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Semantic Representation of General Topology in the Wolfram Language

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Intelligent Computer Mathematics (CICM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10383))

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Abstract

The Wolfram Knowledgebase, powered by the entity framework, contains expertly curated data from thousands of diverse domains. We have begun expanding this framework to include mathematical knowledge, making significant strides in the representation of results pertaining to continued fractions, function spaces, and most recently, topology. This paper will focus on our progress in the representation of general topology. We have curated over 700 entities representing concept definitions, theorem statements, and concrete topological spaces, as well as their corresponding properties, including their formal representations as well as references, computed properties, and other metadata. Virtually every formal representation in this project required extensions to the Wolfram Language, mostly for basic set theory. We will outline all of these design choices by way of examples, as well as present additional functionality for querying, usage messages, formatting, and other computations.

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Notes

  1. 1.

    You can see that the standard form for an entity is a box containing a string called the entity’s common name. The common name is a property that is available for all entities.

  2. 2.

    There is some interest in introducing a symbolic type language within the Wolfram Language suitable for communicating basic information between proof assistants and Mathematica. This could be used, for example, to use Mathematica’s computer algebra functionalities to assist these proof assistants (see Lewis [7]).

  3. 3.

    We say “essentially” because Wolfram Language syntax is very permissive. Syntactically valid expressions can semantically represent statements in a first-order theory, but are not guaranteed to do so in all cases. For example, ForAll, is intended to be used a first-order quantifier, and functions like Resolve use heuristics (such as assuming variables appearing in inequalities are real numbers) to attempt to remove quantifiers, but ForAll is not axiomitized. The argument structure allows for arbitrary expressions to be passed as arguments, in which cases the expression remains in an unevaluated symbolic form.

  4. 4.

    This is accomplished by defining a default function used to calculate the common name from the canonical name, but which can be manually overwritten.

  5. 5.

    One might ask why these properties even have different names from the corresponding concept properties. After all, under the Curry-Howard correspondence, a universally quantified proposition is essentially a function type. For example, \(\forall _{\texttt {x,x}\in \texttt {Reals}} \texttt {x}^2 \ge \texttt {0}\) can be viewed as a function which takes a real number x and outputs a proof that \(\texttt {x}^2\ge \texttt {0}\). This is natural and powerful in languages such as Lean [9] which are based on dependent type theory, but as the Wolfram Language does not have a type system, this equivalence is not made.

  6. 6.

    The \(\pi \)-Base project [3] uses a similar representation of topological property relations and uses them under the hood to deduce topological properties from the properties asserted of their many example topological spaces, but to our knowledge they do not provide an easy way for users to do their own computations as we do here.

References

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Acknowledgements

I thank CICM, Michael Trott, and Wolfram Research for this opportunity, as well as Stephen Wolfram, Michael Trott, James Mulnix, Eric Weisstein, Robert Lewis, and José Martín-García for their support and great work on pure mathematics in the Wolfram Language.

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Ford, I. (2017). Semantic Representation of General Topology in the Wolfram Language. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds) Intelligent Computer Mathematics. CICM 2017. Lecture Notes in Computer Science(), vol 10383. Springer, Cham. https://doi.org/10.1007/978-3-319-62075-6_12

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