From 042addc955daa6b38caef801ca4f87f13fe07ec0 Mon Sep 17 00:00:00 2001 From: troyyyyy Date: Wed, 29 Nov 2023 16:37:06 +0800 Subject: [PATCH] [DOC] add img, add abduction --- docs/Brief-Introduction/Usage.rst | 3 ++ docs/Overview/Abductive Learning.rst | 42 +++++++++++++++++++++------- 2 files changed, 35 insertions(+), 10 deletions(-) diff --git a/docs/Brief-Introduction/Usage.rst b/docs/Brief-Introduction/Usage.rst index dc256ab..672a48b 100644 --- a/docs/Brief-Introduction/Usage.rst +++ b/docs/Brief-Introduction/Usage.rst @@ -3,3 +3,6 @@ Use ABL-Package Step by Step .. contents:: Table of Contents +Intialize the Reasoning Part +---------------------------- + diff --git a/docs/Overview/Abductive Learning.rst b/docs/Overview/Abductive Learning.rst index f97c53d..9a4e4dc 100644 --- a/docs/Overview/Abductive Learning.rst +++ b/docs/Overview/Abductive Learning.rst @@ -1,15 +1,20 @@ Abductive Learning ================== +Integrating the Power of Machine Learning and Logical Reasoning +--------------------------------------------------------------- + Traditional supervised machine learning, e.g. classification, is -predominantly data-driven. 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 +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. -（可能加一张图，比如左边是ML，右边是ML+KB） +.. 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 @@ -30,21 +35,22 @@ The process of ABL is as follows: 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, returning - revised logical facts :math:`\Delta(\mathcal{O})` which are +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 for further learning of the classifier. -6. As a result, the classifier is updated and replaces the previous one + 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 process is repeated until the classifier is no longer updated, or +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: -一张图 +.. 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 @@ -57,4 +63,20 @@ 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.