Marcelo Arenas

JAIR Journal 2025 Journal Article

On Computing Probabilistic Explanations for Decision Trees

• Marcelo Arenas
• Pablo Barcelo
• Alexander Kozachinskiy
• Miguel Romero
• Bernardo Subercaseaux

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of sufficient reasons, a kind of explanation in which given a decision tree T and an instance x, one explains the decision T (x) by providing a subset y of the features of x such that for any other instance z compatible with y, it holds that T (z) = T (x), intuitively meaning that the features in y are already enough to fully justify the classification of x by T. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation. For such a reason, the community has started to study their probabilistic counterpart, in which one requires that the probability of T (z) = T (x) must be at least some value δ ∈ (0, 1], where z is a random instance that is compatible with y. Our paper settles the computational complexity of δ-sufficient-reasons over decision trees, showing that both (1) finding δ-sufficient-reasons that are minimal in size, and (2) finding δ-sufficient-reasons that are minimal inclusion-wise, are computationally intractable. By doing this, we answer two open problems originally raised by Izza et al. (2021), and extend the hardness of explanations for Boolean circuits presented by Wäldchen et al. (2021) to the more restricted case of decision trees. Furthermore, we present sharp non-approximability results under a widely believed complexity hypothesis. On the positive side, we identify structural restrictions of decision trees that make the problem tractable.

AAAI Conference 2025 Conference Paper

Probabilistic Explanations for Linear Models

• Bernardo Subercaseaux
• Marcelo Arenas
• Kuldeep S. Meel

Formal XAI is an emerging field that focuses on providing explanations with mathematical guarantees for the decisions made by machine learning models. A significant amount of work in this area is centered on the computation of ``sufficient reasons''. Given a model M and an input instance x, a sufficient reason for the decision x is a subset S of the features of x such that for any instance z that has the same values as x for every feature in S, it holds that M(x) = M(z). Intuitively, this means that the features in S are sufficient to fully justify the classification of x by M. For sufficient reasons to be useful in practice, they should be as small as possible, and a natural way to reduce the size of sufficient reasons is to consider a probabilistic relaxation; the probability of M(x) = M(z) must be at least some value delta in (0,1], where z is a random instance that coincides with x on the features in S. Computing small delta-sufficient reasons (delta-SRs) is known to be a theoretically hard problem; even over decision trees — traditionally deemed simple and interpretable models — strong inapproximability results make the efficient computation of small delta-SRs unlikely. We propose the notion of (delta, epsilon)-SR, a simple relaxation of delta-SRs, and show that this kind of explanations can be computed efficiently over linear models.

KR Conference 2024 Conference Paper

A Uniform Language to Explain Decision Trees

• Marcelo Arenas
• Pablo Barceló
• Diego Bustamante
• Jose Caraball
• Bernardo Subercaseaux

The formal XAI community has studied a plethora of interpretability queries aiming to understand the classifications made by decision trees. However, a more uniform understanding of what questions we can hope to answer about these models, traditionally deemed to be easily interpretable, has remained elusive. In an initial attempt to understand uniform languages for interpretability, Arenas et al. proposed FOIL, a logic for explaining black-box ML models, and showed that it can express a variety of interpretability queries. However, we show that FOIL is limited in two important senses: (i) it is not expressive enough to capture some crucial queries, and (ii) its model agnostic nature results in a high computational complexity for decision trees. In this paper, we carefully craft two fragments of first-order logic that allow for efficiently interpreting decision trees: Q-DT-FOIL and its optimization variant OPT-DT-FOIL. We show that our proposed logics can express not only a variety of interpretability queries considered by previous literature, but also elegantly allows users to specify different objectives the sought explanations should optimize for. Using finite model-theoretic techniques, we show that the different ingredients of Q-DT-FOIL are necessary for its expressiveness, and yet that queries in Q-DT-FOIL can be evaluated with a polynomial number of queries to a SAT solver, as well as their optimization versions in OPT-DT-FOIL. Besides our theoretical results, we provide a SAT-based implementation of the evaluation for OPT-DT-FOIL that is performant on industry-size decision trees.

JMLR Journal 2023 Journal Article

On the Complexity of SHAP-Score-Based Explanations: Tractability via Knowledge Compilation and Non-Approximability Results

• Marcelo Arenas
• Pablo Barcelo
• Leopoldo Bertossi
• Mikael Monet

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential~ Shap-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, we prove a strong positive result stating that the Shap-score can be computed in polynomial time over deterministic and decomposable Boolean circuits under the so-called product distributions on entities. Such circuits are studied in the field of Knowledge Compilation and generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). Our positive result extends even beyond binary classifiers, as it continues to hold if each feature is associated with a finite domain of possible values. We also establish the computational limits of the notion of Shap-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the Shap-score intractable (namely, $\#P$-hard). It also implies that computing Shap-scores is $\#P$-hard even over the class of propositional formulas in DNF. Based on this negative result, we look for the existence of fully-polynomial randomized approximation schemes (FPRAS) for computing Shap-scores over such class. In stark contrast to the model counting problem for DNF formulas, which admits an FPRAS, we prove that no such FPRAS exists (under widely believed complexity assumptions) for the computation of Shap-scores. Surprisingly, this negative result holds even for the class of monotone formulas in DNF. These techniques can be further extended to prove another strong negative result: Under widely believed complexity assumptions, there is no polynomial-time algorithm that checks, given a monotone DNF formula $\varphi$ and features $x,y$, whether the Shap-score of $x$ in $\varphi$ is smaller than the Shap-score of $y$ in $\varphi$. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

NeurIPS Conference 2022 Conference Paper

On Computing Probabilistic Explanations for Decision Trees

• Marcelo Arenas
• Pablo Barceló
• Miguel Romero Orth
• Bernardo Subercaseaux

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of sufficient reasons, a kind of explanation in which given a decision tree $T$ and an instance $x$, one explains the decision $T(x)$ by providing a subset $y$ of the features of $x$ such that for any other instance $z$ compatible with $y$, it holds that $T(z) = T(x)$, intuitively meaning that the features in $y$ are already enough to fully justify the classification of $x$ by $T$. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation. For such a reason, the community has started to study their probabilistic counterpart, in which one requires that the probability of $T(z) = T(x)$ must be at least some value $\delta \in (0, 1]$, where $z$ is a random instance that is compatible with $y$. Our paper settles the computational complexity of $\delta$-sufficient-reasons over decision trees, showing that both (1) finding $\delta$-sufficient-reasons that are minimal in size, and (2) finding $\delta$-sufficient-reasons that are minimal inclusion-wise, do not admit polynomial-time algorithms (unless P = NP). This is in stark contrast with the deterministic case ($\delta = 1$) where inclusion-wise minimal sufficient-reasons are easy to compute. By doing this, we answer two open problems originally raised by Izza et al. , and extend the hardness of explanations for Boolean circuits presented by W{\"a}ldchen et al. to the more restricted case of decision trees. On the positive side, we identify structural restrictions of decision trees that make the problem tractable, and show how SAT solvers might be able to tackle these problems in practical settings.

STOC Conference 2021 Invited Paper

A polynomial-time approximation algorithm for counting words accepted by an NFA (invited paper)

• Marcelo Arenas
• Luis Alberto Croquevielle
• Rajesh Jayaram
• Cristian Riveros

NeurIPS Conference 2021 Conference Paper

Foundations of Symbolic Languages for Model Interpretability

• Marcelo Arenas
• Daniel Báez
• Pablo Barceló
• Jorge Pérez
• Bernardo Subercaseaux

Several queries and scores have recently been proposed to explain individual predictions over ML models. Examples include queries based on “anchors”, which are parts of an instance that are sufficient to justify its classification, and “feature-perturbation” scores such as SHAP. Given the need for flexible, reliable, and easy-to-apply interpretability methods for ML models, we foresee the need for developing declarative languages to naturally specify different explainability queries. We do this in a principled way by rooting such a language in a logic called FOIL, which allows for expressing many simple but important explainability queries, and might serve as a core for more expressive interpretability languages. We study the computational complexity of FOIL queries over two classes of ML models often deemed to be easily interpretable: decision trees and more general decision diagrams. Since the number of possible inputs for an ML model is exponential in its dimension, tractability of the FOIL evaluation problem is delicate but can be achieved by either restricting the structure of the models, or the fragment of FOIL being evaluated. We also present a prototype implementation of FOIL wrapped in a high-level declarative language and perform experiments showing that such a language can be used in practice.

AAAI Conference 2021 Conference Paper

The Tractability of SHAP-Score-Based Explanations for Classification over Deterministic and Decomposable Boolean Circuits

• Marcelo Arenas
• Pablo Barceló
• Leopoldo Bertossi
• Mikaël Monet

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAPscore, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).

STOC Conference 2021 Conference Paper

When is approximate counting for conjunctive queries tractable?

• Marcelo Arenas
• Luis Alberto Croquevielle
• Rajesh Jayaram
• Cristian Riveros

Conjunctive queries are one of the most common class of queries used in database systems, and the best studied in the literature. A seminal result of Grohe, Schwentick, and Segoufin (STOC 2001) demonstrates that for every class G of graphs, the evaluation of all conjunctive queries whose underlying graph is in G is tractable if, and only if, G has bounded treewidth. In this work, we extend this characterization to the counting problem for conjunctive queries. Specifically, for every class C of conjunctive queries with bounded treewidth, we introduce the first fully polynomial-time randomized approximation scheme (FPRAS) for counting answers to a query in C, and the first polynomial-time algorithm for sampling answers uniformly from a query in C. As a corollary, it follows that for every class G of graphs, the counting problem for conjunctive queries whose underlying graph is in G admits an FPRAS if, and only if, G has bounded treewidth (unless BPP is different from P). In fact, our FPRAS is more general, and also applies to conjunctive queries with bounded hypertree width, as well as unions of such queries.

KR Conference 2018 Conference Paper

On the Progression of Situation Calculus Universal Theories with Constants

• Marcelo Arenas
• Jorge Baier
• Juan Navarro
• Sebastian Sardina

The progression of action theories is an important problem in knowledge representation. Progression is second-order definable and known to be first-order definable and effectively computable for restricted classes of theories. Motivated by the fact that universal theories with constants (UTCs) are expressive and natural theories whose satisfiability is decidable, in this paper we provide a thorough study of the progression of situation calculus UTCs. First, we prove that progression of a (possibly infinite) UTC is always first-order definable and results in a UTC. Though first-order definable, we show that the progression of a UTC may be infeasible, that is, it may result in an infinite UTC that is not equivalent to any finite set of first-order sentences. We then show that deciding whether there is a feasible progression of a UTC is undecidable. Moreover, we show that deciding whether a sentence (in an expressive fragment of first-order logic) is in the progression of a UTC is CONEXPTIME-complete, and that there exists a family of UTCs for which the size of every feasible progression grows exponentially. Finally, we discuss resolutionbased approaches to compute the progression of a UTC. This comprehensive analysis contributes to a better understanding of progression in action theories, both in terms of feasibility and difficulty.

IJCAI Conference 2016 Conference Paper

Incomplete Causal Laws in the Situation Calculus Using Free Fluents

• Marcelo Arenas
• Jorge A. Baier
• Juan S. Navarro
• Sebastian Sardina

We propose a simple relaxation of Reiter's basic action theories, based on fluents without successor state axioms, that accommodates incompleteness beyond the initial database. We prove that fundamental results about basic action theories can be fully recovered and that the generalized framework allows for natural specifications of various forms of incomplete causal laws. We illustrate this by showing how the evolution of incomplete databases, guarded action theories, and non-deterministic actions can be conveniently specified.

AIJ Journal 2016 Journal Article

Knowledge base exchange: The case of OWL 2 QL

• Marcelo Arenas
• Elena Botoeva
• Diego Calvanese
• Vladislav Ryzhikov

In this article, we define and study the problem of exchanging knowledge between a source and a target knowledge base (KB), connected through mappings. Differently from the traditional database exchange setting, which considers only the exchange of data, we are interested in exchanging implicit knowledge. As representation formalism we use Description Logics (DLs), thus assuming that the source and target KBs are given as a DL TBox+ABox, while the mappings have the form of DL TBox assertions. We define a general framework of KB exchange, and study the problem of translating the knowledge in the source KB according to the mappings expressed in OWL 2 QL, the profile of the standard Web Ontology Language OWL 2 based on the description logic DL-Lite R. We develop novel game- and automata-theoretic techniques, and we provide complexity results that range from NLogSpace to ExpTime.

IJCAI Conference 2015 Conference Paper

Bidirectional Constraints for Exchanging Data: Beyond Monotone Queries

• Marcelo Arenas
• Gabriel Di
• eacute; guez
• Jorge P
• eacute; rez

In this paper, we propose to use the language of bidirectional constraints to specify schema mappings in the context of data exchange. These constraints impose restrictions over both the source and the target data, and have the potential to minimize the ambiguity in the description of the target data to be materialized. We start by making a case for the usefulness of bidirectional constraints to give a meaningful closed-world semantics for st-tgds, which is motivated by Clark’s predicate completion and Reiter’s formalization of the closed-world assumption of a logical theory. We then formally study the use of bidirectional constraints in data exchange. In particular, we pinpoint the complexity of the existence-of-solutions and the query evaluation problems in several different scenarios, including in the latter case both monotone and nonmonotone queries.

IJCAI Conference 2013 Conference Paper

Exchanging OWL 2 QL Knowledge Bases

• Marcelo Arenas
• Elena Botoeva
• Diego Calvanese
• Vladislav Ryzhikov

Knowledge base exchange is an important problem in the area of data exchange and knowledge representation, where one is interested in exchanging information between a source and a target knowledge base connected through a mapping. In this paper, we study this fundamental problem for knowledge bases and mappings expressed in OWL 2 QL, the profile of OWL 2 based on the description logic DL-LiteR. More specifically, we consider the problem of computing universal solutions, identified as one of the most desirable translations to be materialized, and the problem of computing UCQrepresentations, which optimally capture in a target TBox the information that can be extracted from a source TBox and a mapping by means of unions of conjunctive queries. For the former we provide a novel automata-theoretic technique, and complexity results that range from NP to EXPTIME, while for the latter we show NLOGSPACE-completeness.

I&C Journal 2013 Journal Article

Solutions and query rewriting in data exchange

• Marcelo Arenas
• Pablo Barceló
• Ronald Fagin
• Leonid Libkin

Data exchange is the problem of taking data structured under a source schema and creating an instance of a target schema. Given a source instance, there may be many solutions – target instances that satisfy the constraints of the data exchange problem. Previous work has identified two classes of desirable solutions: canonical universal solutions, and their cores. Query answering in data exchange amounts to rewriting a query over the target schema to another query that, over a materialized target instance, gives the result that is semantically consistent with the source (specifically, the “certain answers”). Basic questions are then: (1) how do these solutions compare in terms of query rewriting? and (2) how can we determine whether a query is rewritable over a particular solution? Our goal is to answer these questions. Our first main result is that, in terms of rewritability by relational algebra queries, the core is strictly less expressive than the canonical universal solution, which in turn is strictly less expressive than the source. To develop techniques for proving queries non-rewritable, we establish structural properties of solutions; in fact they are derived from the technical machinery developed in the rewritability proofs. Our second result is that both the canonical universal solution and the core preserve the local structure of the data, and that every target query rewritable over any of these solutions cannot distinguish tuples whose neighborhoods in the source are similar. This gives us a first simple tool for checking whether a query is non-rewritable over the canonical universal solution or over the core. We also show that these tools generalize to arbitrary transformations that preserve the local structure of the data, and investigate an alternative semantics of query answering in data exchange.

KR Conference 2012 Short Paper

Exchanging Description Logic Knowledge Bases

• Marcelo Arenas
• Elena Botoeva
• Diego Calvanese
• Vladislav Ryzhikov
• Evgeny Sherkhonov

mappings are sets of DL inclusions. In such a setting, in order to minimize the exchange (and hence transfer and materialization) of explicit (i. e., ABox) information, we are interested in computing translations, from now on referred to as solutions, that contain as much implicit knowledge as possible. This leads us to define the novel notion of representability, which helps us in understanding the capacity of solutions to transfer implicit knowledge. Checking representability and computing a representation of a source TBox under a mapping turn out to be crucial problems in the context of knowledge base exchange. Furthermore, we argue that the right notion of solution, on which to base our investigations, should not be the standard one based on the correspondence between models of source and target KBs. Indeed, we show that such solutions present severe limitations since, on the one hand, they do not allow for the use of implicit target information to represent implicit source information, and on the other hand, they may lead to exponentially large target ABoxes. To overcome these drawbacks, we introduce the weaker notion of Q-solution, for a query language Q, which is based on the correspondence between answers to queries in Q over source and target KBs. Notice that such a notion, though weaker, is in line with the objective of (data and) knowledge base exchange of providing in the target sufficient information to answer queries in Q that could also be posed over the source. We then develop results and techniques for KB exchange and for the Q-representability problem in the case where Q are unions of conjunctive queries (UCQs), and where KBs are expressed in DL-LiteRDFS, a member of the DL-Lite family (Calvanese et al. 2007) that corresponds to the FOL fragment of RDFS (Brickley and Guha 2004), the widely adopted standard Semantic Web language. In this paper, we study the problem of exchanging knowledge between a source and a target knowledge base (KB), connected through mappings. Differently from the traditional database exchange setting, which considers only the exchange of data, we are interested in exchanging implicit knowledge. As representation formalism we use Description Logics (DLs), thus assuming that the source and target KBs are given as a DL TBox+ABox, while the mappings have the form of DL TBox assertions. We study the problem of translating the knowledge in the source KB according to these mappings. We define a general framework of KB exchange, and address the problems of representing implicit source information in the target, and of computing different kinds of solutions, i. e., target KBs with specified properties, given a source KB and a mapping. We develop first results and study the complexity of KB exchange for DL-LiteRDFS, a DL corresponding to the FOL fragment of RDFS, and for DL-LiteR.

CSL Conference 2004 Conference Paper

Game-Based Notions of Locality Over Finite Models

• Marcelo Arenas
• Pablo Barceló
• Leonid Libkin

Abstract Locality notions in logic say that the truth value of a formula can be determined locally, by looking at the isomorphism type of a small neighborhood of its free variables. Such notions have proved to be useful in many applications. They all, however, refer to isomorphism of neighborhoods, which most local logics cannot test for. A more relaxed notion of locality says that the truth value of a formula is determined by what the logic itself can say about that small neighborhood. Or, since most logics are characterized by games, the truth value of a formula is determined by the type, with respect to a game, of that small neighborhood. Such game-based notions of locality can often be applied when traditional isomorphism-based locality cannot. Our goal is to study game-based notions of locality. We work with an abstract view of games that subsumes games for many logics. We look at three, progressively more complicated locality notions. The easiest requires only very mild conditions on the game and works for most logics of interest. The other notions, based on Hanf’s and Gaifman’s theorems, require more restrictions. We state those restrictions and give examples of logics that satisfy and fail the respective game-based notions of locality.

TCS Journal 2003 Journal Article

Scalar aggregation in inconsistent databases

• Marcelo Arenas
• Leopoldo Bertossi
• Jan Chomicki
• Xin He
• Vijay Raghavan
• Jeremy Spinrad

We consider here scalar aggregation queries in databases that may violate a given set of functional dependencies. We define consistent answers to such queries to be greatest-lowest/least-upper bounds on the value of the scalar function across all (minimal) repairs of the database. We show how to compute such answers. We provide a complete characterization of the computational complexity of this problem. We also show how tractability can be improved in several special cases (one involves a novel application of Boyce–Codd Normal Form) and present a practical hybrid query evaluation method.