# Explanations for Monotonic Classifiers

arXiv: https://arxiv.org/abs/2106.00154 · HTML: https://mlsanity.com/paper/2106.00154/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 57 references that are in the map.
- First posted on arXiv 1 Jun 2021.

## Builds on (most cited first, 22 of 22)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Abduction-Based Explanations for Machine Learning Models](https://mlsanity.com/paper/1811.10656/) (2018)
- [A Symbolic Approach to Explaining Bayesian Network Classifiers](https://mlsanity.com/paper/1805.03364/) (2018)
- [On the Tractability of SHAP Explanations](https://mlsanity.com/paper/2009.08634/) (2020)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Explaining Naive Bayes and Other Linear Classifiers with Polynomial Time and Delay](https://mlsanity.com/paper/2008.05803/) (2020)
- [On Explaining Random Forests with SAT](https://mlsanity.com/paper/2105.10278/) (2021)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [On Explaining Decision Trees](https://mlsanity.com/paper/2010.11034/) (2020)
- [On the Computational Intelligibility of Boolean Classifiers](https://mlsanity.com/paper/2104.06172/) (2021)
- [SAT-Based Rigorous Explanations for Decision Lists](https://mlsanity.com/paper/2105.06782/) (2021)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [On Tractable Representations of Binary Neural Networks](https://mlsanity.com/paper/2004.02082/) (2020)
- [Counterexample-Guided Learning of Monotonic Neural Networks](https://mlsanity.com/paper/2006.08852/) (2020)
- [Monotonic classification: an overview on algorithms, performance measures and data sets](https://mlsanity.com/paper/1811.07155/) (2018)
- [Deontological Ethics By Monotonicity Shape Constraints](https://mlsanity.com/paper/2001.11990/) (2020)
- [On Symbolically Encoding the Behavior of Random Forests](https://mlsanity.com/paper/2007.01493/) (2020)
- [On Relating 'Why?' and 'Why Not?' Explanations](https://mlsanity.com/paper/2012.11067/) (2020)
- [Prediction Rule Reshaping](https://mlsanity.com/paper/1805.06439/) (2018)
- [Sufficient reasons for classifier decisions in the presence of constraints](https://mlsanity.com/paper/2105.06001/) (2021)

## Built on by (most cited first, 30 of 39)

- [Logic-Based Explainability in Machine Learning](https://mlsanity.com/paper/2211.00541/) (2022)
- [On Tackling Explanation Redundancy in Decision Trees](https://mlsanity.com/paper/2205.09971/) (2022)
- [On Efficiently Explaining Graph-Based Classifiers](https://mlsanity.com/paper/2106.01350/) (2021)
- [Defining and Quantifying the Emergence of Sparse Concepts in DNNs](https://mlsanity.com/paper/2111.06206/) (2021)
- [On Computing Probabilistic Explanations for Decision Trees](https://mlsanity.com/paper/2207.12213/) (2022)
- [The Inadequacy of Shapley Values for Explainability](https://mlsanity.com/paper/2302.08160/) (2023)
- [Eliminating The Impossible, Whatever Remains Must Be True](https://mlsanity.com/paper/2206.09551/) (2022)
- [Feature Necessity & Relevancy in ML Classifier Explanations](https://mlsanity.com/paper/2210.15675/) (2022)
- [Local vs. Global Interpretability: A Computational Complexity Perspective](https://mlsanity.com/paper/2406.02981/) (2024)
- [On Computing Probabilistic Abductive Explanations](https://mlsanity.com/paper/2212.05990/) (2022)
- [Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory](https://mlsanity.com/paper/2405.05369/) (2024)
- [Explaining, Fast and Slow: Abstraction and Refinement of Provable Explanations](https://mlsanity.com/paper/2506.08505/) (2025)
- [Disproving XAI Myths with Formal Methods -- Initial Results](https://mlsanity.com/paper/2306.01744/) (2023)
- [Explain Yourself, Briefly! Self-Explaining Neural Networks with Concise Sufficient Reasons](https://mlsanity.com/paper/2502.03391/) (2025)
- [On Formal Feature Attribution and Its Approximation](https://mlsanity.com/paper/2307.03380/) (2023)
- [Hard to Explain: On the Computational Hardness of In-Distribution Model Interpretation](https://mlsanity.com/paper/2408.03915/) (2024)
- [Formal Mechanistic Interpretability: Automated Circuit Discovery with Provable Guarantees](https://mlsanity.com/paper/2602.16823/) (2026)
- [Axiomatic Aggregations of Abductive Explanations](https://mlsanity.com/paper/2310.03131/) (2023)
- [Explaining k-Nearest Neighbors: Abductive and Counterfactual Explanations](https://mlsanity.com/paper/2501.06078/) (2025)
- [Anytime Approximate Formal Feature Attribution](https://mlsanity.com/paper/2312.06973/) (2023)
- [Cardinality-Minimal Explanations for Monotonic Neural Networks](https://mlsanity.com/paper/2205.09901/) (2022)
- [A New Class of Explanations for Classifiers with Non-Binary Features](https://mlsanity.com/paper/2304.14760/) (2023)
- [On Computing Relevant Features for Explaining NBCs](https://mlsanity.com/paper/2207.04748/) (2022)
- [On the Complexity of Enumerating Prime Implicants from Decision-DNNF Circuits](https://mlsanity.com/paper/2301.13328/) (2023)
- [Sufficient, Necessary and Complete Causal Explanations in Image Classification](https://mlsanity.com/paper/2507.23497/) (2025)
- [Faster Verified Explanations for Neural Networks](https://mlsanity.com/paper/2512.00164/) (2025)
- [FAME: Formal Abstract Minimal Explanation for Neural Networks](https://mlsanity.com/paper/2603.10661/) (2026)
- [Unifying Formal Explanations: A Complexity-Theoretic Perspective](https://mlsanity.com/paper/2602.18160/) (2026)
- [Finding Minimum-Cost Explanations for Predictions made by Tree Ensembles](https://mlsanity.com/paper/2303.09271/) (2023)
- [Uncovering Bugs in Formal Explainers: A Case Study with PyXAI](https://mlsanity.com/paper/2511.03169/) (2025)

## Abstract (arXiv)

In many classification tasks there is a requirement of monotonicity. Concretely, if all else remains constant, increasing (resp. decreasing) the value of one or more features must not decrease (resp. increase) the value of the prediction. Despite comprehensive efforts on learning monotonic classifiers, dedicated approaches for explaining monotonic classifiers are scarce and classifier-specific. This paper describes novel algorithms for the computation of one formal explanation of a (black-box) monotonic classifier. These novel algorithms are polynomial in the run time complexity of the classifier and the number of features. Furthermore, the paper presents a practically efficient model-agnostic algorithm for enumerating formal explanations.