In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.

Every mathematician will agree that the discovery, analysis, and communication

We present an interface connecting the ACL2 theorem prover with external deduction tools. The logic of ACL2 contains several constructs intended to facilitate structuring of interactive proof development, which complicates the design of such an interface. We discuss some of these complexities and develop a precise specification of the requirements from external tools for sound connection with ACL2. We also develop constructs within ACL2 to enable the developers of external tools to satisfy our specifications. 1

in this document are those of the author(s) and should not be interpreted as representing the official policies, either

this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the Defense Advanced Research Projects Agency or the U.S. Government.

We propose a general method for overwriting theories with model conservative extensions in mechanized mathematics systems. Model conservative extensions, which include the denition of new constants and the introduction of new abstract datatypes, are \safe" because they preserve models as well as consistency. The method employs the notions of theory interpretation and theory instantiation. It is illustrated using many-sorted rst-order logic, but it works for a variety of underlying logics. Supported by the MITRE-Sponsored Research program. 1 1 Introduction Mathematical reasoning is always performed in some mathematical context consisting of vocabulary and assumptions. The formal counterpart of a context is a theory consisting of a formal language plus a set of sentences of the language called axioms. (We will denote a theory T by the pair (L; ) where L is the formal language of T and is the set of axioms of T .) An extension of a theory T is any theory T 0 obtained by ...

To my parents, Henry and Karen Reeber, and my fiancée, Carrie Pankrast, for all their love, guidance, and support. Acknowledgments Most of all, I would like to thank my thesis advisor, Warren Hunt. Warren always has the amazing ability to give me what I need, before I even ask for it. Furthermore, Warren has been a source of constant encouragement and guidance, without which I never would have started this dissertation, let alone completed it. I would also like to thank the rest of my dissertation committee, Allen Emerson, Steve Keckler, J Moore, and Anna Slobodova, for all the time and energy they spent re-viewing my research and for their great feedback both on the dissertation itself and the earlier dissertation proposal. Anna in particular provided me with copious notes that have significantly improved the quality of this dissertation. Thanks also to Sandip Ray, Simha Sethumadhavan, and Jun Sawada for providing excellent feedback on portions of this dis-sertation. A number of professors at the University of Texas have influenced my work. My

in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the

Contents Project Summary . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . Our previous work in mechanized reasoning systems . . . . . . . Existing reasoning systems . . . . . . . . . . . . . . . Existing logical frameworks . . . . . . . . . . . . . . Open mechanized reasoning systems . . . . . . . . . . . . Project Description . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accomplishments of Previous NSF Support . . . . . . . . . . Budget Pages . . . . . . . . . . . . . . . . . . . Biography of McCarthy . . . . . . . . . . . . . . . . Biography of Giunchiglia . . . . . . . . . . . . . . . Biography of Talcott . . . . . . . . . . . . . . . . i 1. Project summary There is a growing interest in the interconnection and integration of reasoning modules and systems. For example, developers of hardware veri

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