Axiomatic Way Research Articles

A formal approach for the development of reactive systems

Context This paper deals with the development and verification of liveness properties on reactive systems using the Event-B method. By considering the limitation of the Event-B method to invariance properties, we propose to apply the language TLA + to verify liveness properties on Event-B models. Objective This paper deals with the use of two verification approaches: theorem proving and model-checking, in the construction and verification of safe reactive systems. The theorem prover concerned is part of the Click_n_Prove tool associated to the Event-B method and the model checker is TLC for TLA + models. Method To verify liveness properties on Event-B systems, we extend first the expressivity and the semantics of a B model (called temporal B model) to deal with the specification of fairness and eventuality properties. Second, we propose semantics of the extension over traces, in the same spirit as TLA + does. Third, we give verification rules in the axiomatic way of the Event-B method. Finally, we give transformation rules from a temporal B model into a TLA + module. We present in particular, our prototype system called B2TLA +, that we have developed to support this transformation; then we can verify liveness properties thanks to the model checker TLC on finite state systems. For the verification of infinite-state systems, we propose the use of the predicate diagrams and its associated tool DIXIT. As the B refinement preserves invariance properties through refinement steps, we propose some rules to get the preservation of liveness properties by the B refinement. Results The proposed approach is applied for the development of some reactive systems examples and our prototype system B2TLA + is successfully used to transform a temporal B model into a TLA + module. Conclusion The paper successfully defines an approach for the specification and verification of safety and liveness properties for the development of reactive systems using the Event-B method, the language TLA + and the predicate diagrams with their associated tools. The approach is illustrated on a case study of a parcel sorting system.

Scalar and fuzzy cardinalities of crisp and fuzzy multisets

We define in an axiomatic way scalar and fuzzy cardinalities of finite multisets over ]0, 1], and we obtain explicit descriptions for them. We show that, for multisets over ]0, 1] associated to finite fuzzy sets, the cardinalities defined in this study are equivalent to the cardinalities of the corresponding fuzzy sets previously introduced in the literature. Finally, we also define in an axiomatic way scalar and fuzzy cardinalities of finite fuzzy multisets over any set X, and we use the descriptions of the cardinalities of finite multisets over ]0, 1] to obtain explicit characterizations of the former. © 2009 Wiley Periodicals, Inc.

Quantum knots and mosaics

In this paper, we give a precise and workable definition of a knot system, the states of which are called knots. This definition can be viewed as a blueprint for the construction of an actual physical system. Moreover, this definition of a knot system is intended to represent the quantum embodiment of a closed knotted physical piece of rope. A knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a knot can exhibit non-classical behavior, such as superposition and entanglement. This raises some interesting and puzzling questions about the relation between topological and entanglement. The knot type of a knot is simply the orbit of the knot under the action of the ambient group. We investigate observables which are invariants of knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

Probabilistic Configuration Theories

We introduce a new framework for describing computations which are both concurrent and probabilistic. This framework is a natural extension of the Configuration Theories of [Pietro Cenciarelli. Configuration theories. In CSL '02: Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic, pages 200–215, London, UK, 2002. Springer-Verlag], and allows to express properties about both the causal and the probabilistic aspect of concurrent computations. Computations are described in an axiomatic way, by using a probabilistic extensions of poset sequents. A scheme of structural rules on these sequents is introduced and shown to be sound, while completeness can be obtained by adding a rule which is not treated in the present work. We also introduce a new probabilistic extension of Configuration Structures, which we use as models for probabilistic sequents.

Geodesics for Dual Connections and Means on Symmetric Cones

Means on self-dual and homogeneous cones (, i.e., symmetric cones) are discussed from a viewpoint of differential geometry with affine connections. We first define means on symmetric cones in an axiomatic way following [8]. Next we consider dualistic differential geometry (, i.e., Riemannian metric and affine connections) [1] naturally introduced on symmetric cones. Elucidating the relation between the geodesics defined by each affine connection, and operator monotone functions that generate means, we show an important class of means are expressed by the (mid)points on geodesics. Related results on the means and submanifolds in a symmetric cone are also presented.

A characterization of generalized concordance rules in multicriteria decision making

This article proposes a principled approach to multicriteria decision making (MCDM) where the worth of decisions along attributes is not supposed to be quantified, as in multiattribute utility theory, or even measured on a unique scale. This approach actually generalizes additive concordance rules a la Electre and is rigorously justified in an axiomatic way by representation theorems. We indeed show that the use of a generalized concordance (GC) rule is the only possible approach when in a purely ordinal framework and that the satisfaction of very simple principles forces the use of possibility theory as the unique way of expressing the importance of coalitions of criteria. © 2003 Wiley Periodicals, Inc.

Linear Dynamical Systems: An Axiomatic Approach

Abstract Linear dynamical systems are introduced in a general axiomatic way, and their development is carried out in great simplicity. The approach is closely related both with the classical transfer function approach and with the Willems behavioral approach.

Divergence and fuzziness measures

In an axiomatic way a divergence between fuzzy sets is introduced which extends the symmetric difference between crisp sets. Any fuzzy measure of the divergence between two fuzzy sets weighs their “distance”. The distance between a fuzzy set and the family of crisp sets is fuzziness measure.

Semantics analysis through elementary meanings

Abstract. Thesaurus is a collection of words classified according to some relatedness measures among them. In this paper, we lay the theoretical foundations of thesaurus construction through elementary meanings of words. The concept of elementary meanings has been advocated and utilized in compiling Webster's Collegiate Thesaurus. If each word is supplied with elementary meanings so that all its meanings are covered by them in a standard fashion, we can define various similarity measures for a given set of words. Here we take an axiomatic way to analyze semantic structure of word groups. Assuming an abstract semantic world, we deduce closed sets as generalized synonym sets. That is, we show that under certain natural axioms, we only need to consider closed sets as far as the semantics are concerned. We also show that the set of generalized synonyms described as a certain pair of closed sets has a lattice structure. In order to have a flexible thesaurus, we also analyze structure changes corresponding to three basic environmental changes: A new word-meaning relation is added, a new word or a new meaning is included with its word-meaning relations. Actually we give algorithms to have updated lattice structure from previous one for the three operations.

Comment on “Arrival time in quantum mechanics” and “Time of arrival in quantum mechanics”

Contrary to claims contained in papers by Grot, Rovelli, and Tate [Phys. Rev. A 54, 4676 1996)] and Delgado and Muga [Phys. Rev. A 56, 3425 (1997)], the ``time operator,'' which I have constructed [Rep. Math. Phys. 6, 361 (1974)] in an axiomatic way, is a self-adjoint operator existing in a usual Hilbert space of (nonrelativistic or relativistic) quantum mechanics.

Ancient Chinese mathematics: the ( Jiu Zhang Suan Shu) vs Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge

The following is a preliminary and relatively brief, exploratory discussion of the nature of early Chinese mathematics, particularly geometry, considered largely in terms of one specific example: the ( Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Western traditions, particularly with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematical thought. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that Chinese mathematicians were by nature primarily concerned with practical problems and their solutions, and, therefore, had no interest in developing a highly theoretical mathematics. Others have stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, which, in turn, worked against the development of abstract mathematics of the Greek sort. While both of these views contain elements of truth, and certainly play a role in understanding why the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a different approach is offered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for how the Chinese were able to conceptualize—and think about—mathematics.

Functions of class $C^k$ without derivatives

We describe a general axiomatic way to define functions of class $C^k$, $k\in \bold N \cup\{\infty \}$ on topological abelian groups. In the category of Banach spaces, this definition coincides with the usual one. The advantage of this axiomatic approach is that one can dispense with the notion of norms and limit procedures. The disadvantage is that one looses the derivative, which is replaced by a local linearizing factor. As an application we use this approach to define $C^\infty$ functions in the setting of graded/super manifolds.

Twenty years of fuzzy preference structures (1978–1997)

The aim of this paper is to summarize the historical development of the concept of a fuzzy preference structure and to present the resulting material in a logical manner. Roughly speaking, three phases can be distinguished. In the first phase, different proposais for constructing fuzzy strict preference and indifference relations emerge. In the second one, this construction is approached in an axiomatic way, by imposing couditions on the fuzzy strict preference, indifference and incomparability relations, leading to functional equations identifying suitabie strict preference, indifference and incomparability generators. Surprisingly, the detinition of a fuzzy preference structure was only given in the third phase, leading to the study of additive fuzzy preference structures. Combined with results from the axiomatic approach, this has linally led to the discovery of characterizable fuzzy preference structures.

On the Relationship between Two Types of Effects Caused by Color Adaptation: Changes in Color Appearance and Color Discriminability

Varying the state of color adaptation not only causes changes in the appearance of color stimuli but also leads to changes in their discriminability. The effects of adaptation on color appearance can be determined experimentally by the method of cross-context matching. If cross-context matches are perfect with respect to all attributes by which color stimuli can be discriminated, it should be possible to use theories for cross-context matching in order to predict color discrimination across contexts. To characterize the empirical situation in which predictions of this type can be made, some conditions for cross-context matching and color discrimination are formulated in an axiomatic way and an elementary representation is derived from them. In order to test predictability of color discriminant across contexts empirically, an experiment was conducted with consists of two parts. In a first step, eight stimuli were matched across contexts. To the adjustments of each of the eight subjects a projective mappingfwas fitted. For a second experiment pairs of stimuli were chosen that, according tof, should correspond to each other. Subjects repeatedly had to match each of these stimuli within the original context taken from the first experiment. Discrimination ellipsoids were calculated, which can be considered as approximations of isodiscriminability contours in color space. For most subjects, corresponding discrimination ellipsoids can be predicted byfwith a sufficient degree of precision.

“Objective” low informative priors for Bayesian inference from totally censored Gaussian data

Consider structural elements with random strength that after a suitable transformation has normal distribution with unknown mean μ and known or unknown standard deviation σ. By proof testing of n of these structural elements to a given load level it is observed that none of the elements fail. Given solely this test information the problem is that in order to state anything about either the value of μ when σ is known or about the values of μ and σ when both parameters are unknown, it is necessary to introduce some more information in the form of a suitable prior distribution of the parameters, that is, to use a Bayesian procedure with an informative prior. The paper considers the problem of defining such a prior in an axiomatic (“objective”) way without extending the information represented by the test results by more than very little extra information based on common physical sense. The solution suggested in the paper implies that the posterior distribution of the mean shifts towards larger values when the sample size n increases. However, convergence to a specific value is not obtained as long as no failures are observed among the tests. Moreover it turns out that the posterior distribution of the standard deviation is invariant to the sample size n, that is, no updating of the standard deviation is obtained as long as there are no failures among the tests.

Mean-risk analysis of risk aversion and wealth effects on optimal portfolios with multiple investment opportunities

In this paper, we first define risk in an axiomatic way and a class of utility functions suitable for the so-called mean-risk analysis. Then, we show that, in a portfolio selection problem with multiple risky investments, an investor who is more risk averse in the Arrow-Pratt sense prefers less risk, in the sense of this paper, with less mean return, and an investor who displays increasing (decreasing) relative risk aversion becomes more conservative (aggressive) as the initial capital increases. The risk aversion effect for diversification on optimal portfolios is also discussed.

Parallel and distributed derivations in the single-pushout approach

Parallel and distributed derivations are introduced and studied in the single-pushout approach, which models rewriting by pushout constructions in appropriate categories of partial morphisms. We present a categorical framework for this approach in an axiomatic way. Models of this categorical framework are among others: graphs, hypergraphs, relational structures, and algebraic specifications with suitable partial morphisms. Several new results concerning parallelism and distributed parallelism are presented which are even new in the example categories.

The simple substitution property of the intermediate propositional logics on finite slices

The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi [4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki [7]. Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is thatL is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki [5]. Also the proof of sufficiency part is improved here. For logics on the ω-th slice, the condition for them to have the simple substitution property is not yet known. We abbreviate the simple substitution property asSSP.

The Kakutani property and the fixed point property of topological spaces with abstract convexity

The paper deals with the usual fixed point property and the following Kakutani property of a space X: for every upper semicontinuous function Φ from X to non-empty closed convex subsets of X, there exists x 0 such that x 0 ϵ Φ( x 0). We derive this property of X from various separation properties of convex subsets of X and a kind of local convexity of X. Convexity in our setup is given in an abstract axiomatic way. Special emphasis is given to the case where X has the form of a product. The obtained results cover several known fixed point theorems: Ky Fan-Glicksberg, Wallace, and special cases of Eilenberg-Montgomery. We also discuss an open problem concerning the fixed point property of finite posets and its role in proving more advanced theorems.

Constructive approach to logics of physical systems: Application to EPR case

A constructive approach to logics of physical systems, according to which families of propositions about physical systems are not defined in an axiomatic way, but are built up in the course of experiments, is proposed. Several ways of joining Boolean algebras of propositions obtained in single experiments are studied. The proposed approach is applied to study families of propositions encountered in EPR-type experiments. Two examples of such experimental families of EPR propositions are studied and they are compared with two theoretical families of EPR propositions in the literature.