arXiv subjects

Martin C. Cooper

Publications and source records attributed to Martin C. Cooper.

At least 19 recordsLinked to original sources

arXiv2026

Fairness of Classifiers in the Presence of Constraints between Features

In Machine Learning, an accepted definition of fairness of a decision taken by a classifier is that it should not depend on protected features, such as gender. Unfortunately, when constraints exist between features, such dependencies can be obscured by the constraints. To avoid this problem, we propose that a decision be considered fair if it has a fair explanation. We define a fair explanation as a prime-implicant reason for the decision that does not contain any protected feature (where the constraints are taken into account in the definition of prime-implicant). Surprisingly, ignoring constraints can completely change the fairness of a decision (according to this definition) even in the absence of constraints between protected and unprotected features. Three possible definitions of fairness of a classifier are that for all its decisions (1) there are only fair explanations, (2) there is at least one fair explanation, or (3) changing protected features does not change the outcome. We identify the relationships between these different definitions of fairness and study the computational complexity of testing fairness of classifiers.

cs.LG↗

arXiv2025

Interpretable DNFs

A classifier is considered interpretable if each of its decisions has an explanation which is small enough to be easily understood by a human user. A DNF formula can be seen as a binary classifier $κ$ over boolean domains. The size of an explanation of a positive decision taken by a DNF $κ$ is bounded by the size of the terms in $κ$, since we can explain a positive decision by giving a term of $κ$ that evaluates to true. Since both positive and negative decisions must be explained, we consider that interpretable DNFs are those $κ$ for which both $κ$ and $\overlineκ$ can be expressed as DNFs composed of terms of bounded size. In this paper, we study the family of $k$-DNFs whose complements can also be expressed as $k$-DNFs. We compare two such families, namely depth-$k$ decision trees and nested $k$-DNFs, a novel family of models. Experiments indicate that nested $k$-DNFs are an interesting alternative to decision trees in terms of interpretability and accuracy.

cs.AI↗

arXiv2024

Axiomatic Characterisations of Sample-based Explainers

Explaining decisions of black-box classifiers is both important and computationally challenging. In this paper, we scrutinize explainers that generate feature-based explanations from samples or datasets. We start by presenting a set of desirable properties that explainers would ideally satisfy, delve into their relationships, and highlight incompatibilities of some of them. We identify the entire family of explainers that satisfy two key properties which are compatible with all the others. Its instances provide sufficient reasons, called weak abductive explanations.We then unravel its various subfamilies that satisfy subsets of compatible properties. Indeed, we fully characterize all the explainers that satisfy any subset of compatible properties. In particular, we introduce the first (broad family of) explainers that guarantee the existence of explanations and their global consistency.We discuss some of its instances including the irrefutable explainer and the surrogate explainer whose explanations can be found in polynomial time.

cs.AI↗

arXiv2024

Backward explanations via redefinition of predicates

History eXplanation based on Predicates (HXP), studies the behavior of a Reinforcement Learning (RL) agent in a sequence of agent's interactions with the environment (a history), through the prism of an arbitrary predicate. To this end, an action importance score is computed for each action in the history. The explanation consists in displaying the most important actions to the user. As the calculation of an action's importance is #W[1]-hard, it is necessary for long histories to approximate the scores, at the expense of their quality. We therefore propose a new HXP method, called Backward-HXP, to provide explanations for these histories without having to approximate scores. Experiments show the ability of B-HXP to summarise long histories.

cs.AI↗

arXiv2024

Homomorphisms and Embeddings of STRIPS Planning Models

Determining whether two STRIPS planning instances are isomorphic is the simplest form of comparison between planning instances. It is also a particular case of the problem concerned with finding an isomorphism between a planning instance $P$ and a sub-instance of another instance $P_0$ . One application of such a mapping is to efficiently produce a compiled form containing all solutions to P from a compiled form containing all solutions to $P_0$. We also introduce the notion of embedding from an instance $P$ to another instance $P_0$, which allows us to deduce that $P_0$ has no solution-plan if $P$ is unsolvable. In this paper, we study the complexity of these problems. We show that the first is GI-complete, and can thus be solved, in theory, in quasi-polynomial time. While we prove the remaining problems to be NP-complete, we propose an algorithm to build an isomorphism, when possible. We report extensive experimental trials on benchmark problems which demonstrate conclusively that applying constraint propagation in preprocessing can greatly improve the efficiency of a SAT solver.

cs.AI↗

arXiv2023

Feature Necessity & Relevancy in ML Classifier Explanations

Given a machine learning (ML) model and a prediction, explanations can be defined as sets of features which are sufficient for the prediction. In some applications, and besides asking for an explanation, it is also critical to understand whether sensitive features can occur in some explanation, or whether a non-interesting feature must occur in all explanations. This paper starts by relating such queries respectively with the problems of relevancy and necessity in logic-based abduction. The paper then proves membership and hardness results for several families of ML classifiers. Afterwards the paper proposes concrete algorithms for two classes of classifiers. The experimental results confirm the scalability of the proposed algorithms.

cs.LG↗

arXiv2022

On Computing Probabilistic Abductive Explanations

The most widely studied explainable AI (XAI) approaches are unsound. This is the case with well-known model-agnostic explanation approaches, and it is also the case with approaches based on saliency maps. One solution is to consider intrinsic interpretability, which does not exhibit the drawback of unsoundness. Unfortunately, intrinsic interpretability can display unwieldy explanation redundancy. Formal explainability represents the alternative to these non-rigorous approaches, with one example being PI-explanations. Unfortunately, PI-explanations also exhibit important drawbacks, the most visible of which is arguably their size. Recently, it has been observed that the (absolute) rigor of PI-explanations can be traded off for a smaller explanation size, by computing the so-called relevant sets. Given some positive δ, a set S of features is δ-relevant if, when the features in S are fixed, the probability of getting the target class exceeds δ. However, even for very simple classifiers, the complexity of computing relevant sets of features is prohibitive, with the decision problem being NPPP-complete for circuit-based classifiers. In contrast with earlier negative results, this paper investigates practical approaches for computing relevant sets for a number of widely used classifiers that include Decision Trees (DTs), Naive Bayes Classifiers (NBCs), and several families of classifiers obtained from propositional languages. Moreover, the paper shows that, in practice, and for these families of classifiers, relevant sets are easy to compute. Furthermore, the experiments confirm that succinct sets of relevant features can be obtained for the families of classifiers considered.

cs.AI↗

arXiv2022

Provably Precise, Succinct and Efficient Explanations for Decision Trees

Decision trees (DTs) embody interpretable classifiers. DTs have been advocated for deployment in high-risk applications, but also for explaining other complex classifiers. Nevertheless, recent work has demonstrated that predictions in DTs ought to be explained with rigorous approaches. Although rigorous explanations can be computed in polynomial time for DTs, their size may be beyond the cognitive limits of human decision makers. This paper investigates the computation of δ-relevant sets for DTs. δ-relevant sets denote explanations that are succinct and provably precise. These sets represent generalizations of rigorous explanations, which are precise with probability one, and so they enable trading off explanation size for precision. The paper proposes two logic encodings for computing smallest δ-relevant sets for DTs. The paper further devises a polynomial-time algorithm for computing δ-relevant sets which are not guaranteed to be subset-minimal, but for which the experiments show to be most often subset-minimal in practice. The experimental results also demonstrate the practical efficiency of computing smallest δ-relevant sets.

cs.AI↗

arXiv2021

Efficient Explanations for Knowledge Compilation Languages

Knowledge compilation (KC) languages find a growing number of practical uses, including in Constraint Programming (CP) and in Machine Learning (ML). In most applications, one natural question is how to explain the decisions made by models represented by a KC language. This paper shows that for many of the best known KC languages, well-known classes of explanations can be computed in polynomial time. These classes include deterministic decomposable negation normal form (d-DNNF), and so any KC language that is strictly less succinct than d-DNNF. Furthermore, the paper also investigates the conditions under which polynomial time computation of explanations can be extended to KC languages more succinct than d-DNNF.

cs.AI↗

arXiv2021

Efficient Explanations With Relevant Sets

Recent work proposed $δ$-relevant inputs (or sets) as a probabilistic explanation for the predictions made by a classifier on a given input. $δ$-relevant sets are significant because they serve to relate (model-agnostic) Anchors with (model-accurate) PI- explanations, among other explanation approaches. Unfortunately, the computation of smallest size $δ$-relevant sets is complete for ${NP}^{PP}$, rendering their computation largely infeasible in practice. This paper investigates solutions for tackling the practical limitations of $δ$-relevant sets. First, the paper alternatively considers the computation of subset-minimal sets. Second, the paper studies concrete families of classifiers, including decision trees among others. For these cases, the paper shows that the computation of subset-minimal $δ$-relevant sets is in NP, and can be solved with a polynomial number of calls to an NP oracle. The experimental evaluation compares the proposed approach with heuristic explainers for the concrete case of the classifiers studied in the paper, and confirms the advantage of the proposed solution over the state of the art.

cs.LG↗

arXiv2020

Explaining Naive Bayes and Other Linear Classifiers with Polynomial Time and Delay

Recent work proposed the computation of so-called PI-explanations of Naive Bayes Classifiers (NBCs). PI-explanations are subset-minimal sets of feature-value pairs that are sufficient for the prediction, and have been computed with state-of-the-art exact algorithms that are worst-case exponential in time and space. In contrast, we show that the computation of one PI-explanation for an NBC can be achieved in log-linear time, and that the same result also applies to the more general class of linear classifiers. Furthermore, we show that the enumeration of PI-explanations can be obtained with polynomial delay. Experimental results demonstrate the performance gains of the new algorithms when compared with earlier work. The experimental results also investigate ways to measure the quality of heuristic explanations

cs.LG↗

arXiv2020

Strengthening neighbourhood substitution

Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard.

cs.AI↗

arXiv2019

Steepest ascent can be exponential in bounded treewidth problems

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.

cs.DM↗

arXiv2019

Variable elimination in binary CSPs

We investigate rules which allow variable elimination in binary CSP (constraint satisfaction problem) instances while conserving satisfiability. We study variable-elimination rules based on the language of forbidden patterns enriched with counting and quantification over variables and values. We propose new rules and compare them, both theoretically and experimentally. We give optimised algorithms to apply these rules and show that each define a novel tractable class. Using our variable-elimination rules in preprocessing allowed us to solve more benchmark problems than without.

cs.DS↗

arXiv2018

Binary Constraint Satisfaction Problems Defined by Excluded Topological Minors

The binary Constraint Satisfaction Problem (CSP) is to decide whether there exists an assignment to a set of variables which satisfies specified constraints between pairs of variables. A binary CSP instance can be presented as a labelled graph encoding both the forms of the constraints and where they are imposed. We consider subproblems defined by restricting the allowed form of this graph. One type of restriction that has previously been considered is to forbid certain specified substructures (patterns). This captures some tractable classes of the CSP, but does not capture classes defined by language restrictions, or the well-known structural property of acyclicity. In this paper we extend the notion of pattern and introduce the notion of a topological minor of a binary CSP instance. By forbidding a finite set of patterns from occurring as topological minors we obtain a compact mechanism for expressing novel tractable subproblems of the binary CSP, including new generalisations of the class of acyclic instances. Forbidding a finite set of patterns as topological minors also captures all other tractable structural restrictions of the binary CSP. Moreover, we show that several patterns give rise to tractable subproblems if forbidden as topological minors but not if forbidden as sub-patterns. Finally, we introduce the idea of augmented patterns that allows for the identification of more tractable classes, including all language restrictions of the binary CSP.

cs.CC↗

arXiv2018

On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

cs.CC↗

arXiv2017

The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns

Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic sub-instances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of simple partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes.

cs.CC↗

arXiv2017

Binarisation for Valued Constraint Satisfaction Problems

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bulin et al. [LMCS'15] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to Minimum-Cost Homomorphism Problems over a fixed digraph.

cs.DM↗