comparing models of the intensional typed l calculus abstract in this paper we compare powerset models of the typed lambda calculus although the interpretation of the type constructor is in general not uniquely determined for intensional lambda calculi we can make a canonical choice in this particular case there exists a minimal interpretation of with respect to a certain class of interpretations which yields models with a maximal theory in that class

on the construction of stable models of untyped λ calculus abstract stable models of untyped calculus have been introduced by berry proceedings of icalp lecture notes in computer science vol 62 springer berlin 1978 pp 72 89 as a refinement of the continuous framework defined by scott unpublished manuscript 1969 53pp in this paper we show that even in the stable case we have a fair amount of freedom during the construction of models we introduce a uniform method to construct stable models that allows to obtain non standard models satisfying certain equational restraints we apply this method in two particular examples first we construct a simple stable model that distinguishes the terms y and second we construct a family of 2 0 stable models with pairwise distinct theories we show that the latter models are sensible using some results of david proceedings of tlca 1999 lecture notes in computer science vol 1581 springer berlin 1999 and kerth j symbolic logic 63 1998 1529 1548

retractions on dl domains as a model for type type abstract we introduce an extension with the type type assumption for a higher order functional language in the framework of p the extension of the type free lambda calculus introduced in r amadio and g longo 1986b in ifip conference formal description of programming concepts ebberup dk august 1986 then we find using the stable functions and berry s di domains a model for p this settles the consistency problem for p up to now open and gives very simple mathematical models for higher order calculi with type type such models are based on the relevant property that the range of a stable retraction on a di domain is still a di domain finally we give the categorical description of the models of p based on moggi asperti s notion of internalization in a asperti 1988 models of higher order calculi internal categories carnegie mellon report in preparation

order incompleteness and finite lambda reduction models many familiar models of the untyped lambda calculus are constructed by order theoretic methods this paper provides some basic new facts about ordered models of the lambda calculus we show that in any partially ordered model that is complete for the theory of or conversion the partial order is trivial on term denotations equivalently the open and closed term algebras of the untyped lambda calculus cannot be non trivially partially ordered our second result is a syntactical characterization in terms of so called generalized mal cev operators of those lambda theories which cannot be induced by any non trivially partially ordered model we also consider a notion of finite models for the untyped lambda calculus or more precisely finite models of reduction we demonstrate how such models can be used as practical tools for giving finitary proofs of term inequalities

set theoretical and other elementary models of the l calculus abstract part i pp 351 373 of this paper is the previously unpublished 1972 memorandum plotkin 1972 with editorial changes and some minor corrections part ii pp 373 409 presents what happened next together with some further development of the material the first part begins with an elementary set theoretical model of the calculus functions are modelled in a similar way to that normally employed in set theory by their graphs difficulties are caused in this enterprise by the axiom of foundation next based on that model a model of the calculus is constructed by means of a natural deduction method finally a theorem is proved giving some general properties of those nontrivial models of the calculus which are continuous complete lattices the second part begins with a brief discussion of models of the calculus in set theories with anti foundation axioms next the model of the calculus of part i and also the closely related but different models of scott 1976 1980 and of engeler 1981 1988 are reviewed then general frameworks in which elementary constructions of models can be given are discussed following longo 1982 one can employ certain scott engeler algebras following coppo et al 1983 one can obtain filter models from their extended applicative type structures an extended discussion is given of various ways of constructing models of the calculus and the connections between them finally an extension of the theorem to complete partial orders is given the theme of the paper is the consideration of means of constructing models there is hardly any analysis of their properties there is no discussion of their application

universal retractions on di domains we prove that every di domain which contains an infinite set of pairwise incompatible elements in particular every reflexive di domain has an infinite set of universal retractions which moreover forms a di domain when ordered by the stable order isomorphic to n where is the divisibility relation

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

the discrete objects in the effective topos the original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two non trivial small i e internal full subcategories which are closed under all small limits in the topos and hence in particular are internally complete the interest in such subcategories arises from the fact that they provide a very natural notion of model for many of the strongly polymorphic type theories the nature of these models is that types are interpreted as objects of the topos and hence as sets in an intuitionistic sense they in fact provide set theoretic models in the sense of reynolds 16 except that the set theory involved is non classical at the end of the paper we indicate briefly how one only of the two full subcategories gives rise to a model of the theory of constructions this sheds some light on how far we can travel with a simple set theoretic picture of data types however our main concern is to present a clean mathematical characterization of the complete subcategories the basic idea due to peter freyd remains very simple it is that the collection of objects orthogonal to any given object is automatically closed under all existing limits and to look at the objects orthogonal to the subobject classifier in the effective topos however one of our original motivations for writing this paper was that we did not quite believe freyd s identification of this category during the course of the paper we shall show that the category of families of objects in orthogonal to q is the category of families of subquotients of n the chain of reasoning now proceeds by saying that this second category is equivalent to the externalisation of an internal category and that the internal category is therefore small complete somewhat to our surprise it was this second stage trivial for an internal category in sfe that we found difficulty in making precise and it is in the discussion of notions of equivalence and completeness for internal categories and fibrations that much of the content of the paper now lies we should warn the reader that even so our treatment is not exhaustive we discuss only two notions weak and strong of equivalence for internal or fibred categories and only two notions of completeness our definitions are welladapted for our present purposes corresponding more or less to the distinction between being told for any instance of a given set of parameters that there is an object with certain properties and being given a function that produces one let c be a small full subcategory of a topos see 3 for details which is closed under arbitrary products then c cannot contain q the object of truth values in fact c cannot contain any object into which q embeds for if a is the disjoint union of the sets in c then q ec and hence q y a which contradicts one of the constructive versions of cantor s theorem this argument runs parallel

hopf algebra of the planar binary trees is a sub hopf algebra there is a basis qn of soln such that the composite map k qn soln k sn has the following property its linear dual k sn k qn is induced by a set theoretic map sn qn in between sn and qn there is an intermediate set yn which is made of planar binary trees with n vertices for instance for n 3 the number of elements in s3 y3 and q3 is 6 5 and 4 respectively the projection from sn to qn is the composite of two maps sn w yn w qn article no ai981759

a theory of sequentiality abstract we show that the notion of sequentiality as presented in concrete data structures may be carried to a more general domain theoretic framework cells are replaced by some linear maps on a domain which turns out to be a di domain in a first phase we do not require the cells to be enabled as they are in cdss and we get a weak form of cartesian closedness then by enriching the structure we get the standard one

maximality and totality of stable functions in the category of stable bifinite domains this paper studies maximality and totality of stable functions in the category of stable bifinite domains we present three main results as follows o 1 every maximum preserving function is a maximal element in the stable function spaces 2 a maximal stable function f d e is maximum preserving if d is maximum separable and e is completely separable and 3 a stable bifinite domain d is maximum separable if and only if for any locally distributive stable bifinite domain e each maximal stable function f d e is maximum preserving

on the completeness of order theoretic models of the λ calculus scott discovered his domain theoretic models of the l calculus isomorphic to their function space in 1969 a natural completeness problem then arises whether any two terms equal in all scott models are convertible there is also an analogous consistency problem whether every equation between two terms consistent with the l calculus has a scott model we consider such questions for wider sets of sentences and wider classes of models the pointed completely partially ordered ones a negative result for a set of sentences shows the impossibility of finding scott models for that class a positive result gives evidence that there might be enough scott models we find for example that the order extensional pointed w cpo models are complete for p 1 sentences with positive matrices whereas the consistency question for s 1 sentences with equational matrices depends on the consistency of certain critical sentences asserting the existence of certain functions analogous to the generalized mal cev operators first considered in the context of the l calculus by selinger

the genericity theorem and parametricity in the polymorphic l calculus abstract this paper focuses on how terms of the polymorphic calculus which may take types as inputs depend on types these terms are generally understood in all models to have an essentially constant meaning on input types we show the proof theory of polymorphic calculus suggests a clear syntactic description of this phenomenon namely under a reasonable condition we show that if two polymorphic functions agree on a single type then they agree on all types equivalently types are generic inputs

robust tracking relation between the numerators mode readability and inverse internal model in the robust multivariable asymptotic tracking problem with two output plants it is shown that if the numerator related to the controlled output is fixed while the other numerator and the denominator are perturbed then there is no solution to the problem if however the whole part of the plant related to the controlled output is stable and fixed while the other part is arbitrarily perturbed the problem has a solution and the compensator that solves the problem incorporates an inverse internal model of the signal to be tracked

on the p calculus and linear logic abstract we detail abramsky s proofs as processes paradigm for interpreting classical linear logic cll girard 1987 into a synchronous version of the calculus recently proposed by milner 1992 1993 the translation is given at the abstract level of proof structures we give a detailed treatment of information flow in proof nets and show how to mirror various evaluation strategies for proof normalization we also give soundness and completeness results for the process calculus translations of various fragments of cll the paper also gives a self contained introduction to some of the deeper proof theory of cll and its process interpretation

uniformity and the taylor expansion of ordinary lambda terms we define the complete taylor expansion of an ordinary lambda term as an infinite linear combination with rational coefficients of terms of a resource calculus similar to boudol s lambda calculus with multiplicities or with resources in our resource calculus all applications are multi linear in the algebraic sense i e commute with linear combinations of the function or the argument we study the collective behaviour of the beta reducts of the terms occurring in the taylor expansion of any ordinary lambda term using in a surprisingly crucial way a uniformity property that they enjoy as a corollary we obtain the main part of a proof that this taylor expansion commutes with bohm tree computation syntactically

relative definability of boolean functions via hypergraphs the aim of this work is to show how hypergraphs can be used as a systematic tool in the classification of continuous boolean functions according to their degree of parallelism intuitively f is less parallel than g if it can be defined by a sequential program using g as its only free variable it turns out that the poset induced by this preorder is as for the degrees of recursion a sup semilattice although hypergraphs have already been used in bucciarelli theoret comput sci to appear as a tool for studying degrees of parallelism no general result relating the former to the latter has been proved in that work we show that the sup semilattice of degrees has a categorical counterpart we define a category of hypergraphs such that every object represents a monotone boolean function finite coproducts in this category correspond to lubs of degrees unlike degrees of recursion where every set has a recursive upper bound monotone boolean functions may have no sequential upper bound however the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs these subsequential functions play a major role in the proof of our main result namely that f is less parallel than g if there exists a morphism between their associated hypergraphs

on a question of h friedman in this paper we answer a question of friedman providing an separable model m of the s calculus there therefore exists an separable model for any 0 the model m permits no non trivial enrichment as a partial order neither does it permit an enrichment as a category with an initial object the open term model embeds in m by way of contrast we provide a model which cannot embed in any non trivial model separating all pairs of distinct elements

the extended calculus of constructions ecc with inductive types abstract luo s extended calculus of constructions ecc is a higher order functional calculus based on coquand s and huet s calculus of constructions but has in addition strong sums and a predicative cumulative type hierarchy in this paper i introduce inductive types on the predicative type levels of ecc i also show how the set model for ecc can be extended to a model for this augmented calculus

a stable programming language it is well known that stable models as di domains qualitative domains and coherence spaces are not fully abstract for the language pcf this fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order which cannot be defined by programs of pcf in this paper a paradigmatic programming language named st pcf is proposed which extends the language pcf with two additional operators the operational description of the extended language is presented in an effective way although the evaluation of one of the new operators cannot be formalized in a pcf like rewrite system since st pcf can define all finite cliques of coherence spaces the above gap with stable models is filled consequently stable models are fully abstract for the extended language

constructing the polynomlal identities and central identities of degree 9 of 3 3 matrices abstract we present an algorithm for computing an independent generating set for the multilinear identities and the multilinear central identities of the m m matrices over a field o of characteristic zero or a large enough prime then we use it to construct all the multilinear identities and all the multilinear central identities of degree m 3 o

categories of embeddings abstract we present a categorical generalisation of the notion of domains which is closed under suitable exponentiation the goal was originally to generalise girard s model of polymorphism to f if we specialise this notion in the poset case we get new cartesian closed categories of domains

what is a model of the lambda calculus an elementary purely algebraic definition of model for the untyped lambda calculus is given this definition is shown to be equivalent to the natural semantic definition based on environments these definitions of model are consistent with and yield a completeness theorem for the standard axioms for lambda convertibility a simple construction of models for lambda calculus is reviewed the algebraic formulation clarifies the relation between combinators and lambda terms

categorical models of polymorphism abstract we present and discuss the relations between two classes of categorical models of the second order or polymorphic lambda calculus namely those based on internal categories internal models and those based on indexed categories external models we start in part i with a detailed introduction to internal categories and their relations to indexed categories the presentation is by means of equations between arrows in an ambient category with finite limits in part ii we recall the definition of the two classes of models and we present the externalization process that given an internal model yields an external model we show how one can go back in a straightforward way and that by making a full round trip from an internal model to an internal model via an external one or vice versa one does obtain equivalent models part iii discusses three major examples of models provable retractions inside a per model per inside set pl categories inside their grothendieck completions the appendix contains an account of internal adjunctions and internal cccs

a full continuous model of polymorphism we introduce a model of the second order lambda calculus such a model is a scott domain whose elements are themselves scott domains and in it polymorphic maps are interpreted by generic continous maps

prime algebraicity a prime algebraic lattice can be characterised as isomorphic to the downwards closed subsets ordered by inclusion of its complete primes it is easily seen that the downwards closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion the converse also holds any completely distributive algebraic lattice is isomorphic to such a set of downwards closed subsets of a partial order the partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes consequently prime algebraic lattices are precisely the completely distributive algebraic lattices the result extends to scott domains several consequences are explored briefly the representation of berry s di domains by event structures a simplified form of information systems for completely distributive scott domains and a simple domain theory for concurrency

easiness in graph models we generalize baeten and boerboom s method of forcing to show that there is a fixed sequence uk k of closed untyped terms satisfying the following properties a for any countable sequence gk k of scott continuous functions of arbitrary arity on the power set of an arbitrary countable set there is a graph model such that x xx x xx uk represents gk in the model b for any countable sequence tk k of closed terms there is a graph model that satisfies x xx x xx uk tk for all k we apply these two results which are corollaries of a unique theorem to prove the existence of 1 a finitely axiomatized theory l such that the interval lattice constituted by the theories extending l is distributive 2 a continuum of pairwise inconsistent graph theories theories that can be realized as theories of graph models 3 a congruence distributive variety of combinatory algebras lambda abstraction algebras respectively

sequential algorithms and strongly stable functions intuitionistic proofs and pcf programs may be interpreted as functions between domains or as strategies on games the two kinds of interpretation are inherently different static vs dynamic extensional vs intentional it is thus extremely instructive to compare and to connect them in this article we investigate the extensional content of the sequential algorithm hierarchy sds introduced by berry and curien we equip every sequential game t sds of the hierarchy with a realizability relation between plays and extensions in this way the sequential game t sds becomes a directed acyclic graph instead of a tree this enables to define a hypergraph t hc on the extensions or terminal leaves of the game t sds we establish that the resulting hierarchy hc coincides with the strongly stable hierarchy introduced by bucciarelli and ehrhard we deduce from this a gametheoretic proof of ehrhard s collapse theorem which states that the strongly stable hierarchy coincides with the extensional collapse of the sequential algorithm hierarchy

program development in constructive type theory abstract we present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications we are interested in two important facts that are the mechanization of the proof construction and the possibility to express in the theory significant concepts for programming like inductively defined types and general recursion we give here a survey on some results and problems appearing in logical frameworks devoted to the programming with proofs approach

robust tracking error feedback and two degree of freedom controllers the purpose of this paper is to establish first that in the absence of error feedback asymptotic tracking is a property that is fragile with respect to the controller parameters this implies that arbitrary small perturbation of the parameters of such a controller will cause asymptotic tracking to fail second we propose a general 2 dof controller architecture we prove that in this structure asymptotic tracking is in general fragile with respect to the controller parameters f sub r f sub y and c while it is robust with respect to the plant parameters as well as the components h and l of the 2 dof controller moreover the only way to realize a 2 dof controller in a nonfragile way is to generate the error signal exactly which amounts to setting f sub r f sub y 1 the results reconcile the requirements of the internal model principle namely the necessity of error feedback with the additional design freedom potentially offered by 2 dof controllers it specifies the controller architecture necessary for robustness and identifies the blocks whose parameters are available for tuning in the design process

on maximal stable functions abstract this note studies maximal elements in di domains especially maximal stable functions first a sufficient condition for a stable function to be maximal is provided the condition requires that the function be total in the sense that it preserves maximal elements examples are then given to demonstrate that being total is not necessary for a stable function to be maximal secondly a characterization is given for di domains for which maximal also implies total finally topological properties of maximal elements are investigated di domains for which maximal is the same as total are those having a certain hausdorff property

stable domains are generalized topological spaces abstract we present a generalization of the notion of topological structure on a set such that di or l domains can be naturally given such topologies in a way making the continuous functions exactly the stable functions as ordinary topological spaces give rise to locales these generalized spaces lead to a generalized form of locale that we define and explore

a relative pcf definability result for strongly stable functions and some corollaries we prove that in the hierarchy of simple types based on the type of natural numbers any finite strongly stable function is equal to the application of the semantics of a pcf definable functional to some strongly stable generally not pcf definable functionals of type two applying a logical relation technique we derive from this result that the strongly stable model of pcf is the extensional collapse of its sequential algorithms model

jordan triple disystems we take an algorithmic and computational approach to a basic problem in abstract algebra determining the correct generalization to dialgebras of a given variety of nonassociative algebras we give a simplified statement of the kp algorithm introduced by kolesnikov and pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras we apply the kp algorithm to the defining identities for jordan triple systems to obtain a new variety of nonassociative triple systems called jordan triple disystems we give a generalized statement of the bso algorithm introduced by bremner and sanchez ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra we apply the bso algorithm to the jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the kp algorithm this provides a large class of examples of jordan triple disystems we formulate a general conjecture expressed by a commutative diagram relating the output of the kp and bso algorithms we conclude by generalizing the jordan triple product in a jordan algebra to operations in a jordan dialgebra we use computer algebra to verify that resulting structures provide further examples of jordan triple disystems for this last result we also provide an independent theoretical proof using jordan structure theory

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

parameterization of decoupling controllers in the generalized plant model a diagonal decoupling problem in linear multi variable systems is formulated for the most general setting a necessary and sufficient condition for the existence of a decoupling controller is presented and when a solution exists the class of all decoupling controllers is characterized in terms of free parameters of stable rational vectors

special identities for the pre jordan product in the free dendriform algebra abstract pre jordan algebras were introduced recently in analogy with pre lie algebras a pre jordan algebra is a vector space a with a bilinear multiplication x y such that the product x y x y y x endows a with the structure of a jordan algebra and the left multiplications l x y x y define a representation of this jordan algebra on a equivalently x y satisfies these multilinear identities x y z u y z x u z x y u z x y u x y z u y z x u x y z u z y x u x z y u z x y u x y z u y z x u the pre jordan product x y x y y x in any dendriform algebra also satisfies these identities we use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree 7 for this product is implied by the identities of degree 4 but that there exist new identities of degree 8 which do not follow from those of lower degree there is an isomorphism of s 8 modules between these new identities and the special identities for the jordan diproduct in an associative dialgebra

quantitative domains and infinitary algebras abstract we describe a class of models of the lambda calculus that generalize and simplify the quantitative domains of girard 1988 their salient feature is that these domains do not have enough points in the sense that a map of domains is not entirely determined by its value on the points i e the morphisms from the terminal object of the category we construct a universal domain whose structure is particularly simple and describe some of the features of the untyped lambda calculus associated to it finally we show how linear logic can be interpreted in these domains and construct another category of domains that supports linear logic

internal models of system f for decompilation this paper considers girard s internal coding of each term of system f by some term of a code type this coding is the type erasing coding definable already in the simply typed lambda calculus using only abstraction on term variables it is shown that there does not exist any decompiler for system f in system f where the decompiler maps a term of system f to its code an internal model of f is given by interpreting each type of f by some type equipped with maps between the type and the code type this paper gives a decompiler normalizer for this internal model in f where the decompiler normalizer maps any term of the internal model to the code of its normal form it is also shown that for any model of f the composition of this internal model and the model produces another model of f whose equational theory is below untyped beta eta equality

comparing hierarchies of types in models of linear logic we show that two models m and n of linear logic collapse to the same extensional hierarchy of types when 1 their monoidal categories c and d are related by a pair of monoidal functors f c d g and transformations idc gf and idd fg and 2 their exponentials m and n are related by distributive laws nf f m and mg g n commuting to the promotion rule the key ingredient of the proof is a notion of back and forth translation between the hierarchies of types induced by m and n we apply this result to compare 1 the qualitative and the quantitative hierarchies induced by the coherence or hypercoherence space model 2 several paradigms of games semantics error free vs error aware alternated vs non alternated backtracking vs repetitive uniform vs nonuniform

a k denotational semantics for map theory in zfc si abstract map theory or mt for short has been designed as an integrated foundation for mathematics logic and computer science by this we mean that most primitives and tools are designed from the beginning to bear the three intended meanings logical computational and set theoretic mt was originally introduced in 17 it is based on calculus instead of logic and sets and it fulfills church s original aim of introducing calculus in particular it embodies all of zfc set theory including classical propositional and classical first order predicate calculus mt also embodies the unrestricted untyped lambda calculus including unrestricted abstraction and unrestricted use of the fixed point operator mt is an equational theory we present here a semantic proof of the consistency of map theory within zfc si where si asserts the existence of an inaccessible cardinal the proof is in the spirit of denotational semantics and relies on mathematical tools which reflect faithfully and in a transparent way the intuitions behind map theory this gives a consistency proof but also for the first time gives a clear presentation of the semantics of map theory in a traditional framework from the metamathematical point of view the strength of mt lies somewhere between zfc and zfc si the lower bound is proved in 17 by means of a syntactical translation of zfc including classical propositional and predicate calculus into map theory and the upper bound by building an exceedingly complex model of map theory within zfc si the present paper confirms the upper bound by providing much simpler models the canonical models of the paper which are in fact the paradigm of a large class of quite natural models of mt that all these models interpret a model of zfc is a consequence of the syntactic translation f which is a difficult theorem of 17 we can however give here a direct proof of a stronger result namely that they interpret some vgs where is an inaccessible cardinal finally we return to the canonical models and show that they are adequate for the notion of computation which underlies mt

manin products koszul duality loday algebras and deligne conjecture in this article we give a conceptual definition of manin products in any category endowed with two coherent monoidal products this construction can be applied to associative algebras non symmetric operads operads colored operads and properads presented by generators and relations these two products called black and white are dual to each other under koszul duality functor we study their properties and compute several examples of black and white products for operads these products allow us to define natural operations on the chain complex defining cohomology theories with these operations we are able to prove that deligne s conjecture holds for a general class of operads and is not specific to the case of associative algebras finally we prove generalized versions of a few conjectures raised by m aguiar and j l loday related to the koszul property of operads defined by black products these operads provide infinitely many examples for this generalized deligne s conjecture

jordan trialgebras and post jordan algebras abstract we compute minimal sets of generators for the s n modules n 4 of multilinear polynomial identities of arity n satisfied by the jordan product and the jordan diproduct resp pre jordan product in every triassociative resp tridendriform algebra these identities define jordan trialgebras and post jordan algebras jordan analogues of the lie trialgebras and post lie algebras introduced by dotsenko et al pei et al vallette loday we include an extensive review of analogous structures existing in the literature and their interrelations in order to identify the gaps filled by our two new varieties of algebras we use computer algebra linear algebra over finite fields representation theory of symmetric groups to verify in both cases that every polynomial identity of arity 6 is a consequence of those of arity 4 we conjecture that in both cases the next independent identities have arity 8 imitating the glennie identities for jordan algebras we formulate our results as a commutative square of operad morphisms which leads to the conjecture that the squares in a much more general class are also commutative

intersection types a la church in this paper we present a fully typed calculus based on the intersection type system discipline which is a counterpart a la church of the type assignment system as invented by coppo and dezani the relationship between and the intersection type assignment system is the standard isomorphism between typed and type assignment system and so the typed language inherits from the untyped system all the good properties like subject reduction and strong normalization moreover both type checking and type reconstruction are decidable

the scott model of linear logic is the extensional collapse of its relational model we show that the extensional collapse of the relational model of linear logic is the model of prime algebraic complete lattices a natural extension to linear logic of the well known scott semantics of the lambda calculus

on functors expressible in the polymorphic typed lambda calculus abstract given a model of the polymorphic typed lambda calculus based upon a cartesian closed category k there will be functors from k to k whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions we show that if t is such a functor then there is a weak initial t algebra and if in addition k possesses equalizers of all subsets of its morphism sets then there is an initial t algebra these results are used to establish the impossibility of certain models including those in which types denote sets and s s denotes the set of all functions from s to s

proff search in type theoretic languages an introduction we introduce the main concepts and problems in the theory of proof search in type theoretic languages and survey some specific connected topics we do not claim to cover all of the theoretical and implementation issues in the study of proof search in type theoretic languages rather we present some key ideas and problems starting from well motivated points of departure such as a definition of a type theoretic language or the relationship between languages and proof objects the strong connections between different proof search methods in logics type theories and logical frameworks together with their impact on programming and implementation issues are central in this context

parallel and serial hypercoherences abstract it is known that the strongly stable functions which arise in the semantics of pcf can be realized by sequential algorithms which can be considered as deterministic strategies in games associated to pcf types studying the connection between strongly stable functions and sequential algorithms two dual classes of hypercoherences naturally arise the parallel and serial hypercoherences the objects belonging to the intersection of these two classes are in bijective correspondence with the so called serial parallel graphs that can essentially be considered as games we show how to associate to any hypercoherence a parallel hypercoherence together with a projection onto the given hypercoherence and present some properties of this construction intuitively it makes explicit the computational time of a hypercoherence

a typed calculus based on a fragment of linear logic abstract linear logic we concisely write ll has been introduced recently by jean yves girard in theoretical computer science 50 1987 born from the semantics of second order lambda calculus ll is more expressive than traditional logic both classical and intuitionistic due to the absence of structural rules and to a particular treatment of negation which is denoted by proofs in ll do not have a directional character the constructive meaning of a proof of a b is a function mapping all proofs of a into proofs of b in ll a b has a similar meaning but b a represents the same formula and has the same proofs so one of such proofs can map a proof of a into one of b as well as a proof of b into one of a in this paper we are interested in the multiplicative second order subsystem ll of linear logic we introduce a calculus called v calculus whose terms are canonical representations of proofs the aim of the calculus is to give a better comprehension of the computational aspects of the process of cut elimination we prove that the v calculus obeys strong normalization and the church rosser properties

a representation of l domains by information systems in this paper we propose the notion of l information systems which provides a concrete representation of l domains in particular we prove that the category of l information systems with approximable mappings as morphisms is equivalent to that of l domains with scott continuous functions as morphisms

information systems revisited the general continuous case in this paper a new notion of continuous information system is introduced it is shown that the information systems of this kind generate exactly the continuous domains the new information systems are of the same logic oriented style as the information systems first introduced by scott in 1982 they consist of a set of tokens a consistency predicate and an entailment relation satisfying a set of natural axioms it is shown that continuous information systems are closely related to abstract bases indeed both categories are equivalent since it is known that the categories of abstract bases and or continuous domains are equivalent it follows that the category of continuous information systems is also equivalent to that of continuous domains in applications mostly subclasses of continuous domains are considered for example the domains have to be pointed algebraic bounded complete or fs conditions are presented that when fulfilled by a continuous information system force the generated domain to belong to the required subclass in most cases the requirements are not only sufficient but also necessary

domain theoretic models of polymorphism abstract we give an illustration of a construction useful in producing and describing models of girard and reynolds polymorphic calculus the key unifying ideas are that of a grothendieck fibration and the category of continuous sections associated with it constructions used in indexed category theory the universal types of the calculus are interpreted as the category of continuous sections of the fibration as a major example a new model for the polymorphic calculus is presented in it a type is interpreted as a scott domain in fact understanding universal types of the polymorphic calculus as categories of continuous sections appears to be useful generally for example the technique also applies to the finitary projection model of bruce and longo and a recent model of girard indeed the work here was inspired by girard s and arose through trying to extend the construction of his model to scott domains it is hoped that by pin pointing a key construction this paper will help towards a deeper understanding of models for the polymorphic calculus and the relations between them

filter models with polymorphic types abstract using ideas and results from barendrecht 1983 and coppo 1984 on intersection types a comparable theory is developed for second order polymorphic types the set of filters constructed with polymorphic type forms with inclusion a continuous lattice which yields a model of what we call expansion i e the value of a term increases under reduction but not of conversion combining intersection with polymorphic types does give filter models but the two standard ways of interpreting terms do not coincide

the semantics of second order lambda calculus abstract in the second order polymorphic typed lambda calculus lambda abstraction over type variables leads to terms denoting polymorphic functions straightforward cardinality considerations show that a naive set theoretic interpretation of the calculus is impossible we give two definitions of semantic models for this language and prove them equivalent our syntactical environment model definition and a more algebraic combinatory model definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus soundness and completeness theorems are proved using the environment model definition we verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense

probabilistic coherence spaces as a model of higher order probabilistic computation we study a probabilistic version of coherence spaces and show that these objects provide a model of linear logic we build a model of the pure lambda calculus in this setting and show how to interpret a probabilistic version of the functional language pcf we give a probabilistic interpretation of the semantics of probabilistic pcf closed terms of ground type last we suggest a generalization of this approach using banach spaces

stability and computability in coherent domains abstract the relation between stability and sequentiality is investigated in the category of girard s coheent domains we introduce and discuss a notion of computability for stable functions based on the recursive enumerability of their traces in a way similar to the definition of computable functions in scott s effectively given domains we then relate this notion of computability to regular sets and relative algorithms oracle machines of the theory of relativized computability the notion of oracle machine is used to formalize the idea of a main sequential program which calls an unspecified external agent o a sort of subroutine call in particular we prove that a function f between two coherent domains x and y is stable and computable if and only if f may be computed by an oracle machine questioning in a positive way a simple class of oracles that supply informations about elements in x

linear logic coherence and dinaturality a general coherence theorem for monoidal closed structures is obtained by modifying the logical approach to coherence questions due to lambek 1969 1990 by making use of linear logic linear logic introduced by girard has many advantages which are of use in studying coherence most notably its resource sensitive nature makes it ideal for studying monoidal closed structures the logical approach is also modified by using natural deduction rather than sequent calculus the natural deduction system in question is proof nets also introduced by girard proof nets have several important properties which are exploited to prove the coherence theorem in particular the cut elimination procedure is confluent and strongly normalizing the approach to coherence is to define a general structure the autonomous deductive system for defining many theories of monoidal closed categories an autonomous deductive system is a deductive system with several added features which are suggested by the properties of proof nets it is then possible to give a straightforward criterion for whether a given theory of monoidal closed categories specified by an autonomous deductive system is coherent finally a relationship is established between coherence and the composition problem for dinatural transformations thus the dinatural approach to modelling polymorphic types due to bainbridge et al 1990 can be extended to linear polymorphism

glueing and orthogonality for models of linear logic we present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic in particular we treat the exponentials in detail we indicate simple applications of the methods and show that they cover familiar examples

a simple existence condition of one degree of freedom block decoupling controllers a block decoupling problem in linear multivariable systems is treated for one degree of freedom controller configuration with unity output feedback the plant transfer matrix which may be non square is assumed to have unstable simple poles and zeros that may coincide a simple existence condition of a block decoupling controller is obtained by directional interpolation approaches

building continuous webbed models for system f we present here a large family of concrete models for girard and reynolds polymorphism system f in a noncategorical setting the family generalizes the construction of the model of barbanera and berardi tech report university of turin 1997 hence it contains complete models for f a complete model for system f preprint june 1998 and we conjecture that it contains models which are complete for f it also contains simpler models the simplest of them e2 being a second order variant of the engeler plotkin model e all the models here belong to the continuous semantics and have underlying prime algebraic domains all have the maximum number of polymorphic maps the class contains models which can be viewed as two intertwined compatible webbed models of untyped calculus in the sense of berline from computations to foundations the calculus and its webbed models revised version theoret comput sci 86 pp to appear but it is much larger than this finally many of its models might he read as two intertwined strict intersection type systems

a higher order calculus and theory abstraction abstract we present a higher order calculus ecc which naturally combines coquand huet s calculus of constructions and martin lof s type theory with universes ecc is very expressive both for structured abstract reasoning and for program specification and construction in particular the strong sum types together with the type universes provide a useful module mechanism for abstract description of mathematical theories and adequate formalization of abstract mathematics this allows comprehensive structuring of interactive development of specifications programs and proofs after a summary of the meta theoretic properties of the calculus an set realizability model of ecc is described to show how its essential properties can be captured set theoretically the model construction entails the logical consistency of the calculus and gives some hints on how to adequately formalize abstract mathematics theory abstraction in ecc is discussed as a pragmatic application

computability on di domains in this paper we investigate a subset of the class of scott computable stable functions called berry computable furthermore we introduce a corresponding notion of an effectively given di domain we obtain the result that the category with these domains as objects and berry computable functions as morphisms has all the important attributes of recursive domains and scott computable continuous functions e g cartesian closedness moreover we investigate the relationship between stable functions and sequential algorithms we show that any berry computable function can be computed in a sequential way by a relative algorithm additionally if the codomain of the considered stable function is an atomic di domain we obtain the result that the reverse holds as well

an algebraic generalization of frege structures binding algebras abstract this paper introduces a kind of second order algebras called binding algebras which is an algebraic generalization of aczel s 1980 frege structure it investigates the universal property of binding algebras by following birkhoff s 1935 method an equational logic eba is introduced we obtain its admissible completeness the relationship between completeness and the admissible completeness is very close and will be discussed

a natural semantics of first order type dependency abstract we develop a natural model of first order type dependency by considering types as stable families of di domains indexed on a di domain or in an equivalent way as stable functors from a di domain to the category of di domains with rigid embedding projections the fundamental idea of the interpretation is to interpret any construction in the type system by an algebraic construction in this category we present here a model of the so called generalised algebraic fragment of martin lof s and we replace it in the d category general framework