yang baxter deformations and rack cohomology abstract in his study of quantum groups drinfeld suggested to consider set theoretic solutions of the yang baxter equation as a discrete analogon as a typical example every conjugacy class in a group or more generally every rack q provides such a yang baxter operator cq x y 7 y x y in this article we study deformations of cq within the space of yang baxter operators infinitesimally these deformations are classified by yang baxter cohomology we show that the yang baxter cochain complex of cq homotopyretracts to a much smaller subcomplex called quasi diagonal this greatly simplifies the deformation theory of cq including the modular case which had previously been left in suspense by establishing that every deformation of cq is gauge equivalent to a quasi diagonal one in a quasi diagonal deformation only behaviourally equivalent elements of q interact if all elements of q are behaviourally distinct then the yang baxter cohomology of cq collapses to its diagonal part which we identify with rack cohomology this establishes a strong relationship between the classical deformation theory following gerstenhaber and the more recent cohomology theory of racks both of which have numerous applications in knot theory

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in computer science vol 62 springer berlin 1978 pp 72 89 as a refinement of the continuous framework defined by scott unpublished manuscript 1969 53pp in this paper we show that even in the stable case we have a fair amount of freedom during the construction of models we introduce a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

categorical formulation of finite dimensional quantum algebras we describe how frobenius monoids give the correct categorical description of certain kinds of finite dimensional quantum algebras we develop the concept of an involution monoid and use it to construct a correspondence between finite dimensional c algebras and certain types of frobenius monoids in the category of hilbert spaces using this technology we recast the spectral theorems for commutative c algebras and for normal operators into an explicitly categorical language and we examine the case that the results of measurements do not form finite sets but rather objects in a finite boolean topos we describe the relevance of these results for topological quantum field theory

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

traces in monoidal categories the main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category this construction is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property

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

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

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

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

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

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

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

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

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

descent methods with linesearch in the presence of perturbations we consider the class of descent algorithms for unconstrained optimization with an armijo type stepsize rule in the case when the gradient of the objective function is computed inexactly an important novel feature in our theoretical analysis is that perturbations associated with the gradient are not assumed to be relatively small or to tend to zero in the limit as a practical matter we expect them to be reasonably small so that a meaningful approximate solution can be obtained this feature makes our analysis applicable to various difficult problems encounted in practice we propose a modified armijo type rule for computing the stepsize which guarantees that the algorithm obtains a reasonable approximate solution furthermore if perturbations are small relative to the size of the gradient then our algorithm retains all the standard convergence properties of descent methods

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

a finite tame wild trichotomy theorem for tensor diagrams abstract in this paper we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis in particular to a tensor or string diagram in a certain monoidal category which we call tensor diagrams we formulate an associated abelian category of representations each representation corresponds to a tensor network on that diagram we then classify which tensor diagrams give rise to categories that are finite tame or wild in the traditional sense of representation theory for those tensor diagrams of finite and tame type we classify the indecomposable representations our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three otherwise it is of tame or finite type

non cyclotomic fusion categories etingof nikshych and ostrik ask in 7 2 if every fusion category can be completely defined over a cyclotomic field we show that this is not the case in particular one of the fusion categories coming from the haagerup subfactor 1 and one coming from the newly constructed extended haagerup subfactor 2 can not be completely defined over a cyclotomic field on the other hand we show that the drinfel d center of the even part of the haagerup subfactor is completely defined over a cyclotomic field we identify the minimal field of definition for each of these fusion categories compute the galois groups and identify their galois conjugates ams classification 18d10 46l37

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

categorical quantum circuits in this paper we extend previous work carried out on the application of the mathematics of category theory to quantum information science specifically we present a realization of a dagger compact category that can model finite dimensional quantum systems and explicitly allows for the interaction of systems of arbitrary possibly unequal dimensions hence our framework can handle generic tensor network states including matrix product states our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science the circuit diagrams themselves now become morphisms in a category making quantum circuits a special case of a much more general mathematical framework we introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information

a synthetic approach to markov kernels conditional independence and theorems on sufficient statistics abstract we develop markov categories as a framework for synthetic probability and statistics following work of golubtsov as well as cho and jacobs this means that we treat the following concepts in purely abstract categorical terms conditioning and disintegration various versions of conditional independence and its standard properties conditional products almost surely sufficient statistics versions of theorems on sufficient statistics due to fisher neyman basu and bahadur besides the conceptual clarity offered by our categorical setup its main advantage is that it provides a uniform treatment of various types of probability theory including discrete probability theory measure theoretic probability with general measurable spaces gaussian probability stochastic processes of either of these kinds and many others

remarks on convergence of the matrix splitting algorithm for the symmetric linear complementarity problem in this paper it is shown how to remove a restriction on the perturbation vectors h i imposed by mangasarian and luo and tseng in their respective convergence analyses of the matrix splitting algorithms

pictures of complete positivity in arbitrary dimension two fundamental contributions to categorical quantum mechanics are presented first we generalize the cpm construction that turns any dagger compact category into one with completely positive maps to arbitrary dimension second we axiomatize when a given category is the result of this construction

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

successive overrelaxation for support vector machines successive overrelaxation sor for symmetric linear complementarity problems and quadratic programs is used to train a support vector machine svm for discriminating between the elements of two massive datasets each with millions of points because sor handles one point at a time similar to platt s sequential minimal optimization smo algorithm 1999 which handles two constraints at a time and joachims svm sup light 1998 which handles a small number of points at a time sor can process very large datasets that need not reside in memory the algorithm converges linearly to a solution encouraging numerical results are presented on datasets with up to 10 000 000 points such massive discrimination problems cannot be processed by conventional linear or quadratic programming methods and to our knowledge have not been solved by other methods on smaller problems sor was faster than svm sup light and comparable or faster than smo

computing the tutte polynomial of lattice path matroids using determinantal circuits we give a quantum inspired o n 4 algorithm computing the tutte polynomial of a lattice path matroid where n is the size of the ground set of the matroid furthermore this can be improved to o n 2 arithmetic operations if we evaluate the tutte polynomial on a given input fixing the values of the variables the best existing algorithm found in 2004 was o n 5 and the problem has only been known to be polynomial time since 2003 conceptually our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems such as those that appear in simulating quantum systems

adjunctions and defects in landau ginzburg models abstract we study the bicategory of landau ginzburg models which has polynomials as objects and matrix factorisations as 1 morphisms our main result is the existence of adjoints in this bicategory and formulas for the evaluation and coevaluation maps in terms of atiyah classes and homological perturbation the bicategorical perspective offers a unified approach to landau ginzburg models we show how to compute arbitrary correlators and recover the full structure of open closed tft including the kapustin li disc correlator and a simple proof of the cardy condition in terms of defect operators which in turn are directly computable from the adjunctions

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

the hom yang baxter equation and hom lie algebras motivated by recent work on hom lie algebras a twisted version of the yang baxter equation called the hom yang baxter equation hybe was introduced by yau j phys a 42 165202 2009 in this paper several more classes of solutions of the hybe are constructed some of the solutions of the hybe are closely related to the quantum enveloping algebra of sl 2 the jones conway polynomial and yetter drinfel d modules under some invertibility conditions we construct a new infinite sequence of solutions of the hybe from a given one

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

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

a 2 categorical pasting theorem in order to facilitate the study of 2 categories with structure we state and prove an n categorical pasting theorem this is based upon a new definition of n pasting scheme that generalises johnson s definition of a well formed loop free pasting scheme by weakening his no direct loops condition we define n pasting prove the theorem and show that for n 3 it incorporates all possible composites of n cells we show that that is not true for higher n we define the horizontal n category of an n 1 category to generalise that of a 2 category we define horizontal and vertical composition for an n 1 category and we state and prove an interchange law we also study further conditions on a pasting diagram and their impact upon how one may evaluate a composite and we express street s free n categories in terms of left adjoints

on computing connected components of line segments it is shown that given a set of n line segments their connected components can be computed in time o n sup 4 3 log sup 3 n a bound of o n sup 4 3 for this problem would imply a similar bound for detecting for a given set of n points and n lines whether some point lies on some of the lines this problem known as hopcroft s problem is believed to have a lower bound of spl omega n sup 4 3 for the special case when for each segment both endpoints fall inside the same face of the arrangement induced by the set of segments we give an algorithm that runs in o nlog sup 3 n time

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

a compositional approach to quantum functions we introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2 category of quantum sets and quantum functions we use this framework to formulate a 2 categorical theory of quantum graphs which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups we show that the 2 categories of quantum sets and quantum graphs are semisimple we analyze dualisable and invertible 1 morphisms in these 2 categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs

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

higher dimensional algebra iv 2 tangles just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions compact surfaces smoothly embedded in r4 can be described as certain 2 morphisms in the 2 category of 2 tangles in 4 dimensions using the work of carter rieger and saito we prove that this 2 category is the free semistrict braided monoidal 2 category with duals on one unframed self dual object by this universal property any unframed self dual object in a braided monoidal 2 category with duals determines an invariant of 2 tangles in 4 dimensions

constructing arrangements of lines and hyperplanes with applications a finite set of lines partitions the euclidean plane into a cell complex similarly a finite set of d 1 dimensional hyperplanes partitions d dimensional euclidean space an algorithm is presented that constructs a representation for the cell complex defined by n hyperplanes in optimal o n d time in d dimensions it relies on a combinatorial result that is of interest in its own right the algorithm is shown to lead to new methods for computing lambda matrices constructing all higher order voronoi diagrams halfspatial range estimation degeneracy testing and finding minimum measure simplices in all five applications the new algorithms are asymptotically faster than previous results and in several cases are the only known methods that generalize to arbitrary dimensions the algorithm also implies an upper bound of 2 cn d c a positive constant for the number of combinatorially distinct arrangements of n hyperplanes in e d

double quantum groups the construction of the drinfeld double d h of a finite dimensional hopf algebra h was one of the first examples of a quasitriangular hopf algebra whose category of modules md h is braided the braided category of yetter drinfeld modules ydhh is the analogue for infinite dimensional hopf algebras it uses a strong dependence between the h module and the h comodule structures we generalize this construction to the category mca of entwined modules that is a modules and c comodules over hopf algebras a and c where the structures are only related by an entwining map c a a c we show that mca is braided iff there is an r map r c c a a satisfying suitable axioms that generalize the axioms of an r matrix for finite dimensional c there is a quasitriangular hopf algebra structure on hom c a the quantum group double generalizing the construction of the drinfeld double

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

generalized counting constraint satisfaction problems with determinantal circuits abstract generalized counting constraint satisfaction problems include holant problems with planarity restrictions polynomial time algorithms for such problems include matchgates and matchcircuits which are based on pfaffians in particular they use gates which are expressible in terms of a vector of sub pfaffians of a skew symmetric matrix we introduce a new type of circuit based instead on determinants with seemingly different expressive power in these determinantal circuits a gate is represented by the vector of all minors of an arbitrary matrix determinantal circuits permit a different class of gates applications of these circuits include proofs of theorems from algebraic graph theory including the chung langlands formula for the number of rooted spanning forests of a graph and computing tutte polynomials of certain matroids they also give a strategy for simulating quantum circuits with closed timelike curves monoidal category theory provides a useful language for discussing such counting problems turning combinatorial restrictions into categorical properties we introduce the counting problem in monoidal categories and count preserving functors as a way to study fp subclasses of problems in settings which are generally p hard using this machinery we show that surprisingly determinantal circuits can be simulated by pfaffian circuits at quadratic cost

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

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

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

modular operads and the nerve theorem abstract we describe a category of undirected graphs which comes equipped with a faithful functor into the category of colored modular operads the associated singular functor from modular operads to presheaves is fully faithful and its essential image can be classified by a segal condition this theorem can be used to recover a related statement due to andre joyal and joachim kock concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full

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

new error bounds and their applications to convergence analysis of iterative algorithms we present two new error bounds for optimization problems over a convex set whose objective function f is either semianalytic or strictly convex with 1 we then apply these error bounds to analyze the rate of convergence of a wide class of iterative descent algorithms for the aforementioned optimization problem our analysis shows that the function sequence f xk converges at least at the sublinear rate of k for some positive constant where k is the iteration index moreover the distances from the iterate sequence xk to the set of stationary points of the optimization problem converge to zero at least sublinearly

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

on linear convergence of iterative methods for the variational inequality problem abstract we present new linear convergence results for iterative methods for solving the variational inequality problem the methods include the extragradient method the proximal point method a matrix splitting method and a certain feasible descent method the proofs of the results are based on certain error bounds related to the algorithmic mappings moreover we show that all these error bounds hold if a projection type error bound holds

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

quantum koszul complexes for majid s braided lie algebras a yang baxter operator was canonically associated to any cocommutative braided lie algebra as defined by majid applying previous results we construct quantum koszul complexes for braided lie algebras with respect to this operator in the usual case the complexes reduce to the chevalley eilenberg complexes for classical lie algebras

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

two dimensional algebra in lattice gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher category theory in doing so we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher cat

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

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 graphical category for higher modular operads abstract we present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy this homotopy theory takes the form of a quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs this new category of undirected graphs denoted u plays a similar role for modular operads that the dendroidal category plays for operads we carefully study properties of u including the existence of certain factorization systems related structures such as cyclic operads and stable modular operads can be similarly treated using categories derived from u

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

on the rosenberg zelinsky sequence in abelian monoidal categories we consider frobenius algebras and their bimodules in certain abelian monoidal categories in particular we study the picard group of the category of bimodules over a frobenius algebra i e the group of isomorphism classes of invertible bimodules the rosenberg zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the picard group which however is typically not surjective we investigate under which conditions there exists a morita equivalent frobenius algebra for which the corresponding homomorphism is surjective one motivation for our considerations is the orbifold construction in conformal field theory

on the geometry and algebra of networks with state in 1 an algebra of automata with interfaces span graph was introduced with main operation being communicating parallel composition a system is represented by an expression in this algebra a system so described has two aspects an informal network geometry arising from the algebraic expression and a space of states and transition given by its evaluation in span graph note that span graph yields purely compositional descriptions of systems in this paper we give an alternative globally non compositional view of the same systems in order to do this we make mathematically precise the network geometry in terms of monoidal graphs and assignment of state in terms of morphisms of monoidal graphs we call such globally described systems networks with state to relate networks with state and span graph systems we use the algebra of cospans of monoidal graphs as an illustration we give a new representation of a non compositionally described class of petri nets in the compositional setting of span graph we also present a tool for calculating with networks with state both algebras of spans and of cospans are symmetric monoidal categories with commutative separable algebra structures on the objects we include a short section following 2 in which we show that a simple modification of the algebra allows the description of networks in which the tangling of connectors may be represented yielding a connection with the theory of knots