A note on the monotone functional interpretation

We prove a result relating the author's monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that (over model of majorizable functionals) largely a solution for the bounded interpretation also is a solution for the monotone functional interpretation although the latter uses the existence of an underlying precise witness. This makes it possible to focus on the extraction of bounds (as in the bounded interpretation) while using the conceptual benefit of having precise realizers at the same time without having to construct them.     

A Behavioral Interpretation of Belief Functions

Shafer’s belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer’s belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature.     

Intuitionistic nonstandard bounded modified realisability and functional interpretation

We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles.The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability.We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.     

一阶模糊谓词逻辑公式的解释模型真度理论及其应用

基于一阶模糊谓词逻辑公式的有限和可数解释真度的理论,引入了一阶模糊谓词逻辑公式的解释模型及解释模型真度的概念,并讨论了它们的一系列性质及其在近似推理中的应用.     

On systems of definitions,induction and recursion

Primitive recursion can be seen as the computational interpretation of induction through the Curry-Howard interpretation of propositions-as-types. In the present paper we discuss what happens if we apply this idea to possible non monotone inductive definitions. We start with a logical interpretation of a class of inductive definitions, thepartial inductive definitions. Then we internalise induction over such definitions as a rule of inference and consider a Curry-Howard interpretation of these definitions as type systems. As a basic example we discuss what meaning this interpretation gives to primitive recursion on a definition likeN=NN.     

A note on NTU convexity

For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with nontransferable utility, however, the literature mainly focuses on two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this paper, we analyse three types of convexity for NTU games that generalise the marginalistic interpretation of convexity. Received: December 2000     

Interpretation of contradictory images by means of systems of linear inequalities

We consider the problem of interpretation of three-dimensional images from their flat projections up to the set of visible faces. For projections of convex polyhedra, we present an interpretation algorithm based on maximal feasible subsystems of a certain infeasible system of linear inequalities modeling the visibility requirement for faces. A number of model examples are given; in particular, the algorithm is applied to the interpretation of the Necker cube.     

Denotational semantics for intuitionistic type theory using a hierarchy of domains with totality

A modified version of Normann's hierarchy of domains with totality [9] is presented and is shown to be suitable for interpretation of Martin-L?f's intuitionistic type theory. This gives an interpretation within classical set theory, which is natural in the sense that -types are interpreted as sets of pairs and -types as sets of choice functions. The hierarchy admits a natural definition of the total objects in the domains, and following an idea of Berger [3] this makes possible an interpretation where a type is defined to be true if its interpretation contains a total object. In particular, the empty type contains no total objects and will therefore be false (in any non-empty context). In addition, there is a natural equivalence relation on the total objects, so we derive a hierarchy of topological spaces (quotient spaces wrt. the Scott topology), and give a second interpretation using this hierarchy. Received: 11 December 1995 / Revised version: 14 October 1996     

Sensitivity analysis in fuzzy linear programming

In this paper we propose a certain interpretation of a partially fuzzy LP-Problem. This LP-is dualized and the corresponding variables of the Dual are analyzed and an economic interpretation is given.     

On the Herbrand functional interpretation

We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation we also strengthen the monotonicity property of the original presentation, and prove a monotonicity property for our alternative definition.     

A Diller-Nahm-style functional interpretation of $\hbox{\sf KP} \omega$

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity () given in [1] uses a choice functional (which is not a definable set function of (). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in . This yields the following characterization: The class of -definable set functions of coincides with the collection of set functionals of type 1 primitive recursive in . Received: 26 August 1998     

The FAN principle and weak König's lemma in herbrandized second-order arithmetic

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.     

A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

We present a geometric interpretation of the operation a⊕b and the gyration on the unit-disc as defined by A.A. Ungar. Using this geometric interpretation we show that the two known generalizations to the n-dimensional unit ball are identical. The interpretation in the plane leads us to the notion of outer-median of a triangle and we discuss some possible properties of this median.     

遗传算法的物理解释

提出了一种理想化的模拟仿生搜索算法——扰动算法 ,以此方法为基础 ,分析了遗传算法的搜索过程和效率问题 ,阐明了遗传算法作为一种次优算法的有效性 .相对于遗传算法的生物解释 ,本文给出了相应的物理解释 .同时 ,本文为遗传算法、进化策略和模拟退火算法找到了一种统一的物理解释 ,揭示了这些重要的仿生类算法实质上的相似性 .     

A note on q-Eulerian numbers

The paper contains a combinatorial interpretation of the q-Eulerian numbers suggested by H. O. Foulkes' combinatorial interpretation of the ordinary Eulerian numbers. Some applications are also given.     

Reflections on the status and nature of visual reasoning - the case of the matches

This paper gives an interpretation of the transformation formulae (in matrix notation) for rotations in the plane in terms of the exponential function. Then the fact that the addition of two rotations corresponds to the multiplication of the corresponding matrices is given a simple interpretation.     

Stable Higgs bundles with trivial Chern classes. Several examples

We propose a new interpretation of Higgs bundles as vector bundles together with actions of the symmetric tangent tensor algebra. Via this interpretation, we construct new nontrivial examples of stable Higgs bundles with vanishing Chern classes. To the memory of Professor V.A. Iskovskikh     

Links between Latin squares,nets, graphs and groups: Work inspired by a paper of A. Barlotti and K. Strambach

Starting from a computer generated example of a transitive Latin square graph over a proper loop we describe a computer free interpretation of this specific strongly regular graph. Moreover, with this interpretation we were able to generalize the result to an infinite series of similar examples.     

The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.     

Venn Diagram with Names of Individuals and Their Absence: A Non-classical Diagram Logic

Venn diagram system has been extended by introducing names of individuals and their absence. Absence gives a kind of negation of singular propositions. We have offered here a non-classical interpretation of this negation. Soundness and completeness of the present diagram system have been established with respect to this interpretation.