24,570 research outputs found
Data Definitions in the ACL2 Sedan
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
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 algebra
An explicit homomorphism that relates the elements of the infinite
dimensional non-Abelian algebra generating currents and the
standard generators of the 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 Onsager
spectrum generating algebra, the infinite 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 orthogonal polynomials are proposed.Comment: 11 page
Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations
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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