[Paper Review] Weight Learning in a Probabilistic Extension of Answer Set Programs

Joohyung Lee, Wang Yi|arXiv (Cornell University)|Aug 14, 2018

Logic, Reasoning, and Knowledge16 references3 citations

TL;DR

This paper introduces weight learning in ${\rm LP}^{\rm{MLN}}$, a probabilistic extension of answer set programming, using gradient ascent and MC-ASP for MCMC sampling via answer set solvers. It enables probabilistic reasoning in knowledge-rich domains like action planning and abductive diagnosis, with learned weights reflecting failure probabilities from observed action histories.

ABSTRACT

LPMLN is a probabilistic extension of answer set programs with the weight scheme derived from that of Markov Logic. Previous work has shown how inference in LPMLN can be achieved. In this paper, we present the concept of weight learning in LPMLN and learning algorithms for LPMLN derived from those for Markov Logic. We also present a prototype implementation that uses answer set solvers for learning as well as some example domains that illustrate distinct features of LPMLN learning. Learning in LPMLN is in accordance with the stable model semantics, thereby it learns parameters for probabilistic extensions of knowledge-rich domains where answer set programming has shown to be useful but limited to the deterministic case, such as reachability analysis and reasoning about actions in dynamic domains. We also apply the method to learn the parameters for probabilistic abductive reasoning about actions.

Motivation & Objective

• To enable parameter learning in ${\rm LP}^{\rm{MLN}}$ programs, extending deterministic answer set programming to probabilistic inference.
• To address the challenge of assigning weights manually in complex ${\rm LP}^{\rm{MLN}}$ programs by learning them from observed data.
• To adapt Markov Logic Network learning techniques—particularly gradient ascent and MCMC sampling—to the stable model semantics of ${\rm LP}^{\rm{MLN}}$.
• To demonstrate the applicability of ${\rm LP}^{\rm{MLN}}$ learning in domains with incomplete or partially observed data, such as action failure diagnosis.
• To integrate probabilistic abductive reasoning into dynamic systems by learning failure probabilities from historical action traces.

Proposed method

• Adapts gradient ascent for Maximum Likelihood Estimation (MLE) to learn rule weights in ${\rm LP}^{\rm{MLN}}$ by maximizing the likelihood of observed stable models.
• Introduces MC-ASP, a Markov Chain Monte Carlo (MCMC) sampler that uses answer set solvers (ASP) instead of SAT solvers, based on a penalty-based formulation of ${\rm LP}^{\rm{MLN}}$.
• Employs a uniform sampler as a black-box component in MCMC to generate candidate models, with acceptance probabilities based on the penalty of false rules.
• Applies the learning framework to both single and multiple stable model settings, and extends it to handle incomplete interpretations.
• Uses the transformation $ P(\text{abnormality}) = \frac{\exp(-w)}{\exp(-w) + 1} $ to convert learned weights into failure probabilities.
• Integrates with existing tools via reductions: ${\rm LP}^{\rm{MLN}}$ learning is shown to generalize both ProbLog and Markov Logic learning as special cases.

Experimental results

Research questions

• RQ1 How can weight learning be formalized in ${\rm LP}^{\rm{MLN}}$ under the stable model semantics to support probabilistic reasoning in knowledge-rich domains?
• RQ2 Can MCMC sampling techniques from Markov Logic be adapted to ${\rm LP}^{\rm{MLN}}$ using answer set solvers, and how does MC-ASP differ from MC-SAT?
• RQ3 How can ${\rm LP}^{\rm{MLN}}$ learning handle partially observed data, such as incomplete action traces with missing fluents or abnormal conditions?
• RQ4 Can the framework support probabilistic abductive reasoning by learning failure probabilities from action execution histories?
• RQ5 What is the relationship between ${\rm LP}^{\rm{MLN}}$ weight learning and existing probabilistic logic learning frameworks like ProbLog and Markov Logic?

Key findings

• The learned weights for action failure rules were -1.084 for enter_failed, -1.064 for drop_book, and -0.068 for pickup_failed, yielding failure probabilities of approximately 0.253, 0.257, and 0.483, respectively.
• MC-ASP enables MCMC sampling in ${\rm LP}^{\rm{MLN}}$ using ASP solvers, overcoming the limitations of ergodicity in traditional MCMC due to deterministic dependencies.
• The system correctly identifies the most probable cause of failure: pickup_failed when the book is not at r2, and enter_failed when the robot is also not at r2.
• The framework supports learning from incomplete interpretations, allowing inference of missing fluents and abnormal conditions based on stable model semantics.
• The prototype implementation successfully learns parameters for abductive reasoning in dynamic systems, demonstrating the method’s utility in real-world diagnostic scenarios.
• The approach generalizes both ProbLog and Markov Logic learning, enabling reuse of existing tools and algorithms for specific subcases of ${\rm LP}^{\rm{MLN}}$.