Inheritance networks are not expressive enough to capture all intuitions behind inheritance. Hence, a number of significantly different semantics have been proposed. Here, a family of local theories of inheritance is presented that are conceptually simple and computationally tractable. A general model-theoretic framework for specifying direct semantics to networks and clarifying relationships among them is developed. It is also shown that the local constraints specifying the semantics are satisfiable when the attention is restricted to acyclic networks. the notion of local semantics is generalized to what is called “ground” local semantics to further understand the relationship between the local theories and the path-based theories of inheritance. A family of polynomialtime set-based inheritance algorithm8 that conforms to these inheritance theories is also presented. © 1995 John Wiley & Sons, Inc.