Automated Theorem Proving: Theory and Practice: A Review

deals with the development of computer programs that show that some statement (the conjecture) is a logical consequence of a set of statements (the axioms and hypotheses). ATP systems are used in a wide variety of domains: A mathematician might use the axioms of group theory to prove the conjecture that groups of order two are commutative; a management consultant might formulate axioms that describe how organizations grow and interact and, from these axioms, prove that organizational death rates decrease with age; or a frustrated teenager might formulate the jumbled faces of a Rubik’s cube as a conjecture and prove, from axioms that describe legal changes to the cube’s configuration, that the cube can be rearranged to the solution state. All these tasks can be performed by an ATP system, given an appropriate formulation of the problem as axioms, hypotheses, and a conjecture. Most commonly, ATP systems are embedded as components of larger, more complex software systems, and in this context, the ATP systems are required to autonomously solve subproblems that are generated by the overall system. To build a useful ATP system, several issues have to carefully be considered, independently and in relation to each other, and addressed in a synergetic manner. These issues include the choice of logic that will be used to represent the problems, the calculus that will be used for deduction, the programming language that will be used to write the ATP system, the data structures that will be used to hold the statements of the problem and the statements deduced by the system, the overall scheme for controlling the deduction steps of the system, and the heuristics that will control the finegrained aspects of the deduction steps (the heuristics are most likely to determine the success or failure of an ATP system because they most directly world champion, Gary Kasparov. It is Newborn’s background in the search issues of computer chess that led to his more recent interest in ATP. His book provides an introduction to some of the basic logic, calculi, heuristics, and practicalities of first-order ATP. Rather than working rigorously through the theory of mathematical logic, the book focuses on just the necessary foundations for understanding how first-order ATP systems operate and links these foundations to the implementation of two ATP systems, HERBY and THEO. The book comes with a CD containing the source code for HERBY and THEO and several suites of ATP problems for HERBY and THEO to attempt.1 Thus, the reader is able to experiment as both a user and a developer of ATP systems. The provision of the ATP systems and problems sets this book apart from most other introductory books on ATP that provide a more in-depth treatment of the theory but fail to get readers over the initial hurdles of using an ATP system (a notable exception is Wos et al.’s book, Automated Reasoning: Introduction and Applications [McGraw-Hill, 1992], which comes with the well-known ATP system OTTER). Newborn’s book is suitable as an introduction to ATP for undergraduate university students and independent, interested readers. After an introductory chapter explaining the structure of the book and software installation, chapters 2 to 4 introduce first-order logic (in the syntax used by HERBY and THEO) and the basic mechanics and semantics of resolution-based ATP. Chapters 5 and 6 then provide the underlying theory and describe the calculi for the two ATP systems, chapter 5 corresponding to HERBY and chapter 6 corresponding to THEO. The architecture, use, and implementation of HERBY are described in chapters 7, 8, and 11, respectively, and the same information is provided for THEO in chapters 9, 10, and 12. The last chapter steps aside to briefly discuss the CADE ATP System Competition (CADE [the Conference on Automated Deduction] is the major forum for the presentation of new research in all aspects of automated deduction). Variants of HERBY and THEO participated in the competitions in 1997 and 1998.