The Logical Complexity of Control

The COCOLOG system is a partially ordered family of rst order logical theories that describe the controlled evolution of the state of a given partially observered-nite machine M. Following the review of the general COCOLOG system and its Markovain fragment system , a mutil-complexity measurement called Joint is deened and further a partial order simplicity is introduced. Finally, some comparisons among various control problems have been given. The COCOLOG system, introduced by P.E. Caines and S. Wang CW 90], CW91] and W91], is a partially ordered family of rst order logical theories which describe the controlled evolution of the state of a given partially observered nite machine M. The initial theory of the system Th 0 gives the general theory of M without any data given on the initial state. Later theories , fTh(o k 1); k 1g; depend upon, through their axiom sets f k ; k 1g, the (partially ordered lists of) observed input-output trajectories fo k 1 ; k 1g, since new data is accepted sequentially into the subsequent theories in the form of the new axioms AXM obs (L k). The inputs U(k) are determined by the solution to control problems posed along with each theory in the form of the conditional control rules CCR(L k). An important class of control problems involve the reachability predicate Rbl(x; y; l), deened axiomatically in each theory , corresponding to the reachability of state y from state x in l steps. The solution to one problem of this type would be, say, the rst control in a sequence of controls giving a minimal length path to the state y from the current state x. The reader is referred to CW90,91,W91,WC92, Cwe93] for a full exposition of all terms and expressions which are not completely explained here. Deenition 1.1 A (partially observed) nite (input-state-output) machine is a quintuple where U is a ((nite) set of inputs, X is a ((nite) set of states, Y is a ((nite) set of outputs, : X U ! X is a transition function, : X ! Y is an output function. We deene a typed rst order language L 0 to describe such machines. Instead of the full scale deenition of L 0 and its syntaxs, we only list its special characteristic below: L 0 contains a nite set of constants symbols that sorted according to the input, state, output and integer. The set …