification of finite monoids the language approach abstract the concept of a variety of congruences is introduced and related to variety of languages and variety of monoids a systematic construction of increasingly complex varieties of congruences is presented this construction is powerful enough to generate all monoids containing solvable groups some hierarchies occurring through this process are shown to correspond to well known hierarchies of monoids thus indicating that our construction is natural from an algebraic point of view some problems that remain open are also discussed

ambiguity and structural ambiguity of symmetric difference nfas abstract okhotin showed an exponential trade off in the conversion from nondeterministic unary automata to unambiguous nondeterministic unary automata we show that the trade off in the case of unary symmetric difference automata to finitely structurally ambiguous unary symmetric difference automata is linear with constant 1 in the number of states in particular for every n state unary nondeterministic symmetric difference automaton there is an equivalent finitely structurally ambiguous n state unary symmetric difference nondeterministic automaton we also consider the complexity of deciding unambiguity for k deterministic finite automata and investigate the interplay between ambiguity and structural ambiguity

deterministic generalized automata a generalized automaton ga is a finite automaton where the single transitions are defined on words rather than on single letters generalized automata were considered by k hashiguchi who proved that the problem of calculating the size of a minimal ga is decidable

some new results about the geometry of sets of finite perimeter we establish that the intrinsic distance d e associated with an indecomposable plane set e of finite perimeter is infinitesimally euclidean namely in e by this result we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential we also provide some applications to complex analysis moreover we present a collection of results that would seem to suggest the possibility of developing a de rham cohomology theory for integral currents

nondeterministic complexity in subclasses of convex languages abstract we study the nondeterministic state complexity of basic regular operations on the classes of prefix suffix factor and subword free closed and convex regular languages and on the classes of right left two sided and all sided ideal regular languages for the operations of concatenation intersection union reversal star and complementation we get tight upper bounds for all considered classes except for complementation on factor and subword convex languages most of our witnesses are described over optimal alphabets the description of a proper suffix convex language over a five letter alphabet meeting the upper bound 2 n for complementation and obtaining an asymptotically tight bound n for complementation of unary prefix free languages are among the most interesting results of this paper

relating the type of ambiguity of finite automata to the succinctness of their representation we consider the problem of how the size of a nondeterministic finite automaton nfa representing a regular language depends on the degree of ambiguity of the nfa we obtain results for the unary and bounded inputs and partial results for the unrestricted inputs one of the main results of this paper shows that for unrestricted inputs deterministic unambiguous and nondeterministic machines form a hierarchy with respect to the number of states solving an open problem of stearns and hunt we also propose a new approach to the study of the succinctness of representation through regularity preserving closure properties and obtain some results in this direction

improved combinatorial group testing algorithms for real world problem sizes we study practically efficient methods for performing combinatorial group testing we present efficient nonadaptive and two stage combinatorial group testing algorithms which identify the at most d items out of a given set of n items that are defective using fewer tests for all practical set sizes for example our two stage algorithm matches the information theoretic lower bound for the number of tests in a combinatorial group testing regimen

unambiguous finite automata over a unary alphabet nondeterministic finite automata nfa with at most one accepting computation on every input string are known as unambiguous finite automata ufa this paper considers ufas over a one letter alphabet and determines the exact number of states in dfas needed to represent unary languages recognized by n state ufas in terms of a new number theoretic function g the growth rate of g n and therefore of the ufa dfa tradeoff is estimated as e q n l n 2 n 3 the conversion of an n state unary nfa to a ufa requires ufas with g n o n 2 e 1 o 1 n l n n states where g n is the greatest order of a permutation of n elements known as landau s function in addition it is shown that representing the complement of n state unary ufas requires ufas with at least n 2 o 1 states in the worst case while the kleene star requires up to exactly n 1 2 1 states

state complexity of the concatenation of regular tree languages we consider the state complexity of basic concatenation operations for regular tree languages we show that the sequential respectively parallel concatenation of tree languages recognized by deterministic bottom up automata with m and n states can be recognized by an automaton with n 1 m 2 n 2 n 1 1 respectively m 2 n 2 n 1 1 states and establish matching state complexity lower bounds the bound for sequential concatenation of tree languages differs by an order of magnitude from the corresponding bound for regular string languages

state complexity of inversion operations the reversal operation is well studied in the literature and the deterministic respectively nondeterministic state complexity of reversal is known to be 2 n respectively n we consider the inversion operation where some substring of the given string is reversed formally the inversion respectively prefix inversion of a language l consists of all strings u x r v such that u x v l respectively all strings u r x where u x l we show that the nondeterministic state complexity of prefix inversion is n 2 and that of inversion is n 3 we show that the deterministic state complexity of prefix inversion is at most 2 n log n n and has lower bound 2 n log n the same lower bound holds for the state complexity of inversion but for inversion we do not have a matching upper bound we also study the state complexity of other variants of the inversion operation

a new algorithm for regularizing one letter context free grammars constructive methods for obtaining regular grammar counterparts for some sub classes of context free grammars cfgs have been investigated by many researchers an important class of grammars for which this is always possible is the one letter cfg we show in this paper a new constructive method for transforming an arbitrary one letter cfg to an equivalent regular expression of star height 0 or 1 our new result is considerably simpler than a previous construction by leiss and we also propose a new normal form for a regular expression with only a single star occurrence through an alphabet factorization theorem we show how to go beyond the one letter cfg in a straight forward way

on the existence of prime decompositions we investigate factorizations of regular languages in terms of prime languages a language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps we give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition we show that there exist non regular unary languages that do not have prime decompositions we also consider infinite factorizations of unary languages

converting nondeterministic automata and context free grammars into parikh equivalent one way and two way deterministic automata we investigate the conversion of one way nondeterministic finite automata and context free grammars into parikh equivalent one way and two way deterministic finite automata from a descriptional complexity point of view we prove that for each one way nondeterministic automaton with n states there exist parikh equivalent one way and two way deterministic automata with e o n l n n and p n states respectively where p n is a polynomial furthermore these costs are tight in contrast if all the words accepted by the given automaton contain at least two different letters then a parikh equivalent one way deterministic automaton with a polynomial number of states can be found concerning context free grammars we prove that for each grammar in chomsky normal form with h variables there exist parikh equivalent one way and two way deterministic automata with 2 o h 2 and 2 o h states respectively even these bounds are tight

on the primality of n 1 and 2 3 5 p 1 for each prime p let p be the product of the primes less than or equal to p we have greatly extended the range for which the primality of n 1 and p 1 are known and have found two new primes of the first form 6380 1 6917 1 and one of the second 42209 1 we supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found

state complexity of permutation on finite languages over a binary alphabet abstract the set of all strings parikh equivalent to a string in a language l is called the permutation of l the permutation of a finite n state dfa deterministic finite automaton language over a binary alphabet can be recognized by a dfa with n 2 n 2 2 states we show that if the language consists of equal length binary strings the bound can be improved to f n n 2 n 1 3 and for every n congruent to 1 modulo 3 there exists an n state dfa a recognizing a set of equal length strings such that the minimal dfa for the permutation of l a needs f n states

on nfas where all states are final initial or both we examine questions involving nondeterministic finite automata where all states are final initial or both initial and final first we prove hardness results for the nonuniversality and inequivalence problems for these nfas next we characterize the languages accepted finally we discuss some state complexity problems involving such automata

oblivious two way finite automata decidability and complexity we investigate the descriptional complexity and decidability of obliviousness for two way finite automata in particular we consider the simulation of two way deterministic finite automata 2dfas by oblivious 2dfas the simulation of oblivious 2dfas by sweeping 2dfas and one way nondeterministic finite automata 1nfas as well as the simulation of sweeping 2dfas by 1nfas in all cases exponential lower bounds on the number of states are obtained for languages over an alphabet with at most four letters finally a procedure for deciding obliviousness of an arbitrary 2dfa is given and moreover the problem is shown to be pspace complete

optimal simulations between unary automata we consider the problem of computing the costs in terms of states of optimal simulations between different kinds of finite automata recognizing unary languages our main result is a tight simulation of unary n state two way nondeterministic automata by o rm e sqrt n ln n state one way deterministic automata in addition we show that given a unary n state two way nondeterministic automaton one can construct an equivalent o n2 state two way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape further results on simulating unary one way alternating finite automata are also discussed

finite automata and unary languages abstract we prove that o e n log n states are sufficient to simulate an n state 1nfa recognizing a unary language by a 1dfa the lower bound is the same similar tight bounds are shown for the simulation of a 2dfa by a 1dfa and a 1nfa we also show that o n 2 states are sufficient and necessary to simulate an n state 1nfa recognizing a unary language by a 2dfa

harnack s theorem for harmonic compact operator valued functions abstract in this paper we show that harmonic compact operator valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis we also give an example which shows that an operator valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator valued function we use our harmonic matrix coefficients characterization to establish a harnack s theorem for an increasing sequence of harmonic compact self adjoint operator valued functions and we show that this harnack s theorem need not hold when the compactness restriction is dropped

operational state complexity of unary nfas with finite nondeterminism we study the state complexity of language operations for unary nfas with limited nondeterminism we consider the boolean operations concatenation and kleene star we give upper bounds for the state complexity of these language operations and lower bounds that are fairly close to the upper bounds our constructions rely on the fact that minimal unary nfas with limited nondeterminism can be found in chrobak normal form for most measures of nondeterminism the measures of nondeterminism which are considered here with the above property are tree width advice and trace

optimal simulation of self verifying automata by deterministic automata self verifying automata are a special variant of finite automata with a symmetric kind of nondeterminism we study the conversion of self verifying automata to deterministic automata from a descriptional complexity point of view the main result is the exact cost in terms of the number of states of such a simulation

magic numbers in the state hierarchy of finite automata a number d is magic for n if there is no regular language for which an optimal nondeterministic finite state automaton nfa uses exactly n states and at the same time the optimal deterministic finite state automaton dfa uses exactly d states we show that in the case of unary regular languages the state hierarchy of dfa s for the family of languages accepted by n state nfa s is not contiguous there are some holes in the hierarchy i e magic numbers in between values that are not magic this solves for automata with a single letter input alphabet an open problem of existence of magic numbers actually most of the numbers is magic in the unary case as an additional bonus we also get a new universal lower bound for the conversion of unary d state dfa s into equivalent nfa s nondeterminism does not reduce the number of states below log 2d not even in the best case

distributions for which div v f has a continuous solution the equation div upsilon f has a continuous weak solution in an open set u subset of r m if and only if the distribution f satisfies the following condition the f phi i converge to 0 for every sequence phi i of test functions such that the support of each i is contained in a fixed compact subset of u and in the l 1 norm phi i converges to 0 and del phi i is bounded c 2007 wiley periodicals inc

deciding determinism of unary languages in this paper we investigate the complexity of deciding determinism of unary languages first we give a method to derive a set of arithmetic progressions from a regular expression e over a unary alphabet and establish relations between numbers represented by these arithmetic progressions and words in l e next we define a problem relating to arithmetic progressions and investigate the complexity of this problem then by a reduction from this problem we show that deciding determinism of unary languages is conp complete finally we extend our derivation method to expressions with counting and prove that deciding whether an expression over a unary alphabet with counting defines a deterministic language is in 2 p we also establish a tight upper bound for the size of the minimal dfa for expressions with counting

the k th prime is greater than k ln k ln ln k 1 for k 2 rosser and schoenfeld have used the fact that the first 3 500 000 zeros of the riemann zeta function lie on the critical line to give estimates on x and x with an improvement of the above result by brent et al we are able to improve these estimates and to show that the k th prime is greater than k ln k ln ln k 1 for k 2 we give further results without proof

lower bounds for the size of deterministic unranked tree automata tree automata operating on unranked trees use regular languages called horizontal languages to define the transitions of the vertical states that define the bottom up computation of the automaton it is well known that the deterministic tree automaton with smallest total number of states that is number of vertical states and number of states used to define the horizontal languages is not unique and it is hard to establish lower bounds for the total number of states by relying on existing bounds for the size of unambiguous finite automata we give a lower bound for the size blow up of determinizing a nondeterministic unranked tree automaton the lower bound improves the earlier known lower bound that was based on an ad hoc construction

the parallel complexity of finite state automata problems the goal of this paper is to study the exact complexity of several important problems concerning finite state automata and to classify the degrees of ambiguity of nondeterministic finite state automata our results are as follows 1 minimization of deterministic finite automata is nc 1 complete for nl 2 testing whether the degree of ambiguity of a nondeterministic finite automaton is exponential or polynomial or bounded is nc 1 complete for nl 3 checking whether a given nondeterministic finite automaton is unambiguous or k ambiguous is nc 1 complete for nl where k is some fixed constant 4 the bounded nonuniversality problem for nondeterministic finite automata which is the problem of deciding whether l m s n s n for a given nondeterministic finite automaton m and a unary integer n is log space complete for np 5 the bounded nonuniversality problem for unambiguous finite automata is in det the class of problems nc 1 reducible to computing the determinants of integer matrices and for deterministic finite automata it is nc 1 complete for nl 6 the inequivalence problems for unambiguous and k ambiguous finite automata are both in det where k is some fixed constant

nondeterministic state complexity of star free languages we investigate the nondeterministic state complexity of several operations on finite automata accepting star free and unary star free languages it turns out that in most cases exactly the same tight bounds as for general regular languages are reached this nicely complements the results recently obtained by brzozowski and liu 2011 8 for the operation problem of star free and unary star free languages accepted by deterministic finite automata

removing nondeterminism in constant height pushdown automata we study the descriptional cost of removing nondeterminism in constant height pushdown automata i e pushdown automata with a built in constant limit on the height of the pushdown we show a double exponential size increase when converting a constant height nondeterministic pushdown automaton into an equivalent deterministic device moreover we prove that such a double exponential blow up cannot be avoided by certifying its optimality as a direct consequence we get that eliminating nondeterminism in classical finite state automata is single exponential even with the help of a constant height pushdown store

minimal nfa problems are hard finite automata fa s are of fundamental importance in theory and in applications the following basic minimization problem is studied given a dfa deterministic fa find a minimum equivalent no

on the power of las vegas ii two way finite automata abstract the investigation of the computational power of randomized computations is one of the central tasks of complexity and algorithm theory while for one way finite automata the power of different computational modes was successfully determined one does not have any nontrivial result relating the power of determinism las vegas and nondeterminism for two way finite automata the main results of this paper are as follows i if for a regular language l there exist small two way nondeterministic finite automata for both l and l then there exists a small two way las vegas finite automaton for l ii there is a quadratic gap between nondeterminism and las vegas for two way finite automata iii for every k n there is a regular language s k such that s k can be accepted by a two way las vegas finite automaton with o k states but every two way deterministic finite automaton recognizing s k has at least k 2 log 2 k states