Complexity of decision problems under incomplete information

Abstract

In the paper we consider a variant of information systems introduced by Pawlak in [1]. In our model, suggested by Skowj:gn, [2] and [3], knowledge about a given domain is represented by a directed graph. Nodes of such a graph are labeled by states and edges by actions. Moreover some nodes of the graph are distinguished as terminal (or accepting). We may think of this model in the following way.~odes correspond to specific situations; states assigned to these nodes represent (incomplete) descriptions of situations; edges represent possible changes of these situations when an a~tion (attached to a given edge) is performed. For example a part of medical knowledge can be represented in this framework: situations correspond to specific diseases (or their complete descriptions); states correspond to partial descriptions of these diseases (symptoms); actions correspond to transitions from one situation to another what reflects the effect of an application of a particular treatment. All nodes which correspond to a specific disease or to a state of good health can be distinguished as terminal ones. A decision problem can be informally described as follows: a knowledge base (graph) and a state s is given; decide whether there is a strategy of applying actions such that each node labeled by s is transformed by this strategy to a terminal one. The main result of this paper states that the above problem is NP-complete (provided a knowledge base is represented by an acyclic graph). Moreover a deterministic polynomial time algorithm is presented for the case of decision problems of a bounded degree; the degree of the decision problem is defined as the number of nodes labeled by the initial state. The paper is organized as follows: in the next section precise definitions of an information system, strategy and decision problem are given; the NP-completeness of the decision problem is shown in section 3 whereas a deterministic polynomial time algorithm for the restricted decision problem is described in section 4. In the last section we argue that the unrestricted decision problem (the graph does not need to be acyclic) is PSPACE-complete.