Applications of Unskolemization

This dissertation describes a novel method for deriving logical consequences of first-order formulas using resolution and unskolemization. A complete unskolemization algorithm is given and its properties analyzed. This method is then applied to a number of different fields, namely program verification, machine learning, and mathematical induction. The foremost problem in automating program verification is the difficulty of mechanizing the generation of inductive assertions for loops in a program. We show that this problem can be viewed as one of generating logical consequences of the conditions which are true at the entry to a loop. A complete and sound algorithm for generating loop invariants in first-order logic is described. All previous techniques given in the literature for deriving loop invariants are heuristic in nature and are not complete in any sense. There are a number of problems associated with machine learning, such as the diversity of representation languages used and the complexity of learning. We present a graph-based polynomial-time algorithm for learning from examples which makes use of the method for generating logical consequences. The representation language used is first-order logic, which enables the algorithm to be applied in a large number of fields where first-order logic is the language of choice. The algorithm is shown to compare favorably with others in the literature, and applications of the algorithm in a number of fields are demonstrated. The difficulty of mechanizing mathematical induction in existing theorem provers is due to the inexpressibility of the principle of induction in first-order logic. In order to handle mathematical induction within the framework of first-order logic, it is necessary to find an induction schema for each theorem. We describe two methods for tackling this problem, one of which makes use of our method for generating logical consequences. Most existing methods for mechanizing induction can only handle equational theorems. Our approach is more general and is applicable to equational as well as non-equational theorems.