Simple Axioms Research Articles

Characterizing the interpretation of set theory in Martin-Löf type theory

Constructive Zermelo–Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice. The paper builds on a self-interpretation of CZF (developed in [M. Rathjen, The formulae-as-classes interpretation of constructive set theory, in: Proof Technology and Computation (Proceedings of the International Summer School Marktoberdorf 2003) IOS Press, Amsterdam, 2004 (in press)]) that provides an “inner” model of CZF which also validates the so-called Π Σ -axiom of choice, ΠΣ – AC . The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis ΠΣ – AC . It is also shown that CZF plus the Π Σ -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae.

The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms

The Lambek calculus introduced in Lambek [6] is a strengthening of the type reduction calculus of Ajdukiewicz [1]. We study Associative Lambek Calculus L in Gentzen style axiomatization enriched with a finite set Γ of nonlogical axioms, denoted by L(Γ).It is known that finite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages (see Buszkowski [2]). Then we confine nonlogical axioms to sequents of the form p → q, where p and q are atomic types. For calculus L(Γ) we prove interpolation lemma (modifying the Roorda proof for L [10]) and the binary reduction lemma (using the Pentus method [9] with modification from [3]). In consequence we obtain the weak equivalence of the Context-Free Grammars and grammars based on L(Γ).

A simple axiomatization of risk-averse expected utility

An agent's preferences exhibit risk aversion with respect to some probabilities of states of the world if she prefers deterministic outcome equal to the expected value of a state-contingent outcome under these probabilities to the state-contingent outcome itself. We show that if preferences exhibit risk aversion with respect to some probabilities and satisfy the independence axiom (sure thing principle), then they have an expected utility representation with respect to these probabilities. Since the independence axiom is necessary and sufficient for state-separable utility representation [Debreu, G., 1959. Topological methods in cardinal utility theory. In: Arrow, K., Karlin, S., Suppes, P. (Eds.), Mathematical Methods in Social Sciences. Standford University Press], our result shows that, given separable utility representation, expected utility follows from risk aversion.

How to reach legitimate decisions when the procedure is controversial

Imagine a group that faces a decision problem but does not agree on which decision procedure is appropriate. In that case, can a decision be reached that respects the procedural concerns of the group? There is a sense in which legitimate decisions are possible even if people disagree on which procedure to use. I propose to decide in favour of an option which maximizes the number of persons whose judged-right procedure happens to entail this decision given the profile. This decision rule is based not only on a profile in the standard sense, but in addition on a profile of judged-right procedures. To justify this decision rule, I present a set of simple axioms leading to it as the only solution.

Teaching axiomatic design to engineers—Theory, applications, and software

Axiomatic design (introduced and described by N.P. Suh in 1990 and 2001 books) shows that the engineering of good designs can be taught as a science. There is a simple set of underlying rules, or design axioms, on which good designs are based. The design axioms provide the opportunity for teaching design as a science, in that it is based on some simple underlying principles. Taking advantage of this opportunity, a systematic approach to teaching how to engineer and critically evaluate good designs has been developed. Three steps are used in introducing axiomatic design. First is providing motivation, which is attempted by observing that design is fundamental to all engineering. The purpose of traditional engineering analysis, which is emphasized in engineering studies, is to support design. Second is developing the concept that there are two simple axioms, independence and information, that govern design, just as Newton's laws govern mechanics. Third is observing that to apply the axioms designs must be decomposed into a hierarchical structure. This leads to stating that there are three essential elements to engineering design: the axioms, the structure, and the process for creating that structure. A barrier to teaching and using this scientific, axiomatic approach to design has been the difficulty of creating, modifying, managing, and communicating anything beyond relatively simple design hierarchies. Any design with more than a few functional requirements was difficult to manage. Recently introduced design software (Acclaro, email: axiomaticdesign.com) has overcome this barrier. It can easily handle many hundreds of functional requirements and facilitates the application of the independence axiom. The impact on the teaching process is that the students learn to construct large design hierarchies quickly and spend more time on mastering applications.

Spatial distribution model for tracking extended objects

A Bayesian filter has been developed for tracking an extended object in clutter based on two simple axioms: (i) the numbers of received target and clutter measurements in a frame are Poisson distributed (so several measurements may originate from the target) and (ii) target extent is modelled by a spatial probability distribution and each target-related measurement is an independent ‘random draw’ from this spatial distribution (convolved with a sensor model). Diffuse spatial models of target extent are of particular interest. This model is especially suitable for a particle filter implementation, and examples are presented for a Gaussian mixture model and for a uniform stick target convolved with a Gaussian error. A rather restrictive special case that admits a solution in the form of a multiple hypothesis Kalman filter is also discussed and demonstrated.

Good-By Old, Hello New in Teaching Economics

Throughout the world, economists have observed student lack of interest in pursuing the study of economics. In response, there has been an increase in academic economists' interest in their teaching, but that does not necessarily imply that they have figured out what is required for good teaching or what should be taught. In this article, I again address what is versus what should be taught, and the way economics is taught versus how it should be taught at the tertiary level. I argue that the dumbing down of economics to the dogmatic preaching of a few simple concepts, principles, and axioms of old misses the excitement of modern day economics. The power of economics can be shown at the tertiary level by instructors updating their lists of concepts, abandoning their reliance on chalk and talk type teaching methods, and changing their examples to reflect current social and political issues.

The sectional category of a map

We study a generalization of the Svarc genus of a fibre map. For an arbitrary collection ɛ of spaces and a map f : X → Y, we define a numerical invariant, the ɛ-sectional category of f, in terms of open covers of Y. We obtain several basic properties of ɛ-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple axioms which characterize the ɛ-sectional category. In the final section, we obtain inequalities for the ɛ-sectional category of a composition and inequalities relating the ɛ-sectional category to the Fadell–Husseini category of a map and the Clapp–Puppe category of a map.

A new axiomatization of Jaśkowski's discussive logic

In 1995 N.C.A. da Costa and F. Doria proposed the modal-type elegant axiomatization of Jaśkowski’s discussive logic D 2 . Yet his own problem which was formulated in 1975 in a following way: Is it possible to formulate natural and simple axiomatization for D 2 , employing classical disjunction and conjunction along with discussive implication and conjunction as the only primitive connectives? — still seems left open. The matter of fact is there are some axiomatizations of D 2 proposed, e.g., by T. Furmanowski (1975), J. Kotas and N.C.A. da Costa (1979), G. Achtelik, L. Dubikajtus, E. Dudek and J. Konior (1981), satisfying da Costa’s conditions, but they are rather looking very complicated and unnatural. An attempt is made to solve da Costa’s problem. The new axiomatization of D 2 is proposed essentially based on da Costa’s-Doria axiomatization from 1995.

A simple axiomatization and constructive representation proof for choquet expected utility

We provide a set of simple and intuitive axioms that allow for a direct and constructive proof of the Choquet Expected Utility representation for decision making under uncertainty.

On the call-by-value CPS transform and its semantics

We investigate continuations in the context of idealized call-by-value programming languages. On the semantic side, we analyze the categorical structures that arise from continuation models of call-by-value languages. On the syntactic side, we study the call-by-value continuation-passing transformation as a translation between equational theories. Among the novelties are an unusually simple axiomatization of control operators and a strengthened completeness result with a proof based on a delaying transform.

A Simple Axiomatization of the Foster, Greer and Thorbecke Poverty Orderings

The paper presents a simple characterization of the poverty orderings which are represented by a poverty measure belonging to the Foster, Greer, and Thorbecke class. All properties introduced are formulated within an ordinal framework. Furthermore, a new concept is proposed: the equivalent societal income which—if given to each individual in society—yields the same level of poverty as the actual income distribution. It is a specific indicator of the underlying poverty ordering, has attractive properties and allows us to prove the main result in a direct way.

Subjective ambiguity, expected utility and Choquet expected utility

Using the Savage set up, this paper provides a simple axiomatization of the Choquet Expected Utility model where the capacity is an inner measure. Two attractive features of the model are its specificity and the transparency of its axioms. The key axiom states that the decision-maker uses unambiguous acts to approximate ambiguous ones. In addition, the notion of ‘ambiguity’ is subjective and derived from preferences.

A simple two-axiom characterization of the Nash solution

The oldest and best-known cooperative bargaining solution concept is the Nash solution. Nash [2] characterized his seminal solution concept by using the axioms of ‘Independence of Irrelevant Alternatives’ (IIA), ‘Weak Pareto Optimality’ (WPO), ‘Symmetry’ (SYM), and ‘Scale Invariance’ (SI). Except for WPO, these axioms have been at the center of controversy (especially the most crucial axiom, IIA). This paper considers a new and simple axiom ‘Focal Relevance of a Pareto-optimal Midpoint’ (FRPM). It turns out that the Nash solution can be characterized by WPO and FRPM only.

Combinatorial intersection cohomology for fans

We investigate minimal extension sheaves on arbitrary (possibly non-rational) fans as an approach toward a combinatorial "virtual'' intersection cohomology. These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the equivariant intersection cohomology sheaves on toric varieties. As in "classical'' intersection cohomology, minimal extension sheaves are models for the pure objects of a "perverse category"; a Decomposition Theorem holds. The analysis of the step from equivariant to non-equivariant intersection cohomology of toric varieties leads us to investigate "quasi-convex" fans (generalizing fans with convex or "co-convex" support), where our approach yields a meaningful virtual intersection cohomology. We characterize such fans by a topological condition and prove a version of Stanley's "Local-Global" formula relating the global intersection Poincaré polynomial to local data. Virtual intersection cohomology of quasi-convex fans is shown to satisfy Poincaré duality. To describe the local data in terms of the global data for lower-dimensional complete polytopal fans as in the rational case, one needs a "Hard Lefschetz" type result. It requires a vanishing condition that is valid for rational cones, but has not yet been proven in the general case.

A Simple Axiomatization of Binary Rank-Dependent Utility of Gains (Losses)

For binary gambles composed only of gains (losses) relative to a status quo, the rank-dependent utility model with a representation that is dense in intervals is shown to be equivalent to ten elementary properties plus event commutativity and a gamble partition assumption. The proof reduces to a (difficult) functional equation that has been solved by Aczél, Maksa, and Páles (in press).

Trading Portfolios Electronically – An Experimental Approach

This paper focuses on a simple axiom – investors prefer to hold diversified combinations of assets. Arguing that pooling vehicles like mutual funds offer approximate solutions, we propose a portfolio trading mechanism that allows individuals to trade diverse combinations of assets through a single order. Given the complexity of the market mechanism, we construct a prototype of this trading system and test it through a series of laboratory experiments with student subjects. We address the quality of the price discovery process, quality of allocations, and efficiency of the market mechanism. Results from a set of laboratory experiments indicate that the performance of such systems is sensitive to design aspects as well as relative experience of participants.

Axiomatizing the subsumption and subword preorders on finite and infinite partial words

We consider two-sorted algebras of finite and infinite partial words equipped with the subsumption preorder and the operations of series and parallel product and omega power. It is shown that the valid equations and inequations of these algebras can be described by an infinite collection of simple axioms, and that no finite axiomatization exists. We also prove similar results for two related preorders, namely for the induced partial subword preorder and the partial subword preorder. Along the way of proving these results, we provide a concrete description of the free algebras in the corresponding varieties in terms of generalized series–parallel partial words.

On the Products of Linear Modal Logics

We study two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4.3, S4.3, GL.3, Grz.3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems posed by Gabbay and Shehtman. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatization for the square K4.3 × K4.3 of the minimal liner logic using non‐structural Gabbay‐type inference rules.

Complexity through nonextensivity

The problem of defining and studying complexity of a time series has interested people for years. In the context of dynamical systems, Grassberger has suggested that a slow approach of the entropy to its extensive asymptotic limit is a sign of complexity. We investigate this idea further by information theoretic and statistical mechanics techniques and show that these arguments can be made precise, and that they generalize many previous approaches to complexity, in particular, unifying ideas from the physics literature with ideas from learning and coding theory; there are even connections of this statistical approach to algorithmic or Kolmogorov complexity. Moreover, a set of simple axioms similar to those used by Shannon in his development of information theory allows us to prove that the divergent part of the subextensive component of the entropy is a unique complexity measure. We classify time series by their complexities and demonstrate that beyond the “logarithmic” complexity classes widely anticipated in the literature there are qualitatively more complex, “power-law” classes which deserve more attention.