Benzmüller, Christoph

Scott, Dana S.

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Some Reflections on a Computer-aided Theory ExplorationStudy in Category Theory (Extended Abstract)Christoph Benzmu¨ller1 and Dana S. Scott21 University of Luxembourg, Luxembourg & Freie Universita¨t Berlin, Berlin, Germanyc.benzmueller@gmail.com2 Visiting Scholar at University of California, Berkeley, USAdana.scott@cs.cmu.eduWe present some reflections on the use of automated theorem proving and model findingtechnology in the context of a recent theory exploration study in category theory [1, 2].In our stepwise development of mutually equivalent axioms sets for category theory westarted out with a generalised notion of monoids. More precisely, the first axiom system in ourstudy was obtained by generalizing the standard axioms for a monoid to a partial compositionoperation. In subsequent development steps we simplified this initial axioms set until we reachedthe axioms as proposed by Scott [15] in the 1970s. We then compared this axioms set with analternative proposal by Freyd and Scedrov [11]. In the course of this comparison we revealeda technical flaw for the axiom set of Freyd and Scedrov: either all operations, e.g. morphismcomposition, are total or their axiom system is inconsistent. The repair for this problem isquite straightforward and it essentially corresponds to the set of axioms proposed by Scott.Our experiments were enabled by a semantical embedding of free logic [14] in classical higher-order logic (HOL), which we implemented in the proof assistant system Isabelle/HOL [12].Free logic was utilised to support an adequate handling of partiality and undefinedness in themodeling of morphism composition, and the domain and codomain operators. Our experimentswere substantially supported by automated reasoning technology, in particular, by the modelfinder Nitpick [7] and by various automated theorem provers (CVC4 [10], E [13], Leo-II [3],Satallax [8], SPASS [6], Z3 [9], etc.) integrated with Isabelle/HOL via Sledgehammer [5].In our presentation at AITP 2018 we particularly want to reflect on the role these systemsplayed in our experiments. This is of practical and also of epistemological relevance, since thesesystems, as we will evidence, can indeed substantially foster the gain of new knowledge. Wewill therefore highlight relevant points in our stepwise development in which these systems, inparticular, the model finder Nitpick, supported the gain of intuition by providing countermodelsto still slightly flawed axioms or definitions. And the theorem provers supported the detectionof the constricted inconsistency, in addition to the important, albeit more traditional, role theyplayed in confirming equivalences between different axioms sets as soon as we arrived at theircorrect formulations.Despite our reassuring overall teamwork experience, which involved a domain expert (Scott),a theorem proving expert (Benzmu¨ller) and the Isabelle/HOL framework, we also collectedseveral critical remarks pointing to a range of improvement opportunities. Some of these im-provement opportunities are of technical nature, others may include theoretical aspects. Forexample, Nitpick should be improved by devising and implementing better readable and even-tually more domain specific representations of models and countermodels. In our experimentssuch conversions were in fact laboriously handled by hand by Benzmu¨ller and the results werethen communicated by email to Scott. In some cases calls of external theorem provers viaSledgehammer resulted in technical error messages, which may demotivate non-expert users,and when the theorem provers succeeded, then their proofs could most of the time not be con-verted into informative Isar style proofs. The constricted inconsistency result, for example, hadSome Reflections on a Computer-aided Theory Exploration Study in Category Theory Benzmller and Scottto be reconstructed by hand to obtain an human-friendly Isar style proof (see [4] for a similarexperience in a different context).Hence, our successful experiments, in which automated reasoning tools integrated in Is-abelle/HOL have demonstrated their capabilities beyond mere proof verification, still requireda close interaction between three players: a domain expert, a theorem proving expert and theIsabelle/HOL proof assistant. The challenge in fact still is to get the second player completelyout of the loop, without requiring the first player to adopt a nearly identical level of technicalexpertise in a resource-intensive, laborious manner.References[1] C. Benzmu¨ller and D. Scott. Automating free logic in Isabelle/HOL. In G.-M. Greuel, T. Koch,P. Paule, and A. Sommese, editors, Mathematical Software – ICMS 2016, volume 9725 of LNCS,pages 43–50, Berlin, Germany, 2016. Springer.[2] C. Benzmu¨ller and D. Scott. Axiomatizing category theory in free logic. CoRR, abs/1609.01493,2016. This work is currently submitted for journal publication.[3] C. Benzmu¨ller, N. Sultana, L. C. Paulson, and F. Theiss. The higher-order prover Leo-II. Journalof Automated Reasoning, 55(4):389–404, 2015.[4] C. Benzmu¨ller and B. Woltzenlogel Paleo. The inconsistency in Go¨del’s ontological argument: Asuccess story for AI in metaphysics. In IJCAI 2016, volume 1-3, pages 936–942. AAAI Press, 2016.[5] J. C. Blanchette, S. Bo¨hme, and L. C. Paulson. Extending Sledgehammer with SMT solvers.Journal of Automated Reasoning, 51(1):109–128, 2013.[6] J. C. Blanchette, A. Popescu, D. Wand, and C. Weidenbach. More SPASS with Isabelle - Super-position with Hard Sorts and Configurable Simplification. In ITP, volume 7406 of LNCS, pages345–360. Springer, 2012.[7] J.C. Blanchette and T. Nipkow. Nitpick: A counterexample generator for higher-order logic basedon a relational model finder. In ITP 2010, number 6172 in LNCS, pages 131–146. Springer, 2010.[8] C. E. Brown. Satallax: An automatic higher-order prover. In Automated Reasoning – IJCAR,volume 7364 of LNCS, pages 111–117. Springer, 2012.[9] L. M. de Moura and N. Bjørner. Z3: An Efficient SMT Solver. In TACAS, volume 4963 of LNCS,pages 337–340. Springer, 2008.[10] M. Deters, A. Reynolds, T. King, C. W. Barrett, and C. Tinelli. A tour of CVC4: how it works,and how to use it. In FMCAD, page 7. IEEE, 2014.[11] P. Freyd and A. Scedrov. Categories, Allegories. North Holland, 1990.[12] T. Nipkow, L. C. Paulson, and M. Wenzel. Isabelle/HOL: A Proof Assistant for Higher-OrderLogic. Number 2283 in LNCS. Springer, 2002.[13] S. Schulz. System description: E 1.8. In LPAR-19, volume 8312 of LNCS, pages 735–743. Springer,2013.[14] D. Scott. Existence and description in formal logic. In R. Schoenman, editor, Bertrand Russell:Philosopher of the Century, pages 181–200. George Allen & Unwin, London, 1967. (Reprinted withadditions in: Philosophical Application of Free Logic, edited by K. Lambert. Oxford UniversitryPress, 1991, pp. 28 - 48).[15] D. Scott. Identity and existence in intuitionistic logic. In M. Fourman, C. Mulvey, and D. Scott,editors, Applications of Sheaves: Proceedings of the Research Symposium on Applications of SheafTheory to Logic, Algebra, and Analysis, Durham, July 9–21, 1977, volume 752 of Lecture Notesin Mathematics, pages 660–696. Springer, 1979.2

Some Reflections on a Computer-aided Theory Exploration Study in Category Theory (Extended Abstract)

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Some Reflections on a Computer-aided Theory Exploration Study in Category Theory (Extended Abstract)

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