Annals of Mathematics and Artificial Intelligence最新文献

This paper introduces and studies the sequential composition and decomposition of propositional logic programs. We show that acyclic programs can be decomposed into single-rule programs and provide a general decomposition result for arbitrary programs. We show that the immediate consequence operator of a program can be represented via composition which allows us to compute its least model without any explicit reference to operators. This bridges the conceptual gap between the syntax and semantics of a propositional logic program in a mathematically satisfactory way.

The successes of Machine Learning, and in particular of Deep Learning systems, have led to a reformulation of the Artificial Intelligence agenda. One of the pressing issues in the field is the extraction of knowledge out of the behavior of those systems. In this paper we propose a semiotic analysis of that behavior, based on the formal model of learners. We analyze the topos-theoretic properties that ensure the logical expressivity of the knowledge embodied by learners. Furthermore, we show that there exists an ideal universal learner, able to interpret the knowledge gained about any possible function as well as about itself, which can be monotonically approximated by networks of increasing size.

Systems of decision rules and decision trees are widely used as a means for knowledge representation, as classifiers, and as algorithms. They are among the most interpretable models for classifying and representing knowledge. The study of relationships between these two models is an important task of computer science. It is easy to transform a decision tree into a decision rule system. The inverse transformation is a more difficult task. In this paper, we study unimprovable upper and lower bounds on the minimum depth of decision trees derived from decision rule systems with discrete attributes depending on the various parameters of these systems. To illustrate the process of transformation of decision rule systems into decision trees, we generalize well known result for Boolean functions to the case of functions of k-valued logic.

There are many real life applications where data can not be effectively represented in Hilbert spaces and/or where the data points are uncertain. In this context, we address the issue of binary classification in Banach spaces in presence of uncertainty. We show that a number of results from classical support vector machines theory can be appropriately generalized to their robust counterpart in Banach spaces. These include the representer theorem, strong duality for the associated optimization problem as well as their geometrical interpretation. Furthermore, we propose a game theoretical interpretation of the class separation problem when the underlying space is reflexive and smooth. The proposed Nash equilibrium formulation draws connections and emphasizes the interplay between class separation in machine learning and game theory in the general setting of Banach spaces.

Nondeterministic planning is the process of computing plans or policies of actions achieving given goals, when there is nondeterministic uncertainty about the initial state and/or the outcomes of actions. This process encompasses many precise computational problems, from classical planning, where there is no uncertainty, to contingent planning, where the agent has access to observations about the current state. Fundamental to these problems is belief tracking, that is, obtaining information about the current state after a history of actions and observations. At an abstract level, belief tracking can be seen as maintaining and querying the current belief state, that is, the set of states consistent with the history. We take a knowledge compilation perspective on these processes, by defining the queries and transformations which pertain to belief tracking. We study them for propositional domains, considering a number of representations for belief states, actions, observations, and goals. In particular, for belief states, we consider explicit propositional representations with and without auxiliary variables, as well as implicit representations by the history itself; and for actions, we consider propositional action theories as well as ground PDDL and conditional STRIPS. For all combinations, we investigate the complexity of relevant queries (for instance, whether an action is applicable at a belief state) and transformations (for instance, revising a belief state by an observation); we also discuss the relative succinctness of representations. Though many results show an expected tradeoff between succinctness and tractability, we identify some interesting combinations. We also discuss the choice of representations by existing planners in light of our study.