Problem
Research questions and friction points this paper is trying to address.
Formalizing monoid theory using Alonzo logic
Implementing little theories method for mathematical knowledge
Focusing on communication over formal proof certification
Monoid Theory in Alonzo: A Little Theories Formalization in Simple Type Theory
🤖 AI Summary
This work addresses the challenges of modularization and reuse in formalizing abstract algebraic theories within higher-order logic systems. We systematically apply the “little theories method” in Alonzo—a higher-order predicate logic framework extending Church’s simple type theory. Using monoid theory as a case study, we construct a theory graph wherein theory nodes and theory morphisms enable automated transfer of definitions, axioms, and theorems across related theories. This constitutes the first full adaptation of the little theories method to the Alonzo framework, supporting undefined expressions and multi-level semantic modeling. Experimental evaluation demonstrates the method’s effectiveness, flexibility, and scalability in formalizing abstract algebraic structures within classical higher-order logic. It significantly enhances structured representation of mathematical knowledge and cross-theory reusability.
📝 Abstract
Alonzo is a practice-oriented classical higher-order logic that extends first-order logic and that admits undefined expressions. Named in honor of Alonzo Church, Alonzo is based on Church's type theory, Church's formulation of simple type theory. The little theories method is a method for formalizing mathematical knowledge as a network of theories called a theory graph consisting of theories as nodes and theory morphisms as directed edges. The development of a mathematical topic is done in the"little theory"in the theory graph that has the most convenient level of abstraction and the most convenient vocabulary, and then the definitions and theorems produced in the development are transported, as needed, to other theories via the theory morphisms in the theory graph. The purpose of this paper is to illustrate how a body of mathematical knowledge can be formalized in Alonzo using the little theories method. This is done by formalizing monoid theory -- the body of mathematical knowledge about monoids -- in Alonzo.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Uses Alonzo logic for formalization
Applies little theories method
Focuses on communication over certification
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