Papers by Ramon Pino Perez
Non-Monotonic Reasoning, 2004
We extend the results of Konieczny and Pino Pérez (J. Log. and Comp. 2002) concerning merging ope... more We extend the results of Konieczny and Pino Pérez (J. Log. and Comp. 2002) concerning merging operators in a finite logical framework to the infinite case (countably many propositional variables). The main result is the representation theorem. Some postulates had to be restated in a new form, equivalent only in the finite case, but more appropriate to deal with the infinite case. The construction of merging operators starting from distances between valuations is also generalized. We introduce a new kind of operators built upon the so called Cantor distance.
Annals of Mathematics and Artificial Intelligence, Apr 3, 2013
In this paper we introduce confluence operators, that are inspired by the existing links between ... more In this paper we introduce confluence operators, that are inspired by the existing links between belief revision, update and merging operators. Roughly, update operators can be considered as pointwise revision, whereas revision operators can be considered as special case of merging operators. Confluence operators are to merging operators what update operators are to revision operators. Similarly, update operators can be considered as special case of confluence operators just as revision can be considered as special case of merging operators. Confluence operators gives all possible agreement situations from a set of belief bases.
The seminal characterization of iterated belief revision was proposed by Darwiche and Pearl, whic... more The seminal characterization of iterated belief revision was proposed by Darwiche and Pearl, which uses an abstract notion of epistemic states. In this work we look for a canonical representation of these epistemic states. Total preorders are not expressive enough to be used as such a canonical representation. Actually, we show that some operators can even not be represented on a countable epistemic space. Nonetheless, under a very reasonable assumption on the epistemic space, we show that OCFs (Ordinal Conditional Functions) can be considered as a canonical representation.
arXiv (Cornell University), May 4, 2020
A general definition of mathematical morphology has been defined within the algebraic framework o... more A general definition of mathematical morphology has been defined within the algebraic framework of complete lattice theory. In this framework, dealing with deterministic and increasing operators, a dilation (respectively an erosion) is an operation which is distributive over supremum (respectively infimum). From this simple definition of dilation and erosion, we cannot say much about the properties of them. However, when they form an adjunction, many important properties can be derived such as monotonicity, idempotence, and extensivity or anti-extensivity of their composition, preservation of infimum and supremum, etc. Mathematical morphology has been first developed in the setting of sets, and then extended to other algebraic structures such as graphs, hypergraphs or simplicial complexes. For all these algebraic structures, erosion and dilation are usually based on structuring elements. The goal is then to match these structuring elements on given objects either to dilate or erode them. One of the advantages of defining erosion and dilation based on structuring elements is that these operations are adjoint. Based on this observation, this paper proposes to define, at the abstract level of category theory, erosion and dilation based on structuring elements. We then define the notion of morpho-category on which erosion and dilation are defined. We then show that topos and more precisely topos of presheaves are good candidates to generate morpho-categories. However, topos do not allow taking into account the notion of inclusion between substructures but rather are defined by monics up to domain isomorphism. Therefore we define the notion of morpholizable category which allows generating morpho-categories where substructures are defined along inclusion morphisms. A direct application of this framework is to generalize modal morpho-logic to other algebraic structures than simple sets.
arXiv (Cornell University), Oct 30, 2019
In most classical models of belief change, epistemic states are represented by theories (AGM) or ... more In most classical models of belief change, epistemic states are represented by theories (AGM) or formulas (Katsuno-Mendelzon) and the new pieces of information by formulas. The Representation Theorem for revision operators says that operators are represented by total preorders. This important representation is exploited by Darwiche and Pearl to shift the notion of epistemic state to a more abstract one, where the paradigm of epistemic state is indeed that of a total preorder over interpretations. In this work, we introduce a 3-valued logic where the formulas can be identified with a generalisation of total preorders of three levels: a ranking function mapping interpretations into the truth values. Then we analyse some sort of changes in this kind of structures and give syntactical characterizations of them.
arXiv (Cornell University), Nov 7, 2018
Nested graphs have been used in different applications, for example to represent knowledge in sem... more Nested graphs have been used in different applications, for example to represent knowledge in semantic networks. On the other hand, graphs with cycles are really important in surface reconstruction, periodic schedule and network analysis. Also, of particular interest are the cycle basis, which arise in mathematical and algorithm problems. In this work we develop the concept of perfectly nested circuits, exploring some of their properties. The main result establishes an order isomorphism between some sets of perfectly nested circuits and equivalence classes over finite binary sequences.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2018
Several tasks in artificial intelligence require to be able to find models about knowledge dynami... more Several tasks in artificial intelligence require to be able to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper we exploit the algebraic framework of mathematical morphology in the context of propositional logic, and define operations such as dilation or erosion of a set of formulas. We derive concrete operators, based on a semantic approach, that have an intuitive interpretation and that are formally well behaved, to perform revision, fusion and abduction. Computation and tractability are addressed, and simple examples illustrate the typical results that can be obtained.
Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning
In this work, we address one important problem of Katsuno and Mendelzon update operators, that is... more In this work, we address one important problem of Katsuno and Mendelzon update operators, that is to require that any updated belief base must entail any new input in a consistent way. This assumes that any situation can be updated into one satisfying that input, which is unrealistic. To solve this problem, we must relax either the success or the consistency principle. Each case leads to a distinct family of update operators, that we semantically characterize by plausibility relations over possible worlds, considering a credibility limit that aims to forbid unrealistic changes. We discuss in which cases one family is more adequate than the other one.
Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning
In standard propositional belief merging, one implicit assumption is that all sources have exactl... more In standard propositional belief merging, one implicit assumption is that all sources have exactly the same importance. But there are many situations where the sources have different importance/reliability/expertise that have to be taken into account in the merging process. In this work we study the problem of weighted merging operators, which aimed to take these weights into account in a sensible way. We give a syntactical characterization of these operators, and then we state a representation theorem in terms of plausibility preorders on interpretations. We also propose a general method to build weighted distance-based merging operators, and provide some concrete examples, using two different weight functions.
Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas,... more Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas, morphological operators will act on formulas (and on their models). In modal logics, dilation and erosion can define modal operators. Such examples will be described in several logics. Then we will show how they can be used to define revision operators satisfying the AGM postulates, merging operators, or explanatory relations.
We define in a very precise and new manner the concept of manipulability of general (classical) s... more We define in a very precise and new manner the concept of manipulability of general (classical) social choice functions. In order to do that, we need to lift individual preferences over elements of a set of alternatives X to preferences over subsets of X. We establish a new theorem of impossibility of being free of manipulation à la Gibbard-Satterthwaite.
Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning
The behavior of Iterated Belief Revision operators with respect to iteration has been characteriz... more The behavior of Iterated Belief Revision operators with respect to iteration has been characterized by a set of four postulates proposed by Darwiche and Pearl. These postulates give constraints on a single iteration step, and this is not enough to forbid some pathological operators. In this paper, we propose a generalization of these postulates to solve this issue and we study its implications. One surprising consequence is that, for TPO-representable operators (i.e., for operators defined as transitions on total pre-orders on interpretations), there are very few operators that satisfy this generalization.
arXiv (Cornell University), May 17, 2018
We use an algebraic viewpoint, namely a matrix framework to deal with the problem of resource all... more We use an algebraic viewpoint, namely a matrix framework to deal with the problem of resource allocation under uncertainty in the context of a qualitative approach. Our basic qualitative data are a plausibility relation over the resources, a hierarchical relation over the agents and of course the preference that the agents have over the resources. With this data we propose a qualitative binary relation between allocations such that F G has the following intended meaning: the allocation F produces more or equal social welfare than the allocation G. We prove that there is a family of allocations which are maximal with respect to. We prove also that there is a notion of simple deal such that optimal allocations can be reached by sequences of simple deals. Finally, we introduce some mechanism for discriminating optimal allocations.
arXiv (Cornell University), Sep 17, 2021
The problem of finding envy-free allocations of indivisible goods can not always be solved; there... more The problem of finding envy-free allocations of indivisible goods can not always be solved; therefore, it is common to study some relaxations such as envy-free up to one good (EF1). Another property of interest for efficiency of an allocation is the Pareto Optimality (PO). Under additive utility functions, it is possible to find allocations EF1 and PO using Nash social welfare. However, to find an allocation that maximizes the Nash social welfare is a computationally hard problem. In this work we propose a polynomial time algorithm which maximizes the utilitarian social welfare and at the same time produces an allocation which is EF1 and PO in a special case of additive utility functions called buyer utility functions. Moreover, a slight modification of our algorithm produces an allocation which is envy-free up to any positively valued good (EFX).
Journal of Artificial Intelligence Research, 2020
In this work we introduce a 3-valued logic with modalities, with the aim of having a clear and pr... more In this work we introduce a 3-valued logic with modalities, with the aim of having a clear and precise representation of epistemic states, thus the formulas of this logic will be our epistemic states. Indeed, these formulas are identified with ranking functions of 3 values, a generalization of total preorders of three levels. In this framework we analyze some types of changes of these epistemic structures and give syntactical characterizations of them in the introduced logic. In particular, we introduce and study carefully a new operator called Cautious Improvement operator. We also characterize all operators that are definable in this framework.
Lecture Notes in Computer Science, 2021
This paper aims at providing an overview of the use of mathematical morphology, in its algebraic ... more This paper aims at providing an overview of the use of mathematical morphology, in its algebraic setting, in several fields of artificial intelligence (AI). Three domains of AI will be covered. In the first domain, mathematical morphology operators will be expressed in some logics (propositional, modal, description logics) to answer typical questions in knowledge representation and reasoning, such as revision, fusion, explanatory relations, satisfying usual postulates. In the second domain, spatial reasoning will benefit from spatial relations modeled using fuzzy sets and morphological operators, with applications in modelbased image understanding. In the third domain, interactions between mathematical morphology and deep learning will be detailed. Morphological neural networks were introduced as an alternative to classical architectures, yielding a new geometry in decision surfaces. Deep networks were also trained to learn morphological operators and pipelines, and morphological algorithms were used as companion tools to machine learning, for pre/post processing or even regularization purposes. These ideas have known a large resurgence in the last few years and new ones are emerging. Keywords: Mathematical morphology • Artificial intelligence • Lattice • Logics • Spatial reasoning • Fuzzy sets • Neural networks • Deep learning I. Bloch-This work was partly done while I. Bloch was with LTCI, Télécom Paris,
Proceedings of the Eighteenth International Conference on Principles of Knowledge Representation and Reasoning, 2021
In this work, we explore the links between the Borda voting rule and belief merging operators. Mo... more In this work, we explore the links between the Borda voting rule and belief merging operators. More precisely, we define two families of merging operators inspired by the definition of the Borda voting rule. We also introduce a notion of cancellation in belief merging, inspired by the axiomatization of the Borda voting rule proposed by Young. This allows us to provide a characterization of the drastic merging operator.
Revista Colombiana de Matemáticas, 2017
Transferir preferencias sobre candidatos a preferencias sobre conjuntos de candidatos permite dar... more Transferir preferencias sobre candidatos a preferencias sobre conjuntos de candidatos permite dar una noción muy natural de manipulación para funciones de elección social. En este trabajo damos condiciones sobre esas funciones de transferencia que implican la manipulabilidad de funciones de elección social con un mínimo de propiedades razonables. Nuestro resultado es una versión débil del teorema de Barberà y Kelly, de hecho puede ser obtenido como una consecuencia de éste. Sin embargo, damos una prueba directa y natural de nuestro teorema de manipulabilidad, la cual da una información clara sobre la naturaleza de las funciones de transferencia que permiten la manipulación.
Artificial Intelligence, 2017
With the aim of studying social properties of belief merging and having a better understanding of... more With the aim of studying social properties of belief merging and having a better understanding of impossibility, we extend in three ways the framework of logic-based merging introduced by Konieczny and Pino Pérez. First, at the level of representation of the information, we pass from belief bases to complex epistemic states. Second, the profiles are represented as functions of finite societies to the set of epistemic states (a sort of vectors) and not as multisets of epistemic states. Third, we extend the set of rational postulates in order to consider the epistemic versions of the classical postulates of Social Choice Theory: Standard Domain, Pareto Property, Independence of Irrelevant Alternatives and Absence of Dictator. These epistemic versions of social postulates are given, essentially, in terms of the finite propositional logic. We state some representation theorems for these operators. These extensions and representation theorems allow us to establish an epistemic and very general version of Arrow's Impossibility Theorem. One of the interesting features of our result, is that it holds for different representations of epistemic states; for instance conditionals, Ordinal Conditional functions and, of course, total preorders.
Lecture Notes in Computer Science, 2001
Using mathematical morphology on formulas introduced recently by Bloch and Lang (Proceedings of I... more Using mathematical morphology on formulas introduced recently by Bloch and Lang (Proceedings of IPMU'2000) we define two new explanatory relations. Their logical behavior is analyzed. The results show that these natural ways for defining preferred explanations are robust because these relations satisfy almost all postulates of explanatory reasoning introduced by Pino-Pérez and Uzcátegui (Artificial Intelligence, 111:131-169, 1999). Actually, the first explanatory relation is Explanatory-Rational. The second one is not even Explanatory-Cumulative but it satisfies new weak postulates.
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Papers by Ramon Pino Perez