Techniques for Measuring the Loss of Inferential Strength in Knowledge Representation when Applying Forgetting Policies

Core Concepts

The core message of this paper is to define loss functions for measuring changes in inferential strength based on intuitions from model counting and probability theory when applying different forgetting policies in knowledge representation.

Abstract

The paper presents techniques for measuring the inferential strength of forgetting policies in knowledge representation. It starts by introducing the concepts of strong (standard) forgetting and weak forgetting, which provide upper and lower bounds on the inferential strength of a theory after forgetting certain symbols.
The paper then defines two types of loss measures:

Model counting-based loss measures: These measure the loss in inferential strength in terms of the number of models of the original theory versus the theories resulting from strong and weak forgetting.
Probabilistic-based loss measures: These generalize the model counting-based measures to allow for arbitrary probability distributions on the worlds (assignments of truth values to propositional variables).

The paper shows that these loss measures have desirable properties, such as being in the range [0, 1], being 0 when forgetting redundant variables, and being monotonic with respect to the number of variables forgotten.
The paper also presents a methodology for computing these loss measures using the probabilistic logic programming language PROBLOG. This involves transforming the original theory and the strong/weak forgetting theories into stratified logic programs, and then using PROBLOG's query mechanism to compute the relevant probabilities.
The techniques are first presented for the propositional case, and then generalized to the first-order case with some restrictions. The paper includes examples demonstrating the application of the theoretical results using PROBLOG.

Stats

There are 64 possible worlds (assignments of truth values to the 6 propositional variables in Tc).
There are 13 worlds that satisfy Tc, so P(Tc) = 13/64 = 0.203125.
After forgetting ecar and jcar, the probability of the resulting theory F NC(Tc; ecar, jcar) is 0.6875.

Quotes

"The goal of this paper is to define loss functions for measuring changes in inferential strength based on intuitions from model counting and probability theory."
"The basis for doing this will be to use intuitions from the area of model counting, both classical and approximate, and probability theory."

Key Insights Distilled From

Techniques for Measuring the Inferential Strength of Forgetting Policies

by Patrick Dohe... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02454.pdf

Deeper Inquiries

How could these loss measures be extended to handle more complex logical formalisms beyond propositional and first-order logic

To extend these loss measures to handle more complex logical formalisms beyond propositional and first-order logic, one could explore incorporating techniques from modal logic, temporal logic, or higher-order logic. For modal logic, one could consider introducing modal operators to capture notions of necessity and possibility, which could then be used to define loss measures based on changes in inferential strength. Temporal logic could be used to model reasoning over time, allowing for the evaluation of how forgetting policies impact reasoning across different time points. Higher-order logic could enable the representation of more intricate relationships between entities and concepts, leading to more nuanced loss measures that capture the impact of forgetting on complex logical structures.

What are some potential applications of these loss measures beyond knowledge representation, such as in areas like database query optimization or program analysis

These loss measures have the potential for various applications beyond knowledge representation. In database query optimization, the measures could be used to evaluate the impact of forgetting certain attributes or relations on query performance. By quantifying the loss in inferential strength, database systems could automatically determine the optimal set of attributes to forget in order to improve query processing efficiency without significantly compromising the accuracy of results. In program analysis, the measures could assist in identifying redundant or irrelevant information in codebases, aiding in program comprehension and optimization efforts. By selecting the most appropriate symbols to forget, developers could streamline codebases and enhance program efficiency.

How could these loss measures be used to automatically select the optimal set of symbols to forget in order to balance inferential strength and performance for a given application

These loss measures could be utilized to automatically select the optimal set of symbols to forget in various applications by employing a systematic evaluation process. One approach could involve defining a set of criteria or constraints based on the specific application requirements, such as performance thresholds or inferential accuracy targets. The loss measures could then be used to quantify the impact of forgetting different symbols on these criteria, allowing for the identification of the symbol set that strikes the best balance between inferential strength and performance. Automated algorithms could iteratively evaluate different symbol combinations and select the one that maximizes performance while minimizing the loss in inferential strength, providing a data-driven approach to symbol selection for forgetting policies.