Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Algebraic Approach to Weighted Answer-set Programming

Francisco Coelho, Bruno Dinis, Dietmar Seipel, Salvador Abreu

TL;DR

This work presents an algebraic framework for propagating weights in Answer-set Programs by introducing WASP, an ASP with weighted facts. It shifts weight propagation from the syntax to the semantic space of total choices, stable models, and events, using an equivalence relation on events to handle non-determinism with algebraic parameters $\theta_{s,t}$. Weights are coherently propagated from total choices to stable models and then to events, enabling event-level probabilities after normalization and supporting posterior data-driven refinement. The approach highlights the separation between syntactic total-choice-based distributions and semantic event-based distributions, and it outlines pathways to connect with Bayesian/network-based methods and real-world data for model selection and scoring.

Abstract

Logic programs, more specifically, Answer-set programs, can be annotated with probabilities on facts to express uncertainty. We address the problem of propagating weight annotations on facts (eg probabilities) of an ASP to its standard models, and from there to events (defined as sets of atoms) in a dataset over the program's domain. We propose a novel approach which is algebraic in the sense that it relies on an equivalence relation over the set of events. Uncertainty is then described as polynomial expressions over variables. We propagate the weight function in the space of models and events, rather than doing so within the syntax of the program. As evidence that our approach is sound, we show that certain facts behave as expected. Our approach allows us to investigate weight annotated programs and to determine how suitable a given one is for modeling a given dataset containing events.

An Algebraic Approach to Weighted Answer-set Programming

TL;DR

This work presents an algebraic framework for propagating weights in Answer-set Programs by introducing WASP, an ASP with weighted facts. It shifts weight propagation from the syntax to the semantic space of total choices, stable models, and events, using an equivalence relation on events to handle non-determinism with algebraic parameters . Weights are coherently propagated from total choices to stable models and then to events, enabling event-level probabilities after normalization and supporting posterior data-driven refinement. The approach highlights the separation between syntactic total-choice-based distributions and semantic event-based distributions, and it outlines pathways to connect with Bayesian/network-based methods and real-world data for model selection and scoring.

Abstract

Logic programs, more specifically, Answer-set programs, can be annotated with probabilities on facts to express uncertainty. We address the problem of propagating weight annotations on facts (eg probabilities) of an ASP to its standard models, and from there to events (defined as sets of atoms) in a dataset over the program's domain. We propose a novel approach which is algebraic in the sense that it relies on an equivalence relation over the set of events. Uncertainty is then described as polynomial expressions over variables. We propagate the weight function in the space of models and events, rather than doing so within the syntax of the program. As evidence that our approach is sound, we show that certain facts behave as expected. Our approach allows us to investigate weight annotated programs and to determine how suitable a given one is for modeling a given dataset containing events.

Paper Structure

This paper contains 17 sections, 4 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. About Propagating Weights from TC.
  4. Syntax and Semantics of Weighted ASP
  5. Syntax.
  6. Semantics.
  7. Evaluation without Grounding.
  8. Derived Program.
  9. Events.
  10. Propagation of Weights.
  11. Propagating Weights
  12. An Equivalence Relation
  13. From Total Choices to Events
  14. Inconsistent class.
  15. Consistent classes.
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\lambda$ be the empty set event (notice that $\lambda = \emptyset \in {\mathcal{E}}$) We adopt the notation '$\lambda$' for empty word, from formal languages, to distinguish '$\emptyset \in {\mathcal{E}}$' from '$\emptyset \subset{\mathcal{E}}$'., and $\Lambda$ the consequence class of (consist

Figures (3)

  • Figure 1: This (partial sub-/super-set) diagram shows some events related to the SM of the program $P_{\mathrm{1}}$. The circle nodes are TC and shaded nodes are SM. Solid lines represent relations with the SM and dashed lines some sub-/super-set relations with other events. The set of events contained in all SM, denoted by $\Lambda$, is $\set{ \lambda }$ in this example, because $\overline{a} \cap ab \cap ac = \emptyset = \lambda$.
  • Figure 2: Classes of (consistent) events related to the SM of $P_{\mathrm{1}}$ are defined through sub-/super-set relations. In this picture we can see, for example, that $\set{\overline{c}ab, ab, b}$ and $\set{a, abc}$ are part of different classes, represented by different fillings. As before, the circle nodes are TC and shaded nodes are SM. Notice that $bc$ is not in a filled area.
  • Figure 3: Lattice of the SC from \ref{['ex:fruitful']}. In this diagram the nodes are the different SC that result from the SM, plus the inconsistent class ($\bot$). The bottom node ($\Diamond$) is the class of independent events, those that have no sub-/super-set relation with any SM and the top node ($\Lambda$) represents events related with all the SM i.e. the consequences of the program. As in previous diagrams, shaded nodes represent the SM.

Theorems & Definitions (11)

  • Example 1
  • Definition 1: Equivalence Relation on Events
  • Proposition 1: Class of the Program's Consequences
  • proof
  • Proposition 2: Two Distributions
  • proof
  • Theorem 4.1: Weight of Certain Facts
  • proof
  • Corollary 1: Probability of Certain Facts
  • proof
  • ...and 1 more