Markov logic network

A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, enabling uncertain inference. Markov logic networks generalize first-order logic, in the sense that, in a certain limit, all unsatisfiable statements have a probability of zero, and all tautologies have probability one.

Work in this area began in 2003 by Pedro Domingos and Matt Richardson, and they began to use the term MLN to describe it.

Briefly, it is a collection of formulas from first-order logic, to each of which is assigned a real number, the weight. Taken as a Markov network, the vertices of the network graph are atomic formulas, and the edges are the logical connectives used to construct the formula. Each formula is considered to be a clique, and the Markov blanket is the set of formulas in which a given atom appears. A potential function is associated to each formula, and takes the value of one when the formula is true, and zero when it is false. The potential function is combined with the weight to form the Gibbs measure and partition function for the Markov network.

The above definition glosses over a subtle point: atomic formulas do not have a truth value unless they are grounded and given an interpretation; that is, until they are ground atoms with a Herbrand interpretation. Thus, a Markov logic network becomes a Markov network only with respect to a specific grounding and interpretation; the resulting Markov network is called the ground Markov network. The vertices of the graph of the ground Markov network are the ground atoms. The size of the resulting Markov network thus depends strongly (exponentially) on the number of constants in the domain of discourse.

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