needed reduction and spine strategies for the lambda calculus abstract a redex r in a lambda term m is called needed if in every reduction of m to normal form some residual of r is contracted among others the following results are proved 1 r is needed in m iff r is contracted in the leftmost reduction path of m 2 let r m0 m1 m2 reduce redexes ri mi mi 1 and have the property that i j i rj is needed in mj then r is normalising i e if m0 has a normal form then r is finite and terminates at that normal form 3 neededness is an undecidable property but has several efficiently decidable approximations various versions of the so called spine redexes

infinitary lambda calculus in a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems in this paper we perform the same task for the lambda calculus from the viewpoint of infinitary rewriting the bohm model of the lambda calculus can be seen as an infinitary term model in contrast to term rewriting there are several different possible notions of infinite term which give rise to different bohm like models which embody different notions of lazy or eager computation

the greatest fixed points and rational omega tree languages abstract rational omega tree languages are defined by extending the notion of rational or regular expressions for languages and their properties are investigated they are characterized by means of greatest fixed points of certain functions and are shown to be precisely the omega tree languages recognized by rabin s special automata a simple normal form for rational omega tree languages is also presented

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

weak linearization of the lambda calculus we identify a restricted class of terms of the lambda calculus here called weak linear that includes the linear lambda terms keeping their good properties of strong normalization non duplicating reductions and typability in polynomial time the advantage of this class over the linear lambda calculus is the possibility of transforming general terms into weak linear terms with the same normal form we present such transformation and prove its correctness by showing that it preserves normal forms

highlights in infinitary rewriting and lambda calculus we present some highlights from the emerging theory of infinitary rewriting both for first order term rewriting systems and l calculus in the first section we introduce the framework of infinitary rewriting for first order rewrite systems so without bound variables we present a recent observation concerning the continuity of infinitary rewriting in the second section we present an excursion to the infinitary l calculus after the main definitions we mention a recent observation about infinite looping l terms that is terms that reduce in one step to themselves next we describe the fundamental trichotomy in the semantics of l calculus bohm trees levy longo trees and berarducci trees we conclude with a short description of a new refinement of bohm tree semantics called clocked semantics

une extension aux mots infinis de la notion de transduction rationelle here is introduced an extension for infinite words of the classical notion of rational transduction we prove that this extension has the important property of mapping the adherence of a language of finite words into the adherence of an other language of finite words the set of such extensions is closed by composition and is exactly the family of the compositions of an inverse faithful sequential mapping and of a faithful sequential mapping

on the longest perpetual reductions in orthogonal expression reduction systems we study perpetual reductions in orthogonal or conflict free fully extended expression reduction systems oers ers is a formalism for rewriting that subsumes term rewriting systems trss and the l calculus we design a strategy that for any given term t in a fully extended oers constructs a longest reduction starting from t if t is strongly normalizing and otherwise constructs an infinite reduction we call this strategy a limit strategy for a large class of oerss a limit strategy is computable the conservation theorem for fully extended oerss follows easily from the properties of the strategy we develop a method for computing the lengths of longest developments in oerss copyright 2001 elsevier science b v

on strong normalization and type inference in the intersection type discipline we introduce a new unification procedure for the type inference problem in the intersection type discipline it is well known that type inference in this case should succeed exactly for the strongly normalizing expressions we give a proof for the strong normalization result in the intersection type discipline which we obtain by putting together some well known results and proof techniques our proof uses a variant of klop s extended l calculus for which it is shown that strong normalization is equivalent to weak normalization this is proved here by means of a finiteness of developments theorem obtained following de vrijer s combinatory technique the main property of this extended calculus is uniformity i e weak and strong normalizabilities coincide the strong normalizability result is therefore a consequence of the fact first established by coppo and dezani for the l calculus that typability implies weak normalizability we then show that the unification process which is the basis for type inference exactly corresponds to reduction in the extended l calculus finally we show that our notion of unification allows us to compute a principal typing for any typable l expression

recursion and topology on 2 ω for possibly infinite computations in the context of possibly infinite computations yielding finite or infinite binary outputs the space 2 2 u 2 appears to be one of the most fundamental spaces in computer science though underconsidered next to 2 this space can be viewed section 3 5 2 as the simplest compact space native to computer science in this paper we study some of its properties involving topology and computability though 2 can be considered as a computable metric space in the sense of computable analysis a direct and self contained study based on its peculiar properties related to words is much illuminating it is well known that computability for maps 2 2 reduces to continuity with recursive modulus of continuity with 2 things get less simple maps 2 2 or 2 2 induced by input output behaviours of turing machines on finite or infinite words which we call semicomputable maps are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 continuity asks for a stronger notion of computability we prove for semi continuous and semi computable maps f f o with f o 2 2 a detailed representation theorem theorem 81 via functions f 2 2 following two approaches bottom up from f to f and top down from f to f

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

bck combinators and linear l terms have types abstract a term l is linear when each abstract x m in it has at most one free x in m and each free variable of l occurs free only once bck combinators correspond exactly to linear terms without free variables the main theorem says that all terms of each kind are stratified i e are typable this result is not deep it merely confirms an intuition that non stratification is caused partly by multiple occurrences of variables for bck combinators it was stated as theorem 1 by bunder and meyer in 1 but the proof in 1 seems to contain a gap it uses induction on a definition of level of a combinator but omits to show that the level concept is defined for all bck combinators the proof below proceeds by translating to linear terms for which the induction becomes simply one on length in fact the proof is essentially just an amalgamation of elements from an unpublished proof of m coppo and m dezani ciancaglini and elements from 1 the type schemes of linear terms form a very primitive linear logic and linear logic is of current interest see girard 3 so we shall set out the basic facts about linear terms and bck combinators in some detail these basic facts are not new some appearin komori 6 for example and some are implicit in curry and feys 2

levels of undecidability in rewriting undecidability of various properties of first order term rewriting systems is well known an undecidable property can be classified by the complexity of the formula defining it this classification gives rise to a hierarchy of distinct levels of undecidability starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy where quantification over function variables is allowed in this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies we consider properties related to normalization strong normalization weak normalization and dependency problems and properties related to confluence confluence local confluence and the unique normal form property for all of these we distinguish between the single term version and the uniform version where appropriate we also distinguish between ground and open terms most uniform properties are p 2 0 complete the particular problem of local confluence turns out to be p 2 0 complete for ground terms but only p 2 0 complete and thereby recursively enumerable for open terms the most surprising result concerns dependency pair problems without minimality flag we prove this problem to be p 1 1 complete hence not in the arithmetical hierarchy but properly in the analytic hierarchy some of our results are new or have appeared in our earlier publications others are based on folklore constructions and are included for completeness as their precise classifications have hardly been noticed previously

the hausdorff kuratowski hierarchy of ω regular languages and a hierarchy of muller automata given a finite alphabet we give a simple characterization of those g subsets of which are deterministic regular i e accepted by buchi automata over and then characterize the regular languages in terms of these rational g sets our characterization yields a hierarchy of regular languages similar to the classical difference hierarchy of hausdorff and kuratowski for 03 sets i e the class of sets which are both f and g we then prove that the hausdorff kuratowski difference hierarchy of 03 when restricted to regular languages coincides with our hierarchy we obtain this by showing that if an regular language k can be separated from another regular language l by the union of alternate differences of a decreasing sequence of g sets of length n then there is a decreasing sequence of length n of rational g sets such that the union of alternate differences separates k from l our results not only generalize a result of landweber 1969 but also yield an effective procedure for determining the complexity of a given muller automaton we also show that our hierarchy does not collapse thus giving a fine classification of regular languages and of muller automata

wadge hierarchy of omega context free languages abstract the main result of this paper is that the length of the wadge hierarchy of omega context free languages is greater than the cantor ordinal e 0 and the same result holds for the conciliating wadge hierarchy defined by duparc j symbolic logic to appear of infinitary context free languages studied by beauquier ph d thesis universite paris 7 1984 in the course of our proof we get results on the wadge hierarchy of iterated counter languages which we define as an extension of classical finitary iterated counter languages to languages

a note on ω regular languages abstract this paper is a continuation of the work by landweber 5 to clarify the relationship between classes of languages accepted by finite automata we study their properties in the topological framework introduced by boasson and nivat 1 we characterize the classes in terms of borel hierarchy and show some interrelation of them

the barendregt cube with definitions and generalised reduction in this paper we propose to extend the barendregt cube by generalisings reduction and by adding definition mechanisms generalised reduction allows contracting more visible redexes than usual and definitions are an important tool to allow for a more flexible typing system we show that this extension satisfies most of the original properties of the cube including church rosser subject reduction and strong normalisation

alternating finite automata on ω words abstract alternating finite automata on words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols the classes of languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions it is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on words have the same power as alternating ones

on omega context free languages which are borel sets of infinite rank this paper is a continuation of the study of topological properties of omega context free languages cfl we proved in topological properties of omega context free languages theoretical computer science 262 1 2 2001 669 697 that the class of cfl exhausts the finite ranks of the borel hierarchy and in borel hierarchy and omega context free languages theoretical computer science to appear that there exist some cfl which are analytic but non borel sets we prove here that there exist some omega context free languages which are borel sets of infinite but not finite rank giving additional answer to questions of lescow and thomas logical specifications of infinite computations in a decade of concurrency j w de bakker et al eds springer lncs 803 1994 583 621

characterization of ω regular languages by first order formulas abstract first order formulas are used to specify various ways of acceptance of languages by deterministic finite automata and we study the relationship between the arithmetic hierarchy of the formulas in prenex normal form and the topological hierarchy of the accepted languages among other things it is proved that the languages accepted by finite automata under the accepting conditions specified by 1 type formulas are precisely open regular languages those accepted under the conditions specified by 2 type formulas coincide with the regular languages which are denumerable unions of closed sets and that as long as the accepting conditions are specified by first order formulas the accepted languages remain to be regular

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

logical definability of fixed points note abstract we define a class of monotonic functions whose least fixed points are weakly definable in the sense of rabin we use this property to give a new proof that v definable sets of trees are recognized by rabin s special automata

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

infinitary lambda calculus and discrimination of berarducci trees we propose an extension of lambda calculus for which the berarducci trees equality coincides with observational equivalence when we observe rootstable or rootactive behavior of terms in one direction the proof is an adaptation of the classical bohm out technique in the other direction the proof is based on confluence for strongly converging reductions in this extension

the accommodation of traveling waves of fisher s type to the dynamics of the leading tail traveling wave solutions for equations of fisher s type are given by a connection from a saddle point to a node and their velocity for large times is known to be determined by the spatial decay rate of the initial conditions far ahead of the wave to understand this phenomenon in simple terms initial conditions are considered that correspond to a traveling wave but with an exponential decay rate far ahead of the wave that varies slowly in the region referred to as the leading tail the wave that results is supersonic in the sense that the speed of the core of the traveling wave is greater than the velocity of the characteristics or group velocity in the leading tail consequently the speed of the core adjusts to accommodate the slowly varying decay rate in the leading tail a generalization is also considered that includes fisher s equation for which traveling waves with a stable exponential tail corresponding to a node are supersonic

infinitary combinatory reduction systems we define infinitary combinatory reduction systems icrss thus providing the first notion of infinitary higher order rewriting the systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary l calculus are special cases furthermore we generalise a number of known results from first order infinitary rewriting and infinitary l calculus to icrss in particular for fully extended left linear icrss we prove the well known compression property and for orthogonal icrss we prove that 1 if a set of redexes u has a complete development then all complete developments of u end in the same term and that 2 any tiling diagram involving strongly convergent reductions s and t can be completed iff at least one of s t and t s is strongly convergent we also prove an ancillary result of independent interest a set of redexes in an orthogonal icrs has a complete development iff the set has the so called finite jumps property

strong normalization from weak normalization in typedλ calculi for some typed calculi it is easier to prove weak normalization than strong normalization techniques to infer the latter from the former have been invented over the last twenty years by nederpelt klop khasidashvili karr de groote and kfoury and wells however these techniques infer strong normalization of one notion of reduction from weak normalization of amore complicatednotion of reduction this paper presents a new technique to infer strong normalization of a notion of reduction in a typed calculus from weak normalization of thesamenotion of reduction the technique is demonstrated to work on some well known systems including second order calculus and the system of positive recursive types it gives hope for a positive answer to the barendregt geuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing the paper also analyzes the relationship between the techniques mentioned above and reviews in less detail other techniques for proving strong normalization

a mathematical model for spatially varying extracellular matrix alignment orientation of extracellular matrix fibers in the skin is a keyingredient of tissue appearance and function and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin in this paper the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them the model extends previous work in which spatial variation was excluded numerical simulations of the model are presented which show spatial variations in alignment over long transients but with spatiallyuniform behavior in the long term this is investigated further via asymptotic analysis using the angular diffusion coefficient as a small parameter this method enables calculation of the form of the steadystate orientation peaks observed numerically by considering behavior at large times the rate of approach to these peaks is shown to be exponential extension of this analysis to the spatially varying model confirms that long time behavior will be spatiallyuniform except in one special and biologicallyunrealistic case the authors conclude that the spatiallyvary ing alignment patterns observed in skin are in fact slow transients and biological implications of the modeling are discussed

a useful l notation abstract in this article we introduce a notation that is useful for many concepts of the calculus the new notation is a simple translation of the classical one yet it provides many nice advantages first we show that definitions such as compatibility the heart of a term and redexes become simpler in item notation second we show that with this item notation reduction can be generalised in a nice way we find a relation which extends which is church rosser and strongly normalising this reduction relation may be the way to new reduction strategies in classical notation it is much harder to present this generalised reduction in a convincing manner third we show that the item notation enables one to represent in a very simple way the canonical type a of a term a in context this canonical type plays the role of a preference type and can be used to split a b into the two parts a and a b this means that the question is a typable with a type b is divided into two questions is a typable and is b in the class of types of a it turns out that calculating this preference type of a in item notation is a straightforward operation one just goes through a from left to right performing very trivial steps on the items till the end variable or heart of a is reached fourth we can with this item notation find the parts of a term t relevant for a variable occurrence x in terms of binding typing and substitution again this part of t t x is very easy to find in item notation just take the part of t to the left of x and remove all unmatched parentheses fifth we reflect on the status of variables and show that indeed it is easy to study this status in item notation finally we show that for a substitution calculus a la de bruijn with open terms it is simpler to describe normal forms using item notation there are further advantages of item notation that are studied elsewhere for example in 9 we show that explicit substitution is easily built in item notation and that global and local strategies of substitution can be accommodated in 10 we show that with item notation one can give a unified approach to type theory an implementation of this item notation with most of the concepts discussed in this paper can be found in 15

characterizations of rational ω languages by means of right congruences abstract by means of right congruences we characterize in a unified way different classical classes of rational languages a new congruence associated with an automaton is introduced named cycle congruence the family of rational languages which have by morphism a unique minimal recognizing automaton is characterized it appears that for such a minimal automaton the cycle congruence coincides with the syntactic congruence of the recognized language we prove that the other rational languages have an infinite number of minimal automaton

borel hierarchy and omega context free languages we give in this paper additional answers to questions of lescow and thomas a decade of concurrency lecture notes in computer science vol 803 springer berlin 1994 pp 583 621 proving topological properties of omega context free languages cfl which extend those of o finkel theoret comput sci 262 1 2 2001 669 697 there exist some cfl which are non borel sets and one cannot decide whether an cfl is a borel set we give also an answer to a question of niwinski problem on powers posed in the proceedings of the workshop logics and recognizable sets 1990 and of simonnet automates et th orie descriptive ph d thesis universit paris 7 1992 about powers of finitary languages giving an example of a finitary context free language l such that l is not a borel set then we prove some recursive analogues to preceding properties in particular one cannot decide whether an cfl is an arithmetical set finally we extend some results to context free sets of infinite trees

perpetuality and uniform normalization in orthogonal rewrite systems we study perpetuality of reduction steps as well as perpetuality of redexes in orthogonal rewrite systems a perpetual step is a reduction step which retains the possibility of infinite reductions a perpetual redex is a redex which when put into an arbitrary context yields a perpetual step we generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal term rewriting systems and the calculus due to bergstra and klop and others we first introduce context sensitive conditional expression reduction systems ccerss and define a concept of orthogonality which implies confluence for them in particular several important calculi and their extensions and restrictions can naturally be embedded into orthogonal ccerss we then define a perpetual reduction strategy which enables one to construct minimal w r t levy s permutation ordering on reductions infinite reductions in orthogonal fully extended ccerss using the properties of the minimal perpetual strategy we prove 1 perpetuality of any reduction step that does not erase potentially infinite arguments which are arguments that may become via substitution infinite after a number of outside steps and 2 perpetuality in every context of any safe redex which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite we prove both these perpetuality criteria for orthogonal fully extended ccerss and then specialize and apply them to restricted calculi demonstrating their usefulness in particular we prove the equivalence of weak and strong normalization whose equivalence is here called uniform normalization for various restricted calculi most of which cannot be derived from previously known perpetuality criteria

two characterizations of rational adherences resume nous presentons deux caracterisations des adherences en termes de langages finis d homomorphismes strictement alphabetiques et d homomorphismes uniformes inverses nous en deduisons une caracterisation similaire pour les langages rationnels

x automata on ω words abstract for any storage type x the languages accepted by x automata are investigated six accepting conditions including those introduced by landweber are compared for x automata the inclusions between the corresponding six families of languages are essentially the same as for finite state automata apart from unrestricted automata also real time and deterministic automata are considered the main tools for this investigation are 1 a characterization of the languages accepted by x automata in terms of inverse x transductions of finite state languages and 2 the existence of topological upper bounds on some of the families of accepted languages independent of the storage type x

transfinite reductions in orthogonal term rewriting systems strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems otrss for these we prove the transfinite parallel moves lemma and the compressing lemma strongness is necessary as shown by counterexamples normal forms which we allow to be infinite are unique in contrast to normal forms strongly converging fair reductions result in normal forms

language theoretical representations of ω languages abstract in this paper we give language theoretical representations for some classes of languages defined by finite automata the classes considered in this paper are proved to be the classes of languages represented as f f pf or p p where p and p are prefix free regular languages and f and f are prefix free finite languages

the three subfamilies of rational ω languages closed under ω transduction abstract the transductions defined by gire as bimorphisms coincide with the compositions of non erasing morphisms and inverse non erasing morphisms we prove that there are exactly three subfamilies of rational languages which are closed under transduction the family of rational adherences the family of rational languages whose complementary is deterministic and the whole family of rational languages

the borel hierarchy is infinite in the class of regular sets of trees abstract for all the borel classes of finite order we construct weakly acceptable sets of infinite trees representing these classes

automata on infinite objects and their applications to logic and programming abstract we introduce various types of automata top down automata and bottom up automata on infinite trees we study the power of determinstic and nondeterministic tree automata and prove that deterministic and non deterministic bottom up tree automata accept the same infinite tree sets we establish a relationship between tree automata logic programs recursive program schemes and the monadic second order theory of the tree we prove that the equivalence of two rational logic programs is decidable

nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion abstract we consider a simplified chemotaxis model of tumor angiogenesis described by a keller segel system on the two dimensional infinite cylindrical domain x y r s where s is the circle of perimeter 0 the domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located the system is known to allow planar traveling wave solutions of an invading type in this paper we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if is sufficiently small the same result for the corresponding system in one dimension was obtained by li li wang 2014 17 our result solves the problem remained open in chae choi kang lee 2018 3 at which only linear stability of the planar traveling waves was obtained under certain artificial assumption

traveling wave solutions of a mathematical model for tumor encapsulation the formation of a capsule of dense fibrous extracellular matrix around a solid tumor is a key prognostic indicator in a wide range of cancers however the cellular mechanisms underlying capsule formation remain unclear one hypothesized mechanism is the expansive growth hypothesis which suggests that a capsule may form by the rearrangement of existing extracellular matrix without new matrix production a mathematical model was recently proposed to study the implications of this hypothesis perumpanani sherratt and norbury nonlinearity 10 1997 pp 1599 1614 the model consists of conservation equations for tumor cells and extracellular matrix and exhibits traveling wave solutions in which a pulse of extracellular matrix corresponding to a capsule moves in parallel with the advancing front of the tumor in this paper the author presents a detailed study of traveling wave behavior in the model deriving conditions for the existence of traveling waves and their key properties numerical meth

accepting conditions for automata on ω languages abstract in this paper we study the languages accepted by finite automata under a new type of accepting conditions which are based on whether the automaton enters all of its accepting states infinitely many times or not we investigate the classes of languages defined thereby in relation to other well known subclasses of the class of regular languages

lazy productivity via termination abstract we present a procedure for transforming strongly sequential constructor based term rewriting systems trss into context sensitive trss in such a way that productivity of the input system is equivalent to termination of the output system thereby automated termination provers become available for proving productivity a trs is called productive if all its finite ground terms are constructor normalizing and all inductive constructor paths through the resulting possibly non wellfounded constructor normal form are finite to our knowledge this is the first complete transformation from productivity to termination the transformation proceeds in two steps i the strongly sequential trs is converted into a shallow trs where patterns do not have nested constructors ii the shallow trs is transformed into a context sensitive trs where rewriting below constructors and in arguments not consumed from is disallowed furthermore we show how lazy evaluation can be encoded by strong sequentiality thus extending our transformation to e g haskell programs finally we present a simple but fruitful extension of matrix interpretations to make them applicable for proving termination of context sensitive trss

an equivalence between lambda terms resume dans ce papier on definit une relation d equivalence sur les lambda termes identifiant les termes qui ne different que par des permutations de radicaux la equivalence on demontre qu aucun des criteres operationnels standards de classification du lambda calcul e g la longueur de la plus longue normalisation ne permet de distinguer deux termes equivalents enfin la equivalence est utilisee pour demontrer une generalisation du theoreme de la strategie perpetuelle

a combinatorial approach to the theory of ω automata a combinatorial lemma is proved and used here to derive new results on automata and to give simpler proofs of known ones in particular we reprove mcnaughton s fundamental theorem characterizing the regular sequence sets without having to construct a sophisticated automaton the theorem is obtained by coding the behaviour of automata in a second order language and a simple application of the lemma in close analogy now referring to a first order language a theory of counter free automata is developed it is shown that these automata are appropriate for characterizing the star free sequence sets finally the lemma is applied in mathematical logic here new normal form theorems and also decidability results are proved for the first order and the monadic second order theory of certain structures over the ordering of natural numbers

a type assignment for λ calculus complete both for fptime and strong normalization one of the aims of implicit computational complexity is the design of programming languages with bounded computational complexity one of the most promising approaches to this aim is based on the use of lambda calculus as paradigmatic programming language and the design of type assignment systems for lambda terms where types guarantee both the functional correctness and the complexity bound along this line we propose a system of stratified types inspired by intersection types where intersection is a non associative operator this system is correct and complete for polynomial time when a uniform coding scheme is considered moreover all the strongly normalizing terms are typed in it thus increasing the typing power with respect to the previous proposals based on light logics

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

productivity of stream definitions we give an algorithm for deciding productivity of a large and natural class of recursive stream definitions a stream definition is called productive if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit whereas productivity is undecidable for stream definitions in general we show that it can be decided for pure stream definitions for every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements called pebbles in a finite pebbleflow net work and the production of a pebbleflow net associated with a pure stream definition that is the amount of pebbles the net is able to produce at its output port can be calculated by reducing nets to trivial nets

projection lemmas for ω languages abstract the paper presents projection lemmas relating the infinite behaviour of deterministic and unboundedly branching nondeterministic infinite automata

topological characterizations of infinite tree languages abstract in this paper we define several types of acceptance of finite automata for infinite trees and study the topological properties of the tree languages accepted thereby as a consequence we show that the two classes of tree languages accepted by nondeterministic tree automata in the following two types of acceptance coincide 1 1 an infinite tree is accepted iff there is a run for such that on each path an accepting state is contained 2 2 an infinite tree is accepted iff every run for contains an accepting state

perpetual reductions in λ calculus this paper surveys a part of the theory ofs reduction in calculus which might aptly be calledperpetual reductions the theory is concerned withperpetual reduction strategies i e reduction strategies that compute infinite reduction paths from terms when possible and withperpetual redexes i e redexes whose contraction in terms preserves the possibility when present of infinite reduction paths the survey not only recasts classical theorems in a unified setting but also offers new results proofs and techniques as well as a number of applications to problems in calculus and type theory

infinite word languages and continuous mappings abstract languages with infinite words have often been studied as subsets of a topological space the topology used for this purpose was always the natural topology of sequences this paper suggests that another topology may be more useful it is obtained by taking lim w as the derived set of w the resulting topological space is shown to be completely regular strongly zero dimensional and not normal the languages appearing in the theory of automata are shown to be closely related to functionally closed sets continuous inverse images of the set 0 as a consequence a number of known results can be expressed and proved in terms of continuous mappings

some remarks on non algebraic adherences we present some properties of non algebraic adherences of languages which can be consistently called context sensitive adherences because it is proved that in the chomsky hierarchy of adherences context sensitive and recursive enumerable adherences coincide concerning the center mapping the traversing from algebraic to non algebraic takes simple recursive to not necessarily recursive enumerable except for the last equality the chomsky hierarchy of adherences is proper radh cfadh csadh readh and moreover it can be refined in the non algebraic case by considering the mcnaughton nivat adherences regarding the programming context adherences occur when we consider the set of divergent computations shapes of a program or a program scheme in this respect the interpretation is powerful enough to translate radh to csadh

non existent statman s double fixed point combinator does not exist indeed we give a negative answer to a problem of r statman concerning the existence in the lambda calculus of a fixed point combinatorysuch thaty si yholds

conservation and uniform normalization in lambda calculi with erasing reductions for a notion of reduction in a calculus one can ask whether a term satisfies conservation and uniform normalization conservation means that single step reductions of the term preserve infinite reduction paths from the term uniform normalization means that either the term will have no reduction paths leading to a normal form or all reduction paths will lead to a normal form in the classical conservation theorem for i the distinction between the two notions is not clear uniform normalization implies conservation and conservation also implies uniform normalization the reason for this is that i is closed under reduction due to the fact that reductions never erase terms in i more generally for nonerasing reductions the two notions are equivalent on a set closed under reduction however when turning to erasing reductions the distinction becomes important as conservation no longer implies uniform normalization this paper presents a new technique for finding uniformly normalizing subsets of a calculus this is done by combining a syntactic and a semantic criterion the technique is demonstrated by several applications the technique is used to present a new uniformly normalizing subset of the pure calculus this subset is a superset of i and thus contains erasing k redexes the technique is also used to prove strong normalization from weak normalization of the simply typed calculus extended with pairs this is an extension of techniques developed recently by sorensen and xi before presenting the technique the paper presents a simple proof of a slightly weaker form of the characterization of perpetual redexes by bergstra and klop this is a step for the later applications of the technique

fixed point characterization of context free languages several concepts of context freeness of sets of finite infinite words are characterized by means of greatest solutions of systems of equations of the form xi gi i 1 n where gi is a not necessarily finite union of monomials consideration of the systems with the components gi context free regular or finite leads to characterizations of the following classes of languages the kleene closure of the family of context free languages algebraic languages infinitely generated by context free grammars in the sense of nivat 1977 and cantor like topological closures of context free languages respectively

shift invariant topologies for the cantor space x the space of one sided infinite words plays a crucial rle in several parts of theoretical computer science usually it is convenient to regard this space as a metric space the cantor space it turned out that for several purposes topologies other than the one of the cantor space are useful e g for studying fragments of first order logic over infinite words or for a topological characterisation of random infinite words it is shown that these topologies refine the topology of the cantor space moreover from common features of these topologies we extract properties which characterise a large class of topologies it turns out that for this general class of topologies the corresponding closure and interior operators respect the shift operations and also to some extent the definability of sets of infinite words by finite automata

various hierarchies of wω regular sets barua 1992 studied a hierarchy rn n 1 2 3 where rn is a class of regular sets which are decomposed into n rational g sets forming a decreasing sequence on the other hand kaminski 1985 defined a hierarchy bm m 1 2 3 where bm is a class of regular sets which are decomposed into 2m rational g sets not necessarily forming a decreasing sequence we prove that r2n bn in spite of the difference of defining conditions

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

strictness analysis via abstract interpretation for recursively defined types abstract in this paper we consider a functional language with recursively defined types and a weak form of polymorphism for this language a strictness analysis is developed based on abstract interpretation in a category of complete algebraic lattices

a group analysis via weak equivalence transformations for a model of tumour encapsulation a symmetry reduction of a pdes system describing the expansive growth of a benign tumour is obtained via a group analysis approach the presence in the model of three arbitrary functions suggests the use of lie symmetries by using the weak equivalence transformations an invariant classification is given which allows us to reduce the initial pdes system to an odes system numerical simulations show a realistic enough description of the physical process

topological properties of omega context free languages abstract this paper is a study of topological properties of omega context free languages cfl we first extend some decidability results for the deterministic ones dcfl proving that one can decide whether an dcfl is in a given borel class or in the wadge class of a given regular language we prove that cfl exhaust the hierarchy of borel sets of finite rank and that one cannot decide the borel class of an cfl giving an answer to a question of lescow and thomas a decade of concurrency springer lecture notes in computer science vol 803 springer berlin 1994 pp 583 621 we give also a partial answer to a question of simmonet automates et theorie descriptive ph d thesis universite paris 7 march 1992 about omega powers of finitary languages we show that buchi landweber s theorem cannot be extended to even closed cfl in a gale stewart game with a closed cfl winning set one cannot decide which player has a winning strategy from the proof of topological properties we derive some arithmetical properties of cfl

fixed point characterization of infinite behavior of finite state systems abstract infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras these terms involve the least and greatest fixed point operators and disjunction as the only logical operation a tight correspondence is established between a hierarchy of rabin indices of automata and a hierarchy induced by alternation of the least and greatest fixed point operators it is shown that in the powerset algebra of trees constructed from a set of functional symbols the fixed point hierarchy is infinite unless all the symbols are unary i e trees are words it is also shown that an interpretation of a closed fixed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees from which it is deduced that the question whether a term denotes always can be answered in polynomial time

local rule distributions language complexity and non uniform cellular automata this paper investigates a variant of cellular automata namely n ca indeed n ca are cellular automata which can have different local rules at each site of their lattice the assignment of local rules to sites of the lattice completely characterizes n ca in this paper sets of assignments sharing some interesting properties are associated with languages of bi infinite words the complexity classes of these languages are investigated providing an initial rough classification of n ca

a note on preservation of strong normalisation in the λ calculus an auxiliary notion of reduction r on the l terms preserves strong normalisation if all strongly normalising terms for b are also strongly normalising for b r we give a sufficient condition for r to preserve strong normalisation as an example of application we check easily the sufficient condition for regnier s s reduction rules and the assoc reduction rule inspired by calculi with let expressions this gives the simplest proof so far that the union of all these rules preserves strong normalisation

an analysis of ml typability we carry out an analysis of typability of terms in ml our main result is that this problem is dexptime hard where by dexptime we mean dtime 2 n 0 1 this together with the known exponential time algorithm that solves the problem yields the dexptime completeness result this settles an open problem of p kanellakis and j c mitchell part of our analysis is an algebraic characterization of ml typability in terms of a restricted form of semi unification which we identify as acyclic semi unification we prove that ml typability and acyclic semi unification can be reduced to each other in polynomial time we believe this result is of independent interest

topology on words we investigate properties of topologies on sets of finite and infinite words over a finite alphabet the guiding example is the topology generated by the prefix relation on the set of finite words considered as a partial order this partial order extends naturally to the set of infinite words hence it generates a topology on the union of the sets of finite and infinite words we consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural we provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion

a hierarchy of sets of infinite trees a hierarchy of sets of infinite valued trees is introduced which has no counterpart in the theory of sets of infinite strings languages as a consequence we obtain that for sets of infinite trees an analogue of mcnaughton s fundamental theorem on languages does not hold

finite acceptance of infinite words abstract in this paper we consider the following two types of finite acceptance of infinite words by finite automata an infinite word is accepted if and only if there is a run on input for which 1 1 an accepting state is visited at least once or 2 2 an accepting state is visited at least once but only finitely often the resulting classes of regular languages are characterized by language theoretic means and they are positioned into the known hierarchies of regular languages

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem