Necessary and Sufficient Conditions for Dynamic Programming of Combinatorial Type
Journal of the ACM (JACM)
Formal languages and their relation to automata
Formal languages and their relation to automata
Finite automata and their decision problems
IBM Journal of Research and Development
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In the earlier papers by Karp and Held and by Ibaraki, the representation of a discrete optimization problem given in the form of a discrete decision process (ddp) by a finite state model called a sequential decision process (sdp) was considered. An sdp is a finite automaton with a cost function associated with each state transition. When the cost function satisfies a certain monotonicity condition, it is called a monotone sdp(msdp). As pointed out by Karp and Held, there is a close relationship between an msdp and the dynamic programming developed by Bellman. These models are further restricted in this paper by assuming that each cost function is a recursive function. The resulting models are called r-ddp, r-sdp, and r-msdp, respectively. Two types of representation theorems and properties of sets of optimal policies are investigated in detail for r-sdp and r-msdp. Various decision problems are also considered, and most of them are proved to be unsolvable. In particular, there exists no algorithm to obtain an optimal policy of an arbitrarily given r-sdp or r-msdp. Since this is quite inconvenient from the view point of practical application, a subclass of r-msdp, r-imsdp, is introduced in the last half of this paper. For an arbitrarily given r-imsdp, there exists an algorithm to obtain an optimal policy if it has at least one optimal policy. Most of other decision problems, however, are proved to be still unsolvable.