Algebraic Totality, towards Completeness

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Certain results in connection with boundedness and precompactness, which are valid in topological vector spaces and locally bounded prelimit vector spaces, are extended to the class of strictly *-equable pre-limit vector spaces. This class contains the class of topological vector spaces. Hahn-Banach extension and separation theorems hold in the class of strictly *-equable prelimit vector spaces.

We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi.

This article shows how individual Petri nets form models of Girard's intuitionistic linear logic. It explores questions of expressiveness and completeness of linear logic with respect to this interpretation. An aim is to use Petri nets to give an understanding of linear logic and give some appraisal of the value of linear logic as a specification logic for Petri nets. This article might serve as a tutorial, providing one in-road into Girard's linear logic via Petri nets. With this in mind we have added several exercises and their solutions. We have made no attempt to be exhaustive in our treatment, dedicating our treatment to one semantics of intuitionistic linear logic.Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. The strongest logic considered is intuitionistic linear logic, with ⊗, ⊸, &, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places). The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. A start is made on decidability issues.

We present three different functional interpretations of intuitionistic linear logic ILL and show how these correspond to well-known functional interpretations of intuitionistic logic IL via embeddings of IL into ILL. The main difference from previous work of the second author is that in intuitionistic linear logic (as opposed to classical linear logic) the interpretations of !A are simpler and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. We then compare our approach in developing these three proof interpretations with the one of de Paiva around the Dialectica category model of linear logic.

This paper is the fourth of a series exposing a systematic combinatorial approach to Girard's Geometry of Interaction (GoI) program. The GoI program aims at obtaining particular realisability models for linear logic that accounts for the dynamics of cut-elimination. This fourth paper tackles the complex issue of defining exponential connectives in this framework. For that purpose, we use the notion of \emph{graphings}, a generalisation of graphs which was defined in earlier work. We explain how to define a GoI for Elementary Linear Logic (ELL) with second-order quantification, a sub-system of linear logic that captures the class of elementary time computable functions.

It is well-known that the algebraic tensor product E ⊗ Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called e-product EeY (a space of continuous linear mappings from Y′ into E). So, whenever EeY is complete, even the completed tensor product is (isomorphic to) a subspace of EeY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding j∶U→E and a continuous linear map u∶F→U such that u=j·u. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

A theory of test function spaces is developed. These are generalizations of test function spaces of type Hμ. The elements of the dual spaces are called ultradistributions, It is shown that the Hankel transformation hμ for μ≧−1/2 is a continuous linear mapping from each of these spaces into certain spaces of the above type. The generalized Hankel transformation hμ′ is defined as a continuous linear mapping between the dual spaces. Hankel transformations of arbitrary order and also of arbitrary growth are studied. The theory thus developed is applied to prove a uniqueness theorem for the Cauchy problem for the operator Sμ.

<abstract><p>In this paper, we show that every continuous linear map between unital $ C^\ast $-algebras is skew Lie triple derivable at the identity is a $ \ast $-derivation and that every continuous linear map between unital $ C^\ast $-algebras which is a skew Lie triple homomorphism at the identity is a Jordan $ \ast $-homomorphism.</p></abstract>

Probabilistic coherence spaces yield a model of linear logic and lambda-calculus with a linear algebra flavor. Formulas/types are associated with convex sets of $${\mathbb R^+}^{}$$ -valued vectors, linear logic proofs with linear functions and $$\lambda $$ -terms with entire functions, both mapping the convex set of their domain into the one of their codomain. Previous results show that this model is particularly precise in describing the observational equivalences between probabilistic functional programs. We prove here that the exponential modality is the free commutative comonad, giving a further mark of canonicity to the model.

Let (X, =) be a topological space. A subset A of X is called pre-open if A ⊆ Int(Cl(A)). Let P O(X) denote the family of all pre-open sets in X. In general, P O(X) does not form a topology on X. Furthermore, in topological vector spaces, it is not always true that P O(L) forms a topology on L where L is a topological vector space. In this note, we prove that the class of strongly preirresolute topological vector spaces is that subclass of topological vector spaces in which P O(L) forms a topology and thereby we see that all proved results in [5] concerning strongly preirresolute topological vector spaces are obvious.

In this chapter we will state some of the most important results on finite spaces which are previous to our work. These results can be summarized by the following three items: 1. The correspondence between finite topological spaces and finite partially ordered sets, first considered by Alexandroff in [1] in 1937. The combinatorial description of homotopy types of finite spaces, discovered by Stong in his beautiful article [76] of 1966. 3. The connection between finite spaces and polyhedra, found by McCord [55] also in 1966.

Achieving total information flow security is feasible through the use of formal mathematical models. A fundamental theorem of a topological theory of Discretionary Access Controls is presented: Let U be a set of UNIX users. The association from each data owner p to its data owners group F(p) defines a generalized topological space (U, F) and a binary relation B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> . Thus, a DAC-system is information-flow-secure (via file movements): if and only if (U, F) is a finite topological space; if and only if (U, B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> ) is a reflexive and transitive graph. An important consequence is the following proved secure system in current UNIX: The information flow (via file movements) of UNIX generated by Read-access is information-flow-secure: if (U, F) is a finite topological space;if (U, B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> ) is a reflexive and transitive graph. This model is based on a modified security principle: Every data owner p has the discretionary authority (Individual Data Owner p's authority, or IDO(p)-authority) over who can access the data. The precise security policy is the integration of IDO(p)-security for All data owner p in All possible data states. Once granted access, information propagation can still be controlled by the topological structure.