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Computer-Supported Exploration of a Categorical Axiomatization of Modeloids
Abstract
A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraïssé games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fraïssé games.
... This work models on one-sorted categories, i.e., it only refers to morphisms without mentioning objects. This approach was first expanded upon by Tiemens who defined inverse categories in order to generalize so-called modeloids [31]. ...
This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing the denotational semantic model of intuitionistic multiplicative linear logic. Next to these meta-logical-investigations, we contribute to building an Isabelle category theory library, with a focus on ease of use in the formalization beyond category theory itself. This work paves the way for future formalizations based on category theory and demonstrates the power of automated reasoning in investigating meta-logical questions.
KeywordsFormalization of mathematicsCategory theoryProof assistantsFormal methodsShallow embeddings
... This work models on one-sorted categories, i.e., it only refers to morphisms without mentioning objects. This approach was first expanded upon by Tiemens who defined inverse categories in order to generalize so-called modeloids [31]. ...
This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing the denotational semantic model of intuitionistic multiplicative linear logic. Next to these meta-logical-investigations, we contribute to building an Isabelle category theory library, with a focus on ease of use in the formalization beyond category theory itself. This work paves the way for future formalizations based on category theory and demonstrates the power of automated reasoning in investigating meta-logical questions.
... Proofs for the stated theorems, propositions and lemmas are presented in the extended preprint [16] of this paper (cf. also [15]); the Isabelle/HOL source files are available online. 7 ...
A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automor-phisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraisse games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fraisse games.
A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automor-phisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fra¨ısséFra¨ıssé games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fra¨ısséFra¨ıssé games.
A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category theory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.
Nitpick is a counterexample generator for Isabelle/HOL that builds on Kodkod, a SAT-based first-order relational model finder.
Nitpick supports unbounded quantification, (co)inductive predicates and datatypes, and (co)recursive functions. Fundamentally
a finite model finder, it approximates infinite types by finite subsets. As case studies, we consider a security type system
and a hotel key card system. Our experimental results on Isabelle theories and the TPTP library indicate that Nitpick generates
more counterexamples than other model finders for higher-order logic, without restrictions on the form of the formulas to
falsify.
This book is an introduction to finite model theory which stresses the computer science origins of the area. In addition to presenting the main techniques for analyzing logics over finite models, the book deals extensively with applications in databases, complexity theory, and formal languages, as well as other branches of computer science. It covers Ehrenfeucht-Fraïssé games, locality-based techniques, complexity analysis of logics, including the basics of descriptive complexity, second-order logic and its fragments, connections with finite automata, fixed point logics, finite variable logics, zero-one laws, and embedded finite models, and gives a brief tour of recently discovered applications of finite model theory.
This book can be used both as an introduction to the subject, suitable for a one- or two-semester graduate course, or as reference for researchers who apply techniques from logic in computer science.
If A is a set and A is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on A which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite (> 5) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.
Free logic is an important field of philosophical logic that first appeared in the 1950s. J. Karel Lambert was one of its founders and coined the term itself. The essays in this collection (written over a period of 40 years) explore the philosophical foundations of free logic and its application to areas as diverse as the philosophy of religion and computer science. Amongst the applications on offer are those to the analysis of existence statements, to definite descriptions and to partial functions. The volume contains a proof that free logics of any kind are non-extensional and then uses that proof to show that Quine's theory of predication and referential transparency must fail. The purpose of this collection is to bring an important body of work to the attention of a new generation of professional philosophers, computer scientists and mathematicians.
It follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories , being a Mackey functor on is equivalent to being extendible to a suitable inverse category containing . We further show that extensions of inverse categories by abelian groups are again inverse categories.
- M V Lawson
Lawson, M.V.: Inverse Semigroups. World Scientific (1998). https://doi.org/10.
1142/3645
Bertrand Russell: Philosopher of the Century
- D Scott
The definition of e! in free logic
- K Lambert
Lambert, K.: The definition of e! in free logic. In: Abstracts: The International
Congress for Logic, Methodology and Philosophy of Science. Stanford University
Press, Stanford (1960)
Extending sledgehammer with SMT solvers
- J C Blanchette
- S Böhme
- L C Paulson
- JC Blanchette
Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending sledgehammer with SMT
solvers. J. Autom. Reasoning 51(1), 109-128 (2013). https://doi.org/10.1007/
s10817-013-9278-5