Abductive Explanations Research Articles

Axiomatic Aggregations of Abductive Explanations

The recent criticisms of the robustness of post hoc model approximation explanation methods (like LIME and SHAP) have led to the rise of model-precise abductive explanations. For each data point, abductive explanations provide a minimal subset of features that are sufficient to generate the outcome. While theoretically sound and rigorous, abductive explanations suffer from a major issue --- there can be several valid abductive explanations for the same data point. In such cases, providing a single abductive explanation can be insufficient; on the other hand, providing all valid abductive explanations can be incomprehensible due to their size. In this work, we solve this issue by aggregating the many possible abductive explanations into feature importance scores. We propose three aggregation methods: two based on power indices from cooperative game theory and a third based on a well-known measure of causal strength. We characterize these three methods axiomatically, showing that each of them uniquely satisfies a set of desirable properties. We also evaluate them on multiple datasets and show that these explanations are robust to the attacks that fool SHAP and LIME.

Delivering Inflated Explanations

In the quest for Explainable Artificial Intelligence (XAI) one of the questions that frequently arises given a decision made by an AI system is, ``why was the decision made in this way?'' Formal approaches to explainability build a formal model of the AI system and use this to reason about the properties of the system. Given a set of feature values for an instance to be explained, and a resulting decision, a formal abductive explanation is a set of features, such that if they take the given value will always lead to the same decision. This explanation is useful, it shows that only some features were used in making the final decision. But it is narrow, it only shows that if the selected features take their given values the decision is unchanged. It is possible that some features may change values and still lead to the same decision. In this paper we formally define inflated explanations which is a set of features, and for each feature a set of values (always including the value of the instance being explained), such that the decision will remain unchanged, for any of the values allowed for any of the features in the (inflated) abductive explanation. Inflated formal explanations are more informative than common abductive explanations since e.g. they allow us to see if the exact value of a feature is important, or it could be any nearby value. Overall they allow us to better understand the role of each feature in the decision. We show that we can compute inflated explanations for not that much greater cost than abductive explanations, and that we can extend duality results for abductive explanations also to inflated explanations.

Explaining Random Forests Using Bipolar Argumentation and Markov Networks

Random forests are decision tree ensembles that can be used to solve a variety of machine learning problems. However, as the number of trees and their individual size can be large, their decision making process is often incomprehensible. We show that their decision process can be naturally represented as an argumentation problem, which allows creating global explanations via argumentative reasoning. We generalize sufficient and necessary argumentative explanations using a Markov network encoding, discuss the relevance of these explanations and establish relationships to families of abductive explanations from the literature. As the complexity of the explanation problems is high, we present an efficient approximation algorithm with probabilistic approximation guarantees.

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.

Explaining black-box classifiers: Properties and functions

Explaining black-box classification models is a hot topic in AI, with the overall goal of improving trust in decisions made by such models. Several works have been done and diverse functions have been proposed. However, their formal properties and links have not been sufficiently studied. This paper presents four contributions: The first consists of investigating global explanations of black-box classifiers. We provide a formal and unifying framework in which such explanations are defined from the whole feature space. The framework is based on two concepts, which are seen as two types of global explanations: arguments in favour of (or pro) predictions and arguments against (or con) predictions. The second contribution consists of defining various types of local explanations (abductive explanations, counterfactuals, contrastive explanations) from the whole feature space, investigating their properties, links and differences, and showing how they relate to global explanations. The third contribution consists of analysing and defining explanation functions that generate (global, local) abductive explanations from incomplete information (i.e., from a subset of the feature space). We start by proposing two desirable properties that an explainer would satisfy, namely success and coherence. The former ensures the existence of explanations while the latter ensures their correctness. We show that in the incomplete case, the two properties cannot be satisfied together. The fourth contribution consists of proposing two functions that generate abductive explanations and which satisfy coherence at the expense of success.

Tractability of explaining classifier decisions

Explaining decisions is at the heart of explainable AI. We investigate the computational complexity of providing a formally-correct and minimal explanation of a decision taken by a classifier. In the case of threshold (i.e. score-based) classifiers, we show that a complexity dichotomy follows from the complexity dichotomy for languages of cost functions. In particular, submodular classifiers allow tractable explanation of positive decisions, but not negative decisions (assuming P≠NP). This is an example of the possible asymmetry between the complexity of explaining positive and negative decisions of a particular classifier. Nevertheless, there are large families of classifiers for which explaining both positive and negative decisions is tractable, such as monotone or modular (e.g. linear) classifiers. We extend the characterisation of tractable cases to constrained classifiers (when there are constraints on the possible input vectors) and to the search for contrastive rather than abductive explanations. Indeed, we show that tractable classes coincide for abductive and contrastive explanations in the constrained or unconstrained settings. We show the intractability of returning a set of k diverse explanations even for linear classifiers and k=2. Finding a minimum-cardinality explanation is tractable for the family of modular classifiers, i.e. when the score function is the sum of unary functions, but becomes intractable when any non-modular function is also allowed.

Computing Abductive Explanations

We study the computation of constrained explanations in the framework of abductive logic programming. A general characteristic of abductive reasoning is the existence of multiple abductive explanations. Therefore, identifying a subclass of “preferred explanations” is a relevant problem. A typical approach is to “prefer” explanations that are, in some sense, simple. Several concepts of simplicity were considered in the literature, most notably those based on minimality with respect to inclusion and cardinality. We adopt, as a measure of the quality of an explanation, its degree of arbitrariness that can be briefly described as the number of arbitrary assumptions that have been made to derive the explanation. The more arbitrary the explanation, the less appealing it is, with explanations having no arbitrariness, called constrained, being the preferred ones. In this article, we present a technique that, for a special class of theories, computes constrained explanations. It is based on a rewriting of the theory and the observation into a disjunctive logic program with negation so that the constrained explanations correspond to a subset of its stable models. The proposed technique lays the foundation for using ASP solvers to compute constrained explanations.

On the explanatory power of Boolean decision trees

Decision trees have long been recognized as models of choice in sensitive applications where interpretability is of paramount importance. In this paper, we examine the computational ability of Boolean decision trees for the explanation purpose. We focus on both abductive explanations (suited to explaining why a given instance has been classified as such by the decision tree at hand) and on contrastive explanations (suited to explaining why a given instance has not been classified by the decision tree as it was expected). More precisely, we are interested in deriving, minimizing, and counting abductive explanations and contrastive explanations. We prove that the set of all irredundant abductive explanations (also known as PI-explanations or sufficient reasons) for an instance given a decision tree can be exponentially larger than the size of the input (the instance and the decision tree). Therefore, generating the full set of sufficient reasons for an instance can be out of reach. In addition, deriving a single sufficient reason, though computationally easy when dealing with decision trees, does not prove enough in general; indeed, two sufficient reasons for the same instance may differ on many features. To deal with this issue and generate synthetic views of the set of all sufficient reasons, we define notions of relevant features and of necessary features that characterize the (possibly negated) features appearing in at least one or in every sufficient reason for an instance, and we show that they can be computed in polynomial time. We also introduce the notion of explanatory importance, that indicates how frequent each (possibly negated) feature is in the set of all sufficient reasons. We show how the explanatory importance of a (possibly negated) feature and the number of sufficient reasons for an instance can be obtained via a model counting operation, which turns out to be practical in many cases. We also explain how to enumerate minimum-size sufficient reasons. We finally show that, unlike sufficient reasons, the set of all contrastive explanations for an instance given a decision tree can be derived, minimized and counted in polynomial time.

On the (Complete) Reasons Behind Decisions

Recent work has shown that the input-output behavior of some common machine learning classifiers can be captured in symbolic form, allowing one to reason about the behavior of these classifiers using symbolic techniques. This includes explaining decisions, measuring robustness, and proving formal properties of machine learning classifiers by reasoning about the corresponding symbolic classifiers. In this work, we present a theory for unveiling the reasons behind the decisions made by Boolean classifiers and study some of its theoretical and practical implications. At the core of our theory is the notion of a complete reason, which can be viewed as a necessary and sufficient condition for why a decision was made. We show how the complete reason can be used for computing notions such as sufficient reasons (also known as PI-explanations and abductive explanations), how it can be used for determining decision and classifier bias and how it can be used for evaluating counterfactual statements such as “a decision will stick even if ...because ... .” We present a linear-time algorithm for computing the complete reasoning behind a decision, assuming the classifier is represented by a Boolean circuit of appropriate form. We then show how the computed complete reason can be used to answer many queries about a decision in linear or polynomial time. We finally conclude with a case study that illustrates the various notions and techniques we introduced.

On the Computation of Necessary and Sufficient Explanations

The complete reason behind a decision is a Boolean formula that characterizes why the decision was made. This recently introduced notion has a number of applications, which include generating explanations, detecting decision bias and evaluating counterfactual queries. Prime implicants of the complete reason are known as sufficient reasons for the decision and they correspond to what is known as PI explanations and abductive explanations. In this paper, we refer to the prime implicates of a complete reason as necessary reasons for the decision. We justify this terminology semantically and show that necessary reasons correspond to what is known as contrastive explanations. We also study the computation of complete reasons for multi-class decision trees and graphs with nominal and numeric features for which we derive efficient, closed-form complete reasons. We further investigate the computation of shortest necessary and sufficient reasons for a broad class of complete reasons, which include the derived closed forms and the complete reasons for Sentential Decision Diagrams (SDDs). We provide an algorithm which can enumerate their shortest necessary reasons in output polynomial time. Enumerating shortest sufficient reasons for this class of complete reasons is hard even for a single reason. For this problem, we provide an algorithm that appears to be quite efficient as we show empirically.

Trading Complexity for Sparsity in Random Forest Explanations

Random forests have long been considered as powerful model ensembles in machine learning. By training multiple decision trees, whose diversity is fostered through data and feature subsampling, the resulting random forest can lead to more stable and reliable predictions than a single decision tree. This however comes at the cost of decreased interpretability: while decision trees are often easily interpretable, the predictions made by random forests are much more difficult to understand, as they involve a majority vote over multiple decision trees. In this paper, we examine different types of reasons that explain "why" an input instance is classified as positive or negative by a Boolean random forest. Notably, as an alternative to prime-implicant explanations taking the form of subset-minimal implicants of the random forest, we introduce majoritary reasons which are subset-minimal implicants of a strict majority of decision trees. For these abductive explanations, the tractability of the generation problem (finding one reason) and the optimization problem (finding one minimum-sized reason) are investigated. Unlike prime-implicant explanations, majoritary reasons may contain redundant features. However, in practice, prime-implicant explanations - for which the identification problem is DP-complete - are slightly larger than majoritary reasons that can be generated using a simple linear-time greedy algorithm. They are also significantly larger than minimum-sized majoritary reasons which can be approached using an anytime Partial MaxSAT algorithm.

Firm-Induced Migration Paths and Strategic Human-Capital Outcomes

Firm-induced migration typically entails firms relocating workers to fill value-creating positions at destination locations. But such relocated workers are often exposed to external employment opportunities at their destinations, possibly triggering turnover. We conceptualize the firm-induced migration path, consisting of the relocated workers’ place of origin and destination, as relevant in determining worker performance and turnover postrelocation. Using a unique data set from a large Indian technology firm that hires talent from both large cities and smaller towns, we document robust econometric patterns by exploiting the firm’s randomized assignment of workers to production centers across the country. These production centers are located in the largest technology cluster in India (Bangalore), smaller technology clusters, and noncluster locations. We find that the firm-induced migration path shapes both worker performance and turnover. Compared with workers from large cities, workers from smaller towns achieve higher performance when relocated to Bangalore than to other production centers, but are also more likely to join competing firms. Fine-grained data on employment and human-capital-augmentation opportunities at workers’ destination locations, and on socioeconomic conditions in workers’ places of origin, help us rule in an abductive explanation: across firm-induced migration paths, differences in external labor-market opportunities between workers’ places of origin and their destinations, as well as intrafirm skill-development opportunities at the destination, are related to heterogeneous human-capital outcomes. This paper was accepted by Alfonso Gambardella, business strategy. Supplemental Material: The e-companion and data files are available at https://doi.org/10.1287/mnsc.2022.4361 .

Abductive Inference Method in Problems of Explaining the Observed

The problems of artificial intelligence, control and decision-making with incomplete or unreliable information include a wide class of problems of abductive explanation of the observed, including cause–effect problems. The study is devoted to the substantiation of the method of logical formation of hypotheses explaining the observed. Means of knowledge representation and derivation of hypotheses are proposed. A language possessing the property of substitutability is introduced. The properties of language and the calculi introduced in it provide hypothesizing by combining deduction and abduction. In contrast to the well-known logical methods of abduction, the proposed techniques make it possible to derive hypotheses (minorants) that are necessary and sufficient for a formal explanation of the observed. Based on minorants in combination with the basic theory of the subject area, reliable causes of the observed or relevant circumstances leading to these causes are formed. In this case, in situations with the availability of empirical data, these causes and circumstances can also be formed in plausible versions. Examples from technology and medicine are considered.

Virtual Watercoolers: A Field Experiment on Virtual Synchronous Interactions and Performance of Organizational Newcomers

Do virtual, yet informal and synchronous, interactions affect individual performance outcomes of organizational newcomers? We report results from a randomized field experiment conducted at a large global organization that estimates the performance effects of “virtual water coolers” for remote interns participating in the firm’s flagship summer internship program. Findings indicate that interns who had randomized opportunities to interact synchronously and informally with senior managers were significantly more likely to receive offers for full-time employment, achieved higher weekly performance ratings, and had more positive attitudes toward their remote internships. Further, we observed stronger results when the interns and senior managers were demographically similar. Secondary results also hint at a possible abductive explanation of the performance effects: virtual watercoolers between interns and senior managers may have facilitated knowledge and advice sharing. This study demonstrates that hosting brief virtual water cooler sessions with senior managers might have job and career benefits for organizational newcomers working in remote workplaces, an insight with immediate managerial relevance.

Indefinite abductive explanations

ABSTRACTThis paper stems from previous works of the authors in which a new measure of the simplicity of an explanation based on its degree of arbitrariness is proposed: The more the explanation is arbitray, the less appealing it is, with explanations having no arbitrariness – called constrained – being the preferred ones. In previous works, as commonly done in the literature of abductive logic programming, a set of hypotheses is not an explanation, unless it is definite, i.e. it explains all the data belonging to the observation. In this paper, we follow a different perspective and define the concept of indefinite constrained explanations, i.e. constrained explanations that are not definite, but admit some indefiniteness. An indefinite constrained explanation captures the intuition of the existence of an explanation (indefinite explanation) that would best explain the given evidence, while not making arbitrary choices (constrained explanation). The main contribution of the paper is the study of abduction in indefinite deductive theories: specifically, the paper studies the framework of abductive logic programming extended with integrity constraints in the setting in which both the initial knowledge base and the abductive explanations are indefinite (may contain occurrences of null values) and the domain is possibly infinite. Furthermore, the paper discusses the complexity of problems concerning indefinite (constrained) explanations.

Obligation as Optimal Goal Satisfaction

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation Oq is a logical consequence of the obligations Op and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation Op is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.

On the complexity of second-best abductive explanations

• Formal definition of second-best abductive solutions. • Complexity depends on the ordering: is complete for Dp or Π 2 . • Checking solution and relevance of hypotheses. When looking for a propositional abductive explanation of a given set of manifestations, an ordering between possible solutions is often assumed. While the complexity of computing optimal solutions is already known, in this paper we consider second-best solutions with respect to different orderings, and different definitions of what a second-best solution is: an optimal solution not already found, or a solution that is optimal among the ones not previously found.

A Measure of Arbitrariness in Abductive Explanations

AbstractWe study the framework of abductive logic programming extended with integrity constraints. For this framework, we introduce a new measure of the simplicity of an explanation based on its degree of arbitrariness: the more arbitrary the explanation, the less appealing it is, with explanations having no arbitrariness — they are called constrained — being the preferred ones. In the paper, we study basic properties of constrained explanations. For the case when programs in abductive theories are stratified we establish results providing a detailed picture of the complexity of the problem to decide whether constrained explanations exist.

Generation and Selection of Abductive Explanations for Non-Omniscient Agents

Among the non-monotonic reasoning processes, abduction is one of the most important. Usually described as the process of looking for explanations, it has been recognized as one of the most commonly used in our daily activities. Still, the traditional definitions of an abductive problem and an abductive solution mention only theories and formulas, leaving agency out of the picture. Our work proposes a study of abductive reasoning from an epistemic and dynamic perspective. In the first part we explore syntactic definitions of both an abductive problem in terms of an agent's information and an abductive solution in terms of the actions that modify the agent's information. We look at diverse kinds of agents, including not only omniscient ones but also those whose information is not closed under logical consequence and those whose reasoning abilities are not complete. In the second part, we look at an existing logical framework whose semantic model allows us to interpret the previously stated formulas, and we define two actions that represent forms of abductive reasoning.

Fuzzy Equilibrium Logic

In this article, we introduce fuzzy equilibrium logic as a generalization of both Pearce equilibrium logic and fuzzy answer set programming. The resulting framework combines the capability of equilibrium logic to declaratively specify search problems, with the capability of fuzzy logics to model continuous domains. We show that our fuzzy equilibrium logic is a proper generalization of both Pearce equilibrium logic and fuzzy answer set programming, and we locate the computational complexity of the main reasoning tasks at the second level of the polynomial hierarchy. We then provide a reduction from the problem of finding fuzzy equilibrium logic models to the problem of solving a particular bilevel mixed integer program (biMIP), allowing us to implement reasoners by reusing existing work from the operations research community. To illustrate the usefulness of our framework from a theoretical perspective, we show that a well-known characterization of strong equivalence in Pearce equilibrium logic generalizes to our setting, yielding a practical method to verify whether two fuzzy answer set programs are strongly equivalent. Finally, to illustrate its application potential, we show how fuzzy equilibrium logic can be used to find strong Nash equilibria, even when players have a continuum of strategies at their disposal. As a second application example, we show how to find abductive explanations from Łukasiewicz logic theories.