Parameterized Formal Graph Systems and Their Polynomial-Time PAC Learnability

抄録

A formal graph system (FGS for short) is a logic program consisting of definite clauses whose arguments are graph patterns instead of first-order terms. The definite clauses are referred to as graph rewriting rules. An FGS is shown to be a useful unifying framework for learning graph languages. In this paper, we show the polynomial-time PAC learnability of a subclass of FGS languages defined by parameterized hereditary FGSs with bounded degree, from the viewpoint of computational learning theory. That is, we consider VH-FGSL_k,_Δ(m, s, t, r, w, d) as the class of FGS languages consisting of graphs of treewidth at most k and of maximum degree at most Δ which is defined by variable-hereditary FGSs consisting of m graph rewriting rules having TGP patterns as arguments. The parameters s, t, and r denote the maximum numbers of variables, atoms in the body, and arguments of each predicate symbol of each graph rewriting rule in an FGS, respectively. The parameters w and d denote the maximum number of vertices of each hyperedge and the maximum degree of each vertex of TGP patterns in each graph rewriting rule in an FGS, respectively. VH-FGSL_k,_Δ(m, s, t, r, w, d) has infinitely many languages even if all the parameters are bounded by constants. Then we prove that the class VH-FGSL_k,_Δ(m, s, t, r, w, d) is polynomial-time PAC learnable if all m, s, t, r, w, d, Δ are constants except for k.