A fuzzy fixed-point theorem and its applications to maximal elements and Nash equilibria in noncompact CAT(0) spaces

In this paper, based on the KKM method, we prove a new fuzzy fixed-point theorem in noncompact CAT(0) spaces. As applications of this fixed-point theorem, we obtain some existence theorems of fuzzy maximal element points. Finally, we utilize these fuzzy maximal element theorems to establish some new existence theorems of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games in noncompact CAT(0) spaces. The results obtained in this paper generalize and extend many known results in the existing literature.

Two Options for the Branching of Histories

The chapter discusses how the histories in a common BST structure are related. By the axioms of the core theory of BST, any two histories share some past, but there are different ways to implement this. These are distinguished by the so-called prior choice principles, which make specific demands on the way in which histories branch. On one option (which yields structures of BST92), histories branch, or remain undivided, at points, which means that there is a maximal element in the overlap of any two histories. The other option (which yields BSTNF structures) prohibits the existence of such maximal elements and works with so-called choice sets. The chapter discusses the pattern of branching in the two theories, BST92 and BSTNF, also with respect to topology. As it turns out, the two theories are are intertranslatable. The chapter ends with a sketch of these translatability results.

Strong admissibility for abstract dialectical frameworks

dialectical frameworks (ADFs) have been introduced as a formalism for modeling argumentation allowing general logical satisfaction conditions and the relevant argument evaluation. Different criteria used to settle the acceptance of arguments are called semantics. Semantics of ADFs have so far mainly been defined based on the concept of admissibility. However, the notion of strongly admissible semantics studied for abstract argumentation frameworks has not yet been introduced for ADFs. In the current work we present the concept of strong admissibility of interpretations for ADFs. Further, we show that strongly admissible interpretations of ADFs form a lattice with the grounded interpretation as the maximal element. We also present algorithms to answer the following decision problems: (1) whether a given interpretation is a strongly admissible interpretation of a given ADF, and (2) whether a given argument is strongly acceptable/deniable in a given interpretation of a given ADF. In addition, we show that the strongly admissible semantics of ADFs forms a proper generalization of the strongly admissible semantics of AFs.

Vacuity in synthesis

In reactive synthesis, one begins with a temporal specification $$\varphi $$ φ , and automatically synthesizes a system $$M$$ M such that $$M\models \varphi $$ M ⊧ φ . As many systems can satisfy a given specification, it is natural to seek ways to force the synthesis tool to synthesize systems that are of a higher quality, in some well-defined sense. In this article we focus on a well-known measure of the way in which a system satisfies its specification, namely vacuity. Our conjecture is that if the synthesized system M satisfies $$\varphi $$ φ non-vacuously, then M is likely to be closer to the user’s intent, because it satisfies $$\varphi $$ φ in a more “meaningful” way. Narrowing the gap between the formal specification and the designer’s intent in this way, automatically, is the topic of this article. Specifically, we propose a bounded synthesis method for achieving this goal. The notion of vacuity as defined in the context of model checking, however, is not necessarily refined enough for the purpose of synthesis. Hence, even when the synthesized system is technically non-vacuous, there are yet more interesting (equivalently, less vacuous) systems, and we would like to be able to synthesize them. To that end, we cope with the problem of synthesizing a system that is as non-vacuous as possible, given that the set of interesting behaviours with respect to a given specification induce a partial order on transition systems. On the theoretical side we show examples of specifications for which there is a single maximal element in the partial order (i.e., the most interesting system), a set of equivalent maximal elements, or a number of incomparable maximal elements. We also show examples of specifications that induce infinite chains of increasingly interesting systems. These results have implications on how non-vacuous the synthesized system can be. We implemented the new procedure in our synthesis tool PARTY. For this purpose we added to it the capability to synthesize a system based on a property which is a conjunction of universal and existential LTL formulas.

New Generalized Ekeland’s Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du’s Abstract Maximal Element Principle

In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland’s variational principle and common (fuzzy) fixed point theorem for essential distances.

The Romance of Modal Superlatives as Degree Descriptions

In this paper, I present a novel compositional analysis of modal predicative superlatives, that is, predicative superlatives accompanied by modal adjectives such as possible, as that in (1). (1) Mary wanted to be the prettiest possible. I argue that they are elliptical bona fide degree-relative clauses denoting maximal degrees and whose semantic contribution is similar to that of Measure Phrases. This account will require a novel composition of the superlative which involves the formation of an ordered set and the selection of a maximal element. I argue that not only is this account able to derive their peculiar semantics (dispensing us from the ad hoc components that previous accounts posited), but it can also capture the morphosyntax of these constructions, especially in Romance languages, which turn out more informative than English in this respect.

Deliberately Stochastic

We study stochastic choice as the outcome of deliberate randomization. We derive a general representation of a stochastic choice function where stochasticity allows the agent to achieve from any set the maximal element according to her underlying preferences over lotteries. We show that in this model stochasticity in choice captures complementarity between elements in the set, and thus necessarily implies violations of Regularity/Monotonicity, one of the most common properties of stochastic choice. This feature separates our approach from other models, e.g., Random Utility. (JEL D80, D81)

The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group.