On Markovian Fragments of COCOLOG for Logic Control Systems

The COCOLOG (Conditional Observer and Controller Logic) system is a partially ordered family of rst order logical theories expressed in the typed rst order languages fL k ; k 0g describing the controlled evolution of the state of a given partially observed nite machine M. The initial theory of the system, denoted Th 0 , gives the theory of M with no data being given on the initial state. Later theories, fTh(o k 1); k 1g, depend upon the (partially ordered lists of) observed input-output trajectories fo k 1 ; k 1g, where new data is accepted in the form of the new axioms AXM obs (L k); k 1. A feedback control input U(k) is determined by the solution of a collection of control problems posed in the form of a set of conditional control rules CCR(L k), such a set being paired with the theory Th(o k 1) for each k 1. In this paper, we introduce a restricted version of COCOLOG, called a system of Markovian fragments of COCOLOG, in which a smaller amount of information is communicated from one theory to the next. Such fragment theories are associated with a restricted set of candidate control problems, denoted CCR(L m k); k 1. It is shown that a Markovian fragment theory MTh(o k 1) contains a large subset of Th(o k 1), which includes, in particular, the state estimation theorems of the corresponding full COCOLOG system, and, for the set of control rules CCR(L m k), has what is termed the same control reasoning power. Next, it is shown that proofs of theorems in the fragment systems are necessarily shorter than their proofs in the full COCOLOG systems; nally some computer generated examples are given illustrating this increased theorem proving eeciency.