automata boolean matrices and ultimate periodicity abstract this paper presents a boolean matrix based method to automata theory with an application to the study of regularity preserving functions a new characterization of such functions is derived in terms of the property of ultimate periodicity with respect to powers of boolean matrices this characterization reveals the intrinsic algebraic nature of regularity preserving functions it facilitates a concise proof of known as well as previously unknown properties of regularity preserving functions leading to the solution of the subtraction problem left open by kosaraju

order sorted model theory for temporal executable specifications abstract an order sorted temporal programming paradigm is presented it consists of a typed modular declarative language its associated order sorted temporal horn clause logic basis and a model theory generalizing order sorted algebras with predicates to temporal order sorted structures the essence of this generalization is in time dependent interpretation of predicates so that a temporal order sorted model amounts to a sequence of order sorted equational models with predicates one per each state the main advantage of the presented paradigm in comparison with paradigms based on horn clause logic with equality is that it is more expressive particularly so in representing properly state transitions and other event oriented temporal behavioral properties of objects at the same time the generalized paradigm is proved to have the initial model semantics the rules for temporal order sorted deduction are established as an appropriate generalization of the rules for order sorted horn clause logic with equality the initial model is a quotient temporal order sorted structure constructed from the initial temporal order sorted structure and a congruence relation derived from a given set of temporal constraints temporal order sorted model theoretic properties are also naturally established for temporal queries the temporal constraint language has an execution model and it is intended to be a basis for a prototyping tool for complex typed modular software systems

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

a modular order sorted equational generalization algorithm generalization also called anti unification is the dual of unification given terms t and t a generalizer is a term t of which t and t are substitution instances the dual of a most general unifier mgu is that of least general generalizer lgg in this work we extend the known untyped generalization algorithm to first an order sorted typed setting with sorts subsorts and subtype polymorphism second we extend it to work modulo equational theories where function symbols can obey any combination of associativity commutativity and identity axioms including the empty set of such axioms and third to the combination of both which results in a modular order sorted equational generalization algorithm unlike the untyped case there is in general no single lgg in our framework due to order sortedness or to the equational axioms instead there is a finite minimal and complete set of lggs so that any other generalizer has at least one of them as an instance our generalization algorithms are expressed by means of inference systems for which we give proofs of correctness this opens up new applications to partial evaluation program synthesis and theorem proving for typed equational reasoning systems and typed rule based languages such as asf sdf elan obj cafe obj and maude

the computational complexity of distance functions of two dimensional domains we study the computational complexity of the distance function associated with a polynomial time computable two dimensional domains in the context of the turing machine based complexity theory of real functions it is proved that the distance function is not necessarily computable even if a two dimensional domain is polynomial time recognizable on the other hand if both the domain and its complement are strongly polynomial time recognizable then the distance function is polynomial time computable if and only if p np

rewriting extended regular expressions we concider an extened algebra of regular events languages with intersection besides the usual operations this algebra has the structure of a distributive lattice with monotonic operations the latter property is crucial for some applications we give a new complete horn equational axiomatiztion of the algebra and develop some termrewriting techniques for constructing logical inferences of valid equations

order sorted algebra i equational deduction for multiple inheritance overloading exceptions and partial operations abstract this paper generalizes many sorted algebra msa to order sorted algebra osa by allowing a partial ordering relation on the set of sorts this supports abstract data types with multiple inheritance in roughly the sense of object oriented programming several forms of polymorphism and overloading partial operations as total on equationally defined subsorts exception handling and an operational semantics based on term rewriting we give the basic algebraic constructions for osa including quotient image product and term algebra and we prove their basic properties including quotient homomorphism and initiality theorems the paper s major mathematical results include a notion of osa deduction a completeness theorem for it and an osa birkhoff variety theorem we also develop conditional osa including initiality completeness and mckinsey malcev quasivariety theorems and we reduce osa to conditional msa which allows lifting many known msa results to osa retracts which intuitively are left inverses to subsort inclusions provide relatively inexpensive run time error handling we show that it is safe to add retracts to any osa signature in the sense that it gives rise to a conservative extension a final section compares and contrasts many different approaches to osa this paper also includes several examples demonstrating the flexibility and applicability of osa including some standard benchmarks like stack and list as well as a much more substantial example the number hierarchy from the naturals up to the quaternions

on classes of tractable unrestricted regular expressions abstract it is known that the equivalence problem for the class ure of unrestricted regular expressions is not elementary recursive this implies that constructing the corresponding reduced automaton is not tractable for some expressions in ure we say these expressions are of intractable complexity however the closure of boolean automata under efficient complementation suggests that there are large nontrivial sub classes of ure with arbitrary level of complementation and tractable complexity in this paper we introduce the class c l corresponding in a natural way to boolean automata we also consider other related subclasses of ure all these classes have tractable complexity these results suggest a revised explanation why certain unrestricted regular expressions are so hard

group axioms for iteration iteration theories provide a sound and complete axiomatization of the equational properties of the iteration or fixed point operation in many models of theoretical computer science including ordered and metric structures trees and synchronization trees all known equational axiomatizations of iteration theories consist of a small set of equational axioms for conway theories and a complicated equation scheme the commutative identity here we associate an identity with each finite semigroup we prove that the set consisting of the conway identities and the group identities associated with the finite simple groups is complete moreover we prove that the conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite simple group divides one of the semigroups in the subclass we also formulate a conjecture and study its consequences the results are a generalization of krob s axiomatization of the equational theory of the regular sets

the multiplicative fragment of the yanov equational theory we give a finite equational axiomatization for free identities of regular languages which contain the empty word the axioms for the whole equational theory of such languages were given by yanov

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

an equational axiomatization for multi exit iteration abstract this paper presents an equational axiomatization of bisimulation equivalence over the language of basic process algebra bpa with multi exit iteration multi exit iteration is a generalization of the standard binary kleene star operation that allows for the specification of agents that up to bisimulation equivalence are solutions of systems of recursion equations of the form x 1 def p 1 x 2 q 1 x n def p n x 1 q n wherenis a positive integer and thepiand theqiare process terms the addition of multi exit iteration to bpa yields a more expressive language than that obtained by augmenting bpa with the standard binary kleene star bpa as a consequence the proof of completeness of the proposed equational axiomatization for this language although standard in its general structure is much more involved than that for bpa an expressiveness hierarchy for the family ofk exit iteration operators proposed by bergstra bethke and ponse is also offered

principles of proof scores in cafeobj this paper describes the theoretical principles of a verification method with proof scores in the cafeobj algebraic specification language the verification method focuses on specifications with conditional equations and realizes systematic theorem proving or interactive verification the method is explained using a simple but instructive example and the necessary theoretical foundations which justify every step of the verification are described with precise mathematical definitions some important theorems that result from the definitions are also presented

completeness of park induction abstract the in equational properties of iteration i e least pre fixed point solutions over cpo s are captured by the axioms of iteration theories all known axiomatizations of iteration theories consist of the conway identities and a complicated equation scheme the commutative identity the results of this paper show that the commutative identity is implied by the conway identities and a weak form of the park induction principle hence we obtain a simple first order axiomatization of the in equational theory of iteration it follows that a few simple identities and a weak form of the scott induction principle formulated to involve only inequations are also complete we also show that the conway identities and the park induction principle are not complete for the universal horn theory of iteration

multimodal logic programming using equational and order sorted logic in previous work 2 we proposed a method for automated modal theorem proving based on algebraic concepts and equational techniques basically it uses the translation of modal logic into a specially tailored multi sorted logic called path logic in this paper we extend the method for multi modal logic and apply it to multi modal logic programming the multi modal systems we consider are arbitrary mixing of first order modal systems of type kd kt kd4 or kt4 with interaction axioms of the form ia ja roughly with each modal subsystem is associated a sort in path logic and a specific set of equations and the interaction axioms are captured by the order relation between sorts hence again all the modal logical rules are coded in the unification algorithm if one considers horn clauses in the usual sense we get a logic programming system pathlog for which standard theoretical results apply in pathlog one can either write programs directly in the language of path logic or in various multi modal logics the modal formulas being automatically translated and put in clausal form for instance we obtain as a particular case a system for temporal logic programming which subsumes templog of 1 and whose completeness results immediately from our general theorems

on generalized language equations a system of generalized language equations over an alphabet a is a set of n equations in n variables xi gi x1 xn i 1 n where the gi are functions from p a n into p a i 1 n p a denoting the set of all languages over a furthermore the gi are expressible in terms of set operations concatenations and stars which involve the variable xi as well as certain mixed languages in this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations furthermore we show that a solution is regular if all fixed languages are regular

axiomatizing the identities of binoid languages we present a nontrivial axiomatization for the equational theory of binoid languages the subsets of a free binoid in doing so we prove that a conjecture given in our previous paper was true the identical laws of ordinary string languages written separately using horizontal and vertical operation symbols form a required complete system of axioms

strict coherence of conditional rewriting modulo axioms conditional rewriting modulo axioms with rich types makes specifications and declarative programs very expressive and succinct and is used in all well known rule based languages however the current foundations of rewriting modulo axioms have focused for the most part on the unconditional and untyped case the main purpose of this work is to generalize the foundations of rewriting modulo axioms to the conditional order sorted case a related goal is to simplify such foundations in particular even in the unconditional case the notion of strict coherence proposed here makes rewriting modulo axioms simpler and easier to understand properties of strictly coherent conditional theories like operational equi termination of the r b and r b relations and general conditions for the conditional churchrosser property modulo b are also studied

unrestricted complementation in language equations over a one letter alphabet recently the complete solution of systems of language equations over a one letter alphabet has been derived where the operators are union concatenation and star this paper addresses the question of adding the complementation operator it is known that if no stars are present and if the concatenation operator is restricted such equations need not have solutions but if they do they can be solved explicitly if the constants are regular in which case the solutions are also regular in this paper we study language equations over a one letter alphabet where unrestricted concatenation interacts with complementation union and star we define a procedure for determining a regular solution of the general type of equation assuming that all constants are regular discuss its properties and show examples we then prove that in general equations with complementation and complementation and concatenation in which all constants are regular may have nonregular solutions this shows that language equations can be used to obtain non context free languages from regular languages using only operators under which the regular languages are closed

specification and proof in membership equational logic this paper is part of a long term effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theorem proving tools can be based in particular our rewriting techniques provide semantic foundations for maude s functional sublanguage where they have been efficiently implemented

relating casl with other specification languages the institution level in this work we investigate various specification languages and their relation to casl the recently developed common algebraic specification language in particular we consider the languages larch obj3 and functional cafeobj act one asf and hep theories as well as various sublanguages of casl all these languages are translated to an appropriate sublanguage of casl the translation mainly concerns the level of specification in the small the logics underlying the languages are formalized as institutions and representations among the institutions are developed however it is also considered how these translations interact with specification in the large thus we obtain on the one hand translations of any of the above mentioned specification languages to an appropriate sublanguage of casl this allows us to take libraries and case studies that have been developed for other languages and re use them in casl on the other hand we set up institution representations going from the casl institution and some of its subinstitutions to simpler subinstitutions given a theorem proving tool for such a simpler subinstitution with the help of a representation it can also be used for a more complex institution thus first order theorem provers and conditional term rewriting tools become usable for casl

equational properties of kleene algebras of relations with conversion abstract we describe a set of equational axioms for the variety generated by all algebras of binary relations with operations of union composition conversion and reflexive transitive closure and neutral elements 0 the empty relation and 1 the identity relation

a causal semantics for ccs via rewriting logic we consider two operational semantics for ccs defined in the literature the first exploits proved transition systems pts and the second rewriting logic rl we show that the interleaving interpretation of both semantics agree in that they define the same transitions and exhibit the same non deterministic structure in addition we study causality in ccs computations we recall its treatment via pts exhibiting the notion of causality presented in the literature and we show how to recast it in the rl semantics via suitable axioms also in this case the two semantics agree

on equations for union free regular languages in this paper we consider the variety uf generated by all algebras of binary relations equipped with the operations of composition reflexive transitive closure and the empty set and the identity relation as constants this variety coincides with the variety generated by the union free reducts of kleene algebras of languages and its free objects are formed by union free regular languages that is regular languages represented by regular expressions having no occurrence of we show that the variety uf is not finitely based the situation does not change if we consider the variety uf generated by the above algebras of binary relations equipped with the conversion operation

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

on dynamic algebras abstract dynamic algebras are the lindenbaum tarski algebras of dynamic logics these algebras can be considered as boolean algebras with some operators indexed by the elements of some kleene algebra in this paper we prove that there are infinitely many finitely generated varieties of dynamic algebras having undecidable equational theories all these varieties are generated by representable dynamic algebras

unions of non disjoint theories and combinations of satisfiability procedures in this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability we describe a general combination method which given a procedure that decides constraint satisfiability with respect to a constraint theory t1 and one that decides constraint satisfiability with respect to a constraint theory t2 produces a procedure that semi decides constraint satisfiability with respect to the union of t1 and t2 we provide a number of model theoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete with special emphasis on the case in which the signatures of the component theories are non disjoint we also describe some general classes of theories to which our combination results apply and relate our approach to some of the existing combination methods in the field

language equations over a one letter alphabet with union concatenation and star a complete solution abstract this paper develops a theory of language equations over a one letter alphabet where the operations involved in the equations are union unrestricted concatenation and star several key theorems are proven and tied together to provide an algorithm which enables one to solve any equation of this type an application of this result is a constructive proof of the well known fact that any context free grammar with only one terminal always generates a regular language

left handed completeness abstract we give a new proof of the completeness of the left handed star rule of kleene algebra the proof is significantly shorter than previous proofs and exposes the rich interaction of algebra and coalgebra in the theory of kleene algebra

semantics of temporal classes a model theory of a typed declarative temporal object oriented language system is presented the declarative nature of the language makes it very different from the dominating procedural strongly typed object oriented programming languages in this declarative system methods are specified in a high level temporal constraint language two fundamental properties of these constraints are that they have an execution model and algebraic semantics the model theory is based on temporal order sorted algebras with predicates a variety of orderings are explored in order to represent various types of inheritance as well as the subtyping discipline temporal classes are viewed as temporal theories and some inheritance relationships as morphisms of temporal theories a model of a temporal class is a temporal order sorted structure with predicates which satisfies a set of temporal constraints specified in that class morphisms of those models are naturally required to preserve type coercions a distinguished model of a temporal theory is constructed as a colimit of a suitably defined functor this colimit construction reflects the temporal nature of the paradigm and generalizes the classical initial algebra semantics in contradistinction to major difficulties in developing a model theory for full fledged typed procedural object oriented languages this paper shows that such a task becomes possible for a suitably defined declarative object oriented language this in particular leads to model theoretic results on the preservation of the behavioral properties in the inheritance hierarchies

the variety of kleene algebras with conversion is not finitely based given an arbitrary set a one obtains the full kleene algebra of binary relations over a by considering the operations of union composition reflexive transitive closure conversion and the empty set and the identity relation as constants such algebras generate the variety of kleene algebras with conversion as a result of a general analysis of identities satisfied by varieties having an involution operation we prove that the variety of kleene algebras with conversion has no finite equational axiomatization in our argument we make use of the fact that the variety of kleene algebras without conversion is not finitely based and that relatively to this variety the variety of kleene algebras with conversion is finitely axiomatized

order sorted inductive types abstract system f is an extension of system f with subtyping and bounded quantification order sorted algebra is an extension of many sorted algebra with overloading and subtyping we combine both formalisms to obtain if a higher order typed calculus with subtyping bounded quantification and order sorted inductive types i e data types with built in subtyping and overloading moreover we show that if enjoys important meta theoretic properties including confluence strong normalization subject reduction and decidability of type checking

on a question of a salomaa the equational theory of regular expression salomaa 1969 p 143 asked whether the equational theory of regular expressions over a singleton alphabet has a finite equational base in this paper we provide a negative answer to this long standing question the proof of our main result rests upon a model theoretic argument for every finite collection of equations that are sound in the algebra of regular expressions over a singleton alphabet we build a model in which some valid regular equation fails the construction of the model mimics the one used by conway 1971 p 105 in his proof of a result originally due to redko to the effect that infinitely many equations are needed to axiomatize equality of regular expressions our analysis of the model however needs to be more refined than the one provided by conway 1971

observer design for max linear systems this technical note deals with the state estimation for max plus linear systems this estimation is carried out following the ideas of the observer method for classical linear systems the system matrices are assumed to be known and the observation of the input and of the output is used to compute the estimated state the observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring

axiomatizing rational power series over natural numbers iteration semi rings are conway semi rings satisfying conway s group identities we show that the semi rings n r a t of rational power series with coefficients in the semi ring n of natural numbers are the free partial iteration semi rings moreover we characterize the semi rings n r a t as the free semi rings in the variety of iteration semi rings defined by three additional simple identities where n is the completion of n obtained by adding a point of infinity we also show that this latter variety coincides with the variety generated by the complete or continuous semirings as a consequence of these results we obtain that the semi rings n r a t equipped with the sum order are free in the class of symmetric inductive semi rings this characterization corresponds to kozen s axiomatization of regular languages

partial derivatives of regular expressions and finite automaton constructions we introduce a notion of a partial derivative of a regular expression it is a generalization to the non deterministic case of the known notion of a derivative invented by brzozowski we give a constructive definition of partial derivatives study their properties and employ them to develop a new algorithm for turning regular expressions into relatively small nfa and to provide certain improvements to brzozowski s algorithm constructing dfa we report on a prototype implementation of our algorithm constructing nfa and present some examples

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

implicit language equations existence and uniqueness of solutions we introduce a new type of language equation namely implicit equations and derive a number of properties which clearly demonstrate that the behavior of implicit equations differs substantially from that of conventional or explicit ones explicit language equations express each variable as a regular expression in terms of their variables and constant languages implicit language equations express constant languages as regular expressions in terms of their variables and constant languages an algorithm is given for determining a solution of a given system of implicit language equations if one exists provided all constant languages are regular and the operations permitted in the regular expressions are unrestricted union and concatenation from the left by a constant a characterization of the uniqueness of a solution of such a system and a method for determining all solutions are also provided

the bisimulation problem for equational graphs of finite out degree the bisimulation problem for equational graphs of finite out degree is shown to be decidable we reduce this problem to the eta bisimulation problem for deterministic rational vectors of boolean series on the alphabet of a deterministic pushdown automaton cal m we then exhibit a complete formal system for deducing equivalent pairs of such vectors

equational theories of tropical semirings this paper studies the equational theories of various exotic semirings presented in the literature exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum and addition as product two prime examples of such structures are the max semiring and the tropical semiring it is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations and that similar results hold for the commutative idempotent weak semirings that underlie them for each of these commutative idempotent weak semirings the paper offers characterizations of the equations that hold in them decidability results for their equational theories explicit descriptions of the free algebras in the varieties they generate and relative axiomatization results

an extension problem related to the fractional branson gover operators abstract the branson gover operators are conformally invariant differential operators of even degree acting on differential forms they can be interpolated by a holomorphic family of conformally invariant integral operators called fractional branson gover operators for euclidean spaces we show that the fractional branson gover operators can be obtained as dirichlet to neumann operators of certain conformally invariant boundary value problems generalizing the work of caffarelli silvestre for the fractional laplacians to differential forms the relevant boundary value problems are studied in detail and we find appropriate sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties

inductive semirings one of the most well known induction principles in computer science is the fixed point induction rule or least pre fixed point rule inductive semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms inductive semirings are extensions of continuous semirings and the kleene algebras of conway and kozen we develop in a systematic way the rudiments of the theory of inductive semirings in relation to automata languages and power series in particular we prove that if s is an inductive semiring then so is the semiring of matrices sn n for any integer n 0 and that if s is an inductive semiring then so is any semiring of power series s a as shown by kozen the dual of an inductive semiring may not be inductive in contrast we show that the dual of an iteration semiring is an iteration semiring kuich proved a general kleene theorem for continuous semirings and bloom and esik proved a kleene theorem for all conway semirings since any inductive semiring is a conway semiring and an iteration semiring as we show there results a kleene theorem applicable to all inductive semirings we also describe the structure of the initial inductive semiring and conjecture that any free inductive semiring may be given as a semiring of rational power series with coefficients in the initial inductive semiring we relate this conjecture to recent axiomatization results on the equational theory of the regular sets

a complete system of identities for one letter rational expressions with multiplicities in the tropical semiring abstract we construct a complete system of rational identities for the equational theory of the algebra of one letter rational expressions with multiplicities in the tropical semiring

computational complexity of two dimensional regions the computational complexity of bounded sets of the two dimensional plane is studied in the discrete computational model we introduce four notions of polynomial time computable sets in r 2 and study their relationship the computational complexity of the winding number problem membership problem distance problem and area problem is characterized by the relations between discrete complexity classes of the np theory

conditional rewriting logic as a unified model of concurrency abstract rewriting with conditional rewrite rules modulo a set e of structural axioms provides a general framework for unifying a wide variety of models of concurrency concurrent rewriting coincides with logical deduction in conditional rewriting logic a logic of actions whose models are concurrent systems this logic is sound and complete and has initial models in addition to general models interpreted as concurrent systems which provide a more operational style of semantics more restricted semantics with an incresingly denotational flavor such as preorder poset cpo and standard algebraic models appear as special cases of the model theory this permits dealing with operational and denotational issues within the same model theory and logic a programming language called maude whose modules are rewriting logic theories is defined and given denotational and operational semantics maude provides a simple unification of concurrent programming with functional and object oriented programming and supports high level declarative programming of concurrent systems

generalized language equations with multiple solutions abstract a generalized language equation x f x is an equation in a variable x over an alphabet a where the language function f is expressible in terms of arbitrary set operations including complementation and concatenation from the left by a constant i e a language in previous work these equations were studied provided an easily verifiable condition was met in this case solutions always exist and are always unique in the present paper the condition is dropped as a consequence our equations may have zero one or more up to uncountably infinitely many solutions we give a representation theorem for these equations and use it to classify them for equations in one variable a complete classification is given including parametric representations of all solutions for systems of m equations in m variables the notion of 1 reducible substitution graph is introduced and employed to reduce the problem of existence of solutions of the entire system to the same problem for a single equation the paper concludes with a discussion of the implications if one assumes that all constants are regular languages and poses some open problems