Abstract
This tutorial presents a statistical relational extension of the answer set programming language called \(\textrm{LP}^\mathrm{{MLN}}\), which incorporates the concept of weighted rules into the stable model semantics following the log-linear models of Markov Logic. An \(\textrm{LP}^\mathrm{{MLN}}\) program defines a probability distribution over “soft” stable models, which may not satisfy all rules, but the more rules with larger weights they satisfy, the higher their probabilities, thus allowing for an intuitive and elaboration tolerant representation of problems that require both logical and probabilistic reasoning. The extension provides a natural way to overcome the deterministic nature of the stable model semantics, such as resolving inconsistencies in answer set programs, associating probability to stable models, and applying statistical inference and learning with probabilistic stable models. We also present formal relations between \(\textrm{LP}^\mathrm{{MLN}}\) and other related formalisms, which produce ways of performing inference and learning in \(\textrm{LP}^\mathrm{{MLN}}\).
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Notes
- 1.
It is straightforward to extend \(\textrm{LP}^\mathrm{{MLN}}\) to allow function constants of positive arity as long as the program is finitely groundable.
- 2.
- 3.
- 4.
Note that although any local maximum is a global maximum for the log-likelihood function, there can be multiple combinations of weights that achieve the maximum probability of the training data.
- 5.
A Markov chain is ergodic if there is a number m such that any state can be reached from any other state in any number of steps greater than or equal to m.
Detailed balance means \(P_{\varPi }(X)Q(X\rightarrow Y) = P_{\varPi }(Y)Q(Y\rightarrow X)\) for any samples X and Y, where \(Q(X\rightarrow Y)\) denotes the probability that the next sample is Y given that the current sample is X.
- 6.
Note that \(\varPi ^{neg}\) is only used in MC-ASP. The output of Algorithm 2 may have positive weights.
- 7.
- 8.
Note that here “=” is just a part of the symbol for propositional atoms, and is not equality in first-order logic.
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This work was partially supported by the National Science Foundation under Grants IIS-1815337 and IIS-2006747.
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Lee, J., Yang, Z. (2023). Statistical Relational Extension of Answer Set Programming. In: Bertossi, L., Xiao, G. (eds) Reasoning Web. Causality, Explanations and Declarative Knowledge. Lecture Notes in Computer Science, vol 13759. Springer, Cham. https://doi.org/10.1007/978-3-031-31414-8_4
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