Category Archives: Mathematical Logic

Extending probabilistic logic programming with continuous r.v.s, and a nice and brief introduction to programming logic and probabilistic inference

Steffen Michels, Arjen Hommersom, Peter J.F. Lucas, Marina Velikova, A new probabilistic constraint logic programming language based on a generalised distribution semantics, Artificial Intelligence, Volume 228, November 2015, Pages 1-44, ISSN 0004-3702, DOI: 10.1016/j.artint.2015.06.008.

Probabilistic logics combine the expressive power of logic with the ability to reason with uncertainty. Several probabilistic logic languages have been proposed in the past, each of them with their own features. We focus on a class of probabilistic logic based on Sato’s distribution semantics, which extends logic programming with probability distributions on binary random variables and guarantees a unique probability distribution. For many applications binary random variables are, however, not sufficient and one requires random variables with arbitrary ranges, e.g. real numbers. We tackle this problem by developing a generalised distribution semantics for a new probabilistic constraint logic programming language. In order to perform exact inference, imprecise probabilities are taken as a starting point, i.e. we deal with sets of probability distributions rather than a single one. It is shown that given any continuous distribution, conditional probabilities of events can be approximated arbitrarily close to the true probability. Furthermore, for this setting an inference algorithm that is a generalisation of weighted model counting is developed, making use of SMT solvers. We show that inference has similar complexity properties as precise probabilistic inference, unlike most imprecise methods for which inference is more complex. We also experimentally confirm that our algorithm is able to exploit local structure, such as determinism, which further reduces the computational complexity.

Study of the explanation of probability and reasoning in the human mind through mental models, probability logic and classical logic

P.N. Johnson-Laird, Sangeet S. Khemlani, Geoffrey P. Goodwin, Logic, probability, and human reasoning, Trends in Cognitive Sciences, Volume 19, Issue 4, April 2015, Pages 201-214, ISSN 1364-6613, DOI: 10.1016/j.tics.2015.02.006.

This review addresses the long-standing puzzle of how logic and probability fit together in human reasoning. Many cognitive scientists argue that conventional logic cannot underlie deductions, because it never requires valid conclusions to be withdrawn – not even if they are false; it treats conditional assertions implausibly; and it yields many vapid, although valid, conclusions. A new paradigm of probability logic allows conclusions to be withdrawn and treats conditionals more plausibly, although it does not address the problem of vapidity. The theory of mental models solves all of these problems. It explains how people reason about probabilities and postulates that the machinery for reasoning is itself probabilistic. Recent investigations accordingly suggest a way to integrate probability and deduction.