Logical Deduction Using the Local Computation Framework

A Introduction Computation in a number of uncertainty formalisms has recently been revolutionized by the notion of local computation. 9] and 6] showed how Bayesian probability could be eeciently propagated in a network of variables; this has already lead to sizeable successful applications, as well as a large body of literature on these Bayesian networks and related issues (e.g., the majority of papers in the Uncertainty in Artiicial Intelligence conferences over the last ten years). In the latèEighties, Glenn Shafer and Prakash Shenoy 14] abstracted these ideas, leading to their Local Computation framework. Remarkably, the propagation algorithms of this general framework give rise to eecient computation in a number of spheres of reasoning: as well as Bayesian probability 12], the Local Computation framework can be applied to the calculation of Dempster-Shafer Belief 14, 8], innnitesimal probability functions 17], and Zadeh's Possibility functions. This paper describes how the framework can be used for the computation of logical deduction. Local Computation is based on a structural decomposition of knowledge into a network of variables, in which there are two fundamental operations, combination and marginaliza-tion. The combination of two pieces of information is another piece of information which gives the combined eeect; it is a little like conjunction in classical logic. Marginalization projects a piece of information relating a set of variables, onto a subset of the variables: it gives the impact of the piece of information on the smaller set of variables. Axioms are given which are suucient for the propagation of these pieces of information in the network. General propagation algorithms can be deened using results in the Bayesian network literature and elsewhere. These algorithms are often eecient, depending, roughly speaking, on topological properties of the network. The reason that Local Computation can be very fast is that the propagation is expressed in terms of much smaller (`local') problems, involving only a small part of the network. Finite sets of possibilities (or constraints) can be propagated with this framework, and so deduction in a nite propositional calculus can be performed by considering sets of possible worlds; this is implemented in, for example, PULCINELLA 11], and described formally in 13]. However, dealing with sets of possible worlds is often not computationally eecient; it is only very recently 5] that it has been shown how to use Local Computation to directly propagate sets of formulae in a nite propositional calculus. In the next …