Convex projection and convex multi-objective optimization

In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations.

A Logic For Decidable Reasoning About Actions

We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that can be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in Reiter’s version of the situation calculus. Using this new regression operator, we show that the projection and executability problems (the important reasoning tasks in the situation calculus) are decidable in the modified version even if an initial knowledge base is incomplete. We also discuss the complexity of solving the projection problem in this modified language in general. Furthermore, we define description logic based sub-languages of our modified situation calculus. They are based on the description logics ALCO(U) (or ALCQO(U), respectively). We show that in these sub-languages solving the projection problem has better computational complexity than in the general modified situation calculus. We mention possible applications to formalization of Semantic Web services and some connections with reasoning about actions based on description logics.

A Logic For Decidable Reasoning About Actions

We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that can be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in Reiter’s version of the situation calculus. Using this new regression operator, we show that the projection and executability problems (the important reasoning tasks in the situation calculus) are decidable in the modified version even if an initial knowledge base is incomplete. We also discuss the complexity of solving the projection problem in this modified language in general. Furthermore, we define description logic based sub-languages of our modified situation calculus. They are based on the description logics ALCO(U) (or ALCQO(U), respectively). We show that in these sub-languages solving the projection problem has better computational complexity than in the general modified situation calculus. We mention possible applications to formalization of Semantic Web services and some connections with reasoning about actions based on description logics.

The Projection Problem for Predicates of Taste

Utterances of simple sentences containing taste predicates (e.g. "delicious", "fun", "frightening") typically imply that the speaker has had a particular sort of first-hand experience with the object of predication. For example, an utterance of "The carrot cake is delicious" would typically imply that the speaker had actually tasted the cake in question, and is not, for example, merely basing her judgment on the testimony of others. According to one approach, this 'acquaintance inference' is essentially an implicature, one generated by the Maxim of Quality together with a certain principle concerning the epistemology of taste (Ninan 2014). We first discuss some problems for this approach, problems that arise in connection with disjunction and generalized quantifiers. Then, after stating a conjecture concerning which operators 'obviate' the acquaintance inference and which do not, we build on Anand and Korotkova 2018 and Willer and Kennedy Forthcoming by developing a theory that treats the acquaintance requirement as a presupposition, albeit one that can be obviated by certain operators.

Mental Space Theory on Presupposition Projections in Advertisements

In this economic society in which information is prevailing, advertisements are found here and there, and play a very important role in our daily life. More and more importance is attached to the research of advertising language. This paper reveals presupposition projection problems in advertising, especially in English advertising. It aims to explore the explanation of mental space theory for cancellation or inheritability of presupposition in advertising, that is, the projection problem of presupposition. Different from how traditional presupposition theory is used in seeking for a reasonable explanation for projection problems, this paper will investigate in detail projection problems in advancing from the perspective of Fauconnier's mental space theory in combination with large quantities of advertisement examples.

Towards an Expressive Practical Logical Action Theory

In the area of reasoning about actions, one of the key computational problems is the projection problem: to find whether a given logical formula is true afterperforming a sequence of actions. This problem is undecidable in the generalsituation calculus; however, it is decidable in some fragments. We considera fragment P of the situation calculus and Reiter's basic action theories (BAT)such that the projection problem can be reduced to the satisfiability problemin an expressive description logic $ALCO(U)$ that includes nominals ($O$),the universal role ($U$), and constructs from the well-known logic $ALC$. It turns outthat our fragment P is more expressive than previously explored description logicbased fragments of the situation calculus. We explore some of the logical properties of our theories.In particular, we show that the projection problem can be solved using regressionin the case where BATs include a general ``static" TBox, i.e., an ontology that hasno occurrences of fluents. Thus, we propose seamless integration of traditionalontologies with reasoning about actions. We also show that the projectionproblem can be solved using progression if all actions have only local effects onthe fluents, i.e., in P, if one starts with an incomplete initial theory thatcan be transformed into an $ALCO(U)$ concept, then its progression resulting fromexecution of a ground action can still be expressed in the same language. Moreover,we show that for a broad class of incomplete initial theories progression can be computed efficiently.

Some problems about geometric lattice

In this paper, the survey about some results of the convex lattice set are given and the invariance of projection problem of convex lattice set is also obtained. And combining a famous result in the graph theory, several conjectures about the convex lattice set are presented.

Presupposition: A Semantic or Pragmatic Phenomenon?

There has been debate among linguists with regards to the semantic view and the pragmatic view of presupposition. Some scholars believe that presupposition is purely semantic and others believe that it is purely pragmatic. The present paper contributes to the ongoing debate and exposes the different ways presupposition was approached by linguists. The paper also tries to attend to (i) what semantics is and what pragmatics is in a unified theory of meaning and (ii) the possibility to outline a semantic account of presupposition without having recourse to pragmatics and vice versa. The paper advocates Gazdar’s analysis, a pragmatic analysis, as the safest grounds on which a working grammar of presupposition could be outlined. It shows how semantic accounts are inadequate to deal with the projection problem. Finally, the paper states explicitly that the increasingly puzzling theoretical status of presupposition seems to confirm the philosophical contention that not any fact can be translated into words.