Some solvable subclasses of structural recognition problems

During the last decades a lot of applied structural recognition problems came to light which might be reduced to consistent labelling problems. The consistent labelling problem and its appropriate fuzzy and probabilistic modifications became a proper formalism for a unified formulation of such problems regardless of their specific applied contents. On the other hand there is no formal construction for a unified solution of these problems because they proved to be not polynomially solvable. Despite of the complexity of the whole class of labelling problems there are rather wide subclasses of applied interest which are polynomially solvable. In this paper we describe several examples of such subclasses of labelling problems. All these subclasses have a common property: It is possible to introduce several mechanisms of equivalent transformations of the problems. The algorithm solving a problem has then the form of succesive equivalent transformations of the initial problem until it turns into a problem with a quite evident solution. 1 Survey of the known problems and their generalized formulation 1.1 Consistent labelling problem Structural recognition is the analysis of complex objects composed of several parts. A comlex object is considered as a finite set T consisting of simple objects t. Every simple object t ∈ T may stay in some state k from a finite set of states K. A complete description of a complex object T is a function f : T → K assigning each simple object t ∈ T its state f(t) where the object stays. The function f will be called a labelling, whereas elements of the set K are called labels. Suppose the labelling f satisfies some a priori restrictions, given by a local-conjunctive predicate of second order [5]. That mean a subset Ω ⊂ T × T of pairs object-object is given and a function g(t1, t2) : K ×K → {0, 1} is assigned to each object pair (t1, t2) ∈ Ω. The value g(t1, t2, k1, k2) determines whether the object t1 may stay in the state k1 when the object t2 stays in the state k2. The whole labelling f : T → K is called consistent if ∧ (t1,t2)∈Ω g ( t1, t2, f(t1), f(t2) ) = 1 . (1) Let us suppose that some observations were made for each object t ∈ T independently, resulting in additional, a posteriori information about the state of each object. This information narrow the set of possible states for each object and is denoted by functions q(t) : K → {0, 1}. The value q(t, k) defines, whether the state k is contained in the reduced set of states or not. The consistent labelling problem [1] consists in answering the question, whether the a posteriori information about the states of the objects is consistent with the a priori information, i.e. the problem consists in calculating the quantity ∨