24,570 research outputs found

    Data Definitions in the ACL2 Sedan

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    We present a data definition framework that enables the convenient specification of data types in ACL2s, the ACL2 Sedan. Our primary motivation for developing the data definition framework was pedagogical. We were teaching undergraduate students how to reason about programs using ACL2s and wanted to provide them with an effective method for defining, testing, and reasoning about data types in the context of an untyped theorem prover. Our framework is now routinely used not only for pedagogical purposes, but also by advanced users. Our framework concisely supports common data definition patterns, e.g. list types, map types, and record types. It also provides support for polymorphic functions. A distinguishing feature of our approach is that we maintain both a predicative and an enumerative characterization of data definitions. In this paper we present our data definition framework via a sequence of examples. We give a complete characterization in terms of tau rules of the inclusion/exclusion relations a data definition induces, under suitable restrictions. The data definition framework is a key component of counterexample generation support in ACL2s, but can be independently used in ACL2, and is available as a community book.Comment: In Proceedings ACL2 2014, arXiv:1406.123

    The Theory Behind TheoryMine

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    Abstract. We describe the technology behind the TheoryMine novelty gift company, which sells the rights to name novel mathematical theorems. A tower of four computer systems is used to generate recursive theories, then to speculate conjectures in those theories and then to prove these conjectures. All stages of the process are entirely automatic. The process guarantees large numbers of sound, novel theorems of some intrinsic merit.

    A note on the Oq(sl2^)O_q(\hat{sl_2}) algebra

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    An explicit homomorphism that relates the elements of the infinite dimensional non-Abelian algebra generating Oq(sl2^)O_q(\hat{sl_2}) currents and the standard generators of the q−q-Onsager algebra is proposed. Two straightforward applications of the result are then considered: First, for the class of quantum integrable models which integrability condition originates in the q−q-Onsager spectrum generating algebra, the infinite q−q-deformed Dolan-Grady hierarchy is derived - bypassing the transfer matrix formalism. Secondly, higher Askey-Wilson relations that arise in the study of symmetric special functions generalizing the Askey-Wilson q−q-orthogonal polynomials are proposed.Comment: 11 page

    Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

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    There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.Comment: Corrected typo
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