Abstract
We present a formalisation of basic topology in simple type theory encoded using the Isabelle/HOL proof assistant. In contrast to related formalisation work, which follows more ‘traditional’ approaches, our work builds upon closure algebras, encoded as Boolean algebras of (characteristic functions of) sets featuring an axiomatised closure operator (cf. seminal work by Kuratowski and McKinsey & Tarski). With this approach we primarily address students of computational logic, for whom we bring a different focus, closer to lattice theory and logic than to set theory or analysis. This approach has allowed us to better leverage the automated tools integrated into Isabelle/HOL (model finder Nitpick and Sledgehammer) to do most of the proof and refutation heavy-lifting, thus allowing for assumption-minimality and less-verbose interactive proofs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Cf. Xena Project (https://xenaproject.wordpress.com/).
- 2.
Unless, of course, she happens to be working on a particular application area related to those.
- 3.
It could be further argued that this fact alone already legitimises their study.
- 4.
HOL is in fact an acronym for (classical) higher-order logic. It is traditionally employed to refer to the sort of simple type theory instantiated in proof assistants like Isabelle/HOL.
- 5.
‘Main’ is imported for technical reasons to enable the execution of Sledgehammer and Nitpick.
- 6.
At the time of writing these include LEO-II, Leo-III, Satallax, Zipperposition, and more recently CVC5 and E (starting with release 3.0); as well as model finders Nitpick and Nunchaku. Most of them are available via System-on-TPTP (https://tptp.org/cgi-bin/SystemOnTPTP).
- 7.
To be fair, this feature is in fact shared to a great extent with other Isabelle/HOL formalisations.
- 8.
We didn’t embed any automatically extracted LaTeX sources because of space constraints.
- 9.
Here we follow Isabelle/HOL’s convention of writing type variables with a leading apostrophe as well as omitting parentheses for parameterised types, so that \((`w)\,\sigma \) becomes \(`w\,\sigma \).
- 10.
Note that by employing iADDI (iMULT) in place of ADDI (MULT) to axiomatise a closure (interior) algebra, we obtain a so-called Alexandrov topology (cf. Sect. 3.8 for a discussion).
- 11.
As noted before, we are abusing terminology, as we do not fix a minimal set of conditions that an operator needs to satisfy to deserve being called a ‘closure’; e.g. Moore/hull closures only satisfy MONO, EXPN, and IDEM, while Čech closure operators may not satisfy IDEM (they are sometimes called ‘preclosures’). Anyhow, we let the context dictate how operators are called.
- 12.
The predicates mapping, injective and surjective are formalised in the usual way; we refer to the Isabelle/HOL sources for these and other miscellaneous definitions and lemmata.
- 13.
We have followed this very same workflow during the formalisation work, noting that thanks to Sledgehammer’s good performance, at the time of writing (May 2022) we have had recourse to interactive proofs (as shown in Fig. 1) only as a fallback in a few cases (hence finishing already at step 2). In a teaching context, students are surely expected to go the extra mile.
- 14.
English translations of Zarycki’s works are now available on the web thanks to Mark Bowron (see https://www.researchgate.net/scientific-contributions/Miron-Zarycki-2016157096).
- 15.
- 16.
- 17.
References
Aiello, M., Pratt-Hartmann, I., Van Benthem, J., et al.: Handbook of Spatial Logics. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4
Benzmüller, C.: Universal (meta-)logical reasoning: recent successes. Sci. Comput. Program. 172, 48–62 (2019)
Benzmüller, C., Parent, X., van der Torre, L.: Designing normative theories for ethical and legal reasoning: LogiKEy framework, methodology, and tool support. Artif. Intell. 287, 103348 (2020)
Benzmüller, C., Paulson, L.C.: Quantified multimodal logics in simple type theory. Logica Universalis (Spec. Issue Multimodal Log.) 7(1), 7–20 (2013)
Benzmüller, C., Andrews, P.: Church’s type theory. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Summer 2019 edn. (2019). https://plato.stanford.edu/archives/sum2019/entries/type-theory-church/
Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formalized Reasoning 9(1), 101–148 (2016)
Blanchette, J.C., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14052-5_11
Esakia, L.: Intuitionistic logic and modality via topology. Ann. Pure Appl. Logic 127(1–3), 155–170 (2004)
Fuenmayor, D.: Topological semantics for paraconsistent and paracomplete logics. Archive of Formal Proofs (2020). https://isa-afp.org/entries/Topological_Semantics.html
Fuenmayor, D., Benzmüller, C.: Normative reasoning with expressive logic combinations. In: De Giacomo, G., et al. (eds.) ECAI 2020–24th European Conference on Artificial Intelligence, 8–12 June, Santiago de Compostela, Spain. Frontiers in Artificial Intelligence and Applications, vol. 325, pp. 2903–2904. IOS Press (2020)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains, vol. 93. Cambridge University Press, Cambridge (2003)
Givant, S.: Duality Theories for Boolean Algebras with Operators. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06743-8
Kuratowski, K.: Sur l’opération Ā de l’analysis situs. Fundam. Math. 3(1), 182–199 (1922)
Kuratowski, K.: Topology: Volume I. Academic Press (1966)
Martin, N.M., Pollard, S.: Closure Spaces and Logic, Mathematics and Its Applications, vol. 369. Springer, New York (1996). https://doi.org/10.1007/978-1-4757-2506-3
McKinsey, J.C., Tarski, A.: The algebra of topology. Ann. Math. 45, 141–191 (1944)
Paulson, L.C.: Computational logic: its origins and applications. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 474(2210), 20170872 (2018)
The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics (2013). https://homotopytypetheory.org/book, Institute for Advanced Study
Vickers, S.: Topology via Logic. Cambridge University Press, Cambridge (1996)
Wenzel, M.: Isabelle/Isar-a generic framework for human-readable proof documents. From Insight to Proof-Festschrift in Honour of Andrzej Trybulec 10(23), 277–298 (2007)
Zarycki, M.: Quelques notions fondamentales de l’analysis situs au point de vue de l’algèbre de la logique. Fundam. Math. 9(1), 3–15 (1927)
Zarycki, M.: Some properties of the derived set operation in abstract spaces. Nauk. Zap. Ser. Fiz.-Mat. 5, 22–33 (1947)
Acknowledgements
The first author acknowledges financial support from the Luxembourg National Research Fund (FNR), under grant CORE C20/IS/14616644.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Fuenmayor, D., Serrano Suárez, F.F. (2022). Formalising Basic Topology for Computational Logic in Simple Type Theory. In: Buzzard, K., Kutsia, T. (eds) Intelligent Computer Mathematics. CICM 2022. Lecture Notes in Computer Science(), vol 13467. Springer, Cham. https://doi.org/10.1007/978-3-031-16681-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-16681-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-16680-8
Online ISBN: 978-3-031-16681-5
eBook Packages: Computer ScienceComputer Science (R0)