ABLkit/docs/Overview/Abductive Learning.rst

Abductive Learning
==================

Integrating the Power of Machine Learning and Logical Reasoning
---------------------------------------------------------------

Traditional supervised machine learning, e.g. classification, is
predominantly data-driven, as shown in the below figure. 
Here, a set of training examples :math:`\left\{\left(x_1, y_1\right), 
\ldots,\left(x_m, y_m\right)\right\}` is given, 
where :math:`x_i \in \mathcal{X}` is the :math:`i`-th training
instance, :math:`y_i \in \mathcal{Y}` is the corresponding ground-truth
label. These data are then used to train a classifier model :math:`f:
\mathcal{X} \mapsto \mathcal{Y}` to accurately predict the unseen data.

.. image:: ../img/ML.jpg
   :width: 600px

In **Abductive Learning (ABL)**, we assume that, in addition to data as
examples, there is also a knowledge base :math:`\mathcal{KB}` containing
domain knowledge at our disposal. We aim for the classifier :math:`f:
\mathcal{X} \mapsto \mathcal{Y}` to make correct predictions on unseen 
data, and meanwhile, the logical facts grounded by
:math:`\left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\}`
should be compatible with :math:`\mathcal{KB}`.

The process of ABL is as follows:

1. Upon receiving data inputs :math:`\left\{x_1,\dots,x_m\right\}`,
   pseudo-labels
   :math:`\left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\}`
   are predicted by a data-driven classifier model.
2. These pseudo-labels are then converted into logical facts
   :math:`\mathcal{O}` that are acceptable for logical reasoning.
3. Conduct joint reasoning with :math:`\mathcal{KB}` to find any
   inconsistencies. If found, the logical facts that lead to minimal 
   inconsistency can be identified.
4. Modify the identified facts through **abductive reasoning** (or, **abduction**), 
   returning revised logical facts :math:`\Delta(\mathcal{O})` which are
   compatible with :math:`\mathcal{KB}`.
5. These revised logical facts are converted back to the form of
   pseudo-labels, and used like ground-truth labels in conventional 
   supervised learning to train a new classifier.
6. The new classifier will then be adopted to replace the previous one
   in the next iteration.

This above process repeats until the classifier is no longer updated, or
the logical facts :math:`\mathcal{O}` are compatible with the knowledge
base.

The following figure illustrates this process:

.. image:: ../img/ABL.jpg

We can observe that in the above figure, the left half involves machine
learning, while the right half involves logical reasoning. Thus, the
entire abductive learning process is a continuous cycle of machine
learning and logical reasoning. This effectively forms a paradigm that
is dual-driven by both data and domain knowledge, integrating and
balancing the use of machine learning and logical reasoning in a unified
model.

What is Abductive Reasoning?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Abductive reasoning, also known as abduction, refers to the process of
selectively inferring certain facts and hypotheses that explain
phenomena and observations based on background knowledge. Unlike
deductive reasoning, which leads to definitive conclusions, abductive
reasoning may arrive at conclusions that are plausible but not conclusively
proven. It is often described as an ‘inference to the best explanation.’

In Abductive Learning, given :math:`\mathcal{KB}` (typically expressed
in first-order logic clauses), one can perform deductive reasoning as
well as abductive reasoning. Deductive reasoning allows deriving
:math:`b` from :math:`a` only where :math:`b` is a formal logical
consequence of :math:`a`, while abductive reasoning allows inferring
:math:`a` as an explanation of :math:`b` (as a result of this inference,
abduction allows the precondition :math:`a` to be abducted from the
consequence :math:`b`). Put simply, deductive reasoning and abductive
reasoning differ in which end, left or right, of the proposition
“:math:`a\models b`” serves as conclusion.