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A Logic of "Black Box" Classifier Systems

Xinghan Liu, Emiliano Lorini

TL;DR

This paper provides a product modal logic called PLC (Product modal Logic for binary input Classifier) in which the notion of "black box" is interpreted as the uncertainty over a set of classifiers.

Abstract

Binary classifiers are traditionally studied by propositional logic (PL). PL can only represent them as white boxes, under the assumption that the underlying Boolean function is fully known. Binary classifiers used in practical applications and trained by machine learning are however opaque. They are usually described as black boxes. In this paper, we provide a product modal logic called PLC (Product modal Logic for binary input Classifier) in which the notion of "black box" is interpreted as the uncertainty over a set of classifiers. We give results about axiomatics and complexity of satisfiability checking for our logic. Moreover, we present a dynamic extension in which the process of acquiring new information about the actual classifier can be represented.

A Logic of "Black Box" Classifier Systems

TL;DR

This paper provides a product modal logic called PLC (Product modal Logic for binary input Classifier) in which the notion of "black box" is interpreted as the uncertainty over a set of classifiers.

Abstract

Binary classifiers are traditionally studied by propositional logic (PL). PL can only represent them as white boxes, under the assumption that the underlying Boolean function is fully known. Binary classifiers used in practical applications and trained by machine learning are however opaque. They are usually described as black boxes. In this paper, we provide a product modal logic called PLC (Product modal Logic for binary input Classifier) in which the notion of "black box" is interpreted as the uncertainty over a set of classifiers. We give results about axiomatics and complexity of satisfiability checking for our logic. Moreover, we present a dynamic extension in which the process of acquiring new information about the actual classifier can be represented.
Paper Structure (12 sections, 12 theorems, 22 equations, 1 figure, 2 tables)

This paper contains 12 sections, 12 theorems, 22 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Language and Semantics
  3. Axiomatics and Complexity
  4. Alternative Kripke Semantics
  5. Finite-Variable Case
  6. Infinite-Variable Case
  7. Complexity Results
  8. Application
  9. An Example of Classification Task
  10. Explanations
  11. Dynamic Extension
  12. Conclusion

Key Result

theorem 1

Let $\varphi \in \mathcal{L}$. Then, $\varphi$ is satisfiable relative to the class $\mathbf{MCM}$ if and only if it is satisfiable relative to the class $\mathbf{MDM}$.

Figures (1)

  • Figure 1: A classifier associating color labels in $\{\textrm{red, yellow, green}\}$ to input instances. We do not know its Boolean formula, since $f_0, f_1, f_2, f_3$ are all compatible with our partial knowledge of it. However, we know that the two input instances $s_0$ and $s_3$ are both classified as green.

Theorems & Definitions (25)

  • definition 1
  • definition 2: Satisfaction relation
  • definition 3
  • definition 4: Satisfaction Relation
  • theorem 1
  • proof
  • definition 5: Logic $\text{$\textsf{PLC}$}$
  • theorem 2
  • corollary 1
  • definition 6: Logic $\textsf{WPLC}$
  • ...and 15 more