An important contemporary debate (going back to (Gödel, 1964)) in the philosophy of mathematics is whether or not mathematics needs new axioms. This paper is an attempt to show how one might go about answering this question. I argue that the role of axioms is to allow mathematicians to stay away from philosophical debates, and thus focus on their primary goal of proving interesting theorems. I ...

Absorptions are generally employed in Description Logics (DL) reasoners in a uniform way regardless of the structure of an input knowledge base. In this paper we present an approach to encode some state-of-the-art absorption techniques into a state space planner, aiming to achieve a better solution. The planner applies appropriate operators to general axioms and produces a solution with a minim...

In support of a recent argument by Deutsch (quant-ph/9906015) an arrangement of “quantum dots” and “channels” is specified which does not anticipate a unitary time evolution, yet is able to transform square-root amplitudes into an integer number of equal amplitudes and, hence, equal probabilities. Exploited are linearity of the time evolution and permutational symmetry. Further, it is postulate...

This chapter presents a case-study of how the abstractions and infrastructure discussed in Part III and Part IV of this dissertation may be applied to source code analysis and interactive code generation. The motivation for the case study prototype, an interactive unit test generator, is that current unit testing methodologies often result in poor test coverage due to rather ad-hoc approaches.

A classical result for crisp choice functions shows the equivalence between Arrow axiom and the property of full rationality. In this paper we study a fuzzy form of Arrow axiom formulated in terms of the subsethood degree and of the degree of equality (of fuzzy sets). We prove that a fuzzy choice function verifies Fuzzy Arrow Axiom if and only if it is (fuzzy) full rational. We also show that t...

Machina and Schmeidler [Machina, M., Schmeidler, D., 1992. A more robust definition of subjective probability. Econometrica 60, 745–780] show that probabilistic sophistication can be obtained in a Savage setting without imposing expected utility by dropping Savage's axiom P2 (sure-thing principle) and strengthening his axiom P4 (weak comparative probability). Their stronger axiom, however, embo...

Various semantics for logic programs with negation are described in terms of a dualized program together with additional axioms, some of which are second order formulas. The semantics of Clark, Fitting, and Kunen are characterized in this framework, and a nite rst-order presentation of Kunen's semantics is described. A new axiom to represent \common sense" reasoning is proposed for logic progra...

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...

We study cooperative transferable utility games with a communication structure represented by an undirected graph, i.e., a group of players can cooperate only if they are connected on the graph. This type of games is called graph games and the best-known solution for them is the Myerson value, which is characterized by the component efficiency axiom and the fairness axiom. Recently the average ...

We provide a new characterization of the Shapley value neither using the e¢ciency axiom nor the additivity axiom. In this characterization, e¢ciency is replaced by the gain-loss axiom (Einy and Haimanko, 2011, Game Econ Behav 73: 615–621), i.e., whenever the total worth generated does not change, a player can only gain at the expense of another one. Additivity and the equal treatment axiom are ...