arXiv subjects

Camilo Sarmiento

Publications and source records attributed to Camilo Sarmiento.

8 recordsLinked to original sources

arXiv2024

An action language-based formalisation of an abstract argumentation framework

An abstract argumentation framework is a commonly used formalism to provide a static representation of a dialogue. However, the order of enunciation of the arguments in an argumentative dialogue is very important and can affect the outcome of this dialogue. In this paper, we propose a new framework for modelling abstract argumentation graphs, a model that incorporates the order of enunciation of arguments. By taking this order into account, we have the means to deduce a unique outcome for each dialogue, called an extension. We also establish several properties, such as termination and correctness, and discuss two notions of completeness. In particular, we propose a modification of the previous transformation based on a "last enunciated last updated" strategy, which verifies the second form of completeness.

cs.AI↗

arXiv2023

Action Languages Based Actual Causality for Computational Ethics: a Sound and Complete Implementation in ASP

Although moral responsibility is not circumscribed by causality, they are both closely intermixed. Furthermore, rationally understanding the evolution of the physical world is inherently linked with the idea of causality. Thus, the decision-making applications based on automated planning inevitably have to deal with causality, especially if they consider imputability aspects or integrate references to ethical norms. The many debates around causation in the last decades have shown how complex this notion is and thus, how difficult is its integration with planning. As a result, much of the work in computational ethics relegates causality to the background, despite the considerations stated above. This paper's contribution is to provide a complete and sound translation into logic programming from an actual causation definition suitable for action languages, this definition is a formalisation of Wright's NESS test. The obtained logic program allows to deal with complex causal relations. In addition to enabling agents to reason about causality, this contribution specifically enables the computational ethics domain to handle situations that were previously out of reach. In a context where ethical considerations in decision-making are increasingly important, advances in computational ethics can greatly benefit the entire AI community.

cs.AI↗

arXiv2019

The $ν$-Tamari lattice via $ν$-trees, $ν$-bracket vectors, and subword complexes

We give new interpretations of the $ν$-Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of $ν$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of $ν$-trees and componentwise order, we provide a simple description of the lattice property. We also show that the $ν$-Tamari lattice is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey's lattice conjecture concerning the poset of pipe dreams defined by chute moves. Finally, this point of view generalizes to multi $ν$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.

math.CO↗

arXiv2018

On power ideals of transversal matroids and their "parking functions"

To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of these monomial basis elements "parking functions" of the corresponding transversal matroid. We highlight the connection between our investigation and Stanley-Reisner theory, and relate our findings to Stanley's conjectured necessary condition on matroid $h$-vectors.

math.CO↗

arXiv2017

Geometry of $ν$-Tamari lattices in types $A$ and $B$

In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $ν$-Tamari lattices. In our framework, the main role of "Catalan objects" is played by $(I,\overline{J})$-trees: bipartite trees associated to a pair $(I,\overline{J})$ of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path $ν=ν(I,\overline{J})$. Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the $ν$-Tamari lattice introduced by Prévile-Ratelle and Viennot. In particular, we obtain geometric realizations of $m$-Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F.~Bergeron. The simplicial complex underlying our triangulation endows the $ν$-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams, whose $h$-vector entries are given by a suitable generalization of the Narayana numbers. Our methods are amenable to cyclic symmetry, which we use to present type $B$ analogues of our constructions. Notably, we define a partial order that generalizes the type $B$ Tamari lattice, introduced independently by Thomas and Reading, along with corresponding geometric realizations.

math.CO↗

arXiv2017

Parity binomial edge ideals

Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.

math.AC↗

arXiv2014

Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices $Δ_{n-1}\timesΔ_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $Δ_{r\ n-1}\timesΔ_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever $m\geq k>n$, any triangulation of $Δ_{m-1}^{(k-1)}\timesΔ_{n-1}$ extends to a unique triangulation of $Δ_{m-1}\timesΔ_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interesting interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.

math.CO↗

arXiv2013

Betti numbers of cut ideals of trees

Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from topological combinatorics, we obtain upper bounds for the Betti numbers of this type of ideals. These take the form of simple formulas on the number of vertices, which arise from the enumeration of induced subgraphs of certain incomparability graphs associated to the edge sets of trees.

math.AC↗